As part of the tactical asset allocation (TAA), investment committees often translate a positive view directly into an overweight. That shortcut can be transparent and defensible when the view is simple, isolated and calibrated to one stable reference portfolio. It becomes unreliable when views interact, change volatilities, correlations or tails, or must be implemented across different mandates. The Institutional View Factory treats implementation as a governed sequence rather than an intuitive jump from belief to weight. The factory metaphor does not remove judgment; it makes inputs, outputs, owners and quality gates explicit. It makes view implementation in portfolio management a consistent and repeatable procedure – well suited for the institutional level.

The framework separates three decisions: what the institution believes, how those beliefs change the joint market distribution, and what each portfolio should hold. On the distributional route, Entropy Pooling updates scenario probabilities before a governed optimizer translates the posterior into mandate-specific candidate weights.

Three key points:

• Direct weight heuristics are controlled approximations—not inherently wrong but they require simple, isolated views and stable risk assumptions.

• Interacting, regime, correlation, and tail-risk views should first update the joint distribution; complete-scenario reweighting also transmits implications to assets without direct views.

• Portfolio weights are downstream outputs: objectives, current holdings, risk budgets, and constraints determine how the same institutional view is implemented in each mandate.

After a long investment committee debate, the institution arrives at a positive investment view on a few assets that it wishes to implement in their portfolios.
What should happen next? How to implement this as a tactical asset allocation?

In many institutions, the answer is familiar: translate the view into overweights of the specific assets. A moderately positive view might produce a two-percentage-point tilt. A strongly positive view might produce four percentage points. We call this the “Heuristic route” of view implementation.

Such a specific heuristic rule is transparent and adequate for simple views in one reference portfolio. But once views interact, change the risk structure, or enter portfolios with different starting weights and constraints, a one-to-one mapping no longer contains enough information.

Figure 1. Two implementation routes connect institutional views to portfolio weights: a direct heuristic shortcut and the full distributional route through portfolio optimization. Source: Stylized illustration; QuantStrategy.

The process must then separate the common market belief from portfolio-specific implementation. Instead, the views must be implemented into the joint return distribution, then enter portfolio optimization to generate portfolio weights. This is what we call the “Distributional Route” of view implementation.

This article provides a one-page operating architecture for deciding which route a view requires.

Simple, isolated views can sometimes map directly to weights. Interacting views must first change the joint return distribution.

The Allocator’s Problem

Consider five common committee statements:

These statements may sound similar. They are not the same kind of view.

A single expected-return or relative-value view can sometimes be mapped heuristically when it is isolated, the risk structure is held fixed and the tilt is small. Regime and tail-risk views are different: they can alter several moments or dependencies at once and have no unique translation into one weight. The same problem appears when several simple views interact: they may share one macro driver, offset one another or create an unintended concentration.

Figure 2. Different view objects alter different features of the modeled joint return distribution. Source: Stylized illustration; QuantStrategy.

Yet many committee processes send all of these statements through the same translation rule:

Positive view
→ higher portfolio weight

The shortcut is fast and can avoid the false precision of poorly constrained optimization.

But it compresses two decisions: what the institution believes and what the portfolio should hold. Portfolio weights also depend on current holdings, the objective, risk budget, cross-asset dependencies, mandate constraints, liquidity, turnover and implementation costs.

A view should therefore update the modeled return distribution before portfolio construction determines candidate weights—unless the institution can defend a simple, pre-calibrated mapping.

Why Direct Weight Heuristics Are Useful

A direct heuristic deserves a fair assessment.

Suppose a committee uses a five-point relative-view scale:

It can be a good rule when the view concerns one asset or spread, the risk structure is assumed stable, the tilt is small, one reference portfolio is used and the mapping is calibrated ex ante.

It is understandable, limits model dependence and prevents every meeting from renegotiating position size. In practice, such discipline can be more stable than feeding noisy expected-return estimates into an unconstrained optimizer (Best and Grauer, 1991; DeMiguel, Garlappi and Uppal, 2009).

The correct argument is not that heuristics are wrong.

Direct weight heuristics are approximations whose validity depends on what the view changes—and what the institution is willing to assume remains unchanged.

Their assumptions are not absent; they are embedded in the pre-calibrated mapping. The key assumption is that the view only impacts the specific assets. But in markets reality, all assets are connected and a view should - in fact - change highly correlated assets as well. This is what heuristic views often ignore entirely.

The One-Page Tool: The Institutional View Factory

The Institutional View Factory separates market beliefs from their distributional projection and portfolio implementation. The factory metaphor does not mechanize judgment; it requires an explicit input, output, owner and quality gate for each transformation.

A Rulebook sits before the Factory, defining the universe, view classes, horizons, reference points, decision rights, risk budgets and defaults.

The Factory has four layers.

Figure 3. The Institutional View Factory separates the institutional belief from its distributional projection and portfolio implementation. Source: Stylized illustration; QuantStrategy.

Layer 1: View Governance

View Governance turns evidence and independent judgments into a documented Institutional View Package :

Evidence
→ Individual View Formation
(→ Pre-commitment)
→ Committee Aggregation
→ Institutional View Package

Later articles address input hygiene, causal translation, pre-commitment and human aggregation. Here, the key requirement is that the model receives a governed, operationalized view package—not a meeting transcript or unresolved opinions. Scherer (2026) likewise treats the IC as a mapping from dispersed beliefs to a central portfolio, although his mechanism aggregates portfolio vectors rather than institutional views. Overall, the view package is a operationalized form of the view ready to be used on the joint return distribution.

Layers 2–4: Routing, Projection and Construction

The Translation Gate chooses between a controlled weight heuristic and an explicit distributional update. On the distributional route, Entropy Pooling projects the governed views onto the prior scenario set, producing a posterior joint return distribution. Portfolio construction then combines the posterior with objectives, holdings and constraints to produce candidate weights, while the institution retains final approval.

The Distributional Complexity Gate

The practical decision point inside the Factory has two routes:

The five tests below determine whether the shortcut remains defensible.

If one fails materially, route the view through the joint distribution and portfolio construction. The Gate defines the problem before it selects the technology; Entropy Pooling is one implementation of the distributional route.

A direct heuristic is a controlled approximation:
Views → Portfolio Weights

A distributional view implementation contains the full architecture:
Views → Distribution → Optimization → Portfolio Weights

Route One: The Predefined Weight Heuristic

The heuristic route is appropriate for a narrow, controlled problem.

Suppose the committee judges European equities moderately more attractive than U.S. equities over six months. A direct mapping is defensible if the view concerns only relative expected return, dependencies are assumed stable, one reference portfolio and fixed active-risk budget apply, and the tilt was calibrated ex ante.

The translation can then be:

Moderately positive Europe-versus-U.S. view
→ Europe +2 percentage points
→ U.S. −2 percentage points

This is a deliberate approximation, not a complete solution to the portfolio problem. It is useful when the ignored interactions are immaterial relative to transparency, stability and governance discipline.

The institution should record that only the relative mean changes, the risk structure stays fixed, no other view interacts materially and active risk remains within budget. The rule therefore assumes that only the relative-return spread changes. That may be acceptable for a small, isolated tilt, but it should not be mistaken for a complete market view.

Where the Shortcut Breaks

Several Expected-Return and Factor Views Interact

Now suppose the committee is simultaneously (i) positive Europe, (ii) negative U.S., (iii) positive duration and (iv) negative credit.

Each score could receive a separate weight adjustment, but four scores need not represent four independent views. Europe and U.S. may be the two legs of one relative-value view; positive government duration and negative credit may both express a disinflationary growth-slowdown view.

Mechanical implementation can therefore amplify two convictions while making them look like four bets. Separate heuristics do not automatically capture offsets, common factor exposures, total risk, diversification loss or concentration.

Once several views interact, the institution must evaluate them jointly. Updating one weight at a time no longer represents the portfolio problem; under these conditions, an asset-by-asset mapping is no longer a defensible representation of the combined view.

Regime and Risk Views Change the Distribution

Now consider the statement:

An inflation-led slowdown has become more likely.

An inflation-led slowdown is not a one-asset view. It can change expected returns, volatilities, correlations and tail dependence together (Guidolin and Timmermann, 2007; Campbell, Sunderam and Viceira, 2017).

It may imply weaker equity returns, persistent yields, higher bond volatility, a less negative stock–bond correlation, wider credit spreads and greater downside concentration.

The same applies to views on volatility, CVaR, drawdowns or stress probabilities: they concern distribution shape or dependence, not only one mean.

Direct rules would then have to reconcile competing actions—more or less duration, less credit, more cash or real assets—without showing how the beliefs interact.

Once several dimensions move together, a simple mapping becomes difficult to calibrate consistently.

The problem is now the joint return distribution, not an individual asset score. A fixed score-to-weight mapping cannot represent these changes coherently without embedding an implicit distributional model and portfolio objective.

The Same View Enters Different Portfolios

Two mandates may share the same Europe-over-U.S. view but start from different exposures, benchmarks, liquidity constraints and active-risk budgets.

The market belief is the same; the appropriate trade is not.

The same institutional view can imply different trades across portfolios.

Define the common view independently; portfolio construction then combines it with mandate-specific holdings, objectives and constraints.

Route Two: Views, Distribution, Optimization

For interacting or distributional views, the coherent sequence is Views → Posterior Joint Distribution → Portfolio Optimization → Candidate Weights.

Views into a Joint Distribution

The Institutional View Package specifies the view object, horizon, reference point, target or bound, confidence treatment and review trigger.

The distributional layer combines that package with a prior representation of market outcomes—in this series, a Scenario Atlas of internally consistent multi-period cross-asset paths and probabilities.

Entropy Pooling reweights those scenarios to satisfy the governed views while minimizing the change from the prior (Meucci, 2008; Meucci, Ardia and Colasante, 2014).

Figure 4. Entropy Pooling can transmit a view to an unviewed asset because complete joint scenarios are reweighted. The direction and magnitude depend on the prior dependence structure. Source: Stylized illustration; not empirical.

Because complete scenarios are reweighted, expected returns, volatilities, correlations, tail probabilities, stress-state probabilities and path-risk measures can change jointly.

The central principle is broader than the specific method:

Views change the modeled market distribution.
They do not directly choose portfolio weights.

Distribution into Portfolio Weights

Portfolio construction combines the posterior return distribution with the objective, current holdings, risk budget, mandate constraints, liquidity, turnover and costs. It produces one or more candidate portfolios.

Optimization should be governed, constrained and tested for stability rather than treated as an automatic answer.

Depending on the mandate, that may mean scenario-based or robust optimization, minimum-change rules, CVaR or drawdown constraints, candidate comparisons or cross-atlas stability tests (Rockafellar and Uryasev, 2000).

Poorly calibrated optimizers can generate unstable or economically unreasonable shifts. That is a reason to constrain and validate optimization—not to collapse complex views directly into weights.

The optimizer is a translation layer, not an institutional authority. The institution still owns the objective, risk tolerance, constraints, overrides and final approval.

For complex views, this architecture is structurally superior because it keeps beliefs, distributional implications and portfolio preferences separate.

A View Can Move Assets You Never Viewed

A joint distribution also means the committee need not formulate a separate view on every asset: Entropy Pooling reweights complete cross-asset scenarios, not isolated series.

Suppose the institution has a positive U.S.-equity view but no U.S.-high-yield view. If equity-favorable scenarios also contain stronger high-yield returns, their higher posterior weights raise the high-yield expected return as an indirect consequence. Negative or state-dependent dependence can produce a different spillover.

The committee can therefore express only the views it holds, while the prior joint structure transmits their implications to unviewed assets. Correlation is only one summary; the full scenario dependence matters.

That benefit is also a model-risk boundary: misspecified dependencies propagate misspecified effects.

Worked Example: An Inflation-Led Slowdown

Assume the prior Scenario Atlas contains a soft landing, disinflationary recession, inflation-led slowdown, persistent growth and liquidity stress.

After reviewing new evidence, the committee approves an Institutional View Package containing three connected statements:

1. An inflation-led slowdown is more likely than in the prior.

2. Stock–bond correlation is likely to be less negative in downside environments.

3. Credit and equity tail risks are likely to be greater than implied by the prior.

A direct heuristic might reduce equities and credit, cut or add duration, increase cash and add real assets.

None of these trades is necessarily wrong.

The problem is that the list does not show how the views interact, which joint paths deserve more probability or how much aggregate risk the positions create.

The distributional route addresses that problem in two steps.

Step 1: Change the Joint Distribution

The distributional layer assigns more probability to paths with weak growth, persistent inflation, stressed credit, less favorable stock–bond dependence and more severe equity downside.

Means, volatilities, correlations and tails change together because the probabilities of internally consistent scenarios change.

Figure 5. A regime view changes probability mass across complete market worlds; portfolio weights are determined only after the posterior is constructed. Source: Stylized illustration; probabilities are not estimates.

Step 2: Construct the Portfolio

Portfolio construction evaluates the posterior under each mandate’s objective, holdings, risk budget, benchmark, liquidity profile and constraints.

Government-bond weights may rise if improved expected return dominates weaker hedging properties—or fall if the loss of diversification matters more.

An LDI mandate, balanced portfolio and high-tracking-error total-return mandate can therefore receive different weights from the same Institutional View Package.

That is not ambiguity in the view; it is portfolio-specific implementation.

Use It When — and When Not To

Use a predefined heuristic when the view is simple and isolated, one mean or spread changes, the risk structure is fixed, the tilt is small and the mapping was calibrated ex ante.

Use the distributional route when views interact, regimes or tail behavior may change, the same view enters different mandates or binding constraints shape implementation.

Do not use the Factory to outsource risk appetite, treat Entropy Pooling as a voting rule, force unsupported views into the prior or auto-approve an optimizer result.

Common Mistakes

Calling an Intended Trade a View

“Increase equities by three percentage points” is a portfolio action – not a view. It does not state what the institution believes about market outcomes. The distinction is critical.

Hiding One Regime View Inside Several Scores

A single macro conviction may affect many assets through the same mechanism. Treating each score independently can multiply one belief into several overlapping positions.

Applying the Mapping After Seeing the Desired Portfolio

A heuristic rule must be calibrated / stated ex ante. Otherwise, it becomes discretionary optimization without an explicit objective or reproducible rule.

Assuming Optimization Eliminates Governance

An optimizer cannot choose the institution’s objective, risk tolerance, decision rights or permitted overrides. Its results require validation and institutional approval.

Limitations

The Factory does not create forecasting skill, complete the scenario atlas or stabilize a poor optimizer. It only makes the translation chain explicit; success still depends on the prior, views, objective, constraints and implementation.

The Complexity Gate is a governance test, not a mathematically exact threshold; it forces the institution to state which interactions it is ignoring.

A small organization with one portfolio and narrow active views may rationally stay with heuristics.

A larger multi-asset institution should treat them as controlled approximations, not a complete architecture.

Bottom line: Use a predefined weight heuristic only when the view is simple and isolated; otherwise update the joint distribution before portfolio construction determines the weights.

The Question to Take to Your Investment Committee

Does this view change one isolated expected return—or does it change the joint distribution from which our portfolio should be built?

The Deeper Version

The Committee Is Not an Investor explains why discussion alone cannot provide a reliable institutional aggregation rule.

The technical version follows in From Assumptions to Portfolios: a governed Institutional View Package changes scenario probabilities through Entropy Pooling, and portfolio construction translates the posterior distribution into decision-ready candidate weights.

Selected Sources and Method Boundary

Scherer, B. (2026). Investment Committees: Governance and Design Choices. CFA Institute Research Foundation. DOI: 10.56227/26.1.13.

Best, M. J., and Grauer, R. R. (1991). “On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means.” Review of Financial Studies 4(2), 315–342;

DeMiguel, V., Garlappi, L., and Uppal, R. (2009). “Optimal Versus Naive Diversification.” Review of Financial Studies 22(5), 1915–1953.

Guidolin, M., and Timmermann, A. (2007). “Asset Allocation Under Multivariate Regime Switching.” Journal of Economic Dynamics and Control 31(11), 3503–3544;

Campbell, J. Y., Sunderam, A., and Viceira, L. M. (2017). “Inflation Bets or Deflation Hedges?” Critical Finance Review 6(2), 263–301.

Meucci, A. (2008). “Fully Flexible Views: Theory and Practice.” Risk, October, 97–102;

Meucci, A., Ardia, D., and Colasante, M. (2014). “Portfolio Construction and Systematic Trading with Factor Entropy Pooling.” Risk 27(5), 56–61.

Rockafellar, R. T., and Uryasev, S. (2000). “Optimization of Conditional Value-at-Risk.” Journal of Risk 2(3), 21–42.

Method boundary: The Institutional View Factory and Distributional Complexity Gate are design choices developed in this article. The cited literature supports their component mechanisms; it does not establish universal empirical superiority of this exact governance architecture.

This article is for research and educational purposes and is not investment advice. All examples are illustrative.

The first article in this series established the primitive objective: a Scenario Atlas of multi-period asset paths rather than a covariance matrix.

This second article addresses the next design problem. How can 782 synchronized weekly observations become thousands of plausible multi-year paths without destroying information that matters for portfolios?

Three key points:

• The empirical results are more nuanced than a simple rejection of i.i.d. returns. Raw weekly returns show limited robust serial dependence. The stronger evidence lies in uneven second-moment dynamics, changing cross-asset dependence, synchronized tail events, and descriptive state dependence.

• Those findings motivate three responses—preserve, condition, and filter—while also clarifying the trade-off between an empirical-first architecture and an invariance-first model.

• Empirical-first and invariance-first are legitimate, complementary routes. One preserves more observed structure; the other imposes more model structure to obtain cleaner innovations. Neither is universally superior. The data decides which one is better suited.

[Update 15.08.26: We changed the data foundation from Monthly to Weekly Data]

The Atlas Needs More Than a Shuffle

The previous article strongly advised to change replace a simple asset covariance in favor of a scenario atlas for the portfolio process. Building possible “worlds” e.g. future asset paths is the major first step of the portfolio process and only afterwards we can place investment views on them second, and eventually optimize the portfolio construction. That architecture makes multi-period paths—not a covariance matrix—the central object.

But choosing the object is easier than constructing it.

The complete panel used here contains 782 synchronized weekly observations from January 2011 through December 2025. Each observation is a 12-dimensional vector: eleven investable asset returns plus EUR/USD as an auxiliary risk driver. The Scenario Atlas will later require thousands of paths extending up to 60 months or 260 weeks. We therefore need to turn a relatively short historical panel into a larger set of possible futures.

The naive answer is to shuffle history. Draw a historical week, then another, then another, until a path is complete. That procedure looks empirical because every return came from the data. Yet the word “historical” does not tell us what the correct resampling unit is. A historical building block could be an individual asset return, a synchronized cross-asset vector, a short sequence of vectors, a return conditional on a market state, or a standardized shock separated from its volatility environment.

Each choice preserves some information and destroys other information - this is the key trade-off between all methods. Article 2 exists to make that choice explicit.

Historical returns are the raw material of the Scenario Atlas. They are not yet the scenario model.

The decision job is therefore narrower than finding the “true” return-generating process. We need to identify which features of the historical record are sufficiently important that a scenario engine should preserve them, condition on them, or filter them before resampling.

The i.i.d. Benchmark Is Useful—but Only a Benchmark

Simple single-week resampling has a clean benchmark. If the synchronized return vectors were independent and identically distributed,

r_t ∼ i.i.d. F

then the probability law of an H-week path would factor into the product of the same one-week distribution across all H dates. Simply put: We draw a return vector (not individual asset returns (!)) H-times from the historical returns and chain them together. Historical weeks would be exchangeable. Their ordering would add no information, and the relevant distribution would not change with the market state.

This benchmark is useful because it tells us what a naive bootstrap assumes. It does not require Gaussian returns: an i.i.d. process may still be skewed or fat-tailed. It does require, however, that the past does not materially change the distribution of the next draw.

But are the assumptions of this benchmark legit? If you are an economist like me, think about it this way: this benchmark has a very strong a priori view on the data world. However, like in economics, a model is only good if its assumptions are valid – and this is in fact the key problem when the model hits data reality.

Stationarity provides a useful alternative, but it is not a free pass. Every i.i.d. invariant is strictly stationary; the reverse is not true. A stationary process can still contain serial dependence, volatility clustering, and nonlinear dynamics. That distinction determines the resampling method. Independent single draws require an i.i.d.-type assumption. Block-based resampling can instead work with stationary, weakly dependent observations because relevant dependence is retained within the blocks, subject to additional regularity conditions. For a full scenario distribution, covariance stationarity alone is not enough: stable means and autocovariances do not guarantee stable tails or multivariate dependence.

The practical question is therefore not whether preprocessing can make the data “stationary.” It is whether the resulting representation is sufficiently stable for the chosen scenario engine—and whether that engine preserves the dependence that remains.

Raw Returns Show Limited Serial Memory

Figure 1A begins with the most familiar diagnostic: weekly return autocorrelation over lags one through 52. The four displayed series show several isolated values outside the approximate pointwise white-noise reference bands - thereby indicating cases of raw return autocorrelation.

Figure 1A. Raw weekly return autocorrelation, lags 1–52. Dashed lines are approximate pointwise 95% white-noise references, not simultaneous confidence bands. The portmanteau and nonlinear-dependence tests indicate that the isolated bars do not translate into robust broad serial dependence outside cash. January 2011–December 2025, weekly EUR log returns where applicable.

Looking at individual bars alone can make the return process appear more persistent than the evidence warrants. The Ljung–Box test aggregates autocorrelation results. After correcting across the twelve driver-level tests, 4 non-cash raw-return series rejects joint zero autocorrelation at the 5% level. The nonlinear result points in the same direction - as measured by the Schweizer–Wolff approach

Table 1. Publication-facing summary of serial-dependence diagnostics. Cash is excluded because its return is mechanically inherited from the persistent short-rate level.

This is a useful result. Autocorrelation in weekly returns is present, but limited. Very strong and stable predictability in raw weekly returns would raise questions about market efficiency, data construction, or stale pricing. Instead, the panel looks closer to weakly serially dependent than to strongly persistent.

The implication for scenario design is precise. Overall, the evidence provides some support for long blocks chosen as weekly returns show some univariate memory. For some asset, it clearly does not justify independent asset-by-asset shuffling. The case for synchronized blocks will come mainly from multivariate market structure and joint stress behavior, with short-run sequencing as a secondary consideration.

Volatility Persistence Is Uneven Across Assets

Even low raw-return autocorrelation does not imply i.i.d. returns. A process can have little predictable direction while the size of its moves remains dependent through time. Squared demeaned returns provide a transparent proxy for that second-moment dependence – often arising as Volatility Clustering.

Figure 1B makes the heterogeneity visible. Euro government bonds exhibit broad positive autocorrelation in squared weekly returns across much of the 52-week horizon. Euro investment-grade credit shows an especially large lag-one response that decays relatively quickly, while Global DM ex-EMU displays a weaker pattern concentrated at the shortest lags. Commodities show little systematic persistence. The Ljung–Box Q(52) statistics are approximately 510.9 for Euro government bonds, 320.4 for Euro investment-grade credit, and 281.9 for EUR-hedged global government bonds; all remain highly significant after false-discovery-rate correction. Global DM ex-EMU is also significant, with Q(52) = 158.4 and an adjusted p-value of approximately zero. Commodities remain the exception, with Q(52) = 57.2 and an adjusted p-value of 0.289.

So, the evidence of volatility persistence is overall overwhealming.

Figure 1B. Autocorrelation of squared demeaned weekly returns, lags 1–52. Magnitude dependence is strong in selected fixed-income series but weak in the displayed equity and commodity series. January 2011–December 2025.

This is not evidence that “all assets cluster volatility.” It is evidence that second-moment dynamics are materially different across the panel. A universal filter applied identically to every asset would therefore be difficult to defend. The diagnostic also does not identify the economic cause. Persistent rate trends, duration exposure, liquidity, or proxy construction may all contribute to the fixed-income patterns. The result tells us where filtering may matter; it does not tell us which filter is correct.

Figure 2 shows the economic magnitude of the same broader problem. 52-week realized volatility is far from constant. Global DM ex-EMU ranges from about 8% to 25.9% annualized. Commodities range from 8% to 25%. Euro government bonds move from roughly 1.8% to 12.3%, and Euro High Yield from 1.5% to 15.9%.
[The lower vola of High Yield relative to Gov. Bonds might be surprising, but its duration is very low thereby reducing the sensitivity to interest rate / Inflation shocks. However, have a look at what is happening to Euro High Yield during Covid 19…there you can see the real risk.]

Figure 2. 52-week rolling annualized volatility. The level of realized risk varies materially across time and assets. The overlapping windows make this an economic state chart, not an independent test of serial dependence. Source: QuantStrategy calculations; January 2011–December 2025.

The rolling windows overlap: adjacent estimates share 51 of 52 observations. Their smoothness is therefore partly mechanical, and the chart is not a formal test of volatility clustering. It shows that the chance of an (absolute) high return depends on the current state: The same numerical return can represent a moderate standardized shock in a calm state or a small standardized shock in a stressed state.

For scenario construction, this creates a genuine choice. One engine may preserve the full historical observation, including its original volatility environment. Another may separate the shock from that environment and rescale it. Filtered Historical Simulation follows the second route—but the evidence suggests that the value of filtering is asset-specific rather than universal.

Cross-Asset Dependence Is Part of the Observation

The strongest argument for preserving history is not univariate autocorrelation. It is the joint market structure contained in each synchronized return vector. Varying correlations and especially correlation tightening during adverse shocks is the true nightmare of any portfolio constructor. But to cope with difficulties such as varying correlation, the mature approach is the handle it in your data set – and not simply ignoring it by using a simple covariance matrix.

Figure 3 shows 104-weeks rolling correlations between Global DM ex-EMU and three other assets. The equity–government-bond relationship ranges from approximately -0.2 to +0.25 and changes sign within the sample. Equity–Euro High Yield moves within tight boundaries. Equity–commodities remains mostly positive but varies from almost zero to about 0.7.

These shifts have an economic interpretation. Correlation is not a permanent property attached to two asset labels; it is an outcome of the shocks dominating the window. A growth-led downturn can push equities lower while high-quality government bonds rally. An inflation or policy-credibility shock can pressure both at the same time. Credit may behave like carry in calm periods and like equity beta when liquidity deteriorates (see my article here).

The chart does not identify formal regimes, and the rolling estimates use overlapping windows. It nevertheless provides descriptive evidence that cross-asset relationships vary materially through time. For scenario construction, the relevant historical resampling unit is therefore not an isolated return from each asset, but the synchronized return vector over multiple periods analysed jointly:

r_t = (r_1,t , … , r_52,t )′

Each vector records how all twelve drivers moved in the same historical week. Resampling the assets independently would break that joint realization and create combinations that were never observed together.

Figure 3. 104-week rolling cross-asset correlations with Global DM ex-EMU. Dependence changes materially through time, and the equity–government-bond relationship changes sign. Source: QuantStrategy calculations; January 2011–December 2025.

Tail behavior makes the same point more directly. Figure 4 counts weeks in which at least two of the twelve series fall below their own empirical 5% lower-tail threshold. March 2020 places eight series in their individual lower tails simultaneously. April 2022 places six there, while December 2022 places five. The broader 2022 tightening cycle also produces several four-series events.

A portfolio crisis is not a collection of independent bad asset returns. It is a joint realization. Resampling each asset separately would fragment precisely the stress structure that the Scenario Atlas is meant to keep visible.

At the same time, the figure exposes the limitation of history. With 782 weekly observations, a 5% empirical tail contains roughly nine observations per driver. The observed events are valuable building blocks, but they cannot provide complete support for every plausible liquidity crisis, inflation shock, or policy error. Preserving observed stress and augmenting missing stress are different tasks. The former belongs to the scenario engines; the latter returns in the Atlas Coverage article.

Figure 4. Historical simultaneous lower-tail events. A point marks a week in which at least two series fall below their own empirical 5% threshold. The labeled episodes show how portfolio stress arrives as a synchronized market event. January 2011–December 2025.

State Dependence Is Suggestive, Not a Law

Conditional resampling asks whether every historical week should receive the same relevance at every starting point. Figure 5 offers a deliberately simple diagnostic. It sorts observations into three buckets using the trailing 52-week volatility of Global DM ex-EMU, then compares the average realized returns over the following 52 weeks.

The results differ across states, but they do not form a monotonic rule. Global DM ex-EMU averages about 18.9% after high-volatility observations, Below 10% after medium-volatility observations, and ~ 12% after high-volatility observations. Euro High Yield averages 2.6%, 1.7%, and 8,5%. Gold and Euro government bonds also show a low–middle–high pattern rather than a simple upward or downward slope.

That non-monotonicity is important. The chart does not say that high volatility forecasts high returns, nor that the low-volatility state is always superior. The 52-week forward windows overlap, and the bucket analysis is descriptive rather than an inference test. It shows only that the conditional historical outcome distribution may vary with the starting state:

F(R_t:t+52 | s_t ) may differ across states

Figure 5. Mean 52-week forward simple returns by observed Global DM ex-EMU volatility state. Conditional outcomes differ across the three buckets, but the relationship is non-monotonic and should not be read as a forecast. overlapping forward windows, January 2011–December 2025.

This is enough to motivate state-aware weighting as a modeling dimension. It is not enough to validate a particular similarity metric, kernel, or neighbor count. Set B must earn those choices through sensitivity analysis and out-of-sample diagnostics. The correct inference from Figure 5 is not “use this signal”, but that the most recent history might affect data distributions. It also warns that “equal relevance for all history is also an assumption” which is made by many practitioners rather implicitly.

Why is this so important? Think about the following case: You want to build up a portfolio for the next 52 weeks. As the current starting point of the market determines the multivariate distribution, the portfolio constructor must take this into account to build a well-behaved portfolio. A market after a strong recession will behave differently than after a multi-year boom. And to acknowledge this correctly, we must embrace state dependence as an important concept.

From Diagnostics to Scenario Design

The empirical findings do not produce one scenario engine. They identify three different modeling responses.

Preserve. Keep historically observed structure intact when breaking it would destroy economically relevant information. In this article, the strongest preserve arguments are synchronized cross-asset vectors, changing dependence, and joint tail events. Short blocks may also retain limited sequencing, but the weak raw-return portmanteau evidence argues against using long blocks merely to manufacture persistence.

Condition. Allow the starting state to affect which historical observations receive more weight. The state analysis suggests that unconditional exchangeability may be too strong, but it does not dictate how similarity should be measured. Conditional Similarity Resampling will therefore treat the state definition as a model choice that must be diagnosed.

Filter. Separate a dynamic component from the empirical shock when that separation makes the building blocks more stable or portable. The squared-return results show a clear use case for volatility filtering in some series and a weaker case in others. Filtered Historical Simulation should respond to this heterogeneity rather than impose identical dynamics everywhere.

We need to know which dependencies to preserve, which states to condition on, and which dynamics to filter before resampling history.

These responses map directly to the next three Scenario Atlas engines:

Different answer to a different feature of the return panel. They are also complements rather than a substitutes. Later articles will compare what each set preserves, what it changes, and where it becomes fragile.

Nothing here is an investment view. Block 1 is still constructing the prior set of possible paths. Probability tilts and institutional views enter only after the Atlas exists.

Are those approaches the only ones you might use? Of course not. The three approaches chosen are based on the results of the data analysis, but also partly due to educational reasons in order to present that different approaches can handle different data issues. A promising approach to combine the elements mentioned here can be found here.

Two Legitimate Routes From History to Scenarios

There is a serious alternative to this empirical-first workflow: model the conditional dynamics first and search for more invariant innovations.

In an invariance-first architecture, one might write

r_t = μ_t + σ_t z_t

model the conditional mean and volatility, transform the observations into standardized shocks zt, estimate the dependence among those shocks—possibly with a copula or factor model—and then reconstruct future returns. If the model is adequate, this route produces statistically cleaner building blocks, a more explicit parameterization, and a disciplined way to extrapolate beyond the exact combinations observed in history.

That structure can improve robustness. It can also create model uncertainty. The analyst must choose the conditional dynamics, distribution, dependence model, estimation window, parameter restrictions, and reconstruction rule. A clean residual series does not make those choices disappear; it shifts the burden from the sample into the model.

Empirical-first pays a different price. It preserves more observed co-movement and makes historical building blocks directly inspectable. But it relies more heavily on the representativeness of a short sample, has limited support in the tails, and introduces tuning choices through block length, state variables, and similarity weights. Its assumptions are often less formally parameterized, even when they are intuitively visible.

Cleaner innovations achieved from a parametric modelling do not mean fewer assumptions. They move assumptions from the historical sample into the model.

Figure 6 summarizes the trade-off. The two routes are complementary, and neither is universally superior. The appropriate choice depends on the amount of data, the dimensionality of the universe, the stability of the economic mechanism, and the decision the scenarios must support.

Set C is deliberately hybrid. It models conditional volatility—an invariance-first step—but resamples the standardized shocks empirically rather than drawing them from a fully specified distribution. That is why the broader design principle of this series is not “nonparametric at all costs.” It is empirical-first, parametric where useful.

Figure 6. Two legitimate routes connect historical returns to scenario paths. Empirical-first methods retain more observed market structure and therefore rely more heavily on the representativeness and support of the historical sample. Invariance-first methods model conditional dynamics to obtain cleaner innovations, but shift more uncertainty into model specification, parameter estimation, and scenario reconstruction. Filtered Historical Simulation sits between the two routes. stylized illustration.

Where This Design Can Fail

The diagnostics improve the architecture only if they are interpreted with discipline.

Non-rejection is not proof of i.i.d. behavior. Ljung–Box focuses on linear autocorrelation, Schweizer–Wolff is applied pairwise across lags, and neither establishes stability of the full multivariate distribution through time. Weekly frequency also hides dependence that may be visible in daily data.

Rolling charts are not regime tests. Rolling volatility and correlation estimates are economically informative but mechanically smooth because their windows overlap. They should motivate design questions, not be treated as latent-state classifications.

Block length can manufacture persistence. The raw-return evidence is weak enough that long historical blocks would require an independent justification. Set A should test block-length sensitivity rather than assume that more preservation is always better.

State conditioning can overfit a small sample. The state buckets use overlapping forward returns and provide no forecast validation. Set B can look convincing while concentrating probability on a small set of historical analogues.

Filtering can create false cleanliness. A volatility model may produce standardized residuals that look better diagnostically while misrepresenting leverage effects, cross-asset dependence, or structural breaks. Set C must diagnose both the residuals and the reconstructed paths.

History has zero weight on worlds it never observed. The sample excludes the global financial crisis and earlier inflation regimes. No resampling method can recover support that does not exist. Coverage augmentation must therefore remain explicit rather than hidden inside a fitted model.

Why Model Choice Is a Governance Decision

Article 2 does not identify a single preferred model. It establishes a governance test for every scenario process:

What does your scenario generating process (i) preserve, what does it (ii) transform, and what does it (iii) assume?

This matters because the scenario engine defines the support of the prior. Entropy Pooling can reweight existing worlds, and portfolio construction can allocate across them. Neither can recover a world omitted upstream or restore dependence destroyed during scenario generation.

Therefore, the choices made during this first stage of your investment process are crucial and will have a major impact on your view implementation and eventually portfolio construction. Based on my experience, especially the element of implicit, not clearly stated assumptions can be detrimental to an investment process

Model choice is therefore part of the investment decision. It belongs in the approval package—not only in the codebase.

Bottom Line & Outlook

Historical weekly returns are not so serially pathological that every observation must first be modeled away. Nor are they ready-made scenarios that can be shuffled without consequence. The evidence supports a selective architecture: preserve synchronized market structure, condition cautiously on observable states, and filter dynamics where second moments require it.

The first engine makes the smallest possible modeling claim. Before inventing more market dynamics, it tries to preserve more of the dynamics history already gave us.

Decision question: Which portfolio-relevant feature of your historical data would your current scenario generator destroy first?

Next in the Series

Synchronized Historical Block Bootstrap — building the empirical baseline from short, synchronized cross-asset blocks and diagnosing what that construction preserves and distorts.

This article is for research and educational purposes only. It does not constitute investment advice. Historical market behavior is not a reliable guide to future outcomes, and all scenario methods described here remain illustrative modeling choices.

Reproducibility and Licensing

The diagnostic code, tests, and metadata contract are designed to be inspectable in the public QuantStrategy repository. Each weekly run records the 782-by-12 W-FRI panel contract from 7 January 2011 through 26 December 2025, autocorrelation lags 1–52, Ljung–Box tests at 26 and 52 weeks, the 999-permutation Schweizer–Wolff reference evaluated at twelve predeclared weekly lags, the Benjamini–Hochberg false-discovery-rate adjustment, 52- and 104-week rolling windows, the 5% tail threshold, and the 52-week state definition. It also records source provenance and a checksum identifying the local input snapshot.

Source-derived panel snapshots and generated run artifacts are kept outside Git. The Yahoo/yfinance and public iShares sources are accessible without a paid subscription but are not Open Data, and their terms do not guarantee redistribution or publication rights. The repository therefore provides the code and documented workflow required to recreate the analysis locally, subject to source availability and provider terms.

Information regarding the data setup you can see here.

Sources and Methodological Notes

1. Ljung, G. M., and G. E. P. Box (1978). “On a Measure of Lack of Fit in Time Series Models.” Biometrika 65(2), 297–303. DOI: 10.1093/biomet/65.2.297.

2. McLeod, A. I., and W. K. Li (1983). “Diagnostic Checking ARMA Time Series Models Using Squared-Residual Autocorrelations.” Journal of Time Series Analysis 4(4), 269–273. DOI: 10.1111/j.1467-9892.1983.tb00373.x.

3. Schweizer, B., and E. F. Wolff (1981). “On Nonparametric Measures of Dependence for Random Variables.” The Annals of Statistics 9(4), 879–885. DOI: 10.1214/aos/1176345528.

4. Benjamini, Y., and Y. Hochberg (1995). “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society: Series B 57(1), 289–300. DOI: 10.1111/j.2517-6161.1995.tb02031.x.

5. Meucci, A. (2005/2007). Risk and Asset Allocation. Springer. DOI: 10.1007/978-3-540-27904-4.

6. Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer. DOI: 10.1007/978-1-4757-3803-2.

7. Vorobets, A. (2024, revised 2026). “Portfolio Construction and Risk Management”.

In the first case, the result may depend too much on who speaks first, how the question is framed, or which narrative dominates. In the second, the previous decision may have become the default against which new evidence is judged. A reliable process therefore has to do two things at once: remain consistent when the investment case is materially unchanged and adapt when it has changed.

A reliable committee should reach the same decision from the same evidence—and change its view when new evidence warrants it.

Consider six investment professionals operating under two decision protocols. They receive the same evidence, work under the same mandate, and face the same portfolio constraints. Under the first protocol, the CIO frames the issue before the other members speak. Under the second, members record their initial assessments independently and the CIO speaks last. The group reaches a different institutional view even though nothing about the investment case changed. The process did.

This is a stylized illustration, not an account of a specific investment committee.

A Committee Is an Aggregation System

Investment committees are often treated as larger, “better” investors that benefit from collective intelligence. The intuition is appealing: bring several experienced professionals together, give them the relevant research, let them challenge one another, and expect a better view than any individual could produce. But a committee is not one investor with more information.

An individual investor must move from evidence to interpretation, from interpretation to a view, and from that view to a decision. A committee must also compare different information sets, reconcile different definitions of the same view, assign weight to different judgments, produce one institutional view, and translate that view into a portfolio allocation. That transition from “view” to portfolio is not administrative. It is the defining task of the institution.

Most committees formalize the inputs and the output (views & portfolio) but leave the mapping between them largely implicit. Research decks are archived, minutes are written, and decisions are recorded, yet the mechanism that turned several judgments into one institutional view may remain hard to reconstruct. The prevailing process often looks like this:

Research deck
→ committee discussion
→ apparent consensus
→ CIO summary
→ portfolio action

The sequence feels reasonable. What is missing is the mapping rule. How were the views weighted? Did evidence quality or confidence matter? What did silence mean? Who converted “cautious on duration” into a specific position? Consensus tells us how the meeting ended; it does not explain how dispersed judgments became the institutional view—or how that view became portfolio risk.

When no explicit mapping rule exists, the meeting begins to supply one. To see how, it helps to separate two stages of the process.

First, the inputs are already filtered. Before the discussion begins, each member has converted a large and ambiguous evidence set into a personal judgment. Supporting information may attract more attention than disconfirming evidence (confirmation bias / selective exposure). A vivid recent development may outweigh a broader historical record (recency bias / availability bias). The existing portfolio can become the reference point for every alternative (status-quo bias / anchoring), while a coherent explanation may feel more convincing than stronger but fragmented facts (narrative bias / coherence bias).

The committee therefore does not receive raw evidence. It receives several selected and interpreted versions of the investment case. López de Prado captures the risk in deliberately blunt practitioner language:

“Each attendee seems obsessed about one particular piece of anecdotal information, and giant argumentative leaps are made without fact-based, empirical evidence.”[2]

His observation concerns meetings of discretionary portfolio managers, not investment committees as a general category; it illustrates the mechanism, not its prevalence.

Second, the meeting reweights those inputs. The first substantive contribution can establish the reference point for the discussion (anchoring / primacy effect). The same argument may carry more influence when voiced by a senior or highly confident member (authority bias / status-weighted influence). Information already known by several members is more likely to be repeated, while unique information may receive less attention (shared-information bias). As the direction of the room becomes visible, later contributions may move toward it (conformity / information cascade). Once a position has been defended publicly, revising it can also carry psychological and reputational costs (commitment / cognitive dissonance / sunk-cost fallacy).

The first layer determines what enters the room; the second determines what counts once it is there. Better discussion cannot recover evidence that was filtered out beforehand, while better individual analysis cannot protect a view from a poorly designed aggregation process. None of these mechanisms must decide the outcome alone. Together, however, they can act as hidden weights.

When conversation is the aggregation rule, speaking order, status, and narrative become hidden portfolio inputs.

Two reliability problems follow. Process-induced variation occurs when materially comparable evidence, mandates, constraints, and starting portfolios lead to different institutional decisions because the meeting unfolds differently. Status-quo inertia is the opposite failure: material evidence changes, but the previous position remains the default and is never genuinely reconsidered. One creates unnecessary change; the other prevents necessary change.

Figure 1. The first layer shapes what enters the room; the second shapes what counts once it is there. Source: Author framework; stylized illustration, no empirical effect sizes implied.

What the Evidence Can—and Cannot—Tell Us

Bernhard Scherer provides the most directly relevant institutional frame. He treats an investment committee as an information-aggregation technology and argues that meetings, minutes, and consensus language can substitute for an explicit mapping from dispersed information to portfolio weights. He describes traditional conversation as nonlinear, opaque, and status weighted, and explains why speaker rotation, formal dissent, and standardized materials can improve inputs without changing the mapping to the implemented portfolio.[1]

That distinction is central here: meeting hygiene and decision architecture are not the same. Scherer’s group-shift results come from controlled LLM simulations rather than observed committees, so they help isolate a mechanism; they do not estimate its prevalence or magnitude in any institution.

Criscuolo and coauthors make sequence risk more concrete. In 588 R&D funding decisions at a large professional-services firm, project order was quasi-random. A proposal considered after a funded proposal received 23% less of the requested funding, and the effect strengthened later in the meeting. The setting is not investment management, but it shows that a formally irrelevant place in the agenda can alter a collective allocation decision.[3]

A classic hidden-profile experiment by Stasser and Titus illustrates a different failure mode. In a political-caucus task, discussion was dominated by information members already held in common and by evidence supporting their initial preferences; groups often selected the plurality’s initial favorite rather than the candidate favored by the full information set. This is general laboratory evidence, not an estimate for investment committees, but it shows how unique information can disappear even when it exists somewhere in the group.[4]

Tetlock and Gardner call the relevant habit “perpetual beta”: treating beliefs as provisional and subject to continuous testing. In Good Judgment Project data, Atanasov and coauthors found that the most accurate forecasters tended to update more frequently, in smaller increments, and to reconfirm their previous forecasts less often.[5] The institutional lesson is not that committees should change views frequently. It is that a changed view should be traceable to new evidence—and an unchanged view should be consciously re-underwritten.

Why Better Meetings Are Not Enough

Common meeting improvements include rotating the first speaker, appointing a devil’s advocate, inviting more diverse views, improving the research materials, allowing more time, and keeping a better record. Each can help: anchors may weaken, counterarguments may surface, inputs may become more comparable, and the institutional record may improve. But none of these measures determines how individual views become the institutional view or how that view becomes portfolio direction, size, and risk. A committee can have thoughtful debate and excellent minutes while still relying on an implicit aggregation rule.[1]

The answer is not to eliminate discussion. Deliberation can reveal information missing from the shared materials, test the causal reasoning behind a view, and expose implementation constraints. Those are valuable functions because they improve the inputs to the decision. What discussion should not do implicitly is determine whose view receives the greatest weight.

Discussion should improve the inputs. It should not silently determine the weights.

The View Is Not Yet the Position

Agreement on a market view is not yet a portfolio decision. “Cautious on duration,” “constructive on equities,” or “concerned about inflation” still has to be translated into an instrument, direction, size, horizon, and risk contribution.

If that mapping remains implicit, the committee can agree on the words while implementing materially different decisions. A strong view may produce a high-risk allocation; a moderate view may create a dominant risk contribution; the chosen instrument may not express the intended thesis at all. Better discussion can improve the view, but it cannot by itself specify how the view becomes portfolio risk.

A committee has not completed the decision when it agrees on the view. It has completed the decision only when the view has been translated into an explicit portfolio action.

The Reliability Test: Did the Investment Case Change?

Before comparing two committee decisions, we first need to establish whether the committee was addressing materially the same investment case. “Same evidence” is useful shorthand, but the relevant decision basis also includes the investment horizon, mandate, starting portfolio, constraints, implementation environment, and the decision rule in force.

The test compares two questions:

1. Did the decision-relevant investment case materially change?

2. Did the institutional decision materially change?

Two meetings can appear inconsistent while answering different questions. A three-month tactical view is not comparable with a three-year strategic assumption, and the same market view can rationally produce a different trade when the starting portfolio or a liquidity constraint has changed. The standard is not perfect repeatability. It is material comparability and explainable deviation.

Figure 2. A changed decision is not automatically inconsistent, and an unchanged decision is not automatically disciplined. Reliability depends on whether the decision moved when the underlying investment case moved. Source: Author framework; diagnostic framework, not a correctness score.

The matrix does not classify a decision as correct or incorrect. A stable decision can still be consistently wrong, a changed decision can be an intelligent update, and an unchanged position can be entirely rational. In every case, however, the committee should be able to show why the evidence, mandate, portfolio state, and decision rules produced the result.

The following questions serve two purposes. First, they assess whether the committee records the critical elements of the decision when it is made. Second, they test whether those records would allow the committee—or a competent outsider—to reconstruct later why the decision was taken, how it became a portfolio position, and what would have justified changing it.

The standard is therefore not whether a plausible explanation can be produced after the event. It is whether that explanation is supported by the contemporaneous decision record.

The questions are not a psychological scale or a compliance score, and they should not be added up mechanically. A single missing link between evidence, aggregation, implementation, or review may matter more than several well-documented inputs.

Eight Questions for Your Next Committee Review

1. Can We Reconstruct a Comparable Investment Case?

Question. Can we reconstruct two materially comparable decisions in a common format—including the evidence available at the time?

Why it matters. A different action may be rational if the evidence has changed. Without a common case definition, apparent inconsistency remains ambiguous.

Red flag. The record does not show whether the investment case changed—or only the meeting did.

2. Would the Process Produce the Same Decision Again?

Question. When the investment case was materially unchanged, did the committee reach a materially comparable decision—and can any deviation be explained without relying on meeting mood, speaker order, or status?

Why it matters. A stable investment case should not produce a materially different position merely because the meeting unfolded differently.

Red flag. The meeting choreography predicts the position better than the investment case.

3. Was the Existing Position Re-Underwritten?

Question. When material facts changed, did the committee explicitly re-underwrite the existing position? If the position remained unchanged, is the reason documented?

Why it matters. Keeping a position is a new decision. When the key facts change, the decision might change – but at least the reasons for keeping a position must be well documented.

Red flag. The main reason for the position is that it was already held last week.

4. Did Members Enter With Their Own Decision-Grade Views?

Question. Before the discussion began, did each member independently record their own view—including the object, horizon, reference point, confidence, key evidence, and intended portfolio implication?

Why it matters. Without independent starting views, the committee cannot distinguish genuine agreement from convergence produced during the meeting.

Red flag. Only the final institutional view is recorded, leaving no evidence of what members believed before the discussion began. That leaves no traceable accountability for the individual inputs that shaped the final view.

5. Can We Trace Individual Updates?

Question. Between the previous and current committee meetings—and, where relevant, during the meeting—can we see which individual views changed, by how much, and because of which evidence or reasoning?

Why it matters. Traceable updates show whether opinions changed because the evidence changed or merely converged during the discussion.

Red flag. The institution says, “We changed our view,” but cannot reconstruct who updated, when, or why.

This does not require a public forecasting league table. Initial views may remain anonymous during deliberation while still being reconstructable in a confidential governance review.

6. Can We Reproduce the View-to-Portfolio Mapping?

Question. Can the committee reproduce how member views became the institutional view—and how that view became direction, instrument, size, and overall portfolio allocation?

Why it matters. A view is not decision-grade until its translation into portfolio allocation can be reproduced.

Red flag. The committee claims the right view, but the implemented portfolio expressed something else. Alternatively: Same views generate different portfolio allocations.

7. Was the Review Contract Defined in Advance?

Question. At approval, did the committee record what should happen if the thesis was right, what would weaken or falsify it, when the position would be reviewed, and who owned the next decision?

Why it matters. An ex-ante review contract prevents the thesis, horizon, and success criteria from moving after the outcome becomes visible.

Red flag. Every adverse development extends the horizon or changes the original rationale.

8. Did We Evaluate the Decision Separately From the Outcome?

Question. Did the committee evaluate the quality of the original decision separately from whether the position ultimately made or lost money?

Why it matters. Investment outcomes reflect both decision quality and uncertainty. A sound decision can produce a loss, while a poorly reasoned decision can produce a favorable outcome.

Red flag. The profit or loss determines the verdict: profitable means good, and unprofitable means bad.

A favorable outcome does not repair a decision that was right for the wrong reasons.

The opposite is equally important. A well-reasoned decision can produce an adverse outcome because markets are uncertain. Review should not excuse every loss; it should prevent the committee from learning the wrong lesson from a single realization.[6]

What Did Your Answers Reveal?

You have now had the opportunity to test the process against eight straightforward questions. The useful question is not whether a plausible “yes” can be constructed after the fact, but which answers the contemporaneous decision record can actually support.

Perhaps every answer was a clear and documented “yes.” But some may have been harder. Individual starting views may not be recorded before the discussion. The committee may be unable to explain why a position survived material new evidence, or the institutional view may be clear while its translation into portfolio risk is not.

A “no” or “partly” does not prove that the current position is wrong, nor does it mean that the committee is dysfunctional. It identifies a part of the decision process that remains implicit, difficult to reproduce, or vulnerable to influences outside the investment case.

Could a competent outsider reconstruct why the available evidence led your investment committee to adopt its current portfolio position—and what would justify changing it?

That is the purpose of the questions: not to score the committee, but to locate the missing architecture. The next articles will address these gaps one by one—how evidence should be structured, how individual views should be formed and updated, how they should become an institutional view, how that view should be translated into portfolio risk, and how the decision should later be reviewed.

If the institution cannot reproduce or explain how evidence became the portfolio position, it does not yet have a reliable decision process—even when the portfolio happens to be right.

Next: The Institutional View Factory—From Evidence to Portfolio Input.

References and Evidence Notes

The sources below support the article’s principal claims. Evidence from non-investment settings is identified as such, and simulated magnitudes are not treated as estimates for real committees.

1. Bernhard (Bernd) Scherer, Investment Committees: Governance and Design Choices (CFA Institute Research Foundation, 2026), pp. 1–2, 13–14, 25–27, and 46–48. The cited sections frame the IC as an information-aggregation technology; describe discussion as nonlinear, opaque, and status weighted; explain why common procedural fixes do not by themselves change the mapping to portfolio weights; and state the external-validity limits of the controlled LLM simulations.

2. Marcos López de Prado, Advances in Financial Machine Learning (Wiley, 2018), p. 4, sec. 1.2.1, “The Sisyphus Paradigm.” The quotation is a practitioner observation about discretionary-PM meetings and is used as illustration rather than prevalence evidence.

3. Paola Criscuolo, Linus Dahlander, Thorsten Grohsjean, and Ammon Salter, “The Sequence Effect in Panel Decisions: Evidence from the Evaluation of Research and Development Projects,” Organization Science 32, no. 4 (2021): 987–1008, doi:10.1287/orsc.2020.1413. The study provides field evidence for sequence effects in a professional allocation panel; it is not an estimate for investment committees.

4. Garold Stasser and William Titus, “Pooling of Unshared Information in Group Decision Making: Biased Information Sampling During Discussion,” Journal of Personality and Social Psychology 48, no. 6 (1985): 1467–1478, doi:10.1037/0022-3514.48.6.1467. This is general laboratory evidence on shared and unique information, not a magnitude estimate for investment committees.

5. Philip E. Tetlock and Dan Gardner, Superforecasting: The Art and Science of Prediction (Crown, 2015), chap. 8, “Perpetual Beta,” p. 191; and Pavel Atanasov, Jens Witkowski, Lyle Ungar, Barbara Mellers, and Philip Tetlock, “Small Steps to Accuracy: Incremental Belief Updaters Are Better Forecasters,” Organizational Behavior and Human Decision Processes 160 (2020): 19–35, doi:10.1016/j.obhdp.2020.02.001.

6. Annie Duke, Thinking in Bets: Making Smarter Decisions When You Don’t Have All the Facts (Portfolio/Penguin, 2018), chap. 1, on “resulting”; and James Montier, Behavioural Investing: A Practitioner’s Guide to Applying Behavioural Finance (Wiley, 2007), pp. 24–25, on distinguishing reasons from outcomes. These sources support the decision-quality versus outcome-quality distinction; the full calibration framework is outside this article.

This article is for research and educational purposes and is not investment advice.

Which macro shock is your diversification designed to survive?

Your risk report may show a negative correlation between equities and government bonds. That number looks like diversification. But it is not a permanent property of the portfolio. It is an average of the market environments contained in the estimation window.

For much of the period from the late 1990s through 2020, US equity and government-bond returns were generally negatively correlated. The relationship moved back into positive territory around mid-2021, as inflation shocks became more prominent. A similarly prolonged positive relationship had last been observed in the 1980s and early 1990s. BIS discussion.

The portfolio weights did not change the relationship. The dominant economic shock did.

When growth is the dominant concern, equities and government bonds can move in opposite directions. When inflation dominates, both can lose together. A single full-sample covariance estimate compresses those states into one unconditional summary.

That estimate may be useful. It is not a complete risk model.

The Allocator’s Problem

Most investment committees receive expected volatility, correlation, covariance, risk-contribution and tracking-error statistics.

These are useful diagnostics. For a linear portfolio, a covariance matrix correctly aggregates second-moment risk, and it does not require returns to be normally distributed.

The stronger assumptions arise when a single historical estimate is treated as a sufficient description of portfolio risk. Covariance is most informative when portfolio exposures are approximately linear, volatilities and correlations are sufficiently stable, and average co-movement is an adequate description of dependence. Those are demanding assumptions for a multi-asset portfolio. And even before reaching those questions, the sample covariance matrix itself is a fragile object — with limited observations and a meaningful number of assets, the raw estimate is noisy and sensitive to outliers, problems that require shrinkage or robust estimation to manage. I discussed these estimation problems in the previous article, Why Most Covariance Estimations Fail.

A covariance estimate therefore has two distinct jobs that should not be confused. As a descriptive statistic, it can summarise a chosen history. As a decision input, it implicitly claims that this history is relevant to the allocation being considered. The first claim is mathematical. The second is economic and requires judgement - and is often false.

A full-sample estimate asks: How did these assets move together, on average, across the states contained in the sample?

The investment committee faces a different question: How are these assets likely to interact under the shocks that deserve the most attention from here?

Historical covariance is an unconditional mixture. Portfolio decisions are conditional.

The visual below maps what a covariance page captures well and where the picture becomes incomplete — specifically for the questions a committee needs to answer.

Figure 1. Keep the covariance page for measurement. Add a conditional-diversification page for the questions the committee still has to answer.

The right response is not to discard the covariance matrix. It remains useful for volatility aggregation, risk contribution, and tracking error. The right response is to recognise what it cannot tell you — and to ask the missing questions explicitly. That is what the rest of this article does.

This distinction is especially important for strategic asset allocation. A long investment horizon does not make current conditions irrelevant. Starting yields, inflation uncertainty and the monetary-policy reaction function affect the near-term path and the role of the portfolio’s diversifiers. At the same time, an SAA should not be built around one short-lived regime. The committee has to condition its analysis without allowing the current state to become the only state it can imagine.

One Relationship, Different Eras

Figure 2. Interpretive timeline of the sign of US equity-bond correlation across three broad eras. Based on Lombardi and Sushko (BIS, 2023); not a calculated return-series chart.

The change is economically intuitive.

When inflation is low and well anchored, growth news tends to carry more weight in expectations for monetary policy. A negative growth surprise hurts expected corporate earnings but also raises the prospect of policy easing and lower bond yields. Equities fall while government bonds rise.

When inflation is high or uncertain, the policy response changes. An upside inflation surprise hurts nominal bonds and can also raise equity discount rates. The central bank may have less room to respond to weaker growth. Bonds can then lose their conventional hedging role.

The point is not that one historical estimate was wrong. The point is that it compressed different mechanisms into one unconditional estimate.

The Simple Model: The Dominant-Shock Diversification Check

Henrik Lumholdt provides a useful way to organise the problem. The key question is whether changes in growth expectations or changes in inflation expectations dominate the market scenario.

The framework deliberately isolates one main impulse while holding the other broadly unchanged. It is not a forecast. It is a simple causal map.

Figure 3. Stylised dominant-shock map. Source: conceptually adapted from Lumholdt (2018), Chapter 9, Figure 9.1.

The stock-bond correlation is not the starting point. It is the result of the type of shock that dominates.

This also clarifies two common mistakes. First, a negative correlation does not show that the bond allocation is large enough to absorb an equity loss. Correlation describes direction, not hedge magnitude.

Second, a positive correlation is not automatically harmful. If inflation falls and both assets rise, positive co-movement is not a downside diversification failure. The relevant question is whether the assets lose together when protection is needed.

Monetary policy is part of this transmission mechanism, not a separate explanation. In a growth-led downturn with anchored inflation, expected easing can reinforce the bond hedge. In an inflation-led shock, tightening or constrained policy can push discount rates against both equities and bonds.

Turning the Shock Map into One Risk Estimate

The committee does not have to choose between conditional analysis and one usable risk number. It can have both. For a defined horizon, limit the downside discussion to two shock families: a growth-led shock and an inflation-led shock. Then ask the committee to assign a relevance weight to each one.

Suppose the committee assigns 40% to a growth-led downside and 60% to an inflation-led downside. A quick dashboard heuristic can weight the conditional correlations and volatilities. If the growth-world stock-bond correlation is -0.40 and the inflation-world correlation is +0.45, the weighted correlation is approximately +0.11. This is a useful summary, but not the exact correlation of a fully combined distribution.

For a more consistent risk estimate, weight the two conditional covariance matrices rather than the correlations alone. The resulting matrix can feed the existing risk report: portfolio volatility, stock-bond correlation, risk contributions and tracking error can all be calculated in the usual way.

Figure 4. Dominant-Shock Weighting Protocol. The committee assigns explicit relevance weights to two downside mechanisms, uses them to combine conditional risk estimates and retains one usable portfolio-risk view.

This remains a practical heuristic. The two shock families do not describe every possible market outcome, and the weights are committee judgements rather than objective probabilities. A full scenario approach is richer: it can represent more worlds, preserve complete return paths and calculate tail losses, drawdowns and recovery periods directly. It is also more technically demanding. That is the role of the flagship series, From Assumptions to Portfolios. The Toolkit version provides a simpler bridge that an investment committee can use now.

One number is still possible. The shock weights behind it should remain visible.

Worked Example

Consider the same stylised portfolio used in Article 1: 43% equities, 28% nominal government bonds, 21% credit, 4% gold, 2% commodities, 2% cash.

The asset-class report describes a balanced portfolio with 49% fixed income. Article 1 showed that these line items contain overlapping growth, rate, credit, liquidity and currency exposures. The next question is whether they offset one another under the shock that matters.

Figure 5. The same asset weights can provide very different defensive value in growth-led and inflation-led downside worlds. The directions are illustrative. They are not forecasts or stress-return estimates.

World 1: Growth disappointment

Assume growth expectations fall while inflation remains broadly anchored. Equities lose as expected earnings weaken. Nominal government bonds benefit from lower growth expectations, falling yields and the expected policy response. In this world, the 28% government-bond allocation performs its intended role. The traditional stock-bond diversification case works.

World 2: Inflation shock

Now assume inflation expectations rise while growth is unchanged or begins to weaken. Nominal government bonds lose as yields rise. Equities face higher discount rates and tighter financial conditions. Commodities (or gold) may help, but their role is conditional and their combined weight is small

In this world, 49% fixed income does not mean 49% portfolio defence.

The portfolio has a growth hedge. It may not have an inflation hedge.

Nothing about the asset weights has changed. The economic value of the diversifiers has.

How to Use the Check in Practice

The check should end in a portfolio decision, not merely in a better explanation of correlation. The example below uses three assets, a one-year horizon and deliberately simple policy rules. All numbers are illustrative.

A three-asset case

Start with 50% equities, 35% nominal government bonds and 15% gold. Assume the policy portfolio allows a maximum five-percentage-point change at one review and sets an annualised portfolio-volatility ceiling of 11.0%. To keep the example focused, expected returns, liquidity and transaction costs are assumed unchanged.

1. Define the two conditional risk matrices

The risk team specifies how the three assets are expected to behave if each downside mechanism dominates. These are conditional risk assumptions, not return forecasts.

2. Apply the committee’s shock weights

The committee assigns 40% relevance to a growth-led downside and 60% to an inflation-led downside. For this first-pass check, the two conditional covariance matrices are blended in the same proportions:

Blended covariance lens = 40% × Growth covariance + 60% × Inflation covariance

The role of this matrix is narrow but important. It does not choose the portfolio and it does not produce buy or sell signals. It converts the committee’s qualitative shock judgement into one coherent risk input. The existing risk engine can then revalue the current portfolio and any candidate portfolio on the same basis.

With the illustrative assumptions above, the 40/60 matrix implies an equity–bond correlation of +0.23. Applied to the current 50/35/15 portfolio, it produces annualised portfolio volatility of 11.9% — above the 11.0% policy ceiling.

3. Produce one risk view and show the sensitivity

One central estimate is useful, but it should never be shown without the surrounding cases. The table below keeps the asset weights fixed at 50/35/15 and changes only the committee’s growth/inflation relevance weights.

The practical message is immediate: as the inflation-led shock receives more weight, the stock–bond relationship becomes more positive and the same portfolio consumes more of the risk budget. Under the committee’s 40/60 central case, an adjustment is required.

4. Translate the risk view into candidate portfolio weights

Use a simple pre-agreed rule: when the central risk view breaches the ceiling, test the smallest permitted change from the main growth asset to a defensive asset. Here the committee tests exactly two five-percentage-point shifts: one from equities to government bonds and one from equities to gold.

Candidate A is slightly stronger in the pure growth-led case, but it still leaves the central 40/60 portfolio just above the 11.0% ceiling. Candidate B brings the central estimate down to 10.96% and provides the lower loss volatility in the inflation-led world, which currently carries 60% relevance.

Final illustrative weights: 45% equities · 35% government bonds · 20% gold

These weights are not an optimiser output. They are the smallest allowed policy-band change that restores the central risk ceiling and improves the higher-weighted downside mechanism. If expected returns, liquidity, transaction costs or wider policy constraints change, those inputs must be considered separately.

Figure 6. The blended covariance matrix maps the committee’s shock judgement into one risk view. A pre-agreed minimum-change rule then translates a risk-budget breach into final portfolio weights.

5. Record the decision and define the escalation rule

The decision record should state the shock weights, the blended risk metrics, the selected portfolio, the policy limit that triggered the change and the conditions for review or reversal. Refresh the analysis during the formal TAA/SAA review, after a material allocation change or when the committee’s assessment of the dominant risk changes.

The weighted-covariance protocol is a practical first pass for risk-budget adjustments inside existing policy bands. Escalate to a full scenario-path analysis when the decision depends on tail loss, drawdown, sequencing, recovery time, interacting shocks or material changes in expected returns. That richer approach is better, but also more technically demanding; it is the role of the main From Assumptions to Portfolios series.

Common mistakes include treating relevance weights as objective probabilities, allowing the blended covariance matrix to choose the portfolio, changing weights without a pre-agreed risk limit, ignoring expected-return and liquidity constraints, and presenting only the central weighting case.

Limitations

Growth versus inflation is a deliberately stylised two-shock framework. Actual outcomes can combine growth, inflation, real-rate, policy, liquidity, currency and valuation shocks. The three-asset example is therefore a decision template, not a universal allocation recommendation.

For this first-pass check, the blended matrix is the relevance-weighted average of the two conditional covariance matrices. If scenario-specific expected returns differ materially, the full mixture covariance also contains a between-scenario component. Complete scenario paths are the better tool for that problem and for tail, drawdown and sequencing risk.

Use the protocol for small, risk-budget-driven changes within established policy bands. Do not treat the final weights as an optimal portfolio or convert the covariance matrix mechanically into a buy or sell signal.

The Question to Take to Your Investment Committee

The Deeper Version

This article stops at a practical implementation bridge: two shock families, explicit relevance weights, a blended covariance risk lens and a minimum-change rule that converts a risk-budget breach into candidate portfolio weights.

The full scenario approach is superior when the decision depends on tail loss, drawdown, sequencing, recovery time or interacting shocks. It also requires more data, scenario construction and diagnostics. In the main series, complete paths — not a blended covariance matrix — remain the primitive object. For the technical implementation, see Scenario Atlases, Not Covariance Matrices

This is not investment advice. All examples are illustrative.

Draft source notes

Lumholdt, Henrik (2018). Strategic and Tactical Asset Allocation: An Integrated Approach, Chapter 9, particularly Figure 9.1.

Lombardi, Marco Jacopo, and Vladyslav Sushko (2023). “The Correlation of Equity and Bond Returns,” BIS Quarterly Review, December.

European Central Bank (2022). “Cross-Asset Correlations in a More Inflationary Environment and Challenges for Diversification Strategies,” Financial Stability Review, November.

Before we can talk about scenario atlases, Entropy Pooling, portfolio construction, risk constraints, or strategic allocation decisions, we need something less glamorous but more important: a clean return panel.

This article documents the data foundation for the “From Assumptions to Portfolios“ series. It explains which assets we use, which data sources feed the model, how returns are transformed, what the final sample window is, and which simplifications are intentional.

The goal is not to pretend that the data layer is perfect. The goal is to make it explicit enough that every later article has a clear empirical base. T

This Article will be updated over time. You can find the implementation in the corresponding Github Repo.

The final panel

· 11 investable assets

· 1 auxiliary EUR/USD risk driver

· 12 weekly driver series in total

· 782 fully observed weekly return vectors

· W-FRI observations from 2011-01-07 through 2025-12-26

Everything is expressed as weekly log returns from a EUR investor’s perspective.

Update note (15.08.2026). The series uses weekly returns.

Why start with the data layer?

Most portfolio articles start too late. They begin with expected returns, a covariance matrix, an optimizer, or a set of views. But before any of that, we need to answer a more basic question: what historical market objects are we actually using?

The following chart presents a detailed map of the data framework used throughout this series

For this series, the primitive object is not a covariance matrix. It is a panel of weekly returns across assets and risk drivers. From that panel we can later build scenario paths, terminal return distributions, drawdown diagnostics, stress clusters, and eventually view-adjusted posterior probabilities.

Core idea: A covariance matrix compresses the world. A scenario atlas needs the world before compression.

The asset universe

The investable universe has 11 assets. It is designed to represent a EUR-based multi-asset allocation problem without becoming unnecessarily complex.

The model contains 12 series but only 11 investable assets. EUR/USD is not allocated to directly. In the current data build, it is used explicitly to convert the USD-quoted Gold proxy into EUR and is retained as an auxiliary risk driver for later scenario analysis. The remaining investable proxies already arrive as EUR-quoted or native EUR-hedged series and must not be converted a second time.

An ETF’s EUR listing does not remove the foreign-currency exposure of its underlying portfolio. It means that this exposure is already reflected in the observed EUR market level; applying EUR/USD again would double-count currency translation.

What we deliberately do not include

The universe is intentionally compact. There is no separate Japan sleeve. Japan remains implicitly inside the broad Global Developed Markets ex-EMU proxy, but it is not modeled with a separate MSCI Japan series or a separate EUR/JPY driver.

There is no separate EUR/GBP driver. UK exposure is also part of the broader developed-market proxy. There are no REITs in this version of the Block 1 universe.

These are not claims that those exposures are unimportant. They are scope decisions. The first objective is a reproducible, explainable scenario foundation. Adding more regional sleeves and FX drivers too early would make the implementation more fragile without necessarily improving the educational value of the series.

The concrete data sources

The daily_proxy_2011 profile combines three source types: nine daily adjusted market-level histories obtained through Yahoo Finance/yfinance, one issuer-reported iShares NAV-performance series for Euro High Yield, and two daily FRED series for EUR/USD and the ECB deposit-facility rate. No MSCI export, paid index file, or manually placed market-data file is required.

The ETF histories are investable proxies rather than official benchmark-index histories. Accumulating share classes are used where available. Euro High Yield is based on the iShares performance-chart series for portfolio 251843 / ISIN IE00B66F4759, reported on a EUR NAV basis with gross income reinvested.

Free technical access should not be confused with an open-data licence. Yahoo/yfinance observations and the public iShares endpoint are free-access research sources, but their terms do not guarantee redistribution, publication rights, historical stability, or continued availability.

Return convention

Asset and FX series are represented as weekly log returns:

where \(L_t^{(w)}\) is the last fresh level observed in the corresponding W-FRI bucket. Levels are sampled first and differenced afterwards, so weekly returns are non-overlapping. If Friday is a holiday, the latest fresh observation on or before Friday is used; a completely missing week remains missing.

Cash is constructed differently. The annual ECB deposit-facility rate is accrued along a daily ACT/360 total-return path using only the last rate known at the beginning of the accrual interval. Weekly cash returns are then calculated from that same period-end level path. This avoids both a mechanical division by 52 and the use of future rate information.

Currency treatment

The investor base currency is EUR. Listing currency—not the fund name, share-class name, or accounting currency—determines whether an additional spot conversion is required.

Gold is the only USD-quoted investable proxy in the current panel. Its daily level is converted before weekly sampling:

with EUR/USD quoted as USD per EUR. In log-return form this corresponds to

Global DM ex-EMU, EM Equities, and Commodities are observed through EUR listings, so their underlying currency effects are already embedded in their EUR market levels. No additional EUR/USD conversion is applied.

Global Government Bonds and EM Hard-Currency Bonds use native EUR-hedged accumulating ETF share classes. No synthetic EURIBOR-minus-USD-rate hedge return is added. Their observed returns already include the provider’s implemented hedge, fees, tracking difference, and share-class behavior.

The final sample window

The strict weekly publication contract requests data from 2011-01-07 through 2025-12-31. Because 31 December 2025 is not a Friday, the effective completed sample ends on 2025-12-26. The resulting panel contains 782 weekly rows and all 12 required drivers, with no missing observations.

weekly = build_return_panel(
    source_profile="daily_proxy_2011",
    frequency="weekly",
    start="2011-01-07",
    end="2025-12-31",
)

The same source profile can also generate a corrected 180-row monthly panel from 2011-01-31 through 2025-12-31.

Why the weekly panel still starts in 2011

The latest-starting source in the common profile is the Euro High Yield NAV-performance series, whose observed history begins on 3 September 2010. This supplies the prior level required for the first weekly return in January 2011. Most other sources begin earlier.

The 2011 boundary is therefore a stable publication and comparison contract, not a claim that Euro High Yield is unavailable throughout 2010. No synthetic splice is used, and provider-specific source floors should be treated as observed availability rather than guaranteed permanent history.

Why raw data are not in the repository

The public repository contains the loader, source definitions, tests, and documentation, but not provider-derived raw observations, downloaded return panels, manifests, or generated article outputs. These files are deliberately excluded from version control.

The data download requires no paid source order and no manually placed MSCI files. A FRED API key is required for the standard live retrieval workflow; the key is an access credential and must never be committed.

Local downloads are written to:

data/downloads/daily_proxy_2011/monthly/
data/downloads/daily_proxy_2011/weekly/

Each snapshot receives a provenance manifest and SHA-256 checksum. A later live download need not be bit-for-bit identical because providers may revise historical observations or endpoints.

What this panel will be used for

The weekly panel is the primary input for the current return diagnostics. Downstream scenario code must state its time unit explicitly: an engine whose horizon and block lengths are defined in months must continue to use the monthly panel or be made frequency-aware before weekly observations are supplied.

· Synchronized Historical Block Bootstrap: resampling historical cross-asset blocks without breaking contemporaneous relationships.

· Conditional Similarity Resampling: giving more weight to historical periods that resemble the current market state.

· Filtered Historical Simulation: separating volatility states from empirical shocks without defaulting to Gaussian simulation.

· Atlas Coverage Diagnostics: checking which worlds history captures well and which plausible portfolio environments remain underrepresented.

Later, in Block 2, views will be applied through Entropy Pooling. In Block 3, portfolio construction will use the resulting scenario distribution. But none of that works cleanly unless the data layer is explicit.

The main limitations

This setup is intentionally transparent, not perfect. The main limitations are:

• Several exposures are represented by ETF proxies rather than official benchmark-index histories.

• Yahoo adjusted market levels are total-return proxies, not guaranteed official index total-return series.

• Euro High Yield is NAV-based while the Yahoo inputs are market-price-based, creating a mixed valuation basis.

• Native EUR-hedged ETF returns reflect the provider’s implemented hedge and tracking behavior; they are not frictionless institutional hedge replications.

• Cash is represented by the ECB overnight deposit-facility policy rate, not 3-month EURIBOR and not a directly investable retail cash product.

• The sample contains 782 weeks but still spans only 15 calendar years; the additional observations do not create 782 independent market regimes.

• W-FRI sampling omits intraweek dynamics and may select a pre-Friday observation around holidays.

• Some proxy histories contain benchmark or index changes.

• Provider histories may be corrected retrospectively, and the public iShares endpoint is undocumented and has no availability guarantee.

• The universe still excludes separate Japan, GBP, JPY, and REIT sleeves.

• The model remains EUR-investor centric.

These limitations are not hidden. They are part of the design. The point of this series is not to claim that public data and ETF proxies produce the final word on strategic asset allocation. The point is to build a disciplined, inspectable research workflow that can be understood, challenged, and extended.

None of these limitations invalidate the framework. They define its scope. A scenario atlas built from real data, with named proxies and documented approximations, is more defensible than one built from a parametric model that hides similar choices inside its distributional assumptions. But the choices are still there, and they should be visible.

The key takeaway

Bottom line: the current research panel contains 11 investable assets, one auxiliary EUR/USD risk driver, and 782 complete W-FRI observations from 7 January 2011 through 26 December 2025. It is built from daily free-access ETF, issuer-NAV, FX, and policy-rate sources; only Gold is explicitly converted from USD, while the two foreign-bond sleeves use native EUR-hedged share classes.

That panel is not the conclusion of the research process. It is the starting point.

From here, we can ask better questions: what does history actually show across assets? Which return properties survive simple diagnostics? How should we resample paths without destroying cross-asset structure? Which market worlds are missing from the historical record? How can investor views be added without rewriting the entire scenario set?

That is where the Scenario Atlas begins.

Three key points:

• The optimizer is not the beginning of the portfolio process.
Before we place views or compute weights, we need a model of what can happen.

• A covariance matrix is a useful diagnostic, but a poor foundation.
It summarizes volatility and linear correlation, but it cannot show drawdowns, sequencing risk, regime shifts, or stress paths.

• Before deciding on which “world” your portfolio will live in the future - by weightening the probability of future paths - it is necessary to build a sufficiently rich set of possible worlds. Entropy Pooling (Block 2) and Portfolio Construction (Block 3) come later

Before you think about performing portfolio construction or even place an investment view on a portfolio, you need a model of what can happen to the assets in the future you want to invest in today. Most processes skip this step. They go straight from historical data to a covariance matrix, from a covariance matrix to expected returns, and from expected returns to portfolio weights. The model of what can happen is implicit — buried inside the Gaussian assumption, invisible in the optimization.

This series makes it explicit.

The broader architecture that connects risk identification, scenario generation, view integration and portfolio construction is described in Why Portfolio Construction Is More Than Optimization which describes the process of building and evaluating portfolios in four subsequent blocks. The article at hand is the starting point and focuses on the first question that architecture has to answer: What are the potential futures your portfolio might have to live in?

The central object of this series is not a covariance matrix. It is a scenario atlas: a structured collection of possible multi-year paths across assets, built from historical market data and extended with stress scenarios that history alone cannot provide. Views come later. Optimization comes later. First, we need to know what worlds we are asking our portfolio to survive.

This first article of my new flagship series explains why — and what we are building as necessary prerequisite for the subsequent steps of view implementation and portfolio constructoin.

A few honest words: While our approach will have a certain level of complexity, we still keep it rather simple in some aspects. Our framework is improvable in many areas, but for us it is intuition and the right mindset that is the heart of this series. The implementation can be found on Github. Find infos about the Technical & Data elements in the separate Technical Article of this series

The Problem with Covariance-First Thinking

First Of All: Using a covariance matrix is not wrong. It is a useful summary of how assets have moved together in the past - on average at least. It captures volatility. It captures linear correlation. It gives an optimizer something to work with.

The problem is not the covariance matrix itself. The problem is treating it as the foundation of the investment process — the primitive object from which everything else follows. When you start with covariance, you are already committed to a set of implicit choices that are rarely made explicit and are highly unrealistic:
You are assuming that second moments are the right summary of risk.
You are assuming that the correlations in your estimation window are a reasonable guide to future correlations.
You are assuming that a portfolio that looks efficient in covariance space will behave acceptably across the range of environments it will actually face.

None of these assumptions are obviously true. And more importantly, none of them are visible in the covariance matrix itself.

To be clear: even before we get to these deeper issues, the sample covariance matrix is a fragile object in its own right. With a limited number of observations and a meaningful number of assets, the raw estimate is noisy, ill-conditioned, and sensitive to outliers — problems that require shrinkage, robust estimation, or factor models to manage. We have written about this in detail here. But here is the more uncomfortable point: even a well-estimated covariance matrix faces a structural limitation that no amount of shrinkage can fix. It answers the wrong question.

Consider what a covariance matrix cannot tell you. It cannot tell you how deep a drawdown might be, or how long it might last. It cannot tell you whether the correlation between equities and government bonds will be negative or positive over the next three years — a question that turns almost entirely on the inflation regime. It cannot tell you what happens to credit spreads and equity volatility simultaneously in a liquidity crisis. It cannot represent the difference between a portfolio that loses thirty percent over six months and recovers, and one that loses thirty percent over six months and does not. These are not edge cases. They are the situations in which portfolio decisions actually matter.

The following table summarizes what a covariance matrix captures well and where a scenario atlas adds what it cannot provide.

This series does not reject covariance. Volatilities and correlations are useful diagnostics, and we will compute them from scenario paths throughout. But they are outputs of the analysis, not its foundation. The foundation is a structured set of possible worlds.

What a Scenario Atlas Actually Is

A scenario atlas is a structured collection of possible futures — not point forecasts, not probability distributions in the parametric sense, but concrete multi-year paths across assets. Each path describes a potential future realisation or “world”: how equities moved, how bonds behaved, how credit spreads evolved, how the currency shifted — month by month, over a horizon of one, three, or five years.

The word atlas is deliberate. An atlas does not tell you where you will end up. It shows you the territory. It makes the range of possible destinations visible so that you can reason about them explicitly — which roads are likely, which are dangerous, which you have not prepared for.1

In this series, the atlas contains thousands of such paths, organized into three sets built with different methods. Each set reflects a different way of asking the same question: Given what history tells us about how markets move together, what worlds should our portfolio be prepared to face?

What makes this different from a covariance matrix is not a question of mathematical complexity. The distinction is one of transparency. A covariance matrix compresses the structure of market behavior into something algebraically convenient — a single object that an optimizer can work with directly. A scenario atlas keeps that structure visible. You can look at an individual path and ask whether it is plausible. You can count how many paths end in a drawdown of more than twenty percent. You can check whether your stress scenario of choice — stagflation, a liquidity crisis, a decade of financial repression — is actually represented, and with what weight.

This visibility is not just pedagogically convenient. It is what makes the framework governable. An investment committee can interrogate a collection of scenarios in a way that it cannot interrogate a covariance matrix. The question “which worlds are we not prepared for?” has a natural answer when the worlds are explicit. It has no answer when risk is summarized in a single matrix. 2

One more clarification before we go further. An atlas of possible worlds is not a set of forecasts. Building the atlas does not require a view on which world is more likely. That comes later, in Block 2 of this series, when we use Entropy Pooling to shift probability mass across the scenarios we have already built. The atlas and the views are kept separate — and that separation is the methodological core of this approach.

The Primitive Object: asset_paths

Every framework makes a choice about what to put at the center. In mean-variance optimization, the central object is the covariance matrix — everything else is derived from it or added to it. In this series, the central object is something different:

asset_paths # shape: S × H × N_assets

This is a three-dimensional array. S is the number of scenarios — in practice, several thousand. H is the horizon in months — we work with paths of up to sixty months. N_assets is the number of investable assets in the portfolio. This series models eleven investable assets and one auxiliary FX risk driver — EUR/USD — which is used to convert USD-denominated returns into EUR perspective.

In plain terms: asset_paths is a collection of possible futures, one slice per scenario, showing how each asset moved month by month over the full horizon.

The reason this is the right primitive object is a simple asymmetry. From paths, you can derive almost everything else you might want:

expected_returns = compute_means(asset_paths)
covariance = compute_covariance(asset_paths)
terminal_returns = compound(asset_paths)
max_drawdowns = compute_max_drawdown(asset_paths)
cvar = compute_cvar(asset_paths)

The reverse is not true. From a covariance matrix, you cannot recover paths. You cannot reconstruct drawdowns, sequencing risk, time-under-water, or the joint behavior of assets in a specific stress environment. The compression is irreversible.

This asymmetry is why we start with paths rather than moments. We are not giving up information to gain tractability. We are keeping the information and accepting a slightly different form of computation.

The paths in this atlas are built from historical market data — monthly returns from September 2010 to the present, giving approximately 189 observations. How exactly those paths are constructed from that data, and what assumptions each construction method makes, is the subject of the next four articles in this series.

Why Empirical-First

This series describes itself as empirical-first. That phrase needs unpacking, because it is easy to misread it as a rejection of models, or as a claim that historical data speaks for itself. Neither is true.

Empirical-first means that the starting point is the historical record of how assets have actually moved together — not a parametric model. We resample and reconstruct from observed market behavior rather than drawing from fitted distributions. History provides the raw material; the methods determine how we use it.

The alternative — parametric-first — would mean starting with a model. Assume returns follow a certain distribution. Estimate its parameters. Generate scenarios by drawing from that distribution. This is a legitimate approach, and in some settings the right one. But it carries a risk that is easy to underestimate: the model becomes a hidden assumption. The scenarios look like they came from data, but they actually came from a Gaussian copula, or a hidden Markov model, or whatever distributional choice was made. That choice shapes everything downstream, often invisibly.

Empirical-first makes the nature of the assumptions more inspectable. The building blocks are actual market observations — you can look at a resampled path and ask whether it is plausible, whether it corresponds to a real historical episode, whether the co-movements it contains make economic sense. That kind of inspection is harder to do with a scenario drawn from a fitted distribution, where the building blocks are model parameters rather than observable events.

But — and this is the point the phrase can obscure — empirical-first is not assumption-free. The assumptions do not disappear. They shift. And here the comparison with parametric approaches deserves more nuance than the empirical-first framing sometimes receives.

With a parametric model, the assumptions are written down explicitly. You specify a distribution, you write out the likelihood, you derive estimators with known statistical properties — consistency, asymptotic normality, confidence intervals. The model is on paper and, at least in principle, falsifiable. This is not a weakness. It is a genuine strength of the parametric approach. The formal explicitness of parametric assumptions is something empirical-first methods cannot fully match.

With empirical-first methods, the core assumption — that the historical sample is sufficiently representative of the range of plausible futures — is harder to make precise. It is less a formal assumption than a judgment call. You cannot write a likelihood for it. You cannot derive its sampling properties in the usual sense. In this specific way, the empirical-first framework is formally less explicit than its parametric counterpart, even as its building blocks are more directly inspectable.

The two dimensions are different and should not be conflated:

Neither column wins unconditionally. The empirical-first choice in this series is a deliberate preference, not a universal verdict. With approximately 180 months of data across eleven assets, we do not have enough observations to estimate a rich parametric model reliably — the curse of dimensionality bites hard in that regime. That tilts the balance toward empirical methods in this specific setting. In a different setting — more data, fewer assets, a well-understood return-generating process — the parametric approach might be the stronger choice.

We will also examine parametric alternatives directly. Hidden Markov models, copula-based generators, and factor-structured approaches each offer something the empirical sets do not — explicit regime structure, tail dependence modeling, cleaner separation of marginal and joint behavior. Understanding what they add, and what they cost, is part of taking the empirical-first choice seriously rather than treating it as the only reasonable one.

One more thing this series is not: anti-parametric. Parametric tools appear throughout — as volatility filters in Set C, as diagnostics throughout. The question is not whether to use models. It is where to put them. In this series, they are tools within an empirical framework, not the foundation of it. Whether parametric alternatives deserve a more thorough treatment of their own is a question we will return to separately.

What This Series Builds

Block 1 of this series produces a scenario atlas with three distinct sets of paths, each constructed using a different method. The sets are not competing alternatives — they are complements, each designed to answer a slightly different question about the range of plausible futures.

Set A — Synchronized Historical Block Bootstrap is the baseline. It resamples blocks of historical returns across all assets simultaneously, preserving the cross-asset structure that existed in each historical episode. No distributional assumptions, no latent states. Just history, resampled and stitched together into multi-year paths.

Set B — Conditional Similarity Resampling asks a refinement of the same question: are all historical periods equally relevant to where we are today? Set B weights historical episodes by their similarity to the current market environment — current yield levels, credit spreads, equity volatility, and other observable conditions. Periods that resemble today receive more weight. This is a state-dependent atlas without hidden Markov states.

Set C — Filtered Historical Simulation addresses volatility clustering. It separates the volatility state from the underlying market shocks, uses a filter to scale shocks to the current volatility environment, and then resamples the standardized shocks empirically. The shocks remain historical — no Gaussian draws — but their scaling reflects where we are in the volatility cycle.

In addition to these three sets, the atlas is augmented with a small collection of stress scenarios — structured multi-year paths representing environments that history alone underrepresents: deep stagflation, severe liquidity crises, prolonged financial repression. These are not point forecasts. They are modelling priors, assigned a small collective weight to ensure the atlas covers worlds it would otherwise miss.

All three sets share the same asset universe and the same horizon structure. Paths run to sixty months, and terminal returns are computed at twelve, thirty-six, and sixty months. This multi-horizon design allows the atlas to support both annual SAA reviews and longer-term strategic views.

The asset universe consists of eleven investable assets and one auxiliary FX risk driver:

The data underlying the atlas runs from September 2010 to the present — approximately 180 monthly observations. The start date is determined by the earliest available proxy for Euro High Yield, the binding constraint in the asset universe. This shorter history than one might ideally want is part of the honest accounting this series tries to maintain throughout.

Limitations

This framework is deliberately transparent about its boundaries.

The historical return panel starts in September 2010. That gives a clean common sample across all assets, but it also means that some important worlds — including the 2008 financial crisis and earlier inflation regimes — are not directly present in the empirical sample. This is one reason the atlas later adds explicit stress clusters rather than pretending that history is complete.

The asset universe also uses practical public-market proxies, and the FX treatment is intentionally parsimonious: EUR/USD is the only explicit currency driver. These choices make the framework reproducible and explainable, but they remain approximations.

The detailed data sources, proxy choices, currency treatment, schema conventions and repricing logic are documented separately in Technical Note: Scenario Atlas Infrastructure.

What This Means for Your SAA

If you run a strategic asset allocation process, this series is relevant to you at three levels.

Diagnostic. Even if you never build a scenario atlas yourself, the framework changes what questions you can ask. Which historical environments does your current portfolio handle well, and which does it handle badly? What is the worst twelve-month path in a plausible set of futures? Which stress scenarios are absent from your current risk process? These questions have natural answers when possible worlds are explicit.

Governance. Investment committees are better at reasoning about scenarios than about covariance matrices. A portfolio that survives stagflation but struggles in a liquidity crisis is a comprehensible statement. A portfolio with a tracking error of 4.2% is not. The scenario atlas produces outputs that translate directly into committee language — not because they are simplified, but because they are structured around the questions committees actually care about.

Interested in ways to improve you decisions of your Investment Committee? Then check out our other series: The Allocator's Toolkit and The Irrational Committee.

Methodological. If you currently build your SAA around a mean-variance optimizer fed by a covariance matrix and a set of capital market assumptions, this series offers an alternative architecture. The optimizer still appears in Block 3, and the capital market assumptions still matter in Block 2. But the order of operations is different. You build the worlds first, place views on them second, and optimize third.

Key Takeaways

A covariance matrix is a useful diagnostic tool, but a poor foundation for strategic asset allocation. It compresses out exactly the information — drawdowns, path dependence, regime changes, stress behavior — that matters most in real portfolio decisions.

The primitive object of this series is asset_paths: a collection of possible multi-year futures across eleven assets, built from historical market data and structured into three complementary scenario sets.

Empirical-first does not mean assumption-free. The core assumption — that history is representative enough to resample from — is less formally explicit than a parametric model’s likelihood, but its building blocks are more directly inspectable. Both approaches involve genuine trade-offs.

The three scenario sets in Block 1 address different aspects of that trade-off: Set A is the synchronous historical baseline, Set B conditions on current market state, Set C accounts for volatility clustering. A small stress layer covers worlds history alone underrepresents.

Views come in Block 2. The atlas contains no investment opinion. Its purpose is to make the space of possible worlds visible before any probability tilts are applied.

Next: From Returns to Scenario Building Blocks

The next article in this series asks a more practical question: what exactly does history give us to work with, and what are its limitations as raw material for a scenario atlas?

Before we can resample historical returns responsibly, we need to understand their properties. Monthly returns are not independent draws from a fixed distribution. They exhibit autocorrelation, volatility clustering, fat tails, and time-varying correlations. These properties do not disqualify them as building blocks — but they determine how those building blocks should be used.

Disclaimer

This article is published for informational and educational purposes only. Nothing in this series constitutes investment advice, a recommendation to buy or sell any security, or an invitation to engage in any investment activity. The views expressed are those of the authors in their personal capacity and do not represent the views of any employer, institution, or affiliated organization.

All scenario construction, data handling, and methodological choices described in this series are illustrative. Past market behavior, however carefully analyzed, is not a reliable guide to future outcomes. Code published on GitHub is provided as-is, without warranty of any kind.

How many independent risks sit underneath your portfolio’s asset-class lines?

Most allocation reports cannot answer that question. They show where capital is invested: equities, government bonds, credit, gold, commodities and cash. They do not show how often the same economic force appears underneath those labels.

Global equities, high yield and emerging-market debt sit in different sections of the report. Yet all can depend on resilient growth, accessible funding and liquid markets. Government bonds, investment-grade credit, emerging-market hard-currency debt, gold and parts of the equity book can all react to changes in real yields. Unhedged global assets, gold and commodities all contain a currency translation layer for a euro-based investor.

None of these exposures is necessarily hidden. The risk itself is not hidden. What is hidden is how often it appears across the portfolio.

The Allocator’s Problem

Investment committees approve capital weights. They rarely approve the second allocation embedded underneath them: the portfolio’s allocation to growth, inflation, real-rate, credit, liquidity and currency risk. This matters because asset classes are bundles of exposures, not isolated risk units. It’s like throwing darts at the wrong target.

A credit allocation contains rate and spread risk. A global equity allocation contains business-cycle, valuation, liquidity and currency risk. Gold can protect against some stresses, but it also has real-rate and currency sensitivities. A foreign bond portfolio hedged into euros may remove most spot-currency translation, but it still carries the underlying yield exposure and a hedge overlay.

A pie chart in an IC separates these asset positions. Markets do not. You can own many asset classes - but still hold only a few underlying risks.

The practical solution is not to abandon the asset-class view. It is to add a second page showing which economic drivers recur across the portfolio.

The Simple Model: A Risk-Driver Exposure Map

Start with a qualitative map. Its purpose is not to estimate risk contribution or forecast returns. It is to identify repeated exposures that deserve a deeper test.

Scoring: - = no material direct exposure; 1 = weak or indirect; 2 = secondary; 3 = meaningful; 4 = material; 5 = dominant.

A score of 5 means that the driver matters greatly to the asset - not that the asset benefits when the driver rises. Scores indicate materiality, not direction, volatility or contribution to portfolio risk. They reflect strategic return sensitivity over an SAA horizon, not only the immediate price reaction to a market shock. Do not add them across columns, average them or multiply them by portfolio weights. Crisis role is shown separately because it is a portfolio function, not a risk driver.

The same score can reflect different transmission mechanisms. A high Real Rates score means that real yields are central to the asset’s outcome, not that the return direction or mechanism is the same. For government bonds, the channel is primarily price sensitivity; for gold, opportunity cost; for cash, the reset of future carry rather than an immediate mark-to-market loss.

How to Read the Map

The map contains six exposure channels and one portfolio role. Each column asks how strongly an asset’s outcome depends on that economic force.

Growth. Sensitivity to changes in economic activity, earnings, demand and the business cycle.

Inflation. Sensitivity to inflation surprises and their effect on cash flows, purchasing power and policy expectations.

Real Rates. Sensitivity to inflation-adjusted yields, whether through asset prices, opportunity cost or future carry.

Credit. Sensitivity to credit spreads, default risk and borrower financing conditions.

Liquidity. Sensitivity to funding conditions, market depth, forced selling and changes in risk appetite.

FX / Hedge. Exposure to currency translation or to the carry, basis and rollover effects created by hedging it.

Crisis Role. The asset’s potential ability to preserve capital, liquidity or optionality in stress; this is a portfolio role, not a risk driver.

Worked Example

Consider a stylized policy portfolio: 15% Eurozone Equities, 20% Global Developed Equities, 8% Emerging-Market Equities, 20% Euro Government Bonds, 12% Euro Investment-Grade Credit, 5% Euro High Yield, 8% Global Government Bonds hedged to euros, 4% EM Hard-Currency Bonds hedged to euros, 4% Gold, 2% Commodities and 2% Cash.

The asset-class page reports 43% equities, 49% fixed income, 6% real assets and 2% cash. That looks balanced. The second map reorganizes exactly the same capital by shared exposure.

What the Second Portfolio Map Reveals

How the clusters are formed: The second table is a diagnostic cluster summary, not a mechanical re-summing of the score table. Each row uses the channel that defines that cluster. For growth and liquidity, core exposure means material exposure to both Growth and Liquidity; Euro IG Credit is shown separately as secondary because the exposure is meaningful but weaker. For rates and real rates, the focus is rate-sensitive price risk, so cash is not included even though short-term real yields matter for its carry. FX separates unhedged currency translation from hedged sleeves with hedge carry, basis and rollover. Intended defence is based on portfolio role, not on a single risk-driver score.

Capital footprints overlap and should not be added together. They are not risk contributions. The table shows where a common shock can travel, not how much portfolio volatility each driver contributes.

Nothing in this map proves that the allocation is wrong. It shows that eleven line items are not eleven independent sources of return or protection. The committee approved an asset-class allocation; the table makes the underlying risk allocation discussable.

How to Use the Map in Practice

1. Start with current weights. Use the policy portfolio for the structural review. Add tactical overlays only when the committee is explicitly reviewing current positioning.

2. Score each line consistently. Use the 1-5 scale and the dash for no material direct exposure. Avoid decimals. The aim is disciplined comparison, not precision.

3. Look for repetition, not totals alone. Focus on drivers scored 4 or 5 across several large allocations, especially where the exposures are likely to have the same sign in the stress you care about.

4. Name the intended offset. For every major driver cluster, identify which holding is expected to diversify it and write down the condition under which that protection could fail.

5. Escalate questions, not scores. The map should determine what the risk team measures or stresses next. It should not be converted mechanically into a buy or sell signal.

Update frequency. Refresh the structural map when the policy portfolio changes, when a new asset class is introduced or during the formal TAA/IC/SAA review. Revisit its interpretation when the dominant macro shock changes, but do not turn the map itself into a tactical timing model.

Common mistakes. The most common errors are treating ordinal scores as quantitative risk contributions, counting line items as independent bets, treating a hedged position as economically FX-free, assuming that anything outside equities and government bonds is automatically a diversifier, and adding overlapping capital footprints together.

Where Regimes Enter

A static map can show that several assets share rate sensitivity. It cannot tell you whether those assets will diversify one another in the next shock. In a disinflationary growth shock - broadly, a recessionary shock - government bonds may offset equity losses; in an inflation shock, both can fall as bond-equity correlation increases. In an inflation shock, not even gold will safe you when real interest rates increase. The map’s role is to identify where that conditionality matters and which relationship should be stressed. This is why gold and government bonds are marked as conditional crisis hedges in the table. Cash is the only holding assigned a strong crisis role. A later Toolkit article will address the time-varying (regime-varying) nature of diversification across regimes directly.

Limitations

This is a screening tool, not a risk model. The scores are ordinal and do not encode sign, beta, volatility, correlation, convexity or portfolio risk contribution. The capital footprints overlap, and driver sensitivities change through time. Hedge outcomes also depend on implementation details and market conditions. Use the map to identify hidden overlap, formulate stress tests and improve committee questions - not to claim a precise measure of diversification.

The Question to Take to Your Investment Committee

Which risk would hurt us most - and which holdings only look like diversifiers?

The Deeper Version

This map identifies where portfolio risks may overlap. The next article in From Assumptions to Portfolios develops the broader technical foundation: why scenario paths, rather than a static covariance matrix, should be the starting point for testing how those risks interact across different market worlds.

This is not investment advice. All examples are illustrative.

Most resources on portfolio optimization focus on one thing: the optimizer.

We think that’s too narrow.

Portfolio construction is a decision architecture — from raw assumptions and market views all the way to an implementable, explainable portfolio. The optimizer is one step. This publication covers the entire chain.

Who is this for? Portfolio managers, quant analysts, and serious investors who want to go beyond textbook mean-variance and understand how modern portfolio construction actually works in practice.

What you’ll find here: Rigorous content on covariance estimation, optimization methods, risk modeling, and view integration — always with Python code on GitHub.

Three series, three audiences

From Assumptions to Portfolios provides a complete portfolio construction framework: starting with scenario generation, showing how investment views are integrated, and finally how both feed into portfolio optimization. This series is especially valuable for quants and quantitative practitioners. Python code accompanies every article on GitHub.

The Allocator’s Toolkit looks at the other side of the process. Beyond the quantitative modeling, it’s investment committees and CIOs who actually make the allocation decisions — and this series gives practical guidance on improving that decision-making in real-world settings. No code required.

The Irrational Committee examines the layer both series presuppose but don’t address directly: the investment committee as a social system that produces decisions. Why well-equipped groups still make poor decisions — and how to change that structurally. Behavioral finance for institutions, not individuals.

The Allocator’s Toolkit looks at the other side of the process. Beyond the quantitative modeling, it’s investment committees and CIOs who actually make the allocation decisions — and this series gives practical guidance on improving that decision-making in real-world settings. It speaks directly to members of investment committees and CIOs.

From Assumptions to Portfolios and The Allocator's Toolkit alternate on a two-week cycle, every Tuesday at 8:00 CET. The Irrational Committee publishes monthly.

— Thomas & Felix

Where to start:

1. Portfolio Construction Architecture — The big picture. Why optimization is only one step in the process.

2. How to Apply Portfolio Optimization Correctly — 13 rules that make optimization work in practice.

3. Why Most Covariance Estimations Fail (Part 1) — The most important input to any optimizer, and why the standard approach breaks down.

In this article, I argue that portfolio construction is more than just obtaining asset weights from an optimization. Robust portfolio construction is a broader decision architecture — from scenario generation and disciplined view integration to optimization and post-processing.

Three key points:

• Optimization is only one step.
A good portfolio process starts before the solver, with forward-looking scenarios, dependencies, regimes, and economic assumptions.

• Views should be integrated systematically.
Tactical views should not simply move weights around. They should transform the prior distribution into a posterior distribution in a disciplined and multivariate way.

• The output must be decision-ready.
A candidate portfolio needs to be tested for robustness, stress behavior, implementation, explainability, and mandate fit before it becomes a usable portfolio decision.

The most difficult part of portfolio construction is often not solving the optimization problem. It is defining the investment problem correctly in the first place.

In many investment discussions, portfolio construction is treated as if it were mainly a solver problem. Estimate expected returns, risks, and correlations. Define an objective function. Add constraints. Run the optimizer. Obtain portfolio weights.

This view is attractive because it is clean. It feels precise, quantitative, and disciplined.

But it is also too narrow.

An optimizer can only solve the problem we have chosen to formulate. It does not decide whether the assumptions are economically sensible, whether the scenarios are rich enough, whether the views have been integrated consistently, whether the risk measure is appropriate, or whether the final portfolio can be implemented, explained, and governed in the real world.

A better way to think about portfolio construction is as an architecture that moves through four layers:

1. Generating long-term scenarios

2. Incorporating views

3. Optimizing portfolios

4. Post-processing the output into decision-ready portfolios

The first three steps are largely sequential. We start with a baseline view of the world, transform it with disciplined views, and then optimize based on the resulting distribution.

But the relationship between optimization and post-processing is different. It is iterative. The optimizer produces a candidate portfolio. Post-processing then tests whether this candidate is stable, explainable, implementable, and fit for the mandate. If it is not, the right response is not simply to manually polish the output. It may mean that the optimization problem itself needs to be reformulated.

In other words: A good optimizer produces weights. A good portfolio construction process produces decisions.

This article is not meant to be a technical deep dive into each component. It is a conceptual map of how these components fit together in a robust portfolio construction process.

This article is meant to serve as the conceptual roadmap for a future series on scenario generation, view integration, optimization, and post-processing.

1. Generating long-term scenarios: building the prior

Why forward-looking assumptions matter

It begins with the question:

What could happen - and with what probability?

Before we can optimize, we need a baseline distribution of possible future outcomes. This is the prior: a structured representation of what we believe the capital market environment could look like.

This prior should not be confused with a simple table of expected returns and volatilities. It is much more than that. It is a view on return distributions, risks, dependencies, tail events, and possible regimes.

The quality of this prior matters enormously. If the underlying scenario set is weak, no optimizer will rescue the result. The optimizer may still produce a mathematically precise answer, but it will be precise conditional on a poor description of the world.

This is particularly important for long-term capital market assumptions.

Historical averages can be a useful reference point, but they are a dangerous substitute for forward-looking assumptions. This is especially true for expected returns. Looking into the rear-view mirror can be misleading when valuations, interest rates, inflation regimes, profit margins, and structural growth drivers differ materially from the historical sample.

A simple example is equity returns. A historical average may include decades of valuation expansion. If one mechanically extrapolates that average, one may implicitly assume that the same valuation tailwind repeats itself. That is not a neutral assumption. It is a strong forward-looking claim disguised as historical objectivity.

Historical averages often mix realized cash flows, changing valuations, and regime-specific tailwinds. Treating them as neutral expected return estimates can therefore be misleading.

Good long-term capital market assumptions therefore need an economic foundation. They should ask which structural forces could shape future returns, risks, and dependencies.

Think about artificial intelligence. The relevant question is not simply whether AI is “positive for equities.” The more important question is how AI may affect productivity, margins, investment demand, energy consumption, inflation, and sectoral dispersion — and how these effects translate into asset-class and factor assumptions.

Demographics can influence labor supply, savings behavior, fiscal pressure, growth, real rates, and demand for safe assets. These are structural forces that may shape long-term distributions rather than short-term market views. Deglobalization and geopolitical fragmentation can affect supply chains, inflation, risk premia, regional growth, and correlation patterns in stress periods.

The main point is simple:

Long-term assumptions are not just forecasts. They are structured beliefs about the economic forces that may shape future return distributions.

Building a rich scenario universe

Once we accept that scenario generation is central to portfolio construction, the next question is how to build the scenario universe.

There are broadly two families of approaches: historical or resampling-based approaches, and parametric or simulation-based approaches.

Historical approaches preserve what has actually happened. They can capture real market movements, empirical co-movements, and observed stress periods. This is valuable. There is a certain discipline in forcing a model to confront real data rather than purely theoretical assumptions.

Resampling-based approaches can be useful in this context because they allow us to generate many possible paths while staying connected to empirical market behavior. A good example of this type of thinking can be found in Kristensen and Vorobets (2025) on scenario-based portfolio construction and resampling approaches.

But historical approaches also have a limitation: the past may not contain the full range of future states that matter. A historical sample may not include enough inflation shocks, geopolitical stress regimes, liquidity crises, or correlation breakdowns. It may also overweight regimes that are no longer representative.

Parametric and simulation-based approaches address a different problem. They allow us to design distributions more explicitly. We can model higher moments, fat tails, skewness, kurtosis, changing volatility, and different dependency structures. We can also introduce regime-based dynamics, for example through Markov-type regime
models.

This matters because portfolio construction is not only about average returns. It is also about what happens in unusual states of the world.

Consider a simple regime example. In a benign growth regime, equities and credit may perform well, volatility may be low, and correlations may look stable. In an inflation shock regime, rates, equities, commodities, and currencies may behave very differently. In a geopolitical stress regime, volatility may rise, correlations among risky assets may increase, and safe-haven assets may react differently than in normal times.

A richer scenario universe can include these different states. That is especially important if the next step is to incorporate views using methods such as Entropy Pooling.

Regardless of historical or parametric scenario generation, the aim is not to predict one future. It is to build a rich, economically grounded representation of plausible future states.

A useful scenario universe should contain different states of the world - including different growth, inflation, risk and correlation regimes. This is essential because Entropy Pooling can only reweight the scenario universe that is already available.

Assets, factors, and dependencies

Another design choice is whether scenarios and views are expressed at the asset level, the factor level, or both.

Factors are often the natural language of views. A view on real rates, inflation, credit spreads, the US dollar, volatility, or equity risk premia is usually easier to express as a factor view than as a direct asset allocation decision. This can make the link between research and view integration cleaner.

Assets, however, are the language of implementation. Final portfolios are built from asset classes, funds, indices, or instruments. Asset-specific characteristics — especially in equities, where region, sector, valuation, currency, and index composition matter — cannot be ignored.

A robust architecture should therefore connect both levels: factors for economic interpretation and view formulation, assets for implementation and portfolio construction.

Scenario generation is not only about individual asset behavior. It is also about joint behavior.

Expected returns may get most of the attention, but dependencies are often just as important. Covariances, correlations, tail dependencies, and regime-dependent co-movements shape the actual diversification properties of a portfolio.

This is why covariance modeling is not a technical side issue. It is part of the scenario architecture.

Correlations are not stable. They can change across regimes. They often rise in stress periods exactly when diversification is needed most. Tail dependencies can matter more than average correlations. If the scenario model fails to represent these features, the optimizer will work with an incomplete picture of risk.

I have written a separate article about covariance modeling and why dependency estimates matter for portfolio construction.

For the purpose of this article, the key point is this:

A scenario model is not only a model of returns. It is also a model of dependencies.

2. Incorporating views: from prior to posterior

A prior scenario distribution is only the starting point. In this context, tactical views are not meant as ad-hoc tilts. They are disciplined modifications of the strategic prior.

Investment teams rarely want to optimize purely on an unconditional baseline distribution. They have views for the short- or medium-term. These views may come from CIO discussions, investment committees, tactical asset allocation decisions, macro analysis, and so on.

The question is not whether views should exist. They always do.

The real question is: how they enter the portfolio construction process.

This is where judgement and discipline meet. Views are often an expression of judgement. But without structure, judgement can easily become inconsistent, biased, or impossible to evaluate. A disciplined portfolio construction process should define how judgement enters the distribution.

Views are not a substitute for structure. They become useful only when they are embedded in a disciplined probabilistic framework, such as Entropy Pooling.

SAA as anchor, TAA through views

One way to think about the relationship between strategic and tactical asset allocation is the following:

The long-term scenario distribution represents the strategic baseline. It reflects structural capital market assumptions and long-term beliefs. Tactical views then transform this prior into a posterior distribution.

In this architecture, strategic asset allocation (SAA) provides the anchor while tactical asset allocation (TAA) enters through views. This matters because tactical allocation should not be a disconnected overlay that simply moves portfolio weights around. It should be a disciplined expression of views within the same broader portfolio construction architecture.

In that sense, the TAA portfolio is not merely “the portfolio after some tactical tilts.” It is the portfolio that reflects tactical views in a structured way.

Black-Litterman and Entropy Pooling

There are different ways to integrate tactical views.

Black-Litterman is especially useful for combining prior return estimates with absolute or relative return views. Entropy Pooling is more general in the context discussed here, because it works directly on the scenario distribution and can incorporate views beyond expected returns.

Entropy Pooling starts with a prior scenario distribution and changes scenario probabilities so that the posterior distribution satisfies the specified views while remaining as close as possible to the prior.

Views may concern:

• expected returns

• relative performance

• volatility

• correlations

• tail risks

• factor outcomes

The key advantage is that Entropy Pooling works directly on the scenario distribution. It can therefore incorporate a broader set of views than expected returns alone.

This does not mean that Entropy Pooling is always superior in every context. But for an architecture that starts with a rich scenario universe and wants to express views in a multivariate way, it is a very powerful tool.

The real value of view integration

The main value of view integration is not merely that expected returns can be changed.

The more important point is that these changes can be made consistently within the multivariate distribution.

A naive view might say: increase the expected return of one asset class. But in reality, an investment view often implies more than that. It may also imply changes in volatility, correlations, tail risks, or regime probabilities.

Consider a geopolitical stress scenario. A view on geopolitical escalation is not only a return view. It may imply higher volatility, changing correlations between equities, bonds, currencies, gold, and commodities, as well as higher tail risks.

In such a case, simply adjusting one expected return is not enough. The view should affect the joint distribution.

This is exactly where a scenario-based view-integration framework becomes valuable.

This is also why the prior scenario universe matters so much. If the prior does not contain enough sufficiently different scenarios, Entropy Pooling has little room to express such views. It can reweight what exists. It cannot fully compensate for a poorly designed scenario universe.

3. Portfolio optimization: from beliefs to candidate portfolios

Only after the prior has been built and views have been integrated does optimization enter.

At this point, we have a posterior distribution. This distribution reflects long-term assumptions and disciplined views. The optimizer can now translate this distribution into a candidate portfolio.

This step is important. Optimization is not the enemy. It is a powerful way to formalize trade-offs between return, risk, constraints, and objectives.

But optimization itself is already full of design choices.

What are we optimizing?

• mean-variance efficiency?

• expected return for a given level of downside risk?

• CVaR or other downside-risk measures?

• tracking error?

• turnover?

• robustness?

• multiple objectives in a sequence?

Which risks matter? Which constraints matter? Which trade-offs are acceptable? Which objective is appropriate for the mandate?

A portfolio for a long-term strategic allocation problem may require a different objective than a tactical implementation portfolio. A UHNWI portfolio with illiquid assets and client-specific constraints may require a different design than a model portfolio without legacy exposures.

In a previous article, I discussed lexicographic optimization as one way to deal with multiple objectives in portfolio construction. The broader point is that optimization itself is already embedded in a set of design choices: what we optimize, in which order, under which constraints, and for which purpose.

The optimizer transforms beliefs into candidate portfolios. But it does not automatically transform candidate portfolios into good decisions.

Optimization is only as useful as the design choices around it: the assumptions, objectives, constraints, risk measures, and quality controls that define the problem.

4. Post-processing: from candidate portfolio to decision-ready portfolio

The output of an optimizer is not the end of portfolio construction.

It is a candidate.

The next question is:

Can this portfolio actually be implemented, explained, monitored, governed, and defended?

This is what I mean by post-processing. Not cosmetic polishing. Not manual tinkering. Not making the output look nicer.

Post-processing is the quality-control layer that turns optimized portfolios into decision-ready portfolios.

In this sense, post-processing is not an override of the optimizer. It is a structured review of whether the optimized candidate portfolio is fit for decision-making.

A decision-ready portfolio requires more than expected return and volatility.

It should be tested against stress scenarios. How does the portfolio behave under relevant adverse conditions? What drives the losses? Which assets or factors dominate the stress exposure?

It should be robust to uncertainty about the future. Does the portfolio only work in the baseline scenario? Or does it remain plausible across alternative regimes?

It should be robust to parameter uncertainty. What happens if expected returns change slightly? What if correlations move? What if volatility assumptions are wrong? If small input changes lead to large allocation changes, the portfolio may not be robust. It may be nervous.

It should be checked for concentration and hidden exposures. Does the optimizer create extreme weights? Are there unintended factor bets? Are diversification benefits real or only apparent?

It should be implementable. Turnover, liquidity, costs, operational constraints, and mandate restrictions matter. A mathematically attractive portfolio that cannot be implemented is not a successful outcome.

It should also be explainable.

This is particularly important in advisory, investment committee, and UHNWI contexts. A portfolio that cannot be explained is fragile, even if it looks attractive in a model. The decision-maker must be able to explain why the portfolio looks the way it does, which assumptions drive the result, what role tactical views play, and how the portfolio behaves under uncertainty.

Client-ready does not mean simplified. It means that the portfolio can be stress-tested, implemented, explained, monitored, and governed in the real world.

The iterative link between optimization and post-processing

The relationship between optimization and post-processing is not purely sequential.

It is iterative.

If post-processing shows that the candidate portfolio is stable, explainable, implementable, and aligned with the mandate, the process can move forward.

But if the portfolio is too concentrated, too sensitive, too difficult to explain, too costly to implement, or too weak under relevant stress scenarios, the solution is not simply to adjust the weights manually.

The right question is deeper:

• Was the objective function appropriate?

• Were the constraints correctly specified?

• Was the view too strong?

• Was the prior scenario universe rich enough?

• Were dependencies modeled properly?

• Was the risk measure aligned with the mandate?

• Did the optimization problem express the real investment problem?

This is why post-processing matters. It can reveal that the investment problem needs to be reformulated.

A serious post-processing layer does not simply polish the output. It challenges the formulation of the problem.

Conclusion: Portfolio construction is an architecture

Portfolio construction is therefore not the act of running an optimizer.

The optimizer is indispensable. But it is only one step in the architecture.

The process starts with a prior: a forward-looking scenario distribution grounded in economic assumptions. It then incorporates views to form a posterior distribution. Optimization translates this distribution into candidate portfolios. Post-processing tests whether these candidates can become decision-ready portfolios.

This is the difference between optimization and portfolio construction:

A good optimizer produces weights.

A good portfolio construction process turns scenarios, views, constraints, and uncertainty into portfolios that can be implemented, explained, monitored, and governed.

That is why portfolio construction is more than optimization.

Further reading / related notes

• My related notes:

  • A Better Portfolio Optimization Framework: Risk Constraints First, Lexicographic Selection Second

  • Why the Optimal Portfolio Is Often Not the Best

  • Why Most Covariance Estimations Fail ... and How to Build One that Doesn't (Part 1)

• Covariance modeling:

  • Ledoit & Wolf (2004): A well-conditioned estimator for large-dimensional covariance matrices

• Scenario generation and resampling:

  • Kristensen, L. & Vorobets, A. (2025) Time- and State-Dependent Resampling

  • Vorobets (2025): Time- and State-Dependent Resampling

• Regime switching:

  • Hamilton et al. (2005): Regime-Switching Models

• View integration

  • Meucci (2008): Fully Flexible Views: Theory and Practice

  • Idzorek (2004): A Step-by-Step Guide to the Black-Litterman Model

• Downside-risk optimization:

  • Rockafellar & Uryasev (2000): Optimization of Conditional Value-at-Risk

Disclaimer

This article is for educational purposes only and does not constitute investment advice or a recommendation. All examples and figures are illustrative and intended to explain the portfolio construction process conceptually. They do not represent investment recommendations. Views expressed are my own and not necessarily those of my employer.

• Why stronger risk constraints already improve portfolio optimization materially

• Why risk-feasible portfolios are not automatically the best final portfolios

• Why lexicographic selection still adds value by making portfolios more stable, interpretable, and benchmark-consistent

After publishing my previous article on lexicographic optimization, Anton Vorobets raised exactly the kind of follow-up question that moves a discussion forward.

If standard portfolio optimization is often too fragile, how much of that fragility can already be removed by specifying the primary optimization problem more carefully — for example by combining a portfolio CVaR constraint, a CVaR tracking-error constraint, and implementation controls such as turnover?

That is the right question.

A stronger primary optimization problem can eliminate a large part of the fragility that makes standard optimization hard to trust in practice. But even then, the final portfolio is not automatically the right portfolio to hold. A portfolio can be risk-feasible and still be difficult to explain, unstable in its weights, or poorly aligned with governance needs.

I increasingly think of robust portfolio construction as a two-step architecture:

• Risk realism first

• Portfolio plausibility second

That distinction matters because the required degree of explainability depends on the audience. For a purely quantitative audience, a strong Stage-1 optimization may already be enough. But for advisers, committees, clients, or less technical stakeholders, explainability becomes central. In those settings, even a highly risk-feasible portfolio may still fail if its final weight structure is not plausible and defensible.

In practice, that means: first define a primary optimization problem that is institutionally realistic and closely aligned with the relevant absolute and relative risk measures. Then, among the near-optimal portfolios that satisfy those constraints, choose the final portfolio using a second-stage lexicographic selection step.

In my view, that is often a better production framework than either a thin classical optimization problem or a purely cosmetic smoothing step after the fact.

To improve understanding and you learning, we have provided the code for every graph and each technique on Github. Additionally, we have developed a Streamlit App to further improve understanding of our approaches.

Why this question matters

In the first article, I focused on a problem many practitioners know well: small changes in expected returns can lead to implausibly large changes in portfolio weights.

That problem becomes especially visible when assets are highly correlated, economically similar, and close substitutes in the portfolio. Regional equity blocks are one example. Bond segments are often another. In those settings, very small differences in expected returns can push the optimizer into aggressive relative bets that are mathematically consistent but economically difficult to justify.

Lexicographic optimization is one practical way to address this. By preserving almost all of the primary objective and then selecting the most stable or benchmark-consistent portfolio among the near-optimal solutions, it often produces weight structures that are easier to defend.

But Anton’s point is important: before adding a second-stage selection step, one should ask whether the original optimization problem itself is simply too weak.

That is often the case.

If the primary optimization is defined only by a return objective and a minimal risk control, much of the resulting fragility should not be surprising. In many practical settings, the optimizer should not only be constrained by absolute portfolio risk. It should also be constrained by active downside risk relative to a benchmark and by implementation frictions.

That leads to a better question than “does optimization work?” The better question is:

What optimization architecture produces portfolios that are both risk-sound and practically usable?

What a stronger primary optimization problem looks like

A more realistic primary optimization problem goes beyond a simple expected return objective with a basic volatility or total-risk constraint.

In an institutional setting, a stronger single-stage formulation often contains four elements:

• a primary return or utility objective

• an absolute portfolio risk constraint

• a benchmark-relative active risk constraint

• implementation constraints such as turnover or transaction costs

A stylized Stage 1 would maximize the portfolio objective subject to:

• a portfolio-level CVaR limit

• a CVaR tracking-error limit relative to the benchmark

• turnover controls

• box, budget, liquidity, and mandate constraints

The logic is straightforward. The portfolio CVaR constraint controls absolute tail risk. The CVaR tracking-error constraint controls active tail risk relative to the benchmark. The turnover constraint introduces implementation realism. Together, these already define a much more realistic optimization problem than a thin return-risk formulation.

This is why Anton’s point matters. A richer primary optimization problem can often eliminate a substantial part of the instability problem before one even thinks about resampling, expected-return uncertainty, Bayesian shrinkage, or a lexicographic second stage.

The issue is often not optimization itself. The issue is that the optimization problem is under-specified relative to the realities of the portfolio mandate.

Risk-feasible is not the same as usable

A portfolio can satisfy all relevant risk constraints and still not be the best final portfolio.

Because risk control and portfolio plausibility are not the same thing.

Risk constraints shape the feasible set. They determine which portfolios are acceptable in absolute and relative risk terms. But they do not necessarily select the most interpretable, stable, or governance-friendly portfolio from within that feasible set.

That difference matters enormously in practice.

Within a set of portfolios that all satisfy portfolio CVaR, CVaR tracking error, turnover constraints, and mandate constraints, you may still have many candidate portfolios that are very similar in expected return and risk metrics, yet very different in their weight structure. Some may be closer to the benchmark, some less concentrated, some more stable through time, and some simply easier to explain.

This is not a cosmetic issue. In real investment processes, a portfolio is judged not only by its location in risk-return space, but also by whether its active tilts are understandable, proportional to the strength of the signal, and stable enough to implement with confidence.

That is why a stronger Stage-1 optimization often still leaves an unresolved portfolio selection problem.

Or put differently:

A risk-feasible portfolio is not automatically the most usable portfolio.

The case for a two-step architecture

This is where, in my view, the strongest practical framework begins to emerge.

Instead of viewing richer risk constraints and lexicographic optimization as competing ideas, it is more useful to view them as complementary layers.

Layer 1: risk realism.
The first stage should define the set of economically and institutionally acceptable portfolios. That means the primary optimization should already contain the risk logic the mandate genuinely cares about: absolute downside risk, relative downside risk to the benchmark, implementation frictions, and portfolio construction constraints.

Layer 2: portfolio selection.
The second stage asks a different question:

Among the portfolios that are already acceptable in risk and implementation terms, which one is the most reasonable final portfolio to hold?

This is no longer primarily a risk-definition problem. It is a selection problem.

And that is exactly where lexicographic optimization remains highly valuable.

The idea is simple. Solve the stronger primary problem first. Then preserve almost all of its objective value. Within that restricted near-optimal set, select the portfolio that best satisfies a secondary practical objective such as benchmark proximity, lower concentration, lower turnover, or proximity to the previous portfolio.

This second stage does not replace the primary optimization. It refines it.

What the empirical comparison suggests

To test this architecture more concretely, I built a comparison across three optimization frameworks:

1. Standard Optimization

2. Risk-Constrained Optimization

3. Lexicographic Risk-Constrained Optimization

I build a Streamlit App for this analysis:
You can find it here

The goal was not only to compare final portfolios at one point estimate, but to compare how each architecture translates changing expected returns into portfolio weights.

That is where the differences become most informative.

The first result is exactly what one would expect: Standard Optimization remains the most fragile setup. Small changes in expected returns still translate into excessively large and abrupt shifts in portfolio weights. That is the familiar behavior that causes many practitioners to distrust optimization in the first place.

The second result is more interesting. Risk-Constrained Optimization clearly improves the situation. Adding portfolio CVaR, benchmark-relative TE-CVaR, and turnover controls reduces the optimizer’s freedom to express small expected-return differences through aggressive active bets.

That is a meaningful improvement, and it supports Anton’s core point: a better Stage-1 architecture already goes a long way.

But the third result is, in my view, the most revealing.

The stabilizing effect of the risk-constrained architecture remains quite sensitive to the choice of the TE-CVaR limit. Stronger risk constraints improve feasibility, but the resulting portfolio behavior still depends materially on parameter calibration. Tightening or loosening the active tail-risk budget can change how strongly the portfolio reacts, which makes the final outcome more tuning-dependent than one might initially hope (you can see this using the Streamlit App). In fact, using a different TE-CVaR Limit can leader to results close to the Standard Optimization. So parameterization matters a lot!

By contrast, Lexicographic Risk-Constrained Optimization appears to provide a more consistently stable final portfolio selection relative to the benchmark. Across the weight sweeps, it produces a smoother and more interpretable mapping from expected-return changes to portfolio weights. Small changes in the expected returns have a reasonably low impact on the final portfolio.

It does not merely reduce risk. It improves selection.

That is the key point.

Small changes in Europe’s expected return produce sharply different behavior across the three architectures. Standard Optimization remains the most fragile. Risk-Constrained Optimization improves stability, but the result remains sensitive to TE-CVaR calibration (hint: check it out using the Streamlit App). Lexicographic Risk-Constrained Optimization delivers the smoothest and most benchmark-consistent adjustment path.

The empirical message is not that Stage 1 does not matter. It matters a great deal. The message is that even after a much better Stage-1 problem is specified, the final portfolio still benefits from an explicit selection step that prioritizes plausibility, benchmark consistency, and stability.

A concise way to put it is:

Risk constraints improve feasibility. Lexicographic optimization improves selection.

That is precisely why the two methods should be seen as layers rather than substitutes.

At a given parameter point, the final portfolios already look meaningfully different. The lexicographic second stage tends to preserve the risk-aware structure of the constrained optimizer while selecting a more stable and interpretable benchmark-relative portfolio.

The summary statistics help separate two effects: stronger Stage-1 constraints improve the feasible set, while the lexicographic second stage improves the final portfolio choice within that set.

If you compare the three portfolios, it becomes apparent that the lexicographic portfolio is much closer to the benchmark without giving up to much of the expected return relative to the other frameworks. In fact, it may give up on a few basis points of “expected” portfolio return, but obtains way more stability and even better risk characteristics.

Why this matters in the real world

One of the recurring mistakes in optimization debates is the assumption that if a portfolio is technically “correct,” then it is automatically fit for use in a real investment process.

That is rarely true.

Real-world portfolio decisions are embedded in investment committees, benchmark frameworks, implementation constraints, adviser communication, client communication, and internal governance. In that world, the transparency of the final weight structure matters.

A portfolio that is tightly controlled in CVaR terms but difficult to explain in weight terms is often still a weak portfolio in practice. This is why I do not see benchmark distance, concentration, and turnover as trivial soft concerns. They are part of the real quality of the portfolio.

What this implies for portfolio design

My broader takeaway is that robust portfolio construction is best thought of as an architecture problem, not a one-trick problem.

The right question is not whether to use optimization, or whether CVaR is “better” than lexicographic optimization. The better question is:

How should I structure the optimization process so that it first enforces risk realism and then selects the portfolio that is most usable in practice?

That framing is much closer to how real portfolio construction works.

A professional portfolio framework should not only define what is risk-feasible. It should also define how the final portfolio is selected among the risk-feasible alternatives.

That is why I increasingly prefer the following production logic:

• Stage 1 defines what is acceptable

• Stage 2 defines what is preferable

Final thought

A stronger primary optimization problem can already solve a lot. In many settings, it should be the first thing to fix. If the risk architecture of the optimization problem is too thin, adding a lexicographic second stage alone will not be enough.

But the opposite is also true.

Even a much stronger Stage-1 optimization does not automatically solve the final portfolio selection problem. A portfolio can be well-controlled in absolute and active tail-risk terms and still not be the most stable, interpretable, or governance-friendly portfolio to hold.

That is why I do not see stronger risk constraints and lexicographic optimization as substitutes.

I see them as layers:
the first layer is about risk realism,
the second layer is about portfolio plausibility.

And based on the comparison so far, there is an additional practical nuance: stronger risk constraints clearly help, but their stabilizing effect can still be quite sensitive to calibration. A lexicographic second stage appears to make the final portfolio choice more consistently stable relative to the benchmark.

In many real-world investment processes, robust portfolio construction therefore requires both.

• Why small return changes can cause disproportionately large portfolio shifts

• How lexicographic optimization leads to smoother, more robust allocations

• Why that makes optimization easier to use, explain, and defend in practice

A 10-basis-point change in expected returns should not flip an entire portfolio.

And yet, that is exactly what classical optimization often does.

In one real-world allocation case, I observed a familiar but revealing pattern. When expected returns for US equities were set at 7.0% and European equities at 6.9%, the optimizer tilted aggressively toward the US. When those assumptions were reversed—US at 6.9% and Europe at 7.0%—the portfolio flipped heavily toward Europe.

That is mathematically consistent. But from a portfolio construction perspective, it is hard to defend.

A 10-basis-point difference between two large, liquid, highly correlated equity regions is not a regime shift. It is certainly not strong enough to justify a dramatic reallocation. And yet many optimizers behave as if it were.

This is one of the main reasons so many practitioners distrust optimization. Not because optimization is inherently flawed, but because they have mostly encountered fragile versions of it—versions that translate tiny, uncertain differences in expected returns into extreme portfolio shifts.

In my view, that criticism often misses the real issue. The problem is not optimization itself. The problem is how literally it is implemented.

When done well, optimization remains one of the most powerful tools in portfolio construction. It imposes discipline, improves consistency, and helps transform investment views into objective portfolio decisions. But if it is implemented mechanically—without enough regard for estimation uncertainty, signal reliability, or portfolio behavior—it can become actively harmful.

That is exactly where lexicographic optimization becomes useful.

It is not the only way to make optimization more robust. But it is a highly practical one. In the right setting, it can make the difference between a portfolio process that looks elegant in code and one that actually works in practice.

To improve understanding and you learning, we have provided the code for every graph and each technique on Github. Reading and using the code while following the article is highly advised.

Why classical optimization often loses practitioners

Most practitioners do not reject optimization because they dislike rigor. They reject it because they have seen what standard optimization often does to real portfolios.

It overreacts.

It takes small differences in estimated returns and expresses them with excessive confidence. It treats marginal model advantages as if they were economically decisive. And when assets are similar in portfolio role, risk profile, and correlation structure—as is often the case across regional equities—this behavior becomes especially visible.

That is why equity allocators frequently struggle with optimization. US equities, European equities, Japanese equities, and emerging markets equities are all distinct markets, but they also share a common role as equity risk exposures. Their correlations are high, their economic functions overlap, and the differences in expected returns are often small and uncertain. In that setting, a standard optimizer can become extremely sensitive.

The result is familiar: weights jump around, portfolios become difficult to explain, and confidence in the entire framework erodes.

This is where many people draw the wrong conclusion. They conclude that optimization itself is useless.

I would argue the opposite. Optimization is extremely useful—but only if it is structured in a way that respects the limits of the input signals.

If tiny differences in expected returns trigger extreme portfolio shifts, the problem is not that optimization has failed. The problem is that the mapping from signals to weights is too brittle.

A practical example: when 10 basis points flip the portfolio

To illustrate the point, consider a simple but highly practical equity allocation setup.

The asset universe consists of four regional equity blocks:

• US equities

• European equities

• Emerging market equities

• Japanese equities

The benchmark allocation is:

• US: 40%

• Europe: 40%

• EM: 15%

• Japan: 5%

The optimization is based on a scenario framework, not a simple mean-variance setup. In this case, I use 10,000 scenarios generated from a copula-based approach with skew-t marginals and optimize expected return subject to a hard CVaR constraint. The details of that scenario machinery are not the focus here. What matters is that this is already a more realistic portfolio construction environment than a toy mean-variance example.

The constraints are the same across all cases:

• maximum CVaR: 25%

• US equities: 20% to 60%

• European equities: 20% to 60%

• EM equities: 0% to 25%

• Japan equities: 0% to 20%

Now consider four cases.

Case 1: Standard optimization
Expected returns: US 7.0%, Europe 6.9%, EM 7.5%, Japan 5.5%

Case 2: Standard optimization
Expected returns: US 6.9%, Europe 7.0%, EM 7.5%, Japan 5.5%

Case 3: Lexicographic optimization
Same expected returns as Case 1, but with a second-stage objective that minimizes distance to the benchmark while preserving almost all of the primary optimum

Case 4: Lexicographic optimization
Same expected returns as Case 2, again with a second-stage benchmark-distance objective under a near-optimality constraint

This is exactly the kind of setup that exposes the weakness of classical optimization.

Under Cases 1 and 2, a tiny 10-basis-point change in relative expected returns between US and European equities leads to a large swing in portfolio weights. The portfolio is internally consistent, but economically too aggressive. It is treating a very subtle expected return difference as if it were highly reliable information.

In practice, that is rarely what the portfolio constructor actually wants.

Standard optimization flips aggressively between US and Europe when expected returns change by only 10 basis points. Lexicographic optimization preserves the directional tilt, but keeps the allocation far more stable and intuitive.

What lexicographic optimization does differently

The core idea behind lexicographic optimization is simple: not all objectives should be optimized at the same time and with the same status.

In standard optimization, one typically defines a single objective and solves for the mathematical optimum. In practice, however, portfolio construction usually involves a hierarchy of goals. There is a primary objective—such as maximizing expected return subject to a risk limit—and there are secondary goals, such as reducing unnecessary concentration, staying closer to a benchmark, or avoiding unstable allocation jumps.

Lexicographic optimization formalizes that hierarchy.

In the first stage, solve the primary optimization problem (e.g. Mean-CVaR):

where f(w) is the main portfolio objective.

In the second stage, solve:

Here, g(w) is the secondary objective and ε defines how much of the primary optimum one is willing to give up—typically only a very small amount.

In plain English: first find the best portfolio according to the main objective. Then keep only portfolios that preserve nearly all of that value. Within that restricted set, choose the portfolio that best satisfies the secondary goal.

That may sound like a subtle change. In practice, it is often transformative.

Instead of forcing the optimizer to chase the last few basis points of estimated objective value at any cost, the process acknowledges that many portfolios near the optimum are economically almost equivalent. Once that is recognized, the remaining degrees of freedom can be used to improve portfolio behavior.

Why this works so well in practice

The real strength of lexicographic optimization is that it changes the intensity with which signals are translated into weights.

In a classical framework, if one region has even a marginal expected return advantage, the optimizer may exploit that edge very aggressively—especially when the competing assets are otherwise similar. But in practice, the reliability of such small differences is limited. A well-designed process should reflect that.

Lexicographic optimization does exactly that.

It still respects the primary objective. It does not abandon return-seeking, utility maximization, or disciplined risk budgeting. But it refuses to treat tiny differences in estimated returns as if they deserved a portfolio revolution.

That is why the method is so useful for practitioners. It preserves the economic logic of optimization while producing results that are far easier to defend, communicate, and implement.

In the US-versus-Europe example, that means the lexicographic portfolios in Cases 3 and 4 still tilt in the intuitive direction of the higher expected return region. But the tilt becomes measured rather than extreme. The portfolio remains close to the original investment logic, yet avoids the violent reallocation triggered by standard optimization.

That is precisely the kind of behavior most investors and committees want to see.

The portfolio should react smoothly, not theatrically

One of the strongest pieces of evidence in this example is the sensitivity sweep.

If one gradually lowers the expected return of European equities from 7.0% to 6.0%, the resulting portfolio weights under a lexicographic framework adjust gradually. The Europe weight declines step by step. The US weight rises correspondingly. The process behaves in a smooth and intuitive way.

That matters more than many optimizers admit.

A portfolio construction process should not respond theatrically to subtle shifts in expected returns. It should react proportionally. Small changes in inputs should usually produce small changes in outputs, especially when those inputs are estimated rather than observed with certainty.

This is not only a statistical issue. It is also a communication issue.

When portfolio weights move smoothly and intuitively, the resulting decisions are much easier to explain to advisers, clients, investment committees, and fellow practitioners. A portfolio that changes by 2 or 3 percentage points in response to a small view adjustment feels credible. A portfolio that flips from one regional block to another does not.

That is one reason the sweep matters so much. It shows that lexicographic optimization improves not just the mathematics of the solution, but also the usability of the portfolio decision.

Under lexicographic optimization, the allocation adjusts gradually as Europe’s expected return declines. Small changes in expected returns lead to small changes in portfolio weights — exactly the kind of behavior practitioners typically want.

Why benchmark distance works well here—but is not the main point

In this example, the second-stage objective minimizes the L2 distance to the benchmark.

That works well because the practical problem here is not a lack of return orientation. It is excessive sensitivity in active regional tilts. Using benchmark distance in the second stage stabilizes the portfolio and keeps it aligned with a sensible strategic anchor, while still allowing the portfolio to reflect the return signal.

But the benchmark is not the deeper story.

The deeper story is that lexicographic optimization lets the portfolio constructor express a clear decision hierarchy. First, preserve the main portfolio objective. Then, among all portfolios that are nearly as good, choose the one that behaves better.

In another setting, the secondary objective could be something else entirely:

• minimizing concentration

• minimizing turnover

• minimizing distance to the previous portfolio

• controlling active risk more explicitly

The exact second-stage objective can vary. The principle does not.

Once the primary objective is preserved, the remaining flexibility should be used to improve robustness—not to pursue mathematical precision for its own sake.

A short note on penalty terms

A natural question is whether the same result could be achieved by simply adding a penalty term to the main objective.

In some cases, yes. But the practical distinction is important. A weighted-sum approach mixes everything into one objective and makes the result depend on the calibration of penalty parameters. Lexicographic optimization is cleaner in one important sense: it preserves the priority structure explicitly. First secure the core investment objective. Then improve the secondary portfolio property within that admissible range.

For the purpose of this article, that distinction is enough.

What this says about optimization more broadly

Many of the harshest critiques of optimization come from practitioners who have seen it behave badly in practice. That skepticism is understandable. But it is often directed at the wrong target.

The issue is rarely optimization itself. The issue is that optimization is often implemented too mechanically, too literally, and without enough respect for uncertainty. A weakly informed expected return difference is fed into a strong optimization engine, and the output is then mistaken for conviction.

That is not a failure of optimization. It is a failure of design.

In good hands, optimization remains one of the best ways to turn investment views into disciplined portfolios. It enforces consistency, forces trade-offs to be made explicitly, and can be a powerful antidote to arbitrary asset allocation decisions.

Lexicographic optimization is one practical way to make that process more robust. It is not the only one. Bayesian methods, shrinkage, resampling techniques, and other robustification approaches also matter. But lexicographic optimization deserves more attention than it usually gets, because it addresses a problem that practitioners encounter constantly: excessive portfolio sensitivity to small and uncertain input differences.

Final thought

The best portfolio is often not the mathematically pure optimum.

It is the portfolio that preserves the core economic objective while reacting sensibly to uncertainty.

That is why I find lexicographic optimization so useful in practice. It does not reject optimization. It makes it more usable. It does not replace investment judgment. It gives judgment a cleaner structure. And it does not pretend that every estimated return difference deserves a dramatic portfolio response.

Tiny differences in expected returns should not blow up a portfolio.

If they do, the problem is not optimization itself. The problem is how literally it has been implemented.

Lexicographic optimization is not a silver bullet. But it is one of the most practical ways I know to make optimization behave more like an investor and less like a machine.

If you rebalance monthly but estimate covariances from daily returns (and then multiply by 21), you often end up comparing apples to oranges. While you are actually interested in the risk forecasts of one month (or even more), the usage of daily data will not be aligned with our horizon. The result is: You get different portfolios and different risk forecasts, even when the estimator is identical. Even worse: You might obtain a portfolio for a “day trader”, while you are actually interested in a diversified low-frequency portfolio.

This post separates horizon (what you want to forecast) from frequency (how you measure returns), shows why “monthly = 21×daily” breaks under real market dynamics, and ends with a practical recipe you can ship into production. With this information, you will be able to decide on the compromise between using “more” data vs “better-aligned” data to your specific investment horizon.

Takeaways:
(1) Rebalancing horizon defines the covariance you need.
(2) Scaling is not a free pass.
(3) Fix the horizon first—then compare estimators (Part 3).

To improve understanding and you learning, we have provided the code for every graph and each technique on Github. Reading and using the code while following the article is highly advised.

1) Introduction: The Trap

In Part 1 we showed that the sample covariance matrix is unstable—and why shrinkage, denoising, and factor structure can help (while matrix norms and condition numbers are not portfolio-quality metrics by themselves).

Part 2 picks up the question that usually gets ignored in practice: how much does data frequency (daily vs. monthly) matter when the portfolio horizon is monthly?

Here’s the trap: Many systems rebalance monthly or even quarterly, but estimate Σ from daily returns and then scale it to monthly. The result is often a different risk model, a different optimizer input, and therefore a different portfolio—even when nothing “improved” except your measurement frequency.

2) Clean definitions: horizon ≠ frequency

Horizon is the holding period your portfolio actually lives on (e.g., monthly rebalancing → 1-month risk). Frequency is how you sample returns (daily, weekly, monthly).
If your rebalancing horizon is one month (and we assume variance as you target measure of risk), then your wanted target risk is:

We strongly advise to use robust approaches to estimate the covariance matrix in general and we also advise to use more intuitive risk measures like the Conditional Value at Risk. Our analysis in this post easily generalises to more complex risk measures and we use variance / volatility primarily for convenience.

3) Why daily ≠ monthly ≠ quarterly in real world:

The Covariance can be subdivded into two elements: (1) Volatilities and (2) Correlations - and both elements are heavily affected by using data of different frequencies:

In the equation above, x is the data set and it can combine data of different frequencies. For the following analysis assume that we focus on monthly target risk.

Estimation of Volatilities:

While many people are pretty aware that their horizon is monthly or even longer, people still tend to use daily data to estimate volatilities. The belief is that “more data is better, so daily volatilities are converted into monthly volatilities. The “Scaling” is done a lot in practice (even in regulatory settings).

Scaling boils down to a very simple equation: “Monthly = 21×Daily”. This works only under conditions that markets rarely meet: IID returns, constant volatility, stable correlations, and no regime changes. As we discussed before, these assumptions are highly unrealistic in practice. The chart below shows how “annualised” volatility clearly depends on the frequency of the initial data.

Estimation of Correlation:

In general, please agree that correlations can strongly vary over time due to the presence of different economic regimes. However, even in the absence of regime changes, the data frequency can heavily impact the correlation estimates.

In the chart below, we present estimates of the correlation between equity and bonds based on different frequencies. Although the measures have converged in the recent period, the differences are still enough to affect portfolio allocation tremendously.

Overall, the assumption that frequency and horizon are decisions that can be made separately is incorrect. The “bridge” known as “Scaling” is more a theoretical than empirically sound approach.

Reality is the opposite: volatility clusters, correlations move with regimes, and tail dependence shows up precisely when you care about risk forecasts the most.

Even if volatility scaling is “approximately” right, dependence is not scale-invariant. Daily comovement can look very different once returns are aggregated to a monthly horizon—especially through stress periods and regime transitions.

4) Three canonical setups

To isolate the frequency/horizon effect, we keep the covariance estimator fixed (Ledoit–Wolf shrinkage) and only change how we align estimation and evaluation horizons.

A) Estimate monthly / evaluate monthly (horizon-clean)

• Estimate Σ from monthly returns (rolling 36 months)

• Evaluate risk on the 1-month horizon

• Pro: target matches the decision horizon

• Con: fewer observations → noisier estimates
  Gleichungen

B) Estimate daily / scale to monthly (common in practice)

• Estimate Σ from daily returns (rolling 756 trading days)

• Scale:

• Pro: more observations, typically lower turnover

• Con: dependence is not guaranteed to scale correctly
  

C) Mixed-scale (daily vol + monthly correlation)

• Estimate vol from daily data (EWMA, lambda = 0.94)

• Estimate dependence (correlation) on the monthly horizon

• Build:
  

• Pro: uses daily data where it helps most (vol), keeps dependence horizon-consistent

• Con: more design choices, can increase turnover if vol is too reactive.

5) Mini case study (real ETF universe)

Universe (same as Part 1): SPY, EFA, EEM, AGG, LQD, HYG, GLD, VNQ, DBC.

Backtest: monthly rebalancing (month-end), long-only minimum-variance,

Constraints for Optimization:
(1) Max weight 60% per asset. (2) Minimum Number of Effective Assets is 3.0.

Sample: 2007-04-11 to 2026-01-09 (based on the daily close history used in the run).

Result 1: Same target horizon, different data frequency
→ different portfolio!

Do different setups produce substantially different portfolios?

Across rebalances, the portfolios from Setup A (monthly), Setup B (daily×21) and Setup C are materially different.

It may not “look” much but remember: (1) The scaling of the figure is in favor of underestimate the portfolio differences and (2) we have already introduced some constraints in the Minimum-Variance Optimization.

The average L1 distance between weight vectors are visualized here, when we perform rolling sample optimization.

We see that especially over time, we obtain vastly “different portfolio” claim using different approaches.

Result 2: Risk forecast quality ≠ economic quality !

Which estimator forecasts a fixed horizon better?

We estimate how well each approach forecasts the risk of its respective portfolio. We find, that the mixed approach is superior based on MALE (Mean Absolute Log Error)…but unfortunately this does not in general lead to higher economic value as measured by Sharpe Ratio. However, this result can (fortunatelly) not be generalized, but it shows that the general idea: “Better Forecasts = Better Portfolios” does not always hold.1

We also analyse the forecast capabilities for the classic 60/40 Portfolio (Equity & Bonds) by analysing the Realized / Predicted Variance Ratio

Setup B often appears more stable (lower turnover), but that does not automatically mean it is more correct for a monthly decision horizon—it can simply mean you optimized on a different dependence structure.

Result 3: …What about the Portfolio Performances?

I know what you are thinking: An analysis without a performance summary and chart is incomplete. Every Substack on Finance does this.

Let me be very clear: In many cases, presenting the performance of an estimator for a specific period (e.g. backtest) does not make sense from a statistical point of view. Judging an estimator by “which portfolio won” is the wrong approach. The experiment is about model consistency, not a horse race of realized returns. If you change the covariance estimator, you change the risk model; if you change the risk model, you change the portfolio. Period.

Those return differences are then heavily regime- and path-dependent noise around the actual point: whether your estimator reflects the horizon you are investing on. Treating a short sample performance spread as proof that one estimator is “better” is category error. The real failure is horizon mismatch, because that is what systematically drives mismeasured risk and unstable allocations.

6) Practical rules of thumb

1. Decide the horizon first (rebalance schedule defines your risk target).

2. Evaluate on that horizon (otherwise you can’t compare models fairly).

3. Treat scaling (“×21”) as a model assumption, not a law of nature.

4. If you must use daily data, use it primarily for volatility, not blindly for dependence.

5. Mixed-scale models are often a good compromise: daily vol + horizon-matched correlation.

6. Watch out for “improvements” that are really just frequency artifacts (lower turnover is not proof of better horizon risk).

7. Write down the horizon contract: “This Σ forecasts 1-month risk under monthly rebalancing.”

7) Summary & Teaser to the next Article

• Horizon ≠ frequency.
  Data frequency is a measurement choice; horizon is the portfolio’s objective.
  Do you want your portfolio to be diversified for the long-run or short-run? Is both possible?2 Estimator choice is a horizon choice; frequency-mismatched covariance produces different portfolios, even with the same optimizer
  

• Scaling is fragile
  Autocorrelation, volatility clustering etc. violate the IID assumption of asset returns. This makes scaling from one frequency to another by using the “scaling equation” very fragile. In addition: Dependence structures (e.g. correlations) are also are not scale invariant in real markets and very with data frequency.
  

• Fix the horizon first—then evaluate estimators in portfolio space.
  Focus on the things that are given first. If a portfolio has a long-term objective, this factor should then be a cornerstone of you approach to estimate parameters and optimize the corresponding portfolio.

Part 3 of the Covariance Series: Now that the horizon is well-defined, we can finally compare estimators where it matters: risk forecast error, turnover, and concentration. Evaluation of estimators inf portfolios space will be part of the next article.

While CVaR appears to be a “silver bullet” for many investors, it is also true that in investing “one size does not fits it all”. Whether you are an investor yourself or perform portfolio optimization in a more professional context, CVaR might not always be the most fitting choice as investors have differences in their preferences and their perception of “risk”.

Therefore, this article will:

• Demonstrate how easy convex optimization allows to define risk measures as optimization objectives which can take into account multiple targets and highly complex risk targets

• Examine the impacts of combining CVaR with other risk measures and its implications on the “optimal portfolio”

• How to use Copula-Marginal Combinations to obtain “more realistic” parametric simulations of asset returns and thereby also portfolio returns - while allowing for excess-kurtosis and skewness.

The outline of the article is as follows: First, I will briefly explain why variance should not be used as a risk measure in portfolio optimization — at least not in isolation. Next, I will highlight the superiority of CVaR over variance, as well as its limitations. The main theoretical focus is to show how CVaR can be adjusted with additional convex risk measures to capture more nuanced aspects of “risk” in optimization. In the empirical section, I will perform portfolio optimization assets to illustrate the impact of supplementing CVaR with two different risk measures.

To help you follow along and deepen your understanding, all code for the graphs and techniques discussed in this article is available on Github. Reading and running the code while following the article is highly recommended.

If you want to learn more about convex optimization, the following sources are of immeasurably quality:

1. Stephan Boyd’s Course on Convex Optimization

2. Dany Cajas’ Book on Portfolio Optimization

I. Theoretical Framework

I.1 Variance is (still) not a viable choice

Although there is still resistance in the community, it is widely recognized that variance is not a useful risk measure for portfolio optimization. In finance and investing, there is always a trade-off between having a robust estimation framework (e.g., assuming a specific distribution for returns) and realistically approximating the real world. While variance as a risk measure satisfies the former, it fails at the latter. As a rule of thumb: if something is too far removed from reality, it should not be used as a viable approximation.2

Using variance in isolation as the only risk measure in your optimization is not a sound approach. The following points highlight the shortcomings of the classic Mean–Variance approach:

1. Symmetric Treatment of Gains and Losses
   Variance penalizes both upside and downside deviations from the mean equally. In the history of investing, there have been few complaints about “variance to the upside.”

2. Ignores Tail Risk
   Most people think of “risk” as extreme — sometimes even permanent — capital loss. Day-to-day or month-to-month fluctuations (i.e., variance) are rarely viewed as “risk” in the strict sense. The key point: portfolios with the same variance can have vastly different exposures to catastrophic loss.

3. No Focus on Investor Goals or Thresholds
   Variance does not consider specific return targets or loss thresholds. This is especially relevant for institutional investors with defined return objectives, liabilities (e.g., pension funds, insurance companies), or strict risk budgets.

4. Assumes Normally Elliptically Distributed Returns
   The final nail in the coffin: while mean–variance optimization does not require returns to be normally distributed, it does require them to be elliptical.3 In practice, asset returns often show skewness and fat tails — features that variance cannot capture.

These points show that using variance in portfolio optimization is like aiming at the wrong target. To borrow some insightful words:
”Variance for Intuition, CVaR for Optimization”

I.2 CVaR - the most prominent alternative

Following the discussion of variance as a risk measure, CVaR has emerged as the leading alternative, because it:

(1) exclusively focuses on losses
(2) Specifically targets the left tail (extreme losses).
(3) Easily incorporates non-elliptical return distributions.
(4) Is a coherent (and therefore convex) risk measure.

Based on these characteristics, CVaR is closer to the “general idea of risk” and more in line with the principles of behavioral finance. Let’s examine the general CVaR equation to see its properties:

Overall, the formula is straightforward: only the elements of the loss distribution beyond the Value-at-Risk (VaR) threshold matter for our portfolio choice. These events are weighted by their probabilities. In this sense, the VaR acts as “the wall”: anything beyond it is ignored, and we focus only on our small world of the tails.4

In the practical approach described in Section II, we will use a scenario-based distribution. In this case, the weighting scheme for each data point is as follows: every return worse than the critical VaR receives the same weight, and every return beyond the “VaR wall” receives zero weight and becomes irrelevant.

This highlights one of the few drawbacks of CVaR: while the tails are very important, CVaR disregards everything else. This raises the question: what if other parts of the loss distribution also matter?5

I.3 Risk measures as convex combination

There are several ways to capture more than just the extreme tails in an optimization that uses CVaR. We can group the options into three categories:

1. Add additional risk measures as constraints into the optimization
   Additional measures — such as Lower Partial Moments (LPM) or tracking error — can be included as constraints with maximum allowed values. These constraints only affect the estimated portfolio weights if the limit is exceeded, meaning they are not part of the objective function. A drawback of this approach is that it can be difficult to determine the appropriate maximum values in advance.

2. Change the Weighting Scheme

   One approach is to adjust the weighting scheme described earlier. This is the idea behind Ordered Weighted Average risk measures, which assign weights directly.6 Another related approach is to use spectral risk measures, where a predefined function assigns weights to different parts (e.g., quintiles) of the loss distribution. While these approaches can be effective, they sometimes obscure the interpretation of what exactly is being minimized, as not all weighting schemes have a clear economic meaning.

3. Create customizable risk measures as objective function of the optimization

   It is possible to combine convex risk measures directly into a custom risk function, which is then minimized using standard convex portfolio optimization routines.

Given the following optimization set up:

At the heart of this paper is the risk function g(x) In convex optimization, the main requirement for an objective function (in a minimization problem) is convexity. Since the sum of convex functions is also convex, we can create a customized weighted sum of risk measures.

For example, we could combine a tail-focused measure like CVaR with a more general downside measure:

Using this approach, we can target the tails with CVaR while also incorporating a broader downside measure into the objective function — capturing a dimension of risk that CVaR largely ignores. The parameter θ can then be used to set the relative importance of each measure.

A Note on Scaling

CVaR and LPM do not share the same scale. CVaR is generally much larger than LPM, so one measure may dominate the optimization. This can be addressed through normalization, for example, by using the risk measures of a benchmark portfolio:

This can be solved by using a normalization routine assuming that you have a benchmark of the portfolio:

The benchmark’s risk measures act as the normalizer and do not interfere with the optimization, as they are positive and deterministic. If no benchmark is available, an equally weighted (1/N) portfolio can serve as a simple baseline. Using this formulation, θ can be interpreted as the relative importance of each measure.

Absolute vs. Relative Risk Measures

So far, we have only included absolute risk measures in the objective function. It is also possible to include relative risk measures, such as tracking error. The challenge is that normalization is less straightforward for relative measures, since we cannot use the benchmark-based method described above. For this reason, relative risk measures should only be included if they are on a similar scale to the other measures used.7

Conclusion

Combining tail risk measures with broader downside measures appears to be a promising way to capture different dimensions of risk. In the following section, we will apply this framework using…

II. Case-Study

II.1 Scenario-Generation of Asset Class Returns

In our case study, we use data from the following asset classes:

1. Government Bonds US

2. Government Bonds EMU

3. Corporate Bonds US

4. Corporate Bonds EMU

5. HY Bonds Global

6. Equity US

7. Equity Europe

8. Equity Emerging Markets

The expected returns, covariances, and degrees of freedom are available in the GitHub repository. As explained earlier, we are leaving variance as the sole risk measure behind. Consequently, we also leave the mean–covariance world — i.e., the pure normal / elliptical distribution — behind. We therefore require an alternative to model the joint distribution of returns.

In our setting, we use a Copula–Marginal Combination, which allows us to separate the modeling of the joint dependencies between asset returns from the modeling of the marginal distributions of each asset. Specifically, we use a Gaussian Copula with Skew-t marginals, where each marginal Skew-t distribution has its own degree of freedom and skewness parameter.

While this is still far from a perfect real-world approximation, it is useful because it flexibly accounts for asset-specific differences in the first four moments across asset classes.8 More realistic approaches could involve mixed copulas and/or historical resampling. 9

A Brief Note on Copula–Marginal Approaches

Let’s assume we have already measured the dependencies — in our Gaussian Copula case, in the form of a correlation matrix — and that we know the expected return, variance, skewness, and degrees-of-freedom parameters for each marginal asset return distribution.

The process of generating simulations of the joint probability distribution involves two steps:10

1. Random sampling from the Copula distribution:
   Aim: Generate “grades” that contain only the interdependencies between asset returns.

   1. Random Draw from MV Normal Distribution & Standardization using normal CDF

      1. We draw from a multivariate normal distribution with the expected return vector set to zero and covariance equal to the estimated correlation matrix.

      2. We “standardize” these draws by mapping them through the standard normal CDF and obtain the Grades

         • The resulting grades are uniformly distributed between 0 and 1.

         • Example: Generating 1,000 samples yields a matrix of shape (1,000 × #assets) containing only the joint information

         • They only reflect the “Joint” Information of the Asset Class Distribution.

2. Inserting the Grades into the Marginal Distributions:

   1. Aim:Introduce asset-class-specific information into the distribution.

   2. Since the grades are uniformly distributed, we insert them into the quantile function of each marginal distribution to obtain simulated returns for each asset class.

he following graph illustrates the difference between the grades and the final joint distribution. On the left side, grades are more dispersed when the correlation is close to zero (e.g., US Equities vs. Gold), reflecting weaker dependence. When correlation is high (e.g., US Equity vs. Europe Equity, US Treasuries vs. US Corporate Bonds), high grades in one asset tend to align with high grades in the other.

Even after inserting the grades into the marginal distributions, the joint dependency structure is preserved. A closer look at the marginal distributions also reveals that some are clearly non-symmetric.

An overview about all marginal and joint distributions looks as follows, where we can see (by zooming in) that some of the marginal distributions show non-symmetric profiles.

II.2 Optimization Results and Properties

The central empirical question is: Does combining CVaR with additional risk measures materially change portfolio characteristics, or does CVaR alone already capture these other aspects of risk?

To keep results realistic and avoid overly concentrated portfolios, we apply additional constraints. By setting the target portfolio’s return equal to the benchmark return, we focus solely on the risk dimension. We set the following constraints:

We define the Benchmark as:

Case 1: Combination of CVaR and Lower Partial Moments

We can observe that the Portfolio Weights show some modest changes as we vary the relevance of CVaR vs. LPM.11 However, the change in the portfolio composition is limited and those changes are primarily happening in asset pairs with high correlations.
In this regard, it is also no surprise that the Risk Measures do not really depend whether we optimize using CVaR or LPM. As expected, when we only optimized on CVaR (LPM), we obtain the minimum in the relevant risk metrics, but the differences are overall very low.

Case 2: Combination of CVaR and Mean Average Deviation

Using the MAD12 as additional risk measure, we observe a higher impact on the portfolios and risk measures - which is still very modest overall. This is not surprising, while LPM is also a downside risk measure (like CVaR), MAD is close to Standard Deviation as it is about the dispersion around the “expected return of the portfolio”.

Overall, the impact of including MAD into the risk function has a limited effect on the optimal portfolio as well as the risk metrics.

III. What This Means — And What It Doesn’t

To answer the key question of this article:
”Is CVaR enough as a Measure of Risk?”

For most long-only investors which focus on broad asset classes CVaR is enough. It is an intuitive, well established risk measure which is very close to how people perceive “risk” in reality.

However, what happens with this conclusion, when we consider more professional investors? Change the setup and adjusting assumptions, and you may get very different results. For example:

• Allowing for short positions

• Including derivatives (tail hedges, options)13

• Add more granular assets (single stocks)

• Use dependency structures with correlation tightening in crises

From experience, once you throw derivatives into the mix, combining risk measures can suddenly matter a lot more.

Don’t take my word for it — test it yourself with the code in our GitHub repo. We are looking forward to obtain your feedback!

The best ideas about markets and timing are worth little, when there is no objective, unbiased routine to implement them into a portfolio. The crucial element to survive as an investor is to measure your risk correctly - in this regard, the covariance is still the main ingredient when portfolio risk is quantified. Therefore, after reading this article you will understand the essential concepts of the covariance and you will know superior approaches to estimate the covariance matrix more accurately in order to obtain better results in portfolio optimization and risk quantification. You will understand why it is so important to Clean Up your Covariance Matrix!

In this article, you will see how to capture the long-term characteristics of financial assets that are typically required for the construction of a strategic asset allocation (SAA). This assumes an investment horizon of at least 5 years. This focus allows the investor to look beyond specific parts of economic cycles and their conditional implications for return characteristics.

This article is Part 1 of our quest to show how to improve the estimation of the covariance matrix, which is still considered the most important parameter for portfolio construction in practice. Several advanced techniques will be presented which are easy to implement, commonly used in practice and highly superior to the simple sample-based estimation of the covariance matrix.

To improve understanding and you learning, we have provided the code for every graph and each technique on Github. Reading and using the code while following the article is highly advised.

The Relevance of the Covariance Matrix for Portfolio Optimization

Everyone who has touched the topic of portfolio optimization is familiar with the covariance matrix. In most cases, the covariance matrix is the key quantity to summarize the risk and (linear) dependencies of assets.1 You need an accurate and robust estimate because the optimized portfolio weights are based on it. If the covariance matrix is estimated poorly, you will over- or underestimate the risks of assets and their diversification effects. This should already be enough motivation to put some interest in its estimation.

However, the covariance matrix offers more than meets the eye. In fact, the biggest criticism of mean-variance optimization (MVO) is the sensitivity of portfolio weights to small changes in the input parameters (especially for varying expected returns). This aspect comes from the so-called “condition number” which is a statistical property of a (positive semi-definite) matrix and defined as follows:

The condition number is equivalent to the ratio of the largest and smallest eigenvalue of the covariance matrix.2 Using financial data, it is common to have some large eigenvalue on which the covariance matrix “loads” on. This is due to the stylized fact of high correlations between returns of similar assets. The “market factor” has often the dominant impact on returns and is mostly represented by the largest eigenvalue. However, you should put the focus on the smallest eigenvalues. The smallest eigenvalue will usually not represent much information, but often contains noise and estimation errors. Therefore, a high condition number will often be a sign that our estimation includes unnecessary elements which further reduces the already low signal-to-noise ratio in finance.

A high condition number leads to a very unstable inverse covariance matrix. This is a problem due to the fact that the optimal portfolio weights (in MVO) do not depend on the covariance itself, but on its inverse. Instability in this context means that small changes in the input parameters (e.g., expected returns) can have a large impact on the optimization output. This issue is not really a problem when using (non-parametric) Mean-CVaR optimization, which does not need a covariance inversion. That is another reason, why non-parametric Mean-CVaR is a more sound choice as an optimization algorithm than the classic mean-variance.

When the covariance matrix is inverted, its eigenvalues become the reciprocal of the original eigenvalues, while the eigenvectors remain unchanged. As a result, directions associated with small eigenvalues - potentially noise - in the original covariance matrix become dominant in the inverse. In the context of MVO, this means that these low-variance directions can disproportionately influence the portfolio weights. This sensitivity to small eigenvalues can lead to unstable solutions, which is the reason why regularization / shrinkage techniques are often used and much attention is warranted - especially if you believe that these eigenvalues are rather connected to noise than information. And portfolio weights driven by noisy elements does not sound right, does it? An example for 7 assets is given below. The smallest eigenvalue (EV 7) of the original covariance matrix has the largest eigenvalue in the inverted covariance matrix. Thus, it will have the largest impact on MVO.3

In short, a high condition number means that your covariance matrix is “on the knife edge” of numerical and statistical instability — any downstream computation that relies on it (not just portfolio weights) will be fragile and potentially misleading. This should be kept in mind not only for mean-variance optimization, but also for certain clustering algorithms, outlier detection, etc.

Financial Time Series Data: More Issues ahead

You have to be honest with youself about one aspect: Financial time series are difficult to handle. Before you can even think about estimation of the covariance, you have to make several assumption regarding the data generation process. At its core, you have to assume that the data is identically and independently distributed (IID)4 or at least ergodic5.

IID-ness is a very, very strong statistical assumption. Just to give you an indication which properties of financial time series can lead to a violation:

• Serial correlation (autocorrelation) bias

• Non‐stationarity / time‐varying moments

• Volatility clustering & heteroskedasticity

• Non-synchronous observation times

• Structural breaks or regime shifts

This article does not focus on these elements, but the user should keep in mind that simply using covariance estimators on raw data might not be the best of idea. For example, using return data from equities or (open-ended) funds to estimate long-run parameters is often valid, as long as there is at least some hope that the returns are ergodic. For other financial instruments, like specific bonds or derivatives, you will need a fully fledged simulation framework - simply using past returns of bonds or derivatives does not work in practice.

Estimated vs. True Parameter

For the sake of simplicity, let’s assume that the data generating process is well-behaved (= at least ergodic). In most applications, the data generating process is not known. Therefore, the distribution and its moments must be estimated from the data. The most basic approach to use is the so-called “sample” covariance. In many cases, the usual decision is to estimate the covariance matrix with the sample covariance estimator like its some kind of commandment in statistics - despite its drawbacks.

The following is important and crucial to understand: There is a distinction between an estimator of a parameter and the true parameter itself. This is very relevant and easier to understand when considering the mean: The sample mean is an estimator of the true mean (unbiased, expected value) of a distribution. Another estimator is e.g. a trimmed mean, which is also an (robust but biased) estimator of the true mean of the distribution cleaned for outliers. This means that there are usually several estimators for a given parameter and those estimators can be tested under reasonable assumptions based on the concepts of estimation theory.

The Baseline: Sample Covariance

Following the argumentation from above, it is important to understand that the sample covariance is only one possible estimator of the true covariance. The estimator is common knowledge and reads as follows:

Using the sample covariance to estimate the true covariance is not the best idea. The following contains the key reasons which every quant should learn by heart. Although the arguments are presented as “distinct” elements, most of them are closely connected with each other.

1. Unbiased on average… but with high estimation error

   • Sample covariance estimation is unbiased on average. But to obtain this property, it is paying a high price, as the estimation variance can be tremendous making it a highly unreliable estimate.

   • In practical settings, when data is limited, the sample covariance has the potential to be highly different from the true parameter.

   • Alternative approaches introduce a slight amount of bias, to reduce the variances significantly - which is often a better trade-off.

   

2. Equal-weighting of data points in estimation

   • Are all data points created equally?
     Should you put more “weight” on some data points and less to others?

   • In our opinion, there might be good reason to weight data point differently in the estimation process - due to some data points being too “extreme” or some just being too far away in the past.

   • Sample covariance (in its original form) implicitly assumes an equal-weighting by simply dividing the quadratic sum of deviations by the number of observations. This is not always appropriate.

3. High Variance in Small Sample, Curse of Dimensionality and Invertibility

   • When estimating a sample covariance, the number of parameter to be estimated from data quickly explodes:

   • As the number of parameter increases and the number of data points is fixed, the estimation becomes more noisy and less accurate.

   • If the number of variables (n) is larger than the number of observations (T), the sample covariance estimate becomes singular and the matrix cannot be inverted. This is an issue for any statistical analysis - but especially for many portfolio optimizers.
     

4. Ledoit-Wolf Effect

   • The Ledoit-Wolf effect highlights that the eigenvalues of the estimated sample covariance matrix tend to be more dispersed than the eigenvalues of the true data generating process.

   • For the sake of the argument, let us suppose the following example:

     • Given you have three financial instruments with identical volatilities and each being uncorrelated with the other.

     • What would you assume regarding the eigenvalue structure of the covariance matrix?

     • The eigenvalues should be identical in size…but due to issues introduced by the estimation approach, the eigenvalues are far from equal.

     • For a practical coding example see Github.

   • The estimation process leads to a too high dispersion among eigenvalues which increases the condition numbers. This in turn negatively affects the stability of many portfolio optimizers.
     

5. Sensitivity to Outliers

   • Sample covariance (and sample mean) are known to be highly sensitive to potential outliers.

   • Sample covariance is based on second‐moments and weights every point quadratically.

   • A single aberrant observation can dominate the estimation process, giving wildly inflated variances or spurious correlations.6

   • Therefore, the sample covariance is a non-robust estimator. It is therefore no surprise that the sample covariance is highly dependent on the specific samples used.

   • The following graph shows the impact of only 4 outliers in an dataset of 200 datapoints. In this example, a positive correlation in the data turns negative by the impact of only a few “bad datapoints”.

   • Let’s assume that those are two of your assets in an optimization: Using this (faulty) result, you will falsely assume high diversification effect between both assets which are not present in reality leading to a much higher portfolio risk than estimated.

     

6. Implicit assumption of normally distributed returns

   • Although, no specific assumption regarding the distribution are made, the sample covariance estimator has a clear connection to the normal distribution.

   • Let’s have a look at the maximum-likelihood estimator of Σ given a multivariate normal distribution is assumed. The estimator is:

   • The equation of the estimator is almost identical to the sample covariance.
     

Decomposition: Covariances, Correlations and Volatilities

Before you will learn about the alternative estimators, it is important to stress the relationship of the covariance to the correlation and volatilities. As you (hopefully) know, combining correlations and volatilities generates the covariance matrix. The decomposition is as follows:

The covariance is therefore the combination of marginal and joint information. The important element to consider is that techniques to estimate the covariance matrix can - if necessary - also be separated into two steps:
1) Estimation of the correlation matrix and
2) Estimation of the marginal information (aka volatilities).

This decomposition is important in practice because in many cases, the difficulties with the covariance matrix arise mainly from the joint behaviour between random variables. Univariate volatilities are easier to estimate than the covariance between two random variables. This is because the covariance is the product of two univariate volatilities and the correlation coefficient which contains therefore more parameters (and uncertainty) that needs to be estimated from noisy data.

The decomposition provides the user with even more flexibility to apply alternative approaches to estimate the covariance “in one go” or - which is preferable in many cases - to focus on the estimation of the correlation matrix. This is also what we will be shown in the examples. As most estimators behave better when data is standardized, this will turn the covariance estimator into a correlation estimator. Afterwards, you can still combine the correlation with the volatilities to obtain the covariance matrix.

The Alternatives

At this point of the article, it is time to introduce alternative and superior techniques to estimate the covariance matrix:

1. Exponentially-weighted Sample Covariance

2. Outlier-Robust Sample Covariance

3. Graphical Lasso

4. Covariance shrinkage: Ledoit Wolf

5. Spectral Denoising

6. Factor models

To make a distinction between the approaches:
(1- 2) are approaches to (actually) estimate the covariance matrix, and the approaches (3 -6) are used to improve an already existing covariance matrix using shrinkage.

As an important remark: The most adequate choice will depend on your data set. Therefore, the “optimal” approach may differ.

1. Exponentially-Weighted Sample Covariance

   The easiest approach is to drop the assumption that each datapoint should have the same “weight” in the estimation of the mean and the covariance matrix. The generalized equation to account for different weighting schemes is as follows:

   How to set the probabilities?
   Many possibilities are available but overall two approaches dominate: (1) by some function of one or more state variable (= values of exogenous variables) or (2) by some function of time.7

   The examples in this articles show you how to perform time-dependent filtering and implicitly assume that more recent data points should have a higher weight. The time-filter is obtained by the following equation::

   The decay depends on the half-life parameter alpha. The smaller the alpha, the faster the decay in probabilities is and, therefore the lower the weight of the more distant data points. This is shown in the graph below. The half-life parameter should not be set to high, to avoid using only the most recent data.

   

   Overall, using this “flexible probability approach” to estimate the covariance matrix will not be enough to reduce the number of conditions. In fact, the condition might increase - therefore this approach should be combined with one of the shrinkage approaches presented below.

2. Outlier-Robust Sample Covariance

   When estimating the mean, a common approach is to drop extreme outliers. While it is straightforward, to find outlier in a univariate setting (z-score), it can be more problematic in a multivariate one. But fear not, because the z-score is also defined in the multivariate case:

   The approach is easy to implement:

   (1) You define a threshold as number of points to be dropped (or sum of probabilities, when a flexible probability approach is used).8
   Then, iterate over the following steps, until enough data points are dropped:

   (2.1) Calculate multivariate z-score for each data point
   (2.2) Drop the datapoint with the highest z-score
   (2.3) Recalculate the covariance matrix and means

   
   This simple approach is the equivalent to the “trimmed” mean in the univariate setting. The approach can be summarizes as follows:

   Obviously, using the Mahalanobis distance implicitly assumes that the data is normally distributed. While it is common to use the function EllipticEnvelope from the library scikit-learn , it can be easily implemented and adjusted based on the general idea presented above. This will be a topic in one of the next articles.

   
   The graph below shows the impact of using robust covariance estimator, when outlier are presented. In the used (well-behaving) example, the robust estimator easily deals with the outliers. However, keep in mind, that this will not work as smoothly when real data is used.

   

3. Graphical Lasso (GLasso)

   The Graphical Lasso (GLasso) is a powerful technique used in statistical modeling to estimate the structure of a Gaussian graphical model—a network where nodes represent variables and edges reflect conditional dependencies. It’s particularly useful when dealing with high-dimensional data where traditional covariance estimates break down.

   At its core, GLasso estimates a sparse inverse covariance matrix (also known as the precision matrix) by applying an L1 regularization penalty.9 The main idea is as follows: If the i, j -th element of an inverse covariance is zero, the variables i and j are conditional independent given the other variables. Shrinking the magnitude of coefficients in the inverse covariance matrix will therefore simplify the structure tremendously. Furthermore, the GLasso is one of the few approaches that directly tackle the inverse covariance matrix which is most relevant then Mean-Variance Optimization is used. In its practical application, Glasso provides a regularized inverse covariance matrix as shown below:

   GLasso helps uncover hidden relationships while avoiding overfitting. If you’re working with datasets where variables outnumber observations—or just want a cleaner, more interpretable network structure—Graphical Lasso is a go-to method.
   

4. Covariance Shrinkage: Ledoit Wolf
   Ledoit and Wolf (2004) suggest to shrink the covariance matrix to some target value in order to avoid perturbing the mean-variance optimizer. In mathematical terms, shrinkage reduces the ratio between the smallest and largest eigenvalue of the sample covariance matrix, so it increases the stability of the covariance matrix. To find the optimal shrinkage intensity, the authors provide the following solution:

   Although shrinkage provide a robust estimate of the covariance matrix which significantly improves the pure sample estimator, it is not free of doubts. As mentioned before, the aim of shrinkage is to handle the Ledoit Wolf effect which increases the condition numbers of a covariance matrix. Therefore, shrinkage makes many portfolio optimizer more robust. However, the key issue is the shrinkage intensity. When set too low, the shrinkage is ineffective to improve the condition number. When set too high, you risk to loose information in the covariance matrix with is needed to generate precise portfolio weights.
   

5. Spectral Denoising

   Using the Ledoit Wolf shrinkage, you attempt to reduce the “noise” in the estimate generated by the estimation process. The problem with Ledoit Wolf is:
   Can anyone be sure that you only eliminate noise? What is the right “degree” of shrinkage? Ideally, you want to get rid of the noise without eliminating information or signal in the data which is already low in financial data.
   
   To give you some motivation: Laloux et al. reports that around 94% of the spectrum of an empirical correlation matrix estimated from the returns of the S&P 500 constituents is indistinguishable from the spectrum of a random correlation matrix. Therefore, denoising can make great sense - especially when the variables are similar (like are from the same asset class)!
   
   In this regard, spectral denoising can help. It uses Random Matrix Theory to eliminate noise, while keeping (most of) information in the data. What this approach does is to separate eigenvalues, which contain information from those eigenvalues which are more likely to contain noise. The separation is done by using the so-called Marchenko-Pasteur distribution (MPD). Eigenvalues, which contain noise follow the MPD and are mostly very small. Eigenvalues, which are significantly different from the distribution are assumed to contain information. The reader can compare spectral denoising with Ledoit-Wolf shrinkage by comparing a scalpel with an axe.

   Therefore, you obtain a separation in the eigenvalues in two sets: The non-random eigenvalues do not follow the MPD - they are important for our analysis. What to do with the non-relevant eigenvalues? Your answer is called Constant Residual Eigenvalue Method. This means that the average across all eigenvalues with are associated with random behaviour is taken. Averaging across the lowest eigenvalues will lift up the lowest eigenvalue and thereby decreasing the conditions numbers:

   Some words of advice:
   (1) It appears tempting to simply drop or eliminate the eigenvalues which looks random. However, this will reduce the rank of the covariance matrix and makes it (nearly) impossible to invert the covariance matrix.10
   (2) This approach works best by using the correlation matrix - not the covariance matrix due to estimation issues of the MPD.
   (3) Estimation of the MPD needs data points. In this framework, the data points are the eigenvalues. When the number of variables is low, the number of eigenvalues will be low, too. This can make it impossible to estimate the MPD n some settings. Therefore, you should use this process only when the number of variables is high (enough).

   Below, we illustrate one example demonstrating the impact of spectral denoising with the use of equity returns. You can see that the smallest eigenvalues are adjusted (increased), which severely reduces the condition numbers - and will ease some of the optimization / estimation difficulties.

   

6. Factor Models

   In many settings, there are strong correlations in the data which are driven by variables in the background. Quite often, theory gives strong motivations that certain variables drive returns and stochastic processes. For example, nearly all stocks are commonly driven by one “market” factor.
   The problem is that quite often you cannot be sure how to “quantify” those variables. Theory assumes they exist, but nobody really knows how to measure them correctly. In those cases “hidden” factor approach can be useful.

   As you can see, the factor approach decompose the covariance into two components: (1) common information across variables with can be represented by “hidden” factors, (2) idiosyncratic information which only corresponds to the specific variable. This decomposition crucially simplifies the structure (and the number of parameters), as you can focus on the small amount of hidden variables that “drives” the relationships.
   
   Based on our experience, this approach highly decreases the condition numbers in many set ups. However, there are two important aspect to consider: How to set the number of hidden variables? How to estimate the factors? Estimation choice directly corresponds to choosing the distance measure. In many applications, the Frobenius Distance is used. In this case, optimization can be solved via principal axis factorization which is available in standard python packages.

   
   The specification of the correct number of hidden variables is more difficult. To our knowledge, there are no clear cut approaches. We personally prefer to estimate several specifications by varying the number of hidden variables and then compare the covariance estimates based on the factor analysis and some another (robust) covariance estimate using the Frobenius norm (see in Github examples). Thereby you can detect when the covariance is highly different from the (robust) estimate. So you can increase the number of hidden variables, if the number is too low to adequately summarize relationships between variables.
   
   Personally, we like to use this approach on the correlation matrix as we believe that focusing exclusively on the correlations appears to generate more robust results. This is the reason why the factor approach will be used in our examples.

Case Study:
Estimating the Covariance Matrix

As the remainder of this article, we will present a showcase of the presented estimators of the covariance matrix. The following setting is used:

1. Large number of Stocks (= Optimizing an Equity Portfolio)

   1. 50 Equities from the U.S. market between 2010 and 2024

   2. Monthly data starting in 2010

2. Small number of Asset Class Indices (= Optimization a Multi-Asset Portfolio)

   1. 5 Asset Class Indices (3 equity indices, 2 bond indices) between 2010 and 2024

   2. Monthly data starting in 2010

To obtain the data, Yahoo Finance is used. The code to download the data is integrated into the coding example on Github.

To evaluate the estimators, we apply the following strategy: The results are compared by looking at the obtained condition number. As highlighted before, a lower condition number is preferable especially for numerical stability. Afterwards, the differences between the estimated covariances are analyzed using the Frobenius norm. While you cannot - at least in this setting - tell which estimator is better or even “the best” in terms of quality and precision, you can at least say when two estimators are similar regarding the information content (e.g. when the Frobenius norm of the difference is low). Under similar estimators, you should prefer that one with the lower condition number.

Equity Example:

The graphs below show the results of using several methods to estimate the covariance matrix. The estimators are effective in reducing the condition numbers. The shrinkage approaches of Ledoit-Wolf, Glasso, Spectral Denoising and Factor Models are helpful in reducing the number of condition making matrix inversion and optimization more stable. Interestingly, although robust estimation can improve the quality of the coefficients / covariances, it fails at reducing the number of conditions. However, you can solve this issue by additionally apply shrinkage on the robust covariance estimator (see: Robust Estimation + Glasso).

Its important to keep in mind that a low (or high) condition number does not say anything about the overall quality, such as a precise estimation of the true coefficients. In fact, you can reduce the condition number by simply setting the non-diagonal elements of the covariance matrix to zero - something that is not sound. However, given that “richness” of information in a covariance is contained while having a smaller number of conditions will lead to better results.

Therefore, having a look at the Frobenius (L2) norm of the differences between all pairs of covariance matrices is helpful. A smaller Frobenius norm (blue) means that two covariances estimates show little difference. You can see that the shrinkage approaches (columns 5-10) are rather similar to the sample covariance while having a far smaller condition number. These results are indicative that shrinkage on the data improves the numerical stability of the covariance estimates while preserving the majority of information.

*The indices represent the estimators as presented in the graph of the number of conditions.

Multi-Asset Example:

For strategical asset allocation, the second data set is more interesting, Once again, the pattern emerges that estimations without shrinkage generate high condition-number estimates. However, some of the shrinkage approaches are “overdoing” it, as the condition numbers are very small. This is a common situation, when the used shrinkage method may note be applicable with the settings / arguments used. A very low condition number (relative to the sample covariance) can reveal that the shrinkage is too strong which means that nearly all dependencies between variables are eliminated. This appears to be the case for Glasso, performed directly on the covariance matrix, and spectral denoising.
Therefore, this example gives insight that you can actually overdo shrinkage. The condition number and the covariance matrix itself should always be checked.

Summary

In this article, you have learnt that using the sample covariance can and will utterly fail - especially in portfolio optimization. It can lead to complete miscalculations of the dependencies and diversification effects between asset returns in some cases and can render portfolio optimization useless due to high sensitivity of the resulting portfolio weights. Please keep this in mind: You have been warned!

You have seen the key concepts of covariances. It is crucial to know that your data need to have certain characteristics. The methods used here can only applied when these characteristics (like ergodicity) are fulfilled. When this is not the case (as with derivatives and bonds) you will need more advanced techniques.

We illustrate several approaches to better estimate the covariance matrix. In practice, you have to consider two key aspects:

1. The best strategy to estimate the covariance matrix will depend on your data sample. There is no silver bullet that works every time - there is none in investing and none in covariance matrix estimation.

2. The various approaches can be combined to improve results. Why not try robust estimation and subsequently the factor approach? It might work and will tackle the issue from two different sides (outlier-reduction + structure simplification).

In Part 2 of this article, even more sophisticated approaches (e.g. marginal-copula estimation) will be presented. Additionally, you will actually see how to determine the quality of an covariance estimator in a coherent framework which you will be able to use in practice afterwards.

Before this article is concluded, we have one last question for the reader: How important is the choice of data frequency? Should monthly and daily data lead to identical correlations? Which frequency should you chose? Answering this question is far from trivial, but we will provide some guidance in Part 2.

Die Portfoliooptimierung als Vehikel für die Allokation von Portfolios wird oft kritisiert, dass sie fern von der Investmentpraxis sei. In diesem Artikel stellen wir 13 nützliche Regeln vor, die die Portfoliooptimierung effektiver sowie benutzerfreundlicher gestaltet und dadurch Fallstricke aus der Realität begegnet. Indem wir gängige Mythen entkräften, bieten wir Strategien an, die sowohl institutionelle als auch private Investoren anwenden können. Die Regeln zielen darauf ab, die Entscheidungsfindung zu verbessern, Risiken zu minimieren und die Präferenzen der Anleger zu berücksichtigen, sodass die Portfoliooptimierung zu einem wertvollen Instrument für das Erreichen langfristiger Anlageziele wird. Zukünftige Artikel werden jede der präsentierten Regeln im Detail beleuchten – ergänzt durch praxisnahe Beispiele mitsamt Python-Code zur eigenen Umsetzung.

Einleitung

Als Portfoliomanager und quantitative Analysten sind wir oft auf Vorbehalte gegenüber den Vorteilen der Portfoliooptimierung gestoßen, wenn es um das Thema der Assetallokation in Portfolios geht. Ein häufig geäußerter Kritikpunkt lautet, dass die Optimierung mehr schadet als nützt. Im Falle einer fehlerhaften Anwendung sind heuristischer Ansatz oft die bessere Alternative.

Das völlige Ignorieren der Portfoliooptimierung allein wegen ihrer Komplexität ist jedoch nicht die Lösung. Ein mangelndes Verständnis oder ein unpassend eingesetzter Optimierungsansatz können zwar zu suboptimalen Ergebnissen führen, aber mit dem richtigen Know-how und der passenden Methodik wird die Portfolio-Optimierung zu einem wirkungsvollen Instrument. Entscheidend ist dabei, den Prozess strategisch anzugehen und ein gutes Bewusstsein zu entwickeln, was in der Praxis wirklich funktioniert.

In diesem Artikel gehen wir auf gängige Kritikpunkte ein und präsentieren 13 praxisnahe Regeln, die die Portfoliooptimierung im realen Anlageumfeld effektiver machen. Diese Regeln zielen darauf ab, die Entscheidungsfindung zu verbessern, Risiken zu minimieren und den Ziele der Anleger / Investoren gerecht zu werden. Dabei beschränken wir uns nicht auf die klassische Mittelwert-Varianz-Optimierung (MVO), sondern formulieren Prinzipien, die unabhängig von der verwendeten Optimierungsmethode gelten.

Zwar gibt es zahlreiche Optimierungsverfahren – jedes mit eigenen Stärken und Schwächen –, doch liegt der wahre Nutzen darin, zu wissen, wie man diese Ansätze kombiniert, um spezifische Anlegerbedürfnisse zu erfüllen. Durch die Anwendung dieser Regeln können sowohl institutionelle als auch private Investoren ihre Optimierungsprozesse modernisieren und veraltete bzw. heuristische Methoden hinter sich lassen.

In zukünftigen Beiträgen werden wir jede dieser 13 Regeln ausführlicher beleuchten – ergänzt durch Praxisbeispiele und Python-Code zur Umsetzung.

Grundvoraussetzung: Konsistenter Investmentprozess

Bevor man sich in die Portfoliooptimierung stürzt, sollten zwei wesentliche Elemente festgelegt werden:

• Konkrete Anlageziele (und Beschränkungen)

• Ein klar definierter und konsistenter Investmentprozess

Obwohl beide Punkte essenziell sind, liegt unser Hauptaugenmerk auf der Anlagestrategie.

Eine Anlagestrategie beschreibt den systematischen Prozess, mit dem ein Investor fundierte, strategische Entscheidungen darüber trifft, wo, wann und wie Kapital eingesetzt wird. Diese disziplinierte Vorgehensweise stellt sicher, dass Investitionen mit den angestrebten finanziellen Zielen, der Risikotoleranz und den aktuellen Marktbedingungen in Einklang stehen. Sie hilft dabei, rationale Entscheidungen von impulsiven, emotionsgetriebenen Handlungen zu trennen.

Jeder gelebte Investmentprozess hat individuelle Nuancen, weshalb die Portfolio-Optimierung stets maßgeschneidert erfolgen muss. Dabei darf die Optimierung nie isoliert betrachtet werden, sondern muss in einen soliden, strukturierten Anlageprozess eingebettet sein. Wesentliche Fragen – wie etwa die Auswahl der zugelassenen Anlageklassen, festgelegte Investitionslimits oder Risikobudgets und die Entscheidungsfindung – müssen geklärt sein, bevor die Optimierung in Angriff genommen wird.

Dies führt uns zu Regel 1:

Regel 1: Entwickle und verfolge eine klare, konsistente Anlagestrategie.
Deine Portfoliooptimierung muss auf einer fundierten und systematischen Strategie basieren. Ohne einen klaren Rahmen (= Investmentansatz) ist die Optimierung wie der Bau eines Hauses auf wackeligem Untergrund – selbst beste Materialien und Fachwissen nützen wenig, wenn die Grundlagen fehlen.

„Portfolio-Optimierung ist zu sensitiv, um in der Praxis eingesetzt zu werden“

Eine der häufigsten Kritiken an der Portfoliooptimierung ist die wahrgenommene Sensitivität der optimierten Portfoliogewichte – und dieser Kritik liegt ein erheblicher Wahrheitsgehalt zugrunde. Es ist gut dokumentiert, dass Optimierungsmodelle, insbesondere traditionelle wie die Mittelwert-Varianz-Optimierung (MVO), äußerst empfindlich gegenüber marginalen Änderungen der Eingabeparameter, wie etwa der erwarteten Asset-Renditen, sein können. Schon geringfügige Anpassungen können zu dramatischen Veränderungen der Portfolio-Gewichtungen führen – manchmal wechselt eine Allokation von 0 % zu 100 % und umgekehrt.

Aber warum existiert diese Empfindlichkeit?

Der Hauptgrund ist die Kovarianzmatrix, die in den meisten Optimierungsansätzen eine zentrale Rolle spielt. Zwei wesentliche Faktoren tragen zu diesem Problem bei:

• Hohe Korrelationen zwischen Vermögenswerten: Wertpapiere in einem Portfolio sind oft stark korreliert, was die exakte Schätzung der Beziehungen zwischen ihnen erschwert.

• “Noisy” Daten: Kovarianzen werden typischerweise aus historischen Daten geschätzt, die oft ungenau und/oder unzuverlässig sind – insbesondere bei kleinen Stichprobengrößen.

Viele Optimierungsmethoden basieren auf der Inversion der Kovarianzmatrix, um die Portfoliogewichtungen zu berechnen. Sind die Korrelationen hoch und enthalten die Daten eine geringe Signal-Noise Ratio, wird die Kovarianzmatrix „schlecht konditioniert“ sein, was zu enormen Fehlern in der Schätzung führt. Dies hat zur Folge, dass die Gewichtungen des Portfolios übermäßig empfindlich auf kleine Änderungen der erwarteten Renditen reagieren und oft zu unrealistischen sowie über die Zeit stark variierenden Allokationen führen.

In der Praxis ist diese Empfindlichkeit besonders problematisch, wenn mit stark korrelierten Vermögenswerten oder mit kurzen bzw. unzuverlässigen und kurzen historischen Datenstichproben gearbeitet wird. Obwohl dies ein häufiges Problem darstellt, liegt es nicht an der Portfoliooptimierung selbst, sondern an den Daten und Eingabeparametern. Glücklicherweise gibt es Ansätze, die robuster gegen diese Problem sind.

Wie können wir also damit umgehen?

Regel 2: Bereinige deine Kovarianzmatrix!
Verwende niemals direkt die reine Stichproben-Kovarianzmatrix für die Optimierung. Die (unbereinigten) Kovarianzen, die aus historischen Zeitreihen geschätzt werden, sind oft zu sehr mit Rauschen belastet und unzuverlässig, um sie direkt zu nutzen. Stattdessen sollten Techniken wie Shrinkage oder Denoising angewandt werden, um die Robustheit dieser Schätzungen zu verbessern. Diese Methoden helfen, die Kovarianzmatrix zu stabilisieren und reduzieren die Empfindlichkeit der Portfolio-Gewichtungen gegenüber kleinen Änderungen der erwarteten Renditen erheblich.

Regel 3: Strukturiere deine Assets!
Vermeide es, eine Optimierung mit stark korrelierten Wertpapieren durchzuführen. Wenn Vermögenswerte stark korreliert sind, versagen traditionelle Optimierungsansätze oft. Eine bessere Lösung ist die Verwendung von Asset-Clustering – also die Gruppierung von Vermögenswerten in Cluster basierend auf ihren Korrelationen. Die Grundidee ist, dass Assets innerhalb eines Clusters tendenziell stark korreliert sind (Intra-Cluster), während die Korrelationen zwischen den Clustern (Inter-Cluster) wesentlich niedriger sind. Durch die Optimierung auf Inter-Cluster-Ebene kannst du die Auswirkungen hoher Korrelationen reduzieren, was zu stabileren und realistischeren Portfoliogewichtungen führt.

„Portfolio-Optimierung generiert Portfolios, die gegenüber gleichgewichteten Portfolios unterdurchschnittlich abschneiden“

Eine häufig geäußerte Kritik – unterstützt durch verschiedene Studien – besagt, dass optimierte Portfolios häufig schlechter als einfache, gleichgewichtete Portfolios (1/N-Portfolios) abschneiden. Das Argument lautet, dass die Allokation eines gleichen Anteils an jedem Asset effektiver sein könnte als der Einsatz komplexer Optimierungstechniken. Aber hat diese Behauptung wirklich Substanz?

Eine Schlüsselstudie, die oft zu diesem Zweck zitiert wird, stammt von DeMiguel, Garlappi und Raman Uppal (2009). Ihre Forschung zeigt, dass 1/N-Portfolios in Out-of-Sample-Tests besser abschneiden als optimierte Portfolios. Allerdings machen die Autoren dabei einen kritischen Fehler – sie extrapolieren die jüngsten Renditen.

In ihrer Studie schätzen die Autoren die erwarteten Renditen, die als Grundlage für die spätere Optimierung dient, auf Basis von rollierenden Schätzungen, bei denen die zuletzt beobachteten Renditen zur Vorhersage zukünftiger Renditen herangezogen werden. Diese Methode geht davon aus, dass zukünftig erwartete Renditen aus den jüngsten Daten einigermaßen zuverlässig geschätzt werden können – eine Annahme, die sowohl theoretisch als auch empirisch nicht haltbar ist.

Wenn man den Finanzmärkten folgt, erscheint diese Annahme oft allzu vereinfacht. Glaubst du, dass Märkte aufgrund positiver Anlegerstimmung „überhitzen“ können, was die zukünftigen Renditeaussichten schmälert? Meinst du, dass eine Aktie mit einem Preisanstieg von 1000 % und einem extrem hohen Kurs-Gewinn-Verhältnis (KGV) in Zukunft eher moderatere Renditen erzielen wird? Und denkst du, dass Marktbedingungen, wie Rezessionen oder Wirtschaftsbooms, unterschiedliche Renditeprofile schaffen können? Wenn du eine dieser Fragen bejahst, bist du nicht allein. Zahlreiche Studien zeigen, dass sich die erwarteten Renditen im Laufe der Zeit ändern – beeinflusst durch Marktbedingungen, Anlegerstimmung und makroökonomische Faktoren. Leider berücksichtigt die Studie von DeMiguel, Garlappi und Raman Uppal diese Dynamiken nicht.

Dies dient als mahnendes Beispiel: Die Genauigkeit deiner Schätzungen von erwarteten Renditen und Kovarianzen ist entscheidend. Wie man so schön sagt: „Garbage in, garbage out.“ Wenn deine Inputparameter schlecht sind, kannst du auch keine guten Ergebnisse erwarten. Glücklicherweise können wir aus diesem Fehler lernen, indem wir uns an folgende essenzielle Regel erinnern:

Regel 4: Extrapoliere nicht die jüngsten Renditen!
Verwende niemals die jüngsten historischen Renditen als Basis für die erwarteten Renditen. Gehe nicht davon aus, dass sich die Performance des letzten Jahres zwangsläufig wiederholt. Ob du nun ein Portfolio breiter Anlageklassen (z. B. Aktien- und Anleiheindizes) oder einzelner Assets (wie Aktien) optimierst – du solltest auf Schätzmethoden zurückgreifen, die auf soliden theoretischen Grundlagen beruhen. Faktormodelle oder fundamentale Bewertungsmodelle sind verlässlichere Ansätze erwartete Renditen zu prognostizieren. Die ausschließliche Verwendung von kurzfristigen historischen Daten erzeugt eine Illusion von Vorhersagbarkeit und birgt das Risiko einer Unterperformance, da sich Märkte zyklisch verhalten. Auch wenn du ein solides theoretisches Modell verwendest, das zeitvariierende erwartete Renditen erfasst, solltest du dir bewusst sein, dass die Vorhersage von Renditen aufgrund von Parameterinstabilität und Modellunsicherheit immer noch ein sehr schwierig Unterfangen darstellt.

„Portfoliooptimierung führt zu unerwünschten Portfolios“

Während erwartete Renditen und Kovarianzen zentrale Bestandteile der Portfoliooptimierung sind, erfassen sie nur einen Teil des Gesamtbildes. Investoren haben in der Regel Präferenzen, die über reine Rendite und Risiko hinausgehen. Diese Präferenzen können ethische Überlegungen (z. B. Umwelt-, Sozial- und Governance-Kriterien, ESG), Liquiditätsanforderungen oder Einschränkungen bei Abweichungen von einem Benchmark beinhalten.

Um Portfolios zu erstellen, die diesen umfassenderen Anliegen gerecht werden, ist es essenziell, zusätzliche Restriktionen in den Optimierungsprozess zu integrieren.

Regel 5: Formuliere Investorpräferenzen als Restriktionen!
Unerwünschte Portfoliozusammenstellungen können durch das Einführen von Restriktionen vermieden werden. Fühlst du dich mit einem reinen Aktien- oder Anleiheportfolios wohl? Möchtest du Transaktionskosten in Grenzen halten? Oder sind dir Nachhaltigkeitsaspekte bei der Geldanlage wichtig? Dann setze Restriktionen in der Optimierung, um diesen Aspekten Rechnung zu tragen. Eine sinnvolle Übung besteht darin, jede wesentliche Anlageklasse zu überprüfen und potenziell extreme Allokationen zu identifizieren, die deinen Präferenzen oder Zielen widersprechen würden. Auf diese Weise kannst du zusätzliche Informationen – wie ESG-Daten – in den Optimierungsprozess einfließen lassen und sicherstellen, dass dein Portfolio sowohl finanziellen als auch nicht-finanziellen Zielen entspricht.

Regel 6: Setze einen Anker!
Lege ein Benchmarkportfolio fest und definiere ein tolerierbares Maß an Abweichung. Jedes Portfolio benötigt eine Benchmark, sei es zur Performance-Bewertung oder zur Steuerung der aktiven Positionierung. Diese Benchmark sollte deine langfristige strategische Asset Allokation widerspiegeln. Mit der Zeit, wenn sich deine Schätzungen der erwarteten Renditen und Kovarianzen verändern, wird sich auch die Portfoliozusammensetzung anpassen. Durch die Nutzung einer Benchmark und das Festlegen eines maximal akzeptablen Abweichung (Tracking-Errors) stellst du sicher, dass dein Portfolio im Einklang mit deinen langfristigen Investmentzielen bleibt und nicht zu weit von der gewünschten Allokation entfernt.

Regel 7: Messe deine Diversifikation!
Verwende zusätzliche Kennzahlen, um die Diversifikation deines Portfolios zu beurteilen. Diversifikation ist für jede Anlagestrategie essenziell – gehe jedoch nicht davon aus, dass die Portfoliooptimierung automatisch ein gut diversifiziertes Portfolio liefert. Dies ist ein weiterer Grund, warum optimierte Portfolios in Out-of-Sample-Tests häufig hinter gleichgewichteten Portfolios zurückbleiben. Um die Diversifikation deines Portfolios wirklich zu verstehen, müssen zusätzliche Metriken zur Bewertung des Portfolios herangezogen werden. Nutze beispielsweise Entropie-Maße, den (inversen) Herfindahl-Hirschman-Index oder den Gini-Koeffizienten, um die Diversifikation zu quantifizieren. Auf einer fortgeschritteneren Ebene kann auch die Diversifikation in Bezug auf zugrunde liegende Risikofaktoren – wie wirtschaftliches Wachstum, Zinssätze oder geopolitische Ereignisse – bewertet werden, die die Renditen der Anlagen beeinflussen. So stellst du sicher, dass dein Portfolio nicht nur hinsichtlich der bloßen Gewichte der Assets gut diversifiziert ist , sondern auch in Bezug auf die Treiber der Renditen - den ökonomischen Faktoren im Hintergrund.

„Portfoliooptimierung baut auf unrealistischen Annahmen auf und ist daher nutzlos.“

Modelle und Schätzer können nur eine begrenzte Menge an realen Informationen aufnehmen. Traditionelle Ansätze wie die Mittelwert-Varianz-Optimierung (MVO) konzentrieren sich ausschließlich auf die ersten beiden Momente einer Verteilung – Mittelwert und Varianz – und/oder gehen davon aus, dass Finanzmärkte einer multivariaten Normalverteilung folgen. Obwohl diese Modelle ihren Platz haben, treffen sie oft unrealistische Annahmen darüber, wie die Finanzwelt tatsächlich funktioniert. Im Laufe der Zeit sind daher Modelle entstanden, um diese Einschränkungen zu adressieren.

MVO ist als bekannteste Methode zur Portfoliooptimierung mit mehreren Annahmen behaftet, die nicht mit der Realität der Märkte übereinstimmen. Dazu gehören:

• Finanzmärkte sind normalverteilt – Tatsächlich weisen Finanzrenditen „fat tails“ auf und weichen erheblich von einer Normalverteilung ab.

• Investoren haben eine quadratische Nutzenfunktion – Investoren treffen Entscheidungen nicht ausschließlich auf Grundlage von Mittelwert-Varianz-Präferenzen. Ihre Entscheidungen werden häufig von komplexeren psychologischen Faktoren beeinflusst, und sie bewerten ihre Ergebnisse relativ zu bestimmten Referenzpunkten.

• Rationale Investoren mit homogenen Erwartungen – Emotionen, Überzeugungen und psychologische Verzerrungen beeinflussen den Entscheidungsprozess, insbesondere in Krisenzeiten. Zudem variiert die Heterogenität der Erwartungen mit dem Grad der Informationsunsicherheit.

• Statischer, einperiodiger Anlagehorizont – Viele Investoren haben keinen vorab fixen Anlagehorizont und sie berücksichtigen auch die Risiken während des Anlagezeitraums.

• Keine Transaktionskosten oder Steuern – In einer friktionslosen Welt wären kontinuierliches Trading und hohe Portfolioumschläge unproblematisch, doch in der Realität sind sie es aufgrund hoher Kosten nicht.

• Investoren interessieren sich nur für die Varianz – In der Realität gewichten Menschen Verlusten stärker, als betragsmäßig gleichhohe Gewinne. Dies ist eine zentrale Erkenntnis der verhaltensorientierte Finanzmarkttheorie (“Behavioral Finance”).

Die verhaltensorientierte Finanzmarkttheorie und die empirische Evidenz zeigen eindeutig, dass diese Annahmen nicht mit dem tatsächlichen Verhalten und Denken von Investoren übereinstimmen. Bedeutet dies jedoch, dass Portfoliooptimierung grundsätzlich nutzlos ist? Wir würden dem widersprechen.

Heute stehen uns fortschrittlichere Methoden zur Verfügung, die das Verhalten von Investoren und die Dynamik der Märkte besser widerspiegeln. Eine Möglichkeit ist die Berücksichtigung adäquaterer Nutzenfunktionen unter Verwendung der “Prospect Theory” von Kahneman und Tversky. Eine andere ist die Verwendung von Modellen wie Mean-CVaR (Conditional Value-at-Risk), welches eine realistischere Alternative darstellt, da es sich auf Tail Risks konzentrieren – ein entscheidender Aspekt für Investoren, die sich stärker vor großen Verlusten als vor täglicher Volatilität schützen wollen. Diese Modelle ermöglichen eine differenziertere Risikosteuerung, allerdings geht eine höhere Modellflexibilität oft mit erhöhter Komplexität und höherem Schätzrisiko einher.

MVO kann als nützliches Instrument dienen, um erste Erkenntnisse zu gewinnen, sollte jedoch nicht als primäre Methode zur Portfoliooptimierung verwendet werden. Für genauere und realistischere Ergebnisse empfehlen wir dringend, über MVO hinauszugehen und Modelle wie Mean-CVaR zu nutzen, die besser widerspiegeln, wie Investoren tatsächlich Risiken wahrnehmen und darauf reagieren. Investoren sorgen sich weniger um tägliche Schwankungen (Varianz) und fokussieren sich stärker darauf, große, katastrophale Verluste zu vermeiden.

Regel 8: Verabschiede dich von der Varianz als Risikomaß!
Setze stattdessen auf höherdimensionale Risikomaße wie CVaR. Die Portfoliooptimierung sollte auf realistischen Annahmen beruhen. Das Mittelwert-Varianz-Optimierungsmodell erfüllt diesen Standard nicht. Wir plädieren für den Einsatz von Mean-CVaR, da es wesentlich besser mit der tatsächlichen Wahrnehmung und Reaktion der Investoren auf Risiken übereinstimmt – insbesondere wenn es um Tail-Risiken und extreme Ereignisse geht, die im Kontext des realen Investierens aussagekräftiger sind als die Standardabweichung.

Regel 9: Akzeptiere die nichtlineare, nicht-gaußsche Realität!
Die Welt der Finanzdaten ist von Natur aus nichtlinear und keineswegs normalverteilt. Während Korrelationen lineare Zusammenhänge unterstellen, unterschätzt die Annahme normalverteilter Renditen die Wahrscheinlichkeit extremer Ereignisse. Wann immer möglich und sinnhaft, sollten Annahmen von Linearität und Normalverteilung zugunsten flexiblerer Ansätze aufgegeben werden, die (übermäßige) Kurtosis, Schiefe und Nichtlinearitäten berücksichtigen.

„Portfolio-Optimierung ist zu restriktiv für meinen “aktiven”, “flexiblen” Investmentprozess“

Gemäß unseren Erfahrung ist die Portfoliooptimierung äußerst flexibel und nahezu jede konventionelle Anlagestrategie lässt sich darin integrieren. Zwar mag es notwendig sein, maßgeschneiderte Lösungen für spezifische Situationen zu entwickeln – wie etwa hierarchisches Clustering von Portfolios (siehe Regel 3) – doch die Flexibilität, unterschiedliche Ansätze zu berücksichtigen, ist definitiv gegeben.

Die Kritik, dass die Portfoliooptimierung „zu restriktiv“ sei, ist zwar weit verbreitet, jedoch meist fehl am Platz und kann problematisch sein. Häufig rührt diese Kritik aus einem Missverständnis: Dem Unterschied zwischen „Flexibilität“ und dem Fehlen eines strukturierten Investmentprozesses. Ein klar definierter Investmentprozess fungiert als dein Schlachtplan in der Finanzwelt. Er stellt sicher, dass Entscheidungen auf Basis eines klaren Systems getroffen werden, was impulsiven, emotionsgesteuerten Fehlern in Marktexzessen – sei es in Bullen- oder Bärenmärkten – vorbeugt. Ohne einen solchen Prozess steigt die Wahrscheinlichkeit langfristiger Underperformance erheblich.

Um es klar zu formulieren: Wenn du diese Kritik als Verteidigung gegen den Einsatz von Portfoliooptimierung hörst, solltest du zwei Möglichkeiten in Betracht ziehen:

• Dem Portfoliomanager fehlt ein ordentlicher Investmentprozess.

• Der Portfoliomanager ist schlichtweg nicht in der Lage, Portfoliooptimierung einzusetzen.

Wenn ein Portfoliomanager dieses Argument gegen den Einsatz von Portfoliooptimierung einsetzt, solltest du vorsichtig sein. Das könnte auf einen fundamentalen Mangel im Investmentansatz oder auf eine unzureichende Kompetenz in den Techniken des Portfoliomanagements hinweisen.

Betrachten wir die Sache aus einem anderen Blickwinkel: Angenommen, du hast kurzfristige Investmentideen oder aktive Ansichten. Kannst du diese in dein langfristiges Portfolio integrieren? Absolut. Tatsächlich gibt es bewährte Methoden, um kurzfristige Meinungen mit langfristigen Strategien zu verknüpfen. Ansätze wie das Black-Litterman-Modell erlauben es, eigene Markterwartungen in ein optimiertes Portfolio einfließen zu lassen – und reichen in den meisten Fällen aus. Für komplexere Fälle können entropiebasierte Methoden helfen, die Integration kurzfristiger Prognosen (auch für höhere Momente deiner Renditeverteilung) in die langfristige Strategie zu integrieren.

Regel 10: Implementiere deine kurzfristigen Views!
Integriere kurzfristige Einschätzungen, ohne jedoch dein langfristiges Portfolio zu vernachlässigen. Viele Investoren empfinden es als zu eintönig, sich ausschließlich auf ihr langfristiges strategisches Portfolio zu konzentrieren. Wenn du kurzfristige Prognosen oder aktive Ideen hast, integriere sie in deine langfristige Strategie. Aktive Abweichungen von deinem Kernportfolio sind akzeptabel, solange sie in einem angemessenen Rahmen bleiben. Glücklicherweise existieren zahlreiche Methoden, um kurzfristige Ansichten nahtlos mit einem soliden, strategischen langfristigen Portfolio zu verbinden.

„Mein optimiertes Portfolio verzeichnete deutliche Verluste, obwohl es ein geringes Risikoprofil hatte.“

Dieses Szenario haben viele Investoren im Jahr 2022 erlebt. Trotz eines hohen Anteils an Staatsanleihen – die gemeinhin als sichere Anlagen gelten – erlitten viele erhebliche Verluste. Das zentrale Problem dabei ist, dass in der Portfoliooptimierung nahezu alle Informationen über das Risiko in der Kovarianzmatrix enthalten sind. Dies kann sowohl ein Vorteil als auch ein Nachteil sein.

Es ist ein Segen, wenn die Kovarianzmatrix die tatsächliche Marktdynamik präzise widerspiegelt. Doch es wird zum Fluch, wenn man ein Portfolio anhand von Kovarianzdaten optimiert, die vielleicht für das letzte Jahrzehnt gültig waren, und dann plötzlich mit einem fundamentalen Wandel der Marktbedingungen konfrontiert wird. Wenn Volatilität und Korrelationen sich dramatisch ändern, können die zuvor funktionierenden Diversifikationseffekte verschwinden, wodurch dein Portfolio extrem anfällig wird.

Was ist also im Jahr 2022 schiefgelaufen?
Zwischen 2000 und 2021 war die Korrelation zwischen Aktien und Anleihen im Allgemeinen negativ, getrieben von einer prozyklischen Inflationsdynamik und expansiven geldpolitischen Maßnahmen in den 2010er-Jahren. In dieser Periode profitierten Portfolios von einer starken Diversifikation zwischen Aktien und Anleihen. Mit dem scharfen Anstieg der Inflation im Jahr 2022 (als negativer Angebotsschock) kehrte sich diese Dynamik jedoch um: Die Korrelation zwischen Aktien und Anleihen wurde positiv, was bedeutete, dass beide Anlageklassen gleichzeitig fielen. Folglich verschwanden die Diversifikationseffekte durch diese Änderungen, was zu erheblichen Verlusten im Portfolio führte.

Dieses Szenario lässt sich durch ein umsichtiges Portfolio- / Risikomanagement vermeiden. Die entscheidende Erkenntnis lautet: Deine Schätzungen der erwarteten Renditen und der Kovarianzen sollten die zukünftigen Erwartungen widerspiegeln und nicht nur auf historischen Daten beruhen.

Regel 11: Prüfe dein Portfolio im Stresstest!
Verwende Stresstests, um dein Portfolio unter alternativen Parametern zu bewerten. Die Portfoliooptimierung endet nicht mit der Berechnung der Gewichtungen. Nachdem du dein Portfolio festgelegt hast, musst du es einem Stresstest unterziehen, indem du alternative Schätzungen von Volatilität und Korrelationen in Betracht ziehst, um seltene, aber folgenreiche Ereignisse abzubilden. Ein szenariobasierter Ansatz ist hier sehr hilfreich: Denke aktiv darüber nach, welche zukünftigen Szenarien dein Portfolio beeinflussen könnten. Dies ist besonders wichtig, wenn du mit begrenzten Daten arbeitest, die gewisse extreme Ereignisse nicht enthalten, obwohl diese durchaus möglich sind.

Stresstests werden die Diversifikation deines Portfolios hart testen und Schwachstellen aufdecken, die unter normalen Marktbedingungen nicht sichtbar wären. Dieser Ansatz hilft sicherzustellen, dass dein Portfolio auch in unerwarteten Marktsituationen robust bleibt.

„Portfoliooptimierung ist zu langsam, um sich aktuellen Entwicklungen anzupassen“

Ich warne eindringlich davor, zu versuchen, den Markt zu „timen“. Selbst erfahrene Experten tun sich oft schwer, kurzfristige Bewegungen vorherzusagen. Als Privatanleger sind deine Chancen auf langfristigen Erfolg gering, wenn du versuchst, den Markt zu timen. Wenn du den Aktienmarkt als Quelle von Nervenkitzel und Spannung betrachtest, solltest du vielleicht umdenken und stattdessen etwas weniger riskantes wählen – wie ein Festgeldkonto. Der Aktienmarkt ist kein Ort zum Glücksspiel; Investieren sollte diszipliniert und oft eher unspektakulär erfolgen. Für Nervenkitzel gibt es ja den Vergnügungspark.

Dennoch ist die Portfoliooptimierung keineswegs statisch. Sie passt sich im Laufe der Zeit den veränderten Marktbedingungen an. Auch wenn dein Portfolio einer langfristigen Strategie folgen sollte, ist es wichtig, die Schlüsselparameter – wie erwartete Renditen und Kovarianzen – regelmäßig zu überprüfen. Ein jährlicher Realitätscheck dieser Annahmen ist in der Regel ausreichend. Doch sei vorsichtig: Wenn du dramatische Veränderungen bei den Einschätzungen über erwartete Renditen identifizierst, frage dich, ob du auf kurzfristige Marktstimmungen reagierst (was oft ein schlechter Grund ist) oder ob deine Anpassungen auf einer fundierten, objektiven Analyse der zukünftigen Aussichten beruhen (was ein solider Ansatz darstellt).

Ein Beispiel, das die Notwendigkeit regelmäßiger Überprüfungen verdeutlicht: In den 2010er-Jahren lagen die Renditen von Anleihen oft nahe bei null, teilweise sogar im negativen Bereich, was Anleihen unattraktiv machte. Heute sind die Renditen deutlich gestiegen – in Europa bieten Anleihen Erträge von 3 % oder mehr. Diese Verschiebung verändert die Attraktivität von Anleihen grundlegend und könnte eine höhere Allokation in diesem Bereich im kommenden Jahrzehnt rechtfertigen als in den vergangenen Dekade. Folglich könnte dein optimales Portfolio im Jahr 2025 ganz anders aussehen als eines, das vor 10 Jahren entworfen wurde.

Zudem wird dein tatsächliches Portfolio im Laufe der Zeit naturgemäß aufgrund von Marktbewegungen von der ursprünglich optimierten Version abweichen. Dies führt uns zu zwei weiteren, entscheidenden Regeln für den Einsatz “optimaler” Portfolios.

Regel 12: Führe regelmäßige Realitätschecks durch!
Bewerte dein Portfolio und seine Parameter regelmäßig. Erwartete Renditen und Kovarianzen sind nicht statisch – sie verändern mit den Marktbedingungen im Laufe der Zeit. Strukturelle Verschiebungen in der Wirtschaft können Risiko- und Renditeprofile signifikant verändern. Es sollte zur Gewohnheit werden, diese Schätzungen mindestens einmal im Jahr zu überprüfen – achte dabei jedoch darauf, dass diese Anpassungen nicht emotionsgeleitet aufgrund kurzfristiger Marktbewegungen erfolgen. Regelmäßige Überprüfungen stellen sicher, dass deine langfristige Strategie relevant und für deine Zwecke optimal bleibt.

Regel 13: Rebalance dein Portfolio!
Lege klare Regeln fest, wann und wie dein Portfolio neu ausgerichtet werden soll. Du hast dein Portfolio aus einem bestimmten Grund mit einer spezifischen strategischen Allokation entworfen. Mit der Zeit werden Marktschwankungen dazu führen, dass dein Portfolio von diesen optimalen Gewichtungen abweicht. Um die beabsichtigte Gewichtung zu erhalten, solltest du regelmäßig dein Portfolio justieren, um marktinduzierte Über- bzw. Untergewichtungen auszugleichen. Zwar hängt der genaue Zeitpunkt von den spezifischen Merkmalen deines Portfolios ab, in den meisten Fällen sind jedoch ein vierteljährliche oder halbjährliches Rebalancings ausreichend. Dieser Prozess kann zudem zusätzliche Erträge generieren, da er häufig das Prinzip „sell high, buy low“ ermöglicht.

Durch die Umsetzung dieser regelmäßigen Überprüfungen und Rebalancing-Routinen stellst du sicher, dass dein Portfolio mit deinen langfristigen Zielen im Einklang bleibt und sich gleichzeitig an die sich verändernden Marktbedingungen anpasst.

Zusammenfassung

In diesem Artikel haben wir 13 praktische Regeln für die Optimierung von Portfolios im realen Umfeld vorgestellt. Indem wir häufige Missverständnisse und Herausforderungen – wie die Sensitivität von Optimierungsmodellen und die Gefahren der Extrapolation jüngster Renditen – adressiert haben, liefern wir umsetzbare Lösungsansätze, die sowohl institutionelle als auch private Investoren nutzen können, um ihre Portfoliomanagement-Prozesse zu verbessern. Zu den zentralen Empfehlungen gehören der Einsatz robuster Kovarianzschätzungen, die Berücksichtigung der Investorenpräferenzen durch Restriktionen und die Anwendung fortschrittlicher Risikomaße wie den Conditional-Value-at-Risk (CVaR) anstelle der traditionellen Varianz.

Diese Regeln stützen sich sowohl auf theoretische Finanzmarkttheorien als auch auf empirische Forschungsergebnisse und bieten einen soliden Rahmen, um Portfoliooptimierung über die Einschränkungen traditioneller Ansätze wie der Mittelwert-Varianz-Optimierung hinaus sinnvoll anzuwenden. In zukünftigen Artikeln werden wir jede Regel noch detaillierter behandeln und praxisnahe Python-Beispiele zur Implementierung vorstellen.

Portfoliooptimierung kann – richtig angewandt – ein mächtiges Instrument sein, um robuste und strategisch ausgerichtete Portfolios zu erstellen. Mit diesen 13 Regeln hoffen wir, dir zu helfen, häufige Fallstricke zu umgehen und fundiertere Investitionsentscheidungen zu treffen. Bleib dran für weitere detaillierte Diskussionen in den kommenden Beiträgen.

Portfolio optimization as instrument to determine the allocation of assets in a portfolio is often criticized for being impractical in real-world investing. In this article, we present 13 useful rules to address these challenges and make portfolio optimization more effective and user-friendly. By debunking common myths, we offer strategies that can be applied by both institutional and individual investors. The rules are designed to enhance decision-making, mitigate risks, and incorporate investor preferences, ensuring that portfolio optimization is a valuable tool for achieving long-term investment goals. Future articles will explore each rule in more detail, with real-world examples and Python code to guide implementation.

Introduction

As portfolio managers and quantitative analysts, we have often encountered reservations about the benefits of portfolio optimization to make decisions regarding the optimal allocation of assets. A common concern is that, when applied incorrectly, portfolio optimization can be more detrimental than beneficial. When used incorrectly, heuristic approaches often prove to be the superior alternative.

However, ignoring portfolio optimization altogether due to its complexity is not the solution. A lack of understanding or a poorly applied strategy can lead to poor outcomes, but with the right knowledge and methodology, portfolio optimization can be a powerful tool. The key is to approach it strategically.

In this article, we will address the common critiques of portfolio optimization and present 13 practical rules to make it more effective in real-world investing. These rules are designed to enhance decision-making, mitigate risks, and align with investor preferences. We will not limit our discussion to traditional Mean-Variance Optimization (MVO); instead, we’ll formulate rules that are valid regardless of the specific optimization approach used.

While there are many different optimization methods available, each with its own strengths and weaknesses, the true power lies in knowing how to combine these approaches to meet specific investor needs. By following these rules, institutional and individual investors alike can improve their portfolio optimization processes and move beyond outdated methods.

In future articles, we will explore each of these 13 rules in more detail, with real-world examples and Python code to guide implementation.

Square One: A plea for a clear investment process

Before diving into portfolio optimization, it is essential to establish two foundational elements:

1. Clearly Defined Investment Objectives

2. A Clear and Consistent Investment Approach

While both are crucial, we’ll focus primarily on the second element: the investment approach.

An investment approach refers to the systematic process an investor follows to make informed, strategic decisions about where, when, and how to allocate capital. This disciplined approach ensures that investments are aligned with specific financial goals, risk tolerance, and market conditions. It separates rational, consistent decision-making from impulsive, emotion-driven choices based on "gut feeling."

Each investment process is unique, and portfolio optimization must be tailored to each case. However, portfolio optimization doesn’t exist in a vacuum. It must be built upon a solid, structured investment process. Key questions like which asset classes are eligible, what investment limits or risk budgets exist, and how decisions will be made need to be answered before diving into optimization.

This brings us to Rule 1:

Rule 1: Have a clear and consistent investment approach!
Your portfolio optimization strategy must be grounded in a well-defined investment approach. Without a clear framework, portfolio optimization is like building a house on sand—it will collapse regardless of the quality of the materials or the expertise of the architect.

“Portfolio optimization is too sensitive to be used in practice”

One of the most common critiques of portfolio optimization is its perceived sensitivity, and there’s a fair amount of truth to this concern. It’s well-documented that portfolio optimization models, especially traditional ones like Mean-Variance Optimization (MVO), can be highly sensitive to small changes in input parameters, such as expected asset returns. Even slight adjustments can lead to dramatic shifts in portfolio weights—sometimes causing allocations to flip from 0% to 100%, and vice versa.

But why does this sensitivity exist?

The main culprit lies (mostly !) in the covariance matrix, which plays a central role in most optimization approaches. Two key factors contribute to the problem:

1. High Correlations Between Assets: Assets in a portfolio are often highly correlated, which increases the complexity of accurately estimating the relationships between them.

2. Noisy Data: Covariances are typically estimated using historical data, which are often noisy and unreliable, especially when the sample size is small. This results in high estimation uncertainty.

Many optimization methods rely on inverting the covariance matrix to calculate portfolio weights. When correlations are high and data is noisy, the covariance matrix becomes “ill-conditioned,” meaning it’s prone to large errors. This results in portfolio weights that are overly sensitive to small changes in expected returns, often leading to unrealistic and highly volatile allocations.

In practice, this sensitivity is particularly problematic when working with highly correlated assets or when the historical data used to estimate the covariance matrix is short or unreliable. While this is a common issue, it’s not a problem with portfolio optimization itself, but rather with the data and input parameters. Fortunately, there are approaches that are more resilient to this issue.

So, how can we address this?

Rule 2: Clean Up your Covariance Matrix!
Never use the raw sample covariance matrix directly in optimization. Sample covariances, estimated from historical time series, are often too noisy and unreliable to be used as-is. Instead, apply techniques like shrinkage or denoising to improve the accuracy of these estimates. These methods help to stabilize the covariance matrix and significantly reduce the sensitivity of portfolio weights to small changes in expected returns.

Rule 3: Structure your Assets!
Avoid using optimization on portfolios with highly correlated assets. When assets are highly correlated, traditional optimization approaches tend to fail. A better solution is to use asset clustering—grouping assets into clusters based on their correlations. The key idea is that assets within a cluster tend to be highly correlated (intra-cluster), while correlations between clusters (inter-cluster) are much lower. By optimizing at the inter-cluster level, you can reduce the impact of high correlations, leading to more stable and realistic portfolio weights.

“Portfolio optimization generates portfolios that underperform against equal-weighted portfolios”

One common critique of portfolio optimization, backed by several studies, is that optimized portfolios often underperform simple equal-weighted portfolios (1/N portfolios). The argument suggests that simply allocating an equal share to each asset might be more effective than using complex optimization techniques. But is there any real merit to this claim?

A key study often cited in support of this view is by DeMiguel, Garlappi, and Raman Uppal (2009). Their research shows that 1/N portfolios outperform optimized portfolios out-of-sample due to large estimation errors of the input parameters. However, the authors make a critical mistake—extrapolating recent returns.

In their study, the authors estimate expected returns (as input for their optimization) based on rolling estimates, which use the latest observed returns to predict future returns. This method assumes that expected returns can be estimated reasonably well from recent return data—an assumption that, both theoretically and empirically, doesn’t hold up.

If you're familiar with financial markets, this assumption might seem overly simplistic. Do you believe that markets can "heat up" due to positive investor sentiment, reducing future return prospects? Do you think that a stock with a 1000% price increase and an extremely high price-to-earnings (P/E) ratio is likely to have more moderate returns in the future? And do you think that market conditions, such as recessions or economic booms, can create different return profiles? If you answered yes to any of these questions, you're in good company. Numerous studies show that expected returns change over time, influenced by market conditions, sentiment, and macroeconomic factors. Unfortunately, the study by DeMiguel, Garlappi, and Raman Uppal overlooks these dynamics entirely.

This serves as a cautionary tale: the accuracy of your expected returns and covariance estimates is critical. As the saying goes, “Garbage in, garbage out.” If you use poor inputs, you’ll get poor results. Fortunately, we can learn from this mistake by remembering the following essential rule:

Rule 4: Do NOT extrapolate recent Returns!
Never use recent historical returns as the basis for expected returns. Don’t assume that last year’s performance will necessarily repeat. Whether you are optimizing a portfolio of broad asset classes (e.g., equity and bond indices) or individual assets (like stocks), you should rely on estimation methods that are grounded in sound theoretical frameworks. Approaches such as factor models or fundamental valuation models offer more reliable ways to forecast expected returns. Relying on recent historical data—especially from short time periods—creates an illusion of predictability and risks underperformance because markets operate in cycles. Even if you use a solid theoretical model that captures time-varying expected returns, you should be aware that predicting returns is still very difficult due to parameter instability and model uncertainty.

“Portfolio optimization leads to undesirable portfolios”

While expected returns and covariances are vital to portfolio optimization, they only capture part of the picture. Investors typically have preferences that extend beyond mere return and risk. These preferences may include ethical considerations (e.g., Environmental, Social, and Governance (ESG) criteria), liquidity requirements, or constraints on deviations from a benchmark.

To build portfolios that reflect these broader concerns, it’s essential to incorporate additional constraints into the optimization process.

Rule 5: Reflect all your Constraints!
Undesirable portfolio compositions can be avoided by imposing constraints. Does the idea of an all-stock or all-bond portfolio make you uncomfortable? Would you like to avoid excessive trading and limit transaction costs? Or, do you have sustainability preferences for your investment? Then, set constraints in the optimization to take these aspects into account. A useful exercise is to review each major asset class and identify potential extreme allocations that would violate your preferences or objectives. By doing so, you can integrate additional information—such as ESG data—into the optimization process, ensuring that your portfolio aligns with both financial and non-financial goals. Portfolio optimization, therefore, can accommodate more than just return time series.

Rule 6: Anchor Yourself!
Establish a benchmark portfolio and set a tolerable level of deviation. Every portfolio needs a benchmark, whether for performance evaluation or to guide active bets. This benchmark should reflect your long-term strategic asset allocation. As your estimates of expected returns and covariances evolve over time, so will your portfolio allocation. By using a benchmark and setting a maximum acceptable tracking error, you can ensure that your portfolio stays aligned with your long-term investment objectives and doesn’t stray too far from your desired allocation.

Rule 7: Measure your Diversification!
Use additional metrics to assess the diversification of your portfolio. Diversification is essential to any investment strategy, but don’t assume that portfolio optimization automatically ensures a well-diversified portfolio. This is another reason why optimized portfolios often underperform equal-weighted portfolios out of sample. To truly understand your portfolio’s diversification, consider metrics beyond traditional portfolio weights. Use Entropy measures, the (inverse) Herfindahl-Hirschman Index, or the Gini coefficient to quantify diversification. On a more advanced level, assess diversification based on underlying risk factors—such as economic growth, interest rates, or geopolitical events—that drive asset returns. This ensures that your portfolio is well-diversified not just in terms of asset allocation but also across the broader economic landscape.

“Portfolio optimization contains unrealistic assumptions and is not useful.”

Models and estimators can only incorporate a limited amount of real-world information. Traditional approaches, such as Mean-Variance Optimization (MVO), focus solely on the first two moments of a distribution—mean and variance—and/or assuming that financial markets follow a multivariate normal distribution. While these models have their place, they often make unrealistic assumptions about how the financial world works. Over time, more sophisticated models have emerged to address these limitations.

MVO, as the most widely recognized portfolio optimization approach, contains several assumptions that are out of step with the realities of the market. These include:

1. Financial markets are normally distributed — In reality, financial returns exhibit fat tails and they are far from a normal distribution.

2. Investors have quadratic utility — Investors do not make decisions based solely on mean-variance preferences. Their decisions are often driven by more complex psychological factors and they evaluate their outcomes relative to one or more reference points.

3. Rational investors with homogeneous expectations — Emotions, beliefs and psychological biases affect the decision process, in particular during crisis. Furthermore, the heterogeneity of expectations varies with the degree of information uncertainty.

4. Static, one-period investment horizon — Many investors do not have a pre-specified fixed investment horizon and they also consider the risk within the investment period.

5. No transaction costs or taxes — In a frictionless world, continuous trading and high portfolio turnover are not a problem, but in reality they are.

6. Investors are only concerned with variance — In practice, people suffer more from losses than they gain from equivalent gains. This is a key insight from behavioral finance, which shows that losses are perceived as more painful than the satisfaction from gains of the same size.

Behavioral finance and empirical evidence have clearly shown that these assumptions do not align with how real-world investors think and behave. However, does this mean that portfolio optimization, in general, is useless? We would argue that it does not.

Today, we have access to more advanced methods that better reflect investor behavior and market dynamics. One possibility is to consider more adequate utility functions using Kahneman and Tversky's "Cumulative Prospect Theory". Another is the use of models such as Mean-CVaR (Conditional Value-at-Risk) which offers a more realistic alternative by focusing on tail risks, which are more relevant to investors concerned with large losses rather than daily volatility. These models allow for more nuanced risk management, but be aware that increased model flexibility often comes with increased complexity and higher estimation risk.

While MVO can serve as a useful tool for developing initial insights, it should not be relied upon as the primary method for portfolio optimization. For more accurate and realistic results, we strongly recommend moving beyond MVO and adopting models like Mean-CVaR, which better reflect how investors truly perceive and react to risk. Investors are generally less concerned with everyday fluctuations (variance) and more focused on avoiding large, catastrophic losses.

Rule 8: Drop Variance as a Measure of Risk!
Use higher-dimensional risk measures, such as CVaR.
Portfolio optimization should be based on realistic assumptions. The Mean-Variance Optimization model does not meet this standard. We advocate for the use of Mean-CVaR, which more closely aligns with how investors actually perceive and react to risk, particularly focusing on tail risks and extreme events that are more meaningful than standard deviation in the context of real-world investing.1

Rule 9: Embrace the non-linear, non-gaussian Reality!
The world of financial data is inherently non-linear and certainly not normally distributed. While correlations assume linear relationships, the assumption of normally distributed returns underestimates the probability of extreme events. Whenever possible and reasonable, abandon linearity and normality assumptions in favor of more flexible approaches that account for (excess) kurtosis, skewness, and non-linearity.

“Portfolio optimization is too restrictive for my “active”and “flexible” investment approach”

Based on our experience, portfolio optimization is highly flexible, and almost any investment strategy can be integrated with it. While you may need to develop tailored solutions for specific situations—such as hierarchical clustering of portfolio elements (see Rule 3)—the flexibility to accommodate diverse approaches is certainly available.

Criticism of portfolio optimization as being “too restrictive” is common, but it’s often misplaced and can be problematic. In many cases, this criticism stems from a misunderstanding: the difference between “flexibility” and the absence of a structured investment process. A well-defined investment process acts as a blueprint in the financial world. It ensures that investment decisions are made based on a system of checks and balances, helping to prevent the impulsive, emotion-driven mistakes that often occur during market extremes—whether bull or bear markets. Without such a process, the likelihood of long-term underperformance increases significantly.

Let me be clear: When you hear this criticism as a defense against using portfolio optimization, you should consider two possibilities:

1. The portfolio manager lacks a proper investment process.

2. The portfolio manager is simply not equipped to implement portfolio optimization.

If a portfolio manager uses this argument to avoid portfolio optimization, you should proceed with caution. It could indicate a fundamental flaw in their investment approach or a lack of proficiency in portfolio management techniques.

Let’s reframe the question: Suppose you have short-term investment ideas or active views. Can you integrate these ideas into your long-term portfolio? Absolutely. In fact, there are well-established methodologies to blend short-term views with long-term strategies. Approaches like Black-Litterman allow you to incorporate your market views into an optimized portfolio, and they’re sufficient in most cases. For more complex cases, entropy-based methods can help you refine the integration of short-term predictions (also for higher moments of your return distribution) into your long-term strategy. Therefore, there’s no valid excuse for not utilizing portfolio optimization.

Rule 10: Implement your Views!
Incorporate short-term views, but don't abandon your long-term portfolio.
Many investors find it too monotonous to only focus on their long-term strategic portfolio. If you have short-term forecasts or active ideas, integrate them with your long-term strategy. Active deviations from your core portfolio are acceptable, as long as they remain within a reasonable range. Fortunately, numerous methodologies are available to seamlessly blend short-term views with a solid, strategic long-term portfolio.

“My optimised portfolio crashed although it had a low risk profile”

This scenario is something many investors experienced in 2022. Despite having portfolios with a high allocation to sovereign bonds—typically seen as safe assets—many suffered significant losses. The key issue here is that, in portfolio optimization, nearly all information about risk is encapsulated in the covariance matrix. This can be both an advantage and a pitfall.

It’s a blessing when the covariance matrix accurately reflects the true market dynamics. However, it becomes a curse when you optimize a portfolio based on covariance data that may have been valid for the last decade, only to face a sudden shift in underlying market conditions. When volatility and correlations change dramatically, the diversification benefits that once worked may disappear, leaving your portfolio highly vulnerable.

So, what went wrong in 2022?
Between 2000 and 2021, the correlation between stocks and bonds was generally negative, driven by procyclical inflation and expansive monetary policies during the 2010s. During this period, portfolios benefited from strong diversification between equities and bonds. However, with inflation rising sharply in 2022 (as negative supply shock), this dynamic shifted. The correlation between stocks and bonds turned positive, meaning that both asset classes fell together. As a result, the diversification benefits evaporated, leading to significant portfolio losses.

This situation can be avoided with careful portfolio management. The key takeaway: Your estimates of expected returns and covariance should reflect future expectations, not historical data.

Rule 11: Stress-Test Your Portfolio!
Use stress tests to assess your portfolio with alternative parameters.
Portfolio optimization doesn’t end once the weights are calculated. After determining your portfolio, you must stress-test it by considering alternative estimates of volatility and correlations to account for low-probability, high-impact events. A scenario-based approach works well here. Actively think about potential future scenarios and how they could affect your portfolio. This is especially critical when working with limited data that may not capture certain extreme events, which—while rare—are still possible.

Stress testing will push your portfolio’s diversification to the limit, revealing vulnerabilities that may not be visible under normal market conditions. This approach helps ensure that your portfolio remains resilient, even during unexpected market shifts.

“Portfolio optimization is too slow to adjust to current developments”

I strongly caution against attempting to "time" the markets. Even seasoned experts often struggle to predict short-term movements, and as a private investor, your chances of long-term success are slim if you try to time the market. If you view the stock market as a source of thrill and excitement, perhaps you should reconsider and opt for something less risky, like a time deposit. The stock market is not a place for gambling; investing should be a disciplined and often dull activity. For thrills, the amusement park is a more affordable option.

That said, portfolio optimization is not static. It adapts over time as market conditions change. While your portfolio should follow a long-term strategy, it's essential to periodically reassess the key parameters—such as expected returns and covariances. Performing a reality check on these assumptions once a year is generally sufficient. But be cautious: when you see dramatic shifts in your expected returns, ask yourself whether you're adjusting based on short-term market sentiment (which is often a poor reason) or on a thoughtful, objective analysis of future prospects (which is a sound approach).

For instance, let's consider an example of why regular checks are necessary: In the 2010s, yields were close to zero, sometimes even negative, which made bonds an unattractive asset. Today, yields have risen significantly, with bonds offering returns of 3% or more in Europe. This shift dramatically alters the attractiveness of bonds and may justify a higher allocation to them in the coming decade compared to the previous one. Therefore, your optimal portfolio in 2025 may look quite different from one designed 10 years ago.

Additionally, as market conditions evolve, your actual portfolio will naturally deviate from the original optimized version. This brings us to two crucial rules for maintaining an “optimal” portfolio.

Rule 12: Conduct regular Reality Checks!
Evaluate your portfolio and its parameters regularly.
Expected returns and covariances are not fixed; they change over time as market conditions evolve. Structural shifts in the economy can significantly alter risk and return profiles. Make it a habit to reassess your estimates—annually is generally enough—but be cautious not to make these adjustments based on emotional reactions to short-term market movements. Regular assessments ensure that your long-term strategy remains relevant and optimal for your purpose.

Rule 13: Rebalance your Portfolio!
Establish clear rules for when and how to rebalance your portfolio.
Remember, you designed your portfolio with a specific strategic allocation for a reason. Over time, market fluctuations will cause your portfolio to deviate from its optimal weights. To maintain the intended weighting, you should regularly adjust your portfolio in order to counteract over- or underweightings caused by market fluctuations. While the exact timing depends on your portfolio’s specifics, in most cases, quarterly or semi-annual rebalancing is sufficient. This process can also generate additional returns, as it often results in "buying low" and "selling high," which can be beneficial in certain market conditions.

By implementing these regular checks and rebalancing routines, you ensure that your portfolio remains aligned with your long-term goals while adapting to changing market realities.

Summary

In this article, we've outlined 13 practical rules for optimizing portfolios in real-world settings. By addressing common misconceptions and challenges—such as the sensitivity of optimization models and the dangers of extrapolating recent returns—we've provided feasible solutions that both institutional and individual investors can follow to improve their portfolio management processes. Key recommendations include using robust covariance estimations, incorporating investor preferences through constraints, and adopting advanced risk measures like the Conditional Value-at-Risk (CVaR) over traditional variance.

These rules are grounded in both theoretical finance and empirical research, offering a solid framework for enhancing portfolio optimization beyond the limitations of traditional approaches like Mean-Variance Optimization. In future articles, we'll delve deeper into each rule, providing real-world Python examples to guide you through implementation.

Portfolio optimization, when done correctly, can be a powerful tool for building resilient and strategically aligned portfolios. With these 13 rules, we hope to help you navigate common pitfalls and make better-informed investment decisions. Stay tuned for more detailed discussions in the upcoming posts.