Portfolio Optimizer

The NAAIM Exposure Index: Contrarian or Continuation Indicator?

2026-09-26T00:00:00-05:00

In a previous blog post, I described the NAAIM Exposure Index, which represents the average exposure to U.S. equity markets as reported by members of the National Association of Active Investment Managers (NAAIM) in a weekly survey.

At the end of August 2026, the NAAIM Exposure Index hits a value of 102.66, meaning that U.S. active investment managers were in aggregate leveraged long in terms of U.S. stocks exposure.

This led to some interesting discussions on X about whether the NAAIM Exposure Index should be used as a contrarian indicator or as a continuation indicator.

In this third post of this series on sentiment indicators, I will show that the contrarian nature of the NAAIM Exposure Index is actually debatable, with high (resp. low) readings not bearish (resp. not bullish) over the medium term.

As an example of usage, I will show how to convert the values of the NAAIM Exposure Index into probabilities of drawdown for the U.S. stock market over the medium term.

Reminders on the NAAIM Exposure Index

The NAAIM Exposure Index is described elsewhere on this blog.

The NAAIM Exposure Index, a contrarian indicator?

The NAAIM Exposure Index, like most sentiment indicators, is typically considered as a contrarian indicator.

In other words, bullish sentiment numbers are often telling us that it is time to sell, not buy as you might think¹.

There are various explanations for this phenomenon, some known to be false¹ - for example the fact that the people polled are always wrong¹ - and some probably closer to the truth like the explanation provided by Hepburn¹:

When you think about this, you will realize that those who have invested their money are no longer potential buyers but have now become potential sellers. When the number of potential sellers is greater than the number of potential buyers, a condition is created from which markets move lower. At a point, all that is needed for the market to decline is some of that daily news that can drive prices down for no apparent reason.

It also works the other way, when everyone is bearish on the market. This indicates a shortage of potential sellers because everyone has already sold, and there is lots of cash on the sidelines. This often acts like rocket fuel for stock prices when a market turns around and buyers invest their cash.

It is then no surprise that the extreme reading of 102.66 on 26th August 2026 triggered some discussions on X, like the one captured in Figure 1.

Figure 1. The NAAIM Exposure Index as a contrarian indicator, August 2026.

From Figure 1, it appears that values of the NAAIM Exposure Index greater than 100 (in red) are usually followed by retracements in the S&P 500 index, which visually² confirms that the NAAIM Exposure Index might be a contrarian indicator.

Outside of X, Rob Hanna from QuantifiableEdges has also studied the NAAIM Exposure Index and found that strongly oversold readings could be indicative that the market is so oversold it is ready to rally³.

Nevertheless, the particular conditions described in Hanna³ exclude nearly all⁴ the occurences of the NAAIM Exposure Index smaller than 20, so that nothing can be concluded about extremely low readings of that indicator from his analysis.

The NAAIM Exposure Index, a continuation indicator?

Contrary to their extremely low counterparts, extremely high readings of the NAAIM Exposure Index greater than 100 are analysed in Hanna³, which concludes that strongly overbought readings are not a contrarian indicator [and] in fact they often suggest strong momentum that is likely to continue³.

As Hanna³ puts it:

As it turns out, when you have a bunch of smart investment managers getting leveraged, there is a good chance that they are right and that there are market gains ahead.

Another perspective is provided by Eric Evans from Trajecta Investment Advisors LLC.

In a LinkedIn post, Evans empirically demonstrates that forward 12-week 10%+ drawdown risk [of the U.S. stock market] falls monotonically as manager exposure rises⁵.

Figure 2 reproduces Evans’s analysis⁵, with a couple of months of additional data.

Figure 2. Forward 12-week 10%+ drawdown risk of the U.S. stock market (SPY ETF) against NAAIM Exposure Index levels, 07th July 2006 - 05th June 2026.

Figure 2 clearly shows that extremely low levels of the NAAIM Exposure Index are associated with a much higher frequency of 10%+ drawdown risk over the subsequent 12 weeks than extremely high levels of that index.

From this perspective, the NAAIM Exposure Index is a continuation indicator, which Evans⁵ interprets as follows:

Crashes don’t launch from euphoria. They launch from already-nervous positioning that gets more nervous. By the time active managers are fully exposed, the trend driving them in usually has further to run.

Which means a rising NAAIM reading isn’t a warning. It’s a confirmation. More bullish managers ahead means the trend has further to run, not less.

In addition to Hanna’s³ and Evans’⁵ work, Figure 3 depicts the relationship between the NAAIM Exposure Index and the forward 12-week average U.S. stock market return.

Figure 3. Forward 12-week average U.S. stock market return (SPY ETF) against NAAIM Exposure Index levels, 07th July 2006 - 05th June 2026.

On Figure 3, 3 different patterns are visible:

A contrarian behavior in the bulk of the observations

NAAIM Exposure Index levels between 20 and 100 are monotonously decreasing, which suggests a contrarian behaviour.
A continuation behavior in the left-tail of the observations

NAAIM Exposure Index levels lower than 20 break the monotonous pattern above and are associated to an average forward 12-week U.S. stock market return of 0%.
A continuation behavior in the right-tail of the observations

NAAIM Exposure Index levels greater than 100 break the monotonous pattern above and are associated to an average forward 12-week U.S. stock market return on par with that of 40-60 levels.

All in all, and at least over the medium term, it appears that data strongly support calling the NAAIM Exposure Index a continuation indicator rather than a contrarian indicator!

Examples of usage

As a qualitative indicator

The previous section showed that extremely low levels of the NAAIM Exposure Index have historically been associated with a degraded environment in terms of U.S. stock market returns⁶ over the subsequent 12 weeks.

Thus, it makes sense to become cautious for one’s portfolio when the NAAIM Exposure Index reaches levels lower than 20.

As a quantitative indicator

More sophisticated investors might want to go beyond a simple⁷ qualitative usage and integrate the NAAIM Exposure Index directly into their quantitative trading system.

Hepburn’s adaptive rebalancing strategy revisited

From the previous section, one of the simplest idea would be to convert the NAAIM Exposure Index into its historical percentile rank among the observations available at that time and use that percentile rank to dynamically scale the exposure of a portfolio to the U.S. stock market.

For example, at the end of each month, one could:

Like in the previous blog posts of this series⁸, compute the 12-week moving average of the NAAIM Exposure Index
Compute the percentile rank $x$% of that moving average among all the 12-week moving averages of the NAAIM Exposure Index already computed up to that point in time
Allocate $x$% of the portfolio to the SPY ETF and $1-x$% of the portfolio to cash⁹

Figure 4 compares such a dynamic NAAIM Exposure Index-based trading strategy to a fixed 60% allocation trading strategy, over the period October 2007 - August 2026.

Figure 4. Dynamic NAAIM Exposure Index-based v.s. fixed 60% allocation trading strategies, SPY ETF, October 2007 - August 2026.

Associated figures:

Trading Strategy	Average Stocks Exposure	CAGR	Annualized Sharpe Ratio (0)	Maximum (Weekly) Drawdown
Fixed 60% allocation	60%	6.88%	0.66	36.52%
Dynamic NAAIM Exposure Index-based	62%	7.40%	0.76	27.41%

A couple of remarks on this trading strategy:

It is a direct extension of the trading strategy proposed by Hepburn¹⁰.
It is built solely on survey results, and does not take into account price, or trend, or volume, or breadth, or anything else³, which really demonstrates the value of the NAAIM Exposure Index³.
Because it focuses on the offensive sleeve of a 60/40 Portfolio, it might be a good complement to Tom Carlson’s Adaptive 60/40 Portfolio strategy¹¹.

Probability of a 10%+ drawdown of the U.S. stock market in the subsequent 12 weeks

Given the direct relationship between the NAAIM Exposure Index and the forward 12-week drawdown risk of the U.S. stock market exhibited in Figure 2, one additional example of usage of that index could be to build a drawdown forecasting model for the U.S. stock market.

For this, and for the same reasons as in the previous sub-section, I propose to use the 12-week moving average of the NAAIM Exposure Index rather than its (raw) value.

Figure 5 shows that introducing such a smoothing procedure does not degrade the strong relationship between extremely low levels of the NAAIM Exposure Index and the frequency of 10%+ drawdown risk over the subsequent 12 weeks.

Figure 5. Forward 12-week 10%+ drawdown risk of the U.S. stock market (SPY ETF) against 12-week moving average NAAIM Exposure Index levels, 22th September 2006 - 05th June 2026.

If anything, Figure 5 shows that the left-tail behaviour of that relationship is actually magnified by the usage of a moving average!

Because the relationship exhibited in both Figure 2 and Figure 5 is monotonous, it should in theory be possible to use an isotonic regression to extract explicit probabilities of a forward 12-week 10%+ drawdown of the U.S. stock market from the values of the 12-week moving average of the NAAIM Exposure Index.

Figure 6 depicts these probabilities, extracted from the percentile rank of the 12-week moving average of the NAAIM Exposure Index over the full history (in-sample analysis).

Figure 6. Probability of a 10%+ drawdown of the U.S. stock market (SPY ETF) over the next 12 weeks as a function of the percentile rank of the 12-week moving average NAAIM Exposure Index, in-sample isotonic regression, 22th September 2006 - 05th June 2026.

Figure 7 depicts the associated in-sample ROC curve.

Figure 7. ROC curve associated to the in-sample forecasting model of Figure 6.

With the associated in-sample detailed forecasting figures and interpretation:

Metric (in-sample)	Value	Interpretation
Base Rate	0.146	The model baseline
ROC-AUC	0.772	The model generally ranks events (drawdowns >= 10%+) above non-events (drawdowns < 10%+)
PR-AUC	0.434	The model is 3x better than the baseline
Log Loss	0.316	The model predicted probabilities are relatively close to the observed probabilities
Brier	0.096	The model predictions have relatively low mean squared probability error

Now moving from in-sample to walk-forward out-of-sample¹² forecast evaluation, forecasting performances obviously degrade, as illustrated in Figure 8, but the resulting model is still very much usable.

Figure 8. ROC curve associated to the walk-forward out-of-sample forecasting model of Figure 6.

Associated out-of-sample detailed forecasting figures:

Metric (out-of-sample)	Value
Base Rate	0.144
ROC-AUC	0.652
PR-AUC	0.373
Log Loss	0.736
Brier	0.109

In any cases, as already noted by Hanna³, even if relying solely on [that] index may be overenthusiastic³, utilizing it as one input within a larger model seems completely reasonable and could very possibly strengthen it³.

For obvious reasons, such an exercice is out of scope of this blog post.

As an important side note for the forecasting exercice at hand, the selected 10%+ drawdown threshold has nothing magical.

This is confirmed by Figure 9 and Figure 10.

Figure 9. Forward 12-week 5%+ drawdown risk of the U.S. stock market (SPY ETF) against 12-week moving average NAAIM Exposure Index levels, 22th September 2006 - 05th June 2026.

Figure 10. Forward 12-week 20%+ drawdown risk of the U.S. stock market (SPY ETF) against 12-week moving average NAAIM Exposure Index levels, 22th September 2006 - 05th June 2026.

Indeed, on these figures, the 10%+ drawdown threshold of Figure 5 is replaced by a 5%+ (Figure 9) and a 20%+ (Figure 10) threshold and the continuation behaviour in both tails of the observations remains clearly visible.

Conclusion

Sentiment indicators are typically called contrarian indicators because more often than not, the market will move against the sentiment of the majority¹³.

In the specific case of the NAAIM Exposure Index, though, it seems the common rule inverts the actual signal⁵, at least within the context studied in this blog post.

As usual, feel free to connect with me on LinkedIn or follow me on Twitter.

–

See Proactive Advisor Magazine, Industry insights, Will Hepburn, The backward nature of sentiment indicators. ↩ ↩² ↩³ ↩⁴
Depending on the time horizon, visuals might be misleading though… ↩
See Rob Hanna, Modeling with the NAAIM Exposure Index. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹
The first condition listed in Hanna³ filters out 40 of the 45 occurences where the NAAIM Exposure Index is smaller than 20 over the period 05th July 2006 - 16th September 2026. ↩
See Evans, LinkedIn post. ↩ ↩² ↩³ ↩⁴ ↩⁵
Actually both in terms of raw returns and in terms of the number of major drawdowns. ↩
Beware of the midwit meme though; essentially, any quantitative framework built on the NAAIM Exposure Index will ultimately translate very low readings into “bad” and very high reading into “good”. ↩
Here and there. ↩
Cash is assumed to have a 0% return. ↩
See William T. Hepburn, Hepburn Capital Management, Using Adaptive Rebalancing to Bridge the Gap between Strategic Asset Allocation and Tactical Asset Allocation, October 2009. ↩
See Tom Carlson, Adaptive 60/40: Rethinking the Defensive Sleeve, January 20, 2026. ↩
Walk-forward out-of-sample forecast with an expanding learning window. ↩
See American Association of Individual Investors, Contrarian Indicators, January 2012. ↩

Tom Carlson’s Adaptive 60/40 Portfolio: Momentum-based Selection of Stock Diversifiers

2026-09-14T00:00:00-05:00

Welcome to 2026, where bonds no longer diversify stocks, volatility remains elevated and inflation isn’t going anywhere¹.

This is the dramatic, although factual, opening of Basis Points’s video Morgan Stanley’s asset allocation playbook for the next 5 years.

As a consequence, the 60/40 Portfolio - invested 60% in stocks and 40% in bonds - which for decades has stood as the institutional benchmark for ‘balanced’ portfolio allocation² is showing signs of strain² because bonds can not longer be counted on for downside protection during volatile equity markets³.

In this blog post, I propose to review an alternative to the 60/40 Portfolio recently⁴ proposed by Tom Carlson and called the Adaptive 60/40 Portfolio².

After confirming that a regime change took place in 2022 by comparing the recent behaviour of bonds to their history, I will detail the construction rules of the Adaptive 60/40 Portfolio, analyze its performances and study a variation based on the Probabilistic Sharpe Ratio introduced in Bailey and de Prado⁵.

2022 - The year bonds changed

Figure 1, inspired by a LinkedIn post from Andreas Steiner, depicts two mean-variance efficient frontiers within a two-asset class universe made of:

U.S. stocks, represented by the SPY ETF⁶
Long-term U.S. Treasuries, represented by the TLT ETF⁷

Figure 1. SPY/TLT ETFs Mean-Variance Efficient Frontiers, 1989 - 2021 v.s. 2022 - 2026.

From Figure 1 and its underlying data, it is clear that the two periods 1989-2021 and 2022-2026 correspond to two different regimes, with the most recent period exhibiting:

A huge increase in the correlation between stocks and bonds (from -0.12 to 0.57)
A slight increase in stocks mean return (from 9.09% to 10.5%) and volatility (from 14.50% to 15.73%)
A huge decrease in bonds mean return (from 7.23% to -0.64%) and a significant increase in their volatility (from 11.93% to 14.66%)

Figure 1 also confirms that the 60/40 Portfolio underwent marked changes⁸ in terms of risk-adjusted returns over these two periods, which explains the many reports about its death since 2022.

That being said, all portfolios actually underwent similar changes, so that there is nothing specific to the magic number duo 60/40⁹.

The follow-up question - will this regime change last? - is harder to answer since it’s difficult to make predictions, especially about the future, but if history is any guide, it could last several years.

That prediction is supported by Figure 2, which shows that the positive correlation between stocks and bonds experienced since 2022 is not abnormal at all.

As Asness et al.¹⁰ puts it:

Even though this correlation was usually negative from 2000-2020, it was more often than not positive from 1900-2000. So a positive stock-bond correlation is hardly unprecedented; in fact, it’s been the historical norm.

Figure 2. Rolling correlation between U.S stocks and US Treasury returns, 01 January 1900 – 30 September 2022. Source: Brixton et al.

As a consequence, while reports of [the 60/40 Portfolio] death are greatly exaggerated, that portfolio could use some fine-tuning to make it more compatible with the current regime.

The Adaptive 60/40 Portfolio

Rationale

The main observation from the previous section is that the traditional bond sleeve [of the 60/40 Portfolio] is showing signs of strain² in the current market regime characterized by positively correlated stock and bond returns, hightened bond returns volatility and lowered bond returns.

In this new regime, the 40% intended to act as ballast during periods of market stress² needs to be refined in two directions:

In terms of composition, incorporating alternative asset classes across both public and private markets is essential³

Indeed, since bonds are less reliable than in the (recent) past to diversify stock market exposure, other alternative investments need to be incorporated into the 40% defensive sleeve.

Beware of false diversifiers, though, for which returns can be explained by exposure to stock market risk¹⁰.

As Asness et al.¹⁰ puts it:

Too often [diversifiers] include stuff that has even more equity market risk than bonds – we’ve seen (among others) privates, buffer funds, and yes, even crypto.

In examples like these, investors trying to improve portfolio diversification end up making it worse.
In terms of dynamicity, it needs to be made adaptive

Here, Carlson¹¹ highlights that most portfolios […] rely on static diversification¹¹.

However, during periods of market panic, asset correlations tend to rise and fixed allocations can become vulnerable¹¹.

For example, bonds, often seen as a hedge, have sometimes failed to protect capital during dislocations driven by inflation or policy shocks¹¹.

Thus, the 40% defensive sleeve must be made adaptative, which serves multiple roles¹¹:
- Macro Diversifier: Responds to shifting economic regimes
- Crisis Mitigator: Dynamically limits drawdowns
- Risk-Off Core: Generates positive expected return with lower volatility
- Anti-Correlation Sleeve: Low correlation to both equities and bonds, especially during systemic stress

Those two refinements are explained in very simple terms by Alexandre Ventelon, Morgan Stanley Wealth Management Australia Head of Investment Strategy and Solutions¹:

So our recommendation has been with our clients, you need to diversify your diversifiers. You need to have a broader range. They will not all work at the same time.

In this context, Carlon’s² Adaptive 60/40 Portfolio² proposes to replace the static 40% bonds allocation of the 60/40 Portfolio by a dynamic 40% allocation into whichever defensive asset displays the strongest multi-period momentum among [three] liquid macro hedges²:

Bonds, which are still useful as part of an overall portfolio, even with recently higher correlations¹⁰, because there’s little else out there that performs as consistently-well during recessions and related environments of falling growth¹⁰
Gold, that responds positively to monetary instability and declining real rates¹¹
Broad commodities, that perform during stagflation or commodity supply shocks¹¹

This [dynamic] multi-asset structure ensures that [the 40% defensive sleeve] adapts to different economic shocks rather than relying on a single hedge¹¹.

Construction rules

In exact details, the Adaptive 60/40 Portfolio construction rules are the following.

At the end of each month:

Invest 60% of the portfolio into U.S. stocks, represented by the SPY ETF
Invest 40% of the portfolio into the best ranked defensive asset based on the equally weighted average of absolute returns over 1, 3, 6, and 12-month lookbacks among:
- Long-term U.S. Treasuries, represented by the TLT ETF
- Gold, represented by the GLD ETF
- Commodities, represented by the DBC ETF

From these construction rules, a couple of comments:

The Adaptive 60/40 Portfolio has a static 60% allocation to stocks

There is no adaptivity here.
The defensive sleeve of the Adaptive 60/40 Portfolio relies on a relative strength momentum framework²

This makes it adaptive to current market conditions, whatever they are, but such a framework does not prevent from investing into a diversifier with paltry returns¹⁰.

For this, an absolute momentum filter would be needed, which is not present in the vanilla version of the Adaptive 60/40 Portfolio.

As a side note, such a filter exists in another portfolio allocation framework proposed by Carlson¹¹ called Defense First¹¹.

The interested reader is refered to Moskowitz et al.¹² and the subsequent litterature for more information about absolute and relative momentum.

Performances

Figure 3 compares the evolution of the Adaptive 60/40 Portfolio over the period¹³ February 1989 - August 2026 to that of:

The 60/40 Portfolio
The Non-adaptive 60/40 Portfolio, which is the static portfolio invested 60% in stocks, 13.3% in long-term U.S. Treasuries, 13.3% in gold and 13.3% in commodities¹⁴
The Adaptive 60/40 Portfolio

Figure 3. Adaptive 60/40 Portfolio v.s. other portfolios, February 1989 - August 2026.

Figures below.

Portfolio	CAGR	Ann. Standard Deviation	Ann. Sharpe Ratio (0)	Maximum Drawdown (monthly, end of month)
60/40 Portfolio	9.1%	10.0%	0.92	28.4%
Non-adaptive 60/40 Portfolio	9.3%	10.0%	0.94	34.6%
Adaptive 60/40 Portfolio	12.3%	10.7%	1.14	31.2%

Those performances seem pretty compelling.

Neverthtless, as long-time readers of this blog might know, comparing a tactical(ish) asset allocation strategy to a static portfolio is difficult to justify…

For a fairer comparison, the Adaptive 60/40 Portfolio should rather be evaluated against its own true opportunity set¹⁵, which is the set of portfolios statisfying the following construction rules:

Monthly rebalance at the end of each month
Investment of 60% of the portfolio into the SPY ETF
Investment of 40% of the portfolio into any combination of the TLT, GLD and DBC ETFs

The associated methodology - described elsewhere in this blog - leads to Figure 4, which contrasts the Sharpe Ratio of the Adaptive 60/40 Portfolio to that of 500,000 random portfolios representative of the true opportunity set mentioned above.

Figure 4. Adaptive 60/40 Portfolio v.s. 500,000 random portfolios, Sharpe Ratio, February 1989 - August 2026.

From Figure 4, the Sharpe Ratio of the Adaptive 60/40 Portfolio is much higher than nearly all the Sharpe Ratios achiveable within the considered true opportunity set, which is usually a sign of significant overfitting for a trading stragegy…

On the other hand, there is no apparent sign of overfitting in the construction rules of the Adaptive 60/40 Portfolio:

The 60% allocation into U.S. stocks is inherited from the 60/40 Portfolio
The 40% allocation into a defensive asset is a mathematical consequence¹⁶ of the 60% allocation into U.S. stocks
The chosen defensive assets - long-term U.S. Treasuries, gold and commodities - all are standard stock diversifiers
The tactical asset allocation strategy used in the defensive sleeve is backed by an abundant litterature on relative momentum across asset classes

So, ultimately, time will tell!

As a last remark, Carlson² highlights that the correlation between the defensive sleeve of the Adaptive 60/40 Portfolio and U.S. stocks is nearly 0%¹⁷, which also makes it an interesting standalone stock diversifier.

Implementation

It is possible to follow live signals for Carlson’s Adaptive 60/40 Portfolio strategy on BestFolio (no affiliation).

Variation on the Adaptive 60/40 Portfolio

The tactical asset allocation strategy used in the defensive sleeve of the Adaptive 60/40 Portfolio is a monthly relative momentum strategy based on several momentum formation periods.

In details, at each monthly portfolio rebalancing date, multiple rankings are computed

Asset	1-month momentum	3-month momentum	6-month momentum	12-month momentum
TLT ETF	$\text{TLT}_{1}$	$\text{TLT}_{3}$	$\text{TLT}_{6}$	$\text{TLT}_{12}$
GLD ETF	$\text{GLD}_{1}$	$\text{GLD}_{3}$	$\text{GLD}_{6}$	$\text{GLD}_{12}$
DBC ETF	$\text{DBC}_{1}$	$\text{DBC}_{3}$	$\text{DBC}_{6}$	$\text{DBC}_{12}$

and are then converted into an aggregated ranking

Asset	Aggregated momentum
TLT ETF	$\text{TLT}_{agg}$
GLD ETF	$\text{GLD}_{agg}$
DBC ETF	$\text{DBC}_{agg}$

in order to select the best overall defensive asset to hold for the next month.

Similar to Wouter et al.¹⁸, Carlson² proposes to average the momentum observed over 1, 3, 6, and 12 months in order to span the optimal space and stabilize […] estimates¹⁸ so that formulas for the asset aggregated momentum signals are:

$\text{TLT}_{agg} = \frac{1}{4} \left( \text{TLT}_{1} + \text{TLT}_{3} + \text{TLT}_{6} + \text{TLT}_{12} \right)$
$\text{GLD}_{agg} = \frac{1}{4} \left( \text{GLD}_{1} + \text{GLD}_{3} + \text{GLD}_{6} + \text{GLD}_{12} \right)$
$\text{DBC}_{agg} = \frac{1}{4} \left( \text{DBC}_{1} + \text{DBC}_{3} + \text{DBC}_{6} + \text{DBC}_{12} \right)$

I am personally not a big fan of such an aggregation mechanism for three reasons:

Equally averaging momentum signals (here, arithmetic returns) over different formation periods corresponds to an implicit weighting scheme

Typically, asset returns over a 12-month formation period are much higher in amplitude than asset returns over a 1-month formation period.

So, by equally averaging asset returns over different formation periods, much more weight is actually given to the longuest formation period v.s. the other formation periods.

This might be a desirable behavior, but if so, I would prefer it to be explicit.
Averaging momentum signals over different formation periods leads to uncontrolled cancellation of positive and negative asset returns

On top of the previous point, averaging asset returns over different formation periods makes positive and negative asset returns cancelling each other in a rather unpredictable and uncontrollable way.

This might also be a desirable behavior, but in that case, there must be ways to better control that behaviour.
Momentum signals have no built-in adaptability

The different lookback periods chosen in Carlson² are quite common¹⁸ in the momentum litterature, but if - let’s say - momentum over a 6-month formation period did not work in the specific context of the Adaptive 60/40 Portfolio, it would have been excluded from its construction rules with a perfectly reasonable ex-post explanation.

That’s for the past, i.e., the backtest.

In addition, if at one point in time, momentum over - let’ say again - a 6-month formation period stops being predictive of relative asset returns over the next month, the Adaptive 60/40 Portfolio would not adapt¹⁹ to that evolution.

That’s for the future, i.e., live trading.

As an alternative aggregation mechanism, I would propose the following:

For each momentum lookback (1, 3, 6 and 12 months), transform the asset momentum signals into a cross-sectional percentile ranking²⁰
Aggregate the resulting 4 cross-sectional percentile rankings into an overall percentile ranking through a statistical significance-based weighted average

Here, some comments are in order.

If - let’s say - the 6-month momentum signal has an history of being extremely predictive of next month’s relative asset returns v.s. the other lookbacks, the overall percentile ranking should reflect that in a way or another.

One possibility to achieve this objective is to compute the (normalized) average pairwise Probabilistic Sharpe Ratio of each $x$-month momentum signal - considered as a standalone trading strategy - against all the other $y$-month momentum signals with $x \ne y$.

Figure 5 compares the evolution of the Adaptive 60/40 Portfolio over the period March 1992 - August 2026 to that of the resulting portfolio²¹, called the PSR Adaptive 60/40 Portfolio.

Figure 5. Adaptive 60/40 Portfolio v.s. PSR Adaptive 60/40 Portfolio, March 1992 - August 2026.

Figures below.

Portfolio	CAGR	Ann. Standard Deviation	Ann. Sharpe Ratio (0)	Maximum Drawdown (monthly, end of month)
Adaptive 60/40 Portfolio	12.4%	10.7%	1.15	31.2%
PSR Adaptive 60/40 Portfolio	11.9%	10.8%	1.09	36.0%

While there is a slight performance degradation in-sample²², I would be more confident in trading the PSR Adaptive 60/40 Portfolio out-of-sample.

In any cases, Figure 6 confirms that the weights assigned to the different momentum lookbacks vary a lot, so that an evolutive weighting scheme is definitely something missing in the vanilla Adaptive 60/40 Portfolio.

Figure 6. Evolution of the 3-month momentum signal weight, March 1992 - August 2026.

As a side note, with this alternative aggregation mechanism, it is relatively harmeless to add other momentum formation periods - even apparently silly ones like 24 months - as well as to extend the universe of defensive assets.

One remaining caveat, though, is that there is still no absolute momentum filter in the defensive sleeve.

Perfect tinkering opportunity for the geeks out there :-) !

Conclusion

I leave the conclusion of this blog post to Carlson²:

Adaptive 60/40 is a simple framework for updating the defensive sleeve of balanced portfolios.

It’s transparent, easy to implement, and designed for allocators who need reliable tools — not complexity — to navigate evolving market regimes.

It’s not perfect, but it’s robust, repeatable, and built for the environments where static diversification breaks down.

To discover more tactical asset allocation strategies, feel free to connect with me on LinkedIn or follow me on Twitter.

–

See Morgan Stanley’s asset allocation playbook for the next 5 years, Basis Points by Equity Mates Media, 17 June, 2026 ↩ ↩²
See Tom Carlson, Adaptive 60/40: Rethinking the Defensive Sleeve, January 20, 2026. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³
See Remi Olu-Pitan, Building portfolio resilience: why today’s approach needs to be different, Schroders, July 2025. ↩ ↩²
At the time of publication of this blog post. ↩
See Bailey, David H. and Lopez de Prado, Marcos, The Sharpe Ratio Efficient Frontier (April 1, 2012). Journal of Risk, Vol. 15, No. 2, Winter 2012/13. ↩
Extended with Fama and French data. ↩
Extended with the total returns associated to the U.S. 30-Year Treasury Constant Maturity Rates, c.f. a previous blog post for the associated methodology. ↩
An AI would have called these changes “structural”, but I did not use an AI to write this blog post. ↩
See Andreas Steiner’s LinkedIn post. ↩
See Cliff Asness, Daniel Villalon, Antti Ilmanen, A Positive Stock-Bond Correlation Is a Terrible Reason to Add More Equity Risk to Your Portfolio, April 8, 2026 ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶
See Carlson, Thomas, Defense First: A Multi-Asset Tactical Model for Adaptive Downside Protection (July 01, 2025). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰
See Tobias J. Moskowitz, Yao Hua Ooi, Lasse Heje Pedersen, Time series momentum, Journal of Financial Economics, Volume 104, Issue 2, 2012, Pages 228-250. ↩
Compared to Carlson², this is an extended backtest made possible by the extension of the SPY and the TLT ETFs as already described in this blog post, and by the extension of the GLD and DBC ETFs thanks to the associated index data. ↩
In other words, the Non-adaptive 60/40 Portfolio corresponds to the non-tactical flavor of the Adaptive 60/40 Portfolio with 60% invested in stocks and 40% equally invested in all the defensive assets. ↩
See Roberto Stein, Not fooled by randomness: Using random portfolios to analyse investment funds, Investment Analysts Journal, 43:79, 1-15. ↩
Having 100% of a portfolio invested with 60% of that portfolio invested into an asset mathematically requires the remaining 40% of that portfolio to be invested into another asset. ↩
Exacty around 0.25% with the dataset used in this blog post. ↩
See Keller, Wouter J. and Butler, Adam and Kipnis, Ilya, Momentum and Markowitz: A Golden Combination (May 16, 2015). ↩ ↩² ↩³
Pun intended. ↩
See Cakici, Nusret and Zaremba, Adam, Getting the Target Right in Return Prediction (April 20, 2026). ↩
I used a lookback of 24 months for computing the Probabilistic Sharpe Ratio; while this choice seems to be robust (i.e., using 18 months or 36 months does not materially affect the results), I would prefer to use 1 year of daily data in a real implementation. ↩
Which was predictable from Figure 4. ↩

The Sharpe Stability Ratio: Evaluating the Sharpe Ratio Temporal Consistency

2026-08-24T00:00:00-05:00

The Sharpe Ratio¹, one of the most commonly used measure of risk-adjusted performance², is usually reported as a point estimate (Morningstar, Quantalys, etc.).

Thanks to the work of Lo³, Opdyke⁴ and more recently⁵ de Prado et al.⁶, it is nevertheless well understood that such a point estimate […] does not convey information about statistical significance⁶, so that a more meaningful way to measure and compare investment efficiency⁶ is rather to report the Probabilistic Sharpe Ratio (PSR)⁷, which expresses observed Sharpe Ratios in probability space, adjusting for skewness, kurtosis, serial correlation, and sample length⁶.

However, these two measures cannot distinguish persistent skill from episodic outperformance⁸ because they both evaluate performance at a single temporal aggregation without quantifying how risk-adjusted returns evolve within the sample period⁸.

In order to overcome this limitation, Bajo Traver and Rodriguez Dominguez⁸ introduces the Sharpe Stability Ratio⁸, a measure of the temporal consistency of the Sharpe Ratio across overlapping subperiods.

In this blog post, I will first describe the Sharpe Stability Ratio in details and then, as examples of usage, I will:

Show that it provides additional information compared to the Sharpe Ratio
Show that it also provides additional information compared to another measure of risk-adjusted performances called the Ulcer Performance Index⁹
Experiment with it as a portfolio optimization objective and construct its associated efficient frontier

Reminders

The Sharpe Ratio

Let be:

$r_1,…,r_T$ the observed arithmetic returns of an investment (asset/fund/portfolio/trading strategy/etc.) over $T$ periods of time
$r_{f,1},…,r_{f,T}$, the observed risk-free arithmetic returns¹⁰ over the same $T$ periods of time
$ r_1 - r_{f,1},…,r_T - r_{f,T}$, the associated investment observed excess arithmetic returns over the risk-free rate

Then, the observed Sharpe Ratio¹ $\widehat{SR}$ of the investment over the period $1..T$ is defined as:

\[\widehat{SR} = \frac{\widehat{\mu}_{r - r_f}}{\widehat{\sigma}_{r - r_f}}\]

, where:

$\widehat{\mu}_{r - r_f}$ is the sample mean of the excess arithmetic returns
$\widehat{\sigma}_{r - r_f}^2$ is the sample¹¹ variance of the excess arithmetic returns

The Sharpe Ratio provides a scale-free summary of expected excess return per unit of risk⁸ and can be interpreted as a measure of […] signal over noise, or skill over luck⁶.

Prado et al.⁶ notes that the Sharpe ratio satisfies several important properties⁶, among which:

It is rooted in Markowitz’s mean-variance analysis model.

For example, as illustrated in Figure 1, the tangency portfolio is the mean-variance efficient portfolio with maximum Sharpe Ratio and all efficient portfolios are linear combinations of this portfolio and the risk-free asset¹².

Figure 1. The Sharpe Ratio in Mean-Variance Space. Source: de Prado.

As another example, ranking portfolios by their Sharpe Ratios yields the same ordering as expected-utility maximization⁶ under three (sufficient) conditions:
- When portfolio returns follow a normal, or more generally an elliptical distribution
- When portfolio returns satisfy the LS property¹³, that is, when portfolio returns differ from one another only by location and scale parameters¹³
- When investor preferences admit a mean–variance representation, either exactly or as a second-order approximation⁶
It is actually well defined and usable for any return process with finite mean and variance, regardless of higher-order moments or temporal dependence⁶.

This property is of practical importance because a common misunderstanding regarding [the Sharpe Ratio] is that […] [its users] must implicitly assume that returns are i.i.d. Normal⁶.

As explained in de Prado et al.⁶, this is incorrect:

When returns are non-normal or serially correlated, variance may no longer fully characterize downside or tail risk, but it continues to measure dispersion, and the Sharpe Ratio retains its economic interpretation as expected excess return per unit of standard deviation.

Moreover, while several alternative risk-adjusted performance measures have been proposed over the years - whether partial moments-based (Sortino Ratio, Omega Ratio…), drawdown-based (Calmar Ratio…) or Value-at-Risk-based (Rachev Ratio…) - empirical studies have regularly shown that portfolio rankings based on most¹⁴ of these measures are nearly identical to those determined by the Sharpe ratio¹³.

This behavior is partially explained in Schuhmacher and Elingby¹³, which demonstrates that if portfolio returns satisfy the LS property¹⁵, then any “reasonable” risk-adjusted performance measure must be a strictly increasing function in the Sharpe ratio¹³.

The Probabilistic Sharpe Ratio

The usage of hats in the previous subsection for the definition of the Sharpe Ratio emphasises that $\widehat{\mu}_{r - r_f}$ , $\widehat{\sigma}_{r - r_f}^2$ and $\widehat{SR}$ are quantities computed from the observed excess returns $r_1 - r_{f,1}$,…,$r_T - r_{f,T}$ while the true mean excess return $\mu_{r - r_f}$, the true variance $\sigma_{r - r_f}^2$ and the true Sharpe ratio $ SR = \frac{\mu_{r - r_f}}{\sigma_{r - r_f}} $ are quantities associated with the unobservable excess return generating process.

It follows that the observed Sharpe Ratio $\widehat{SR}$ is a statistical estimator of its true counterpart $SR$, subject to an estimation error quantified by Lo³, Opdyke⁴ and de Prado et al.⁶ under various assumptions on the excess return generating process¹⁶.

In order for practitioners to evaluate the credibility of [an] observed Sharpe Ratio⁸ without explicitely referring to the underlying statistical machinery¹⁷, Bailey and de Prado⁷ introduces the Probabilistic Sharpe Ratio $\widehat{PSR}(SR_0)$, a risk-adjusted performance measure defined as the probability that the Sharpe Ratio estimator $\widehat{SR}$ is greater than a benchmark Sharpe Ratio $SR_0$¹⁸ over the period $1..T$:

\[\widehat{PSR}(SR_0) = \Phi\left( \frac{\widehat{SR} - SR_0}{ \sqrt{\frac{1 - \widehat{\kappa}_{r - r_f} \widehat{SR} + (\widehat{\gamma}_{r - r_f} - 1) \frac{\widehat{SR}^2}{4}}{T}} } \right )\]

, where:

$\kappa_{r - r_f}$ and $\gamma_{r - r_f}$ are the true skewness and the true kurtosis of the unobservable excess return generating process and $\widehat{\kappa}_{r - r_f}$ and $\widehat{\gamma}_{r - r_f}$ their empirical counterparts
$\Phi$ is the cumulative distribution function of the standard normal distribution

This makes the Probabilistic Sharpe Ratio one of the very few¹⁹ alternative risk-adjusted performance measures that explicitly penalizes uncertainty due to short samples, downside, tail and drawdown-related risks⁶ while still being grounded in modern portfolio theory⁶.

As a side note, de Prado et al.⁶ further improves the Probabilistic Sharpe Ratio by taking into account the return generating process’s autocorrelation, which enables Sharpe Ratio inference under substantially more general assumptions than prior closed-form treatments⁶, including the original treatment in Bailey and de Prado⁷.

The Sharpe Stability Ratio

Definition

Bajo Traver and Rodriguez Dominguez⁸ notes that neither the Sharpe Ratio nor the Probabilistic Sharpe Ratio enables statistical inference on temporal stability of risk-adjusted performance⁸, because they are both risk-adjusted performance measures computed over the full period $1..T$.

Naturally, practitioners routinely compute rolling Sharpe Ratios to visualize [risk-adjusted performance] dynamics⁸, but no formal framework exists for statistical inference on temporal stability when these rolling estimates are constructed from overlapping windows that induce autocorrelation⁸.

To resolve this shortcoming, Bajo Traver and Rodriguez Dominguez⁸ introduces the Sharpe Stability Ratio $\widehat{SSR}(SR_0)$, defined as the mean rolling Sharpe Ratio minus a benchmak Sharpe Ratio $SR_0$¹⁸ divided by its long-run (or asymptotic²⁰) standard deviation:

\[\widehat{SSR}(SR_0) = \frac{\widehat{\mu}_{\hat{z}} - SR_0}{\widehat{\sigma}_{\infty, \hat{z}}}\]

, where:

$\hat{z}_1 = \widehat{SR}_{1:w}$, $\hat{z}_2 = \widehat{SR}_{2:w+1}$, …, $\hat{z}_{T-w+1} = \widehat{SR}_{T-w+1:T}$ are the Sharpe Ratios over the time periods $1..w$, $2..w+1$, …, $T-w+1…T$

Here, because the time periods $1..w$, $2..w+1$, …, $T-w+1…T$ are overlapping, the associated Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$ are usually called rolling Sharpe Ratios.
$\widehat{\mu}_{\hat{z}} = \frac{1}{T-w+1} \sum_{k=1}^{T-w+1} \hat{z}_k$ is the sample mean of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$
$\widehat{\sigma}_{\infty, \hat{z}}^2$ is an estimator of the long-run variance of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$

Contrary to the standard variance $\sigma^2$, the long-run variance $\sigma^2_{\infty}$ is a measure of uncertainty in the mean that accounts for serial correlation.

In the specific case of the Sharpe Stability Ratio, using the long-run variance instead of the variance is made necessary by⁸:
- The usage of overlapping windows in the computation of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$
  
  Indeed, successive windows share $w-1$ observations, implying first-order autocorrelation close to one when $w$ is large relative to the sampling frequency⁸, which would lead the naive variance estimator [to understate] the long-run variance by a factor of order $w$⁸.
- The (potential) serial dependence in the excess returns $ r_1 - r_{f,1},…,r_T - r_{f,T}$
As a practical estimator for $\sigma^2_{\infty}$, Bajo Traver and Rodriguez Dominguez⁸ chooses the Newey–West heteroskedasticity and autocorrelation consistent (HAC) estimator²⁰ with the triangular kernel, also known as the Bartlett kernel²¹:
\[\widehat{\sigma}_{\infty, \hat{z}}^2 = \widehat{\gamma}_{0, \hat{z}} + 2 \sum_{k=1}^L \left( 1 - \frac{k}{L + 1} \right) \widehat{\gamma}_{k, \hat{z}}\]
, where:
- $\widehat{\gamma}_{0, \hat{z}}$ is equal to the sample variance of the rolling Sharpe Ratios, that is, $\frac{1}{T-w+1} \sum_{k=1}^{T-w+1} \left( \hat{z}_k - \widehat{\mu}_{\hat{z}} \right)^2$
- $\widehat{\gamma}_{k, SRs}, k \geq 1$ is equal to the sample autocovariance of the rolling Sharpe Ratios at lag $k$, that is, $ \frac{1}{T-w+1} \sum_{t=k+1}^{T-w+1} \left( \hat{z}_t - \widehat{\mu}_{\hat{z}} \right) \left( \hat{z}_{t-k} - \widehat{\mu}_{\hat{z}} \right) $
- $L \geq 0$ is a lag truncation parameter, estimated by the nonparametric data-dependent procedure of Newey and West²² based on both theoretical asymptotic and empirical Monte Carlo results²²

Interpretation

Similar to the Sharpe Ratio that is a signal-to-noise ratio for the excess returns $ r_1 - r_{f,1},…,r_T - r_{f,T}$, the Sharpe Stability Ratio is by definition a signal-to-noise ratio for the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$.

In terms of values:

A strongly positive Sharpe Stability Ratio is desirable because it indicates stable superior performances v.s. the benchmark
Conversely, a strongly negative Sharpe Stability Ratio is usually not desirable because it indicates stable inferior performances v.s. the benchmark
Last, a zero-ish Sharpe Stability Ratio is usually not desirable either because it typically indicates instability of performances²³ - whether positive of negative - v.s. the benchmark

A handful of reference values for 17 Barclay Hedge Fund Indices, representing real-world strategies with pronounced non-normality, serial dependence and regime-switching behavior⁸, are provided in Bajo Traver and Rodriguez Dominguez⁸.

For convenience, they are reproduced in Figure 2 (full sample analysis) and Figure 3 (subperiods analysis).

Figure 2. Misc. statistics inc. SSR(0) over the full period, 17 Barclay Hedge Hedge Fund strategy indices, January 1997 - December 2025. Source: Bajo Traver and Rodriguez Dominguez.

Figure 3. Analysis of SSR(0) over subperiods, 17 Barclay Hedge Hedge Fund strategy indices, January 1997 - December 2025. Source: Bajo Traver and Rodriguez Dominguez.

Two comments from these figures:

It seems that reaching a Sharpe Stability Ratio of 1 is challenging but achievable²⁴, so that this value could be considered as a (high) reference value to have in mind when interpreting Sharpe Stability Ratios.

By extension, an interesting question is how high a Sharpe Stability Ratio could be or could not be?

To answer that question, I propose to use Bernard Madoff’s Fairfield Sentry fund.

Because it is now established that this fund was a fraud, studying it generally helps to find implausible risk-adjusted performance metrics.

Figure 4 compares the performance of the S&P500 to that of the Fairfield Sentry fund²⁵ over a little less than 20 years.

Figure 4. S&P500 v.s. Fairfield Sentry, November 1990 - October 2008.

Using 36 months²⁶ as a rolling window length for the Sharpe Stability Ratio computation, figures are provided below.

Instrument	Monthly Sharpe Ratio (0)	Probabilistic Sharpe Ratio (0)	Sharpe Stability Ratio (0)
S&P500 (Total Return)	0.19	0.99	0.37
Fairfield Sentry fund	1.19	1	1.55

From this table, an investment exhibiting a Sharpe Stability Ratio of about 1.5 over a 20-year period should then be considered as highly suspicious.

Now, it is possible to go a little further and compute the Sharpe Stability Ratio of the Fairfield Sentry fund on an expanding window basis²⁷, which will provide reference values for an implausible Sharpe Stability Ratio over all standard evaluation periods for an investment.

Results are provided in Figure 5, on which the apparently huge Sharpe Stability Ratio of 4.31 reported for the Global Macro Hedge Fund Strategy Indice (Figure 3) is not so huge anymore when compared to the Sharpe Stability Ratio of 12 computed for the airfield Sentry fund over an equivalent period of 5 years!

Figure 5. Fairfield Sentry Sharpe Stability Ratio, expanding window from November 1994 to October 2008.

Temporal stability, as measured by [the Sharpe Stability Ratio], is inherently conditional on the prevailing market regime rather than an intrinsic, regime-invariant attribute of each strategy⁸.

As a consequence, the Sharpe Stability Ratio also captures how consistently a strategy performs within a given macro-financial environment and how its stability profile shifts across environments⁸.

Asymptotic distribution and statistical inference

Bajo Traver and Rodriguez Dominguez⁸ establishes that under stationarity and $\alpha$-mixing assumptions, the Sharpe Stability Ratio satisfies

\[\sqrt{T-w+1} \widehat{SSR}(SR_0) \overset{a}{\sim} N \left( 0, 1 \right)\]

That asymptotic normality provides a theoretical basis for making inference about the Sharpe Stability Ratio, although Bajo Traver and Rodriguez Dominguez⁸ emphasises that in practice, due to the strong autocorrelation induced by overlapping windows⁸, bootstrap inference should be prefered to asymptotic approximations.

A couple more words on asymptotic results, though.

If we introduce the “normalized” Sharpe Stability Ratio $\widehat{nSSR}(SR_0)$ defined by

\[\widehat{nSSR}(SR_0) = \sqrt{T-w+1} \widehat{SSR}(SR_0)\]

, it then becomes possible to interpret the associated (normalized) Sharpe Stability Ratio values in statistical terms, which might be useful to become familiar with this new risk-adjusted performance measure.

For example:

A normalized Sharpe Stability Ratio value of about 1.96 would imply that $SR_0$ lies inside the 95% confidence interval around the mean rolling Sharpe Ratio
A normalized Sharpe Stability Ratio value of about 2.576 would imply that $SR_0$ lies inside the 99% confidence interval around the mean rolling Sharpe Ratio

As other examples:

A strongly positive normalized Sharpe Stability Ratio value would imply that the mean rolling Sharpe Ratio is substantially greater than $SR_0$
A strongly negative normalized Sharpe Stability Ratio value would imply that the mean rolling Sharpe Ratio is substantially lower than $SR_0$
A normalized Sharpe Stability Ratio value close to zero would imply that the mean rolling Sharpe Ratio is close to $SR_0$

Comparison with other Sharpe Ratio-based measures

With the introduction of the Sharpe Stability Ratio, there are now at least²⁸ 3 Sharpe Ratio-based risk-adjusted performance measures to choose from when analyzing an investment:

The Sharpe Ratio
The Probabilistic Sharpe Ratio
The Sharpe Stability Ratio

Figure 6 summarizes the main question answered by each of these measures.

Figure 6. Different Sharpe Ratio Dimensions: SR, PSR, SSR. Source: Bajo Traver and Rodriguez Dominguez.

As can be seen in Figure 6, each of these measures capture a different dimension of performance evaluation⁸.

Among them, the Sharpe Stability Ratio focuses on the persistence of risk-adjusted performances over time, which allows it to distinguish between investments with a similar Sharpe Ratio and Probabilistic Sharpe Ratio.

As an example, Figure 7 represents 3 different synthetic strategies with the same²⁹ Sharpe Ratios and Probabilistic Sharpe Ratios but different Sharpe Stability Ratios.

Figure 7. Three synthetic strategies with identical Sharpe Ratios and Probabilistic Sharpe Ratios (0, 0.5 and 1.0). Source: Bajo Traver and Rodriguez Dominguez.

From a visual inspection of Figure 7, Strategy A would be the preferred strategy in terms of temporal stability, which is confirmed by its higher Sharpe Stability Ratio³⁰.

As another example³¹, Figure 8 represents 2 different synthetic strategies with again the same Sharpe Ratios but this time very different Sharpe Stability Ratios.

Figure 8. Two synthetic strategies (consistent v.s. episodic) with identical Sharpe Ratios.

On Figure 8, the episodic strategy can be considered as a “crisis alpha” strategy while the consistent strategy can be considered as a “structural alpha” strategy.

In this case, relying on the Sharpe Ratio alone would not allow to distinguish between these two strategies and would actually favor the episodic strategy that exhibits a slightly higher Sharpe Ratio.

On the contrary, the Sharpe Stability Ratio properly favors³² the consistent strategy that generates persistent alpha⁸.

Practical considerations

How to choose the rolling window length?

Bajo Traver and Rodriguez Dominguez⁸ notes that the rolling window length $w$ is intrinsically a smoothing decision, analogous to selecting a bandwidth in nonparametric estimation: shorter windows produce noisy, high‑variance Sharpe estimates, whereas longer windows yield smoother but more “averaged‑out” dynamics⁸.

Thus, there is no universally applicable rolling window length.

That being said, standard rolling window sizes typically used in the litterature are the following:

252 days³³, for daily returns
52 weeks³⁴, for weekly returns
12 months³³, 36 months³⁵ or 60 months³³, for monthly returns

How to choose the benchmark Sharpe Ratio $SR_0$?

The benchmark Sharpe Ratio that appears in the Sharpe Stability Ratio formula plays a similar role to that of the benchmark Sharpe Ratio that appears in the Probabilistic Sharpe Ratio formula.

Nevertheless, the Sharpe Stability Ratio is somewhat unstable near the benchmark boundary⁸, which leads Bajo Traver and Rodriguez Dominguez⁸ to recommend using $SR_0 = 0$ as a base benchmark in empirical applications⁸.

Implementation in Portfolio Optimizer

The Portfolio Optimizer Web API allows to compute both the Sharpe Stability Ratio and the normalized Sharpe Stability Ratio as described in this blog post, c.f. the documentation.

Examples of usage

The Sharpe Stability Ratio v.s. the Sharpe Ratio

In the previous section, it has been empirically shown that the Sharpe Stability Ratio is complementary to the Sharpe Ratio, c.f. for example Figure 7 and Figure 8.

To illustrate the extent of that difference on real-world strategies with pronounced non-normality, serial dependence and regime-switching behavior⁸, Figure 9 plots the Sharpe Ratio v.s. the Sharpe Stability Ratio of the same 17 Barclay Hedge Fund Indices as used in Bajo Traver and Rodriguez Dominguez⁸ over the period January 1997 - December 2025.

As a side note, Figure 9 is the same as Figure 8 in Bajo Traver and Rodriguez Dominguez⁸, but independently reproduced using the Portfolio Optimizer Web API³⁶.

Figure 9. Sharpe Ratio v.s. Sharpe Stability Ratio, 17 Barclay Hedge Fund Indices, January 1997 - December 2025.

Figure 9 empirically shows again that there is not a one-to-one relationship between the Sharpe Ratio and the Sharpe Stability Ratio.

Thus, there is potentially an informational advantage in using the latter instead of the former!

The Sharpe Stability Ratio v.s. the Ulcer Performance Index

Martin⁹’s Ulcer Performance Index (UPI) - discussed elsewhere on this blog - is a prominent alternative³⁷ to the Sharpe Ratio, even though it does not fit within the mean-variance framework.

The Ulcer Performance Index replaces the standard deviation by the Ulcer Index (UI) as the risk measure, which makes it more directly comparable to the Sharpe Stability Ratio than the Sharpe Ratio because the Ulcer Index takes into account the temporal aspect of an investment returns.

In order to compare those two measures:

Figure 10 plots the Sharpe Ratio v.s. the Ulcer Performance Index of the same 17 Barclay Hedge Fund Indices and over the same period as in the previous subsection.

Figure 10. Sharpe Ratio v.s. Ulcer Performance Index, 17 Barclay Hedge Fund Indices, January 1997 - December 2025.

Figure 10 appears similar to Figure 9, which confirms that the Ulcer Performance Index also captures a different dimension of performance evaluation⁸ than the Sharpe Ratio.
Figure 11 plots the Ulcer Performance Index v.s. the Sharpe Stability Ratio of the same 17 Barclay Hedge Fund Indices and over the same period as in the previous subsection.

Figure 11. Ulcer Performance Index v.s. Sharpe Stability Ratio, 17 Barclay Hedge Fund Indices, January 1997 - December 2025.

The points displayed on Figure 11 are much closer to a straight line than in Figure 9 or in Figure 10, which highlights that the Ulcer Performance Index and the Sharpe Stability Ratio are close cousins.

Not extremely close ones, though, because the correlation between these points is “only” of about 80%.

In other words, people relying on the Ulcer Performance Index for risk-adjusted performance measurement might still find the Sharpe Stability Ratio to be an interesting addition to their arsenal!

The Sharpe Stability Ratio as a portfolio optimization objective

Following the same logic and for mostly the same reasons as in Bailey and de Prado⁷, it seems natural to extend [the Sharpe Stability Ratio] to a portfolio optimization or capital allocation context: rather than a mean-variance frontier of portfolio returns on capital, we will build a mean-variance frontier of portfolio [mean rolling] returns on risk⁷.

Figure 12 illustrates the idea of a Mean Rolling Sharpe Ratio Efficient Frontier, defined as the set of portfolios that deliver the highest expected mean rolling excess return on risk (as expressed by their mean rolling Sharpe Ratios) subject to the level of uncertainty surrounding those portfolios’ mean rolling excess return on risk (the long-run standard deviation of the mean rolling Sharpe Ratios).

Figure 12. Mean Rolling Sharpe Ratio Efficient Frontier, 10 multi-asset class ETFs, December 2006 - August 2023.

On Figure 12:

Each blue point represents a portfolio within a universe of 10 ETFs representative of misc. asset classes³⁸, with each its mean rolling Sharpe Ratio and its long-run standard deviation of that mean rolling Sharpe Ratio
The black line represents the Mean Rolling Sharpe Ratio Efficient Frontier, which is the set of portfolios whose mean rolling Sharpe Ratio is maximal for a given level of the long-run standard deviation of that mean rolling Sharpe Ratio
The black cross represents the maximum Sharpe Stability Ratio portfolio, which plays a similar role on the Mean Rolling Sharpe Ratio Efficient Frontier than the maximum Sharpe Ratio portfolio on the usual mean-variance efficient frontier
The black square represents the usual maximum Sharpe Ratio portfolio, for reference

It is interesting to note that the maximum Sharpe Stability Ratio portfolio and the maximum Sharpe Ratio portfolio seem pretty far appart in terms of their long-run standard deviation of their mean rolling Sharpe Ratio.

Do their holdings differ that much though?

Figure 13 answers that question.

Figure 13. Maximum Sharpe Stability Ratio v.s. maximum Sharpe Ratio portfolio holdings, 10 multi-asset class ETFs, December 2006 - August 2023.

From Figure 13, these two portfolios share the same holdings³⁹ - U.S. stocks (SPY ETF), intermediate U.S. Treasuries (IEF ETF) and Gold (GLD ETF) - but with very different loadings, which is explained by their different opjectives:

The maximum Sharpe Stability Ratio portfolio seeks to maximize its mean rolling Sharpe Ratio stability by diversifying its massive U.S. stocks exposure (60%) with intermediate U.S. Treasuries (10%) and Gold (30%)
The maximum Sharpe Ratio portfolio seeks to maximize its point-in-time Sharpe Ratio by diversifying its massive intermediate U.S. Treasuries exposure (60%) with Gold (10%) and U.S. stocks (30%)

These nearly opposite loadings naturally lead to very different performances over time, depicted in Figure 14:

Figure 14. Maximum Sharpe Stability Ratio v.s. maximum Sharpe Ratio portfolio evolution, 10 multi-asset class ETFs, December 2006 - August 2023.

Additional figures below.

Portfolio	CAGR	Ann. Sharpe Ratio (0)	Sharpe Stability Ratio (0)	Maximum Drawdown
Maximum Sharpe Stability Ratio	8.44%	0.78	1	29%
Maximum Sharpe Ratio	5.90%	0.90	0.74	18%

Conclusion

This blog post detailled the Sharpe Stability Ratio, a statistically grounded method to incorporate temporal stability into performance evaluation by quantifying how much of a strategy’s Sharpe Ratio reflects persistent performance versus concentrated episodic strength⁸.

I hope you will find many applications for it in your trading!

While waiting to discover other performance measures, feel free to connect with me on LinkedIn or follow me on Twitter.

Notes:

A big thanks to Mr Bajo Traver and Mr Rodriguez Dominguez who kindly provided me their implementation of the Sharpe Stability Ratio for comparison with my own.

–

See Sharpe, W., 1966, Mutual Fund Performance, Journal of Business, Vol. 39, No. 1, pp. 119–138. ↩ ↩²
C.f. de Prado et al.⁶, Sharpe ratios are used to report actual and backtested performance; identify new investment factors; rank, hire and fire portfolio managers; filter investment strategies (e.g., in due diligence questionnaires); define targets and constraints in portfolio optimization programs, etc³. ↩
See Lo, Andrew W. 2002. The Statistics of Sharpe Ratios. Financial Analysts Journal 58 (4): 36–52. ↩ ↩² ↩³
See Opdyke, J.D., Comparing Sharpe ratios: So where are the p-values?. J Asset Manag 8, 308–336 (2007). ↩ ↩²
At the date of publication of this blog post. ↩
See Marcos Lopez de Prado, Alexander Lipton, Vincent Zoonekynd, Sharpe Ratio Inference: A New Standard for Decision Making and Reporting, The Journal of Portfolio Management April 2026, 52 (6) 6-50. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸
See Bailey, David H. and Lopez de Prado, Marcos, The Sharpe Ratio Efficient Frontier (April 1, 2012). Journal of Risk, Vol. 15, No. 2, Winter 2012/13. ↩ ↩² ↩³ ↩⁴ ↩⁵
See Bajo Traver, Mario and Rodriguez Dominguez, Alejandro, The Sharpe Stability Ratio: Temporal Consistency of Risk-Adjusted Performance. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹ ↩²⁰ ↩²¹ ↩²² ↩²³ ↩²⁴ ↩²⁵ ↩²⁶ ↩²⁷ ↩²⁸ ↩²⁹ ↩³⁰ ↩³¹ ↩³² ↩³³ ↩³⁴
See Martin, P. G. and B. B. McCann (1989). The Investor’s Guide to Fidelity Funds. Wiley. ↩ ↩²
As noted in Bajo Traver and Rodriguez Dominguez⁸, in empirical applications, $r_f$ is often replaced by its sample mean over the window without material effects, given the low variance of risk-free rates⁸. ↩
Depending on the authors, $\sigma_{r - r_f}^2$ is sometimes bias-corrected; in this blog post - and more generally on this website - bias correction is not used. ↩
See Lopez de Prado, Marcos, Sharpe Ratio Inference: A New Standard for Decision-Making and Reporting (Seminar Slides) (December 21, 2025). ↩
See Frank Schuhmacher, Martin Eling, A decision-theoretic foundation for reward-to-risk performance measures, Journal of Banking & Finance, Volume 36, Issue 7, 2012, Pages 2077-2082. ↩ ↩² ↩³ ↩⁴ ↩⁵
Emphasis on “most”, not “all”, which is also applicable to the Ulcer Performance Index, with Martin⁹ for example commenting that there is little difference between the [Ulcer Performance Index] and Traynor Index rankings of the funds⁹; nevertheless, that comment is not applicable when comparing strategies that seek to avoid major market downturns⁹. ↩
And, c.f. Schuhmacher and Elingby¹³, many distributions commonly used in finance satisfy the LS property¹³. ↩
Whatever the specific assumptions, the general idea is that the higher the non-normality/iid-ness of the excess return generating process, the higher the variance of its Sharpe Ratio. ↩
For the interested reader, the statistical inferential framework - much less practionner-friendly than the simple Probabilistic Sharpe Ratio formula - is detailled in a couple of previous blog posts (here and there). ↩
$SR_0$ is usually taken equal to 0 (no skill), 0.5 (annualized) or 1 (annualized). ↩ ↩²
To its author’s knowledge⁶, it is even the only one! ↩
See Newey, W. K., and K. D. West. A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica. Vol. 55, No. 3, 1987, pp. 703-708. ↩ ↩²
While secondary to this blog post, but as an interesting side note, this choice is justified by the fact that the [first-order] Bartlett kernel is [typically] preferred in finite samples to the asymptotically optimal second-order kernel (the Quadratic Spectral (QS) kernel)⁴⁰; furthermore, Kolokotrones et al.⁴⁰ finds that no first-order kernel improves upon the Bartlett kernel⁴⁰. ↩
See Newey, W. K., and K. D. West. Automatic Lag Selection in Covariance Matrix Estimation. The Review of Economic Studies. Vol. 61, No. 4, 1994, pp. 631-653. ↩ ↩²
Rather than equivalent performances v.s. the benchmark. ↩
At least achievable over 10 years, and in all cases clearly more challenging than reaching a Sharpe Ratio greater than 1 over the same period of time. ↩
See Bernard, Carole and Boyle, Phelim P., Mr. Madoff’s Amazing Returns: An Analysis of the Split-Strike Conversion Strategy (May 14, 2009). Journal of Derivatives, Vol. 17, No. 1, 2009. ↩
Consistent with Bajo Traver and Rodriguez Dominguez⁸. ↩
Other variations are of course possible, like computing the Sharpe Stability Ratio of Fairfield Sentry over all subperiods of a given number of months and averaging them, etc. ↩
Other measures like the Deflated Sharpe Ratio (DSR)⁶ exist, but are better suited to other contexts; for example, when evaluating existing portfolios without proprietary backtesting (e.g., mutual funds, hedge fund indices, published ETFs), [the Deflated Sharpe Ratio] is less relevant — the evaluator performs no ex-ante selection over a large trial universe⁶. ↩
For reference, the 3 strategies share a Sharpe Ratio of 1.184 and Probabilistic Sharpe Ratios of 0.998 (0), 0.952 (0.5) and 0.673 (1.0). ↩
For reference, the Sharpe Stability Ratios (0) of the 3 strategies are 0.401 (A), 0.290 (B) and 0.321 (C). ↩
This one is inspired by Macrosynergy. ↩
General investors should also favor the consistent strategy, for many reasons: the single event around which the episodic strategy delivered nearly all its returns might never re-occur, 17 years is an extremely long time to wait for a strategy to deliver returns, etc. ↩
See Andrea Frazzini, Lasse Heje Pedersen, Betting against beta, Journal of Financial Economics, Volume 111, Issue 1, 2014, Pages 1-25. ↩ ↩² ↩³
See Xinyu Huang, Weihao Han, David Newton, Emmanouil Platanakis, Dimitrios Stafylas & Charles Sutcliffe (2023) The diversification benefits of cryptocurrency asset categories and estimation risk: pre and post Covid-19, The European Journal of Finance, 29:7, 800-825. ↩
See for example Burnside, Craig, et al. “Carry Trade: The Gains of Diversification.” Journal of the European Economic Association, vol. 6, no. 2/3, 2008, pp. 581–88. ↩
And using an independent source for Hedge Fund indices data. ↩
See Russ McBride, Alireza Dastan, Ulcer Index 2.6: A Better Risk Measure?, The Journal of Wealth Management Winter 2022, 25 (3) 59-71. ↩
U.S. stocks (SPY ETF), European stocks (EZU ETF), Japanese stocks (EWJ ETF), Emerging markets stocks (EEM ETF), U.S. REITs (VNQ ETF), International REITs (RWX ETF), U.S. 7-10 year Treasuries (IEF ETF), U.S. 20+ year Treasuries (TLT ETF), Commodities (DBC ETF), Gold (GLD ETF) ↩
Incidentaly, these are the ETFs that should compose a permanent portfolio… ↩
See Thomas Kolokotrones, James H. Stock, Christopher D. Walker, Is Newey–West optimal among first-order kernels?, Journal of Econometrics, Volume 240, Issue 2, 2024, 105399. ↩ ↩² ↩³

The NAAIM-AAII Equities Allocation Spread: Another Smart Money Relative Sentiment Indicator

2026-07-27T00:00:00-05:00

In this second post of this series on sentiment indicators, I will show how that survey of professional money managers can be turned into a relative sentiment indicator thanks to the information provided by the AAII Asset Allocation Survey, which details the exposure to U.S. equity and bond markets as reported by individual investors members of the American Association of Individual Investors (AAII) in a monthly survey.

As an example of usage, I will describe how to use this relative sentiment indicator - that I will call the NAAIM-AAII Equities Allocation Spread for the lack of a better name - to scale the market exposure of a portfolio of U.S. equities, following the original work of Kevin Zatloukal in his Medium article Revisiting the Smart Money.

The NAAIM Exposure Index

The NAAIM Exposure Index is already described elsewhere on this blog.

Figure 1 illustrates the evolution of respondents’ average allocation to U.S. stocks since the inception of the NAAIM Exposure Index.

Figure 1. Evolution of respondents' average allocation to stocks, NAAIM Exposure Index, July 2006 - June 2026. Source: NAAIM.

The AAII Asset Allocation Survey

The AAII Asset Allocation Survey is a monthly polling organized by the AAII among its members since 1987, in which participants are asked to communicate their percentage allocations¹ to U.S. stocks, bonds, and cash.

For the historical note², the AAII originally mailed questionnaires at the beginning of each month and tallied the questionnaires received during the month. It later transitioned to online polling in 2000, which allows the results of that survey to be made available at the very beginning of the following month on the AAII website.

As a side note, the AAII also organizes another survey³ - the weekly AAII Investor Sentiment Survey - this time to collect information about the percentage of individual investors who are bullish, bearish, and neutral on the stock market short term⁴.

Figure 2 depicts the evolution of respondents’ allocation to U.S. stocks, bonds and cash since the inception of the AAII Asset Allocation Survey.

Figure 2. Evolution of respondents' asset allocation, AAII Asset Allocation Survey, November 1987 - June 2026. Source: AAII.

The NAAIM-AAII Equities Allocation Spread as a relative sentiment indicator

Institutional-versus-individual relative sentiment indicators

Micaletti⁵ defines institutional-versus-individual relative sentiment (IIRS)⁶ as a factor that compares the positions, flows, and attitudes of institutional investors to those of retail investors⁵.

Such relative sentiment indicators comparing smart money⁶ to dumb money have - at least in theory - potential for exhibiting market timing abilities, because there exists a large and growing body of research […] that shows institutions tend to have better outcomes in the financial markets than retail traders⁵.

As Micaletti⁶ puts it:

Although the concept of market timing is often derided, studies have repeatedly shown institutional investors tend to outperform individual investors over intermediate time horizons.

Thus, should one attempt to time the market, mimicking the actions of institutions seems to be a better bet than siding with individuals.

The NAAIM-AAII Equities Allocation Spread

In the context at hand, Zatloukal⁷ proposes a positions-based IIRS indicator that compares:

Smart money positionning - The allocation of professional investors⁸ to U.S. stocks, thanks to the value of the NAAIM Exposure Index

with

Dumb money positionning - The allocation of individual investors to U.S. stocks, thanks to the value of the total stocks allocation component of the AAII Asset Allocation Survey

Now, in its raw form, there are two immediate problems with that indicator:

It is ill-undefined, due to different reporting frequencies between the NAAIM Exposure Index and the AAII Asset Allocation Survey.

To circumvene this issue, one possible solution that introduces no look-ahead bias is to match the last value of the NAAIM Exposure Index of a given month with the total stocks allocation component of the AAII Asset Allocation Survey of said month.
Even aligning the reporting dates, there is a huge difference in the volatility of the two reported allocations to U.S. stocks.

As an illustration:
- The NAAIM Exposure Index increased from -2.11 on 29th August 2007 to 39.63 on 5th September 2007, which represents a 41.74% weekly change in U.S. stocks allocation.
- The total stocks allocation component of the AAII Asset Allocation Survey increased from 51.8 on May 2003 to 64.0 on June 2003, which represents a 12.2% monthly change in U.S. stocks allocation.
In order to slow these fluctuations and thereby prevent frequent reversals in the directional positions ultimately prescribed by the [NAAIM-AAII equities allocation spread indicator]⁶, Zatloukal⁷ uses a double moving average of the NAAIM Exposure Index, defined as follows:
- A first average at a monthly-ish level, defined by a 5-week moving average of the weekly NAAIM Exposure Index values.
- A second average at a half-year-ish level, defined by a 6-month moving average of the monthly-ish NAAIM Exposure Index values.
On my side, I propose to use⁹:
- A 26-week moving average of the weekly NAAIM Exposure Index values.
- A 2-month moving average of the monthly total stocks allocation component of the AAII Asset Allocation Survey.

For ease of reference, I will call the resulting relative sentiment indicator the NAAIM-AAII Equities Allocation Spread, whose evolution is depicted in Figure 3.

Figure 3. Evolution of the NAAIM-AAII Equities Allocation Spread, July 2006 - June 2026.

Readings in Figure 3 should be interpreted as follows:

When the NAAIM-AAII Equities Allocation Spread is positive (above the dashed horizontal line), active investment managers are relatively bullish U.S. equities v.s. retail intestors.
When the NAAIM-AAII Equities Allocation Spread is negative (below the dashed horizontal line), active investment managers are relatively bearish U.S. equities v.s. retail intestors.

Comparison with the Smart Money Indicator

The NAAIM-AAII Equities Allocation Spread is similar in spirit to the Smart Money Indicator⁶, a positions-based IIRS indicator introduced in Micaletti⁶.

While the construction of the Smart Money Indicator is very different from that of the NAAIM-AAII Equities Allocation Spread, both aim to measure the aggregate positioning in equities of institutions relative to individuals⁶ and both are designed to be large in magnitude when institutions and individuals have strongly divergent positioning […] and small in magnitude when […] their relative positioning is similar⁶.

What differentiates the two indicators is that:

The Smart Money Indicator relies on the Commitment of Traders (COT) report for its construction.

Unlike the NAAIM Exposure Index or the AAII Asset Allocation Survey, that report is not an informal members survey but an official list of the (futures) positions of all market participants.
The Smart Money Indicator relies on multiple pieces of information to estimate the relative positionning of institutions v.s. individuals in U.S. stocks, including information about investors positioning in assets other than the S&P 500 (aka, “cross-hedging pressure”)⁶.

In contrast, the NAAIM-AAII Equities Allocation Spread is rather fairly naive…

For the interested reader, Ray Micaletti joined Excess Returns in the episode Relative Sentiment Investing with Ray Micaletti to discuss the Smart Money Indicator and more generally all about institutional-versus-individual relative sentiment indicators (domestic equities, international equities, U.S. dollar, bonds, gold…).

Incorporating the NAAIM-AAII Equities Allocation Spread into asset allocation

Zatloukal’s original strategy

More than 25 years ago, Fisher and Statman² noted that combining the level of sentiment of [different] groups [of investors like individuals, newsletter writers and Wall Street strategists] provides forecasts of future S&P 500 returns that can be used in a tactical asset allocation program².

It is thus no surprise that it may make sense to adjust allocations to [a portfolio of U.S. equities] based on whether institutions are more or less bullish […] relative to individuals⁵, and this is exactly what Zatloukal⁷ proposes: a tactical asset allocation strategy for U.S. equities based on the NAAIM-AAII Equities Allocation Spread.

My interpretation of that strategy is the following:

At the end of each month
- Compute the current NAAIM-AAII Equities Allocation Spread $S_{NAAIM-AAII}$.
- Compute the average value of the NAAIM-AAII equities allocation spread $\overline{S_{NAAIM-AAII}}$ using all data until the current month.
- Compute the standard deviation of the NAAIM-AAII Equities Allocation Spread $\sigma_{S_{NAAIM-AAII}}$ using all data until the current month.
If $S_{NAAIM-AAII} \ge \overline{S_{NAAIM-AAII}} - \sigma_{S_{NAAIM-AAII}}$, invest the portfolio in U.S. equities, otherwise leave the portfolio in cash.

Whatever the exact details, Zatloukal⁷ finds that relative sentiment outperforms the S&P 500 by 0.7% per annum since mid-2007⁷ and that more importantly, it decreases the maximum drawdown to just 21% versus 56% for the S&P 500⁷.

My own twist

There are two main points that I find can be improved with Zatloukal’s original strategy¹⁰:

It is a binary stragegy.

In other words, that strategy is either fully invested or fully in cash.

Regular readers of this blog known that I am not a big fan of binary signals and that I usually prefer percentile ranks and similar statistical operators.
Its self-normalization mechanism.

I would be more in favor of a self-normalization mechanism like the one used in Micaletti’s Smart Money Indicator, which incorporates z-scores¹¹ and not some -0.25 standard deviation (why not -0.26 or -0.33)?

As a possible alternative, I propose the following tactical asset allocation strategy based on the NAAIM-AAII Equities Allocation Spread:

At the end of each month¹²
- Compute the current NAAIM-AAII Equities Allocation Spread $S_{NAAIM-AAII}$
- Compute the percentile rank $r_{S_{NAAIM-AAII}} \in [0, 100]$ of $S_{NAAIM-AAII}$ using all the NAAIM-AAII equities allocation spreads until the current month.
Invest $r_{S_{NAAIM-AAII}}$% of the portfolio in U.S. equities and $1 - r_{S_{NAAIM-AAII}}$% in cash

Using the SPY ETF as a proxy for U.S. stocks and the SHY ETF as a proxy for U.S. cash, Figure 4 compares the evolution of that new strategy to a fully invested portfolio.

Figure 4. SPY ETF v.s. NAAIM-AAII Equities Allocation Spread strategy, July 2006 - June 2026.

Figures:

Portfolio Management Strategy	Average Stocks Exposure	CAGR	Annualized Volatility	Annualized Sharpe Ratio	Maximum (Monthly) Drawdown
SPY ETF	100%	11.31%	15.31%	0.78	50.80%
NAAIM-AAII Equities Allocation Spread-based	64%	9.19%	8.92%	1.03	13.30%

From these first results, which include more than 3 years of out-of-sample testing compared to Zatloukal⁷, it would appear that the NAAIM-AAII Equities Allocation Spread is somewhat able to predict the U.S. equity market⁶.

Naturally, heavy data torturing - similar to what is done in Micaletti⁶ - would be required before reaching a definitive conclusion, but I will leave it at that for this blog post.

Conclusion

Thanks to the original work of Zatloukal⁷, we have a brand new positions-based relative sentiment indicator to add in our quantitative arsenal.

Beyond the results of the previous section, Micaletti⁵⁶ and Zatloukal⁷ also show that such a relative sentiment indicator might be useful as a complement to a trend following tactical asset allocation strategy.

So, feel free to experiment with that avenue!

To discover more uncommon indicators, connect with me on LinkedIn or follow me on Twitter.

–

For stocks and bonds, the survey distinguishes direct holdings v.s. indirect holdings through funds. ↩
See Kenneth L. Fisher, and Meir Statman. Investor Sentiment and Stock Returns. Financial Analysts Journal, vol. 56, no. 2, 2000, pp. 16–23. ↩ ↩² ↩³
Whose analysis might be the subject for a future blog post. ↩
See AAII Member Surveys. ↩
See Micaletti, Raymond, The Complementarity of Trend Following and Relative Sentiment (October 2, 2022). ↩ ↩² ↩³ ↩⁴ ↩⁵
See Micaletti, Raymond, Want Smart Beta? Follow the Smart Money: Market and Factor Timing Using Relative Sentiment (April 27, 2018). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹²
See Kevin Zatloukal, Revisiting the Smart Money, Medium article. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹
At the very least, of active investment managers. ↩
To be noted that the specific values used for the smoothing do not seem to really matter, as long as they are sensible. For example, a 13-week moving average of the weekly NAAIM Exposure Index values together with the raw total stocks allocation component of the AAII Asset Allocation Survey lead to similar results. To be consistent with both Zatloukal⁷ and a NAAIM-based trading strategy described elsewhere, I settled on a 2x13 = 26-week moving average of the weekly NAAIM Exposure Index values, which made me choose a 2x1 monthly moving average of the raw total stocks allocation component of the AAII Asset Allocation Survey. ↩
Or, to be more precise, with my interpretation of Zatloukal’s original strategy. ↩
Micaletti explains the underlying rationale in Relative Sentiment Investing with Ray Micaletti. ↩
To be noted that the result of the AAII Asset Allocation Survey is published on the first day of the month (excluding holidays, etc.) so that this strategy is not implementable in real time; that being said, using the close price of the SPY ETF with a one-day or a two-day lag does not materially impact the results, which is understandable given the nature of the strategy. ↩

Value-at-Risk Estimation: Improved Historical and Simulation-based Estimates with the Harrell-Davis Quantile Estimator

2026-06-16T00:00:00-05:00

In a previous blog post of this series, the main univariate Value-at-Risk (VaR) estimation methods were described.

Among these, and for scenario-based VaR estimation like historical VaR or Monte Carlo VaR, the most widely used [non-parametric] estimator is the corresponding order statistic of the empirical quantile of the portfolio return distribution, or a linear combination of two subsequent order statistics¹.

Problem is, because they use too few sample elements¹, these estimators exhibit high variability and provides little information about the distribution of losses around the tail².

That problem is further exacerbated in a multivariate Value-at-Risk setting, under which these estimators might even yield inaccurate estimates³ of VaR sensitivities to portfolio positions (marginal VaR, component VaR…), irrespective of the number of scenarios used.

One possible solution is to use the Harrell-Davis quantile estimator⁴ instead, which is an estimator based on a weighted average of a range of order statistics³ substantially more efficient than the traditional estimators based on one or two order statistics⁴.

In this blog post, I will detail that quantile estimator and I will review its practical performances.

Reminders on Value-at-Risk

Value-at-Risk

The Value-at-Risk $VaR_{\alpha}$ of a portfolio over a time horizon $T$ (1 day, 10 days, 20 days…) and at a confidence level $\alpha$% $\in ]0,1[$ (95%, 97.5%, 99%…) can be defined as the opposite of the $1 - \alpha$ quantile of the portfolio return distribution over the time horizon $T$

\[\text{VaR}_{\alpha} = - F_X^{-1}(1 - \alpha)\]

, where:

$X$ is a random variable representing the portfolio return over the time horizon $T$
$F_X^{-1}$ is the inverse cumulative distribution function, also called the quantile function, of the random variable $X$

Univariate Value-at-Risk estimation through empirical quantile estimators

In a scenario-based approach to VaR estimation, whether relying on historical scenarios or simulated scenarios⁵, a sample of portfolio returns $r_1,…,r_n$ over a given time horizon $T$ is available.

The Value-at-Risk of that portfolio over the horizon $T$ at a confidence level $\alpha \in ]0,1[$ is then simply the opposite of the $(1 - \alpha)$% quantile of the empirical return distribution $r_1,…,r_n$, which essentially turns the problem of VaR estimation into a statistical quantile estimation problem³.

As noted in Akinshin¹, the traditional way to [estimate that $(1 - \alpha)$% quantile] is to use a single order statistic or a linear combination of two subsequent order statistics¹, although Hyndman and Fan⁶ actually describes nine types of traditional quantile estimators which are used in statistical computer packages¹.

For reference, with $r_{(1)} \leq r_{(2)} \leq … \leq r_{(n-1)} \leq r_{(n)}$ the order statistics of the sample of portfolio returns, the following two empirical portfolio VaR estimators, based on two traditional empirical quantile estimators, have been introduced in a previous blog post:

The opposite of the $\lfloor n (1 - \alpha) \rfloor + 1$-th highest portfolio return
\[\text{VaR}_{\alpha} = - r_{\left( \lfloor n (1 - \alpha) \rfloor + 1 \right)}\]
The linear interpolation between the opposite of the $\lfloor (n+1) \left( 1 - \alpha \right) \rfloor $-th and $ \lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1$-th highest portfolio returns
\[\text{VaR}_{\alpha} = - \left( 1 - \gamma \right) r_{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor \right)} - \gamma r_{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1 \right)}\]
, with $\gamma$ $= (n+1) \left( 1 - \alpha \right)$ $- \lfloor (n+1) \left( 1 - \alpha \right) \rfloor$.

These two estimators are examples of non-parametric approaches to VaR estimation, because they do not make any specific distributional assumptions on the portfolio return distribution and they do not depend on any auxiliary parameter.

Value-at-Risk sensitivities

Beyond univariate VaR estimation, managing risk also requires the ability to identify the sources of risk and to assess the impacts of potential trades³.

For this, one must be able to evaluate the Value-at-Risk sensitivity to the positions that comprise a portfolio³, which is done thanks to the partial derivatives of VaR with respect to portfolio allocation⁷:

\[\text{MVaR}_{\alpha,i}(w) = \frac{\partial \text{VaR}_{\alpha} }{\partial w_i} (w)\]

, where

$m$ is the number of portfolio assets
$w \in [0,1]^{m}$ is the vector of portfolio asset weights
$\text{MVaR}_{\alpha,i}(w)$ is the partial derivative of the portfolio Value-at-Risk at a confidence level $\alpha$ with respect to the $i$-th asset of the portfolio, called the $i$-th marginal Value-at-Risk.

The properties of marginal VaRs are studied under general conditions in Hallerbach⁵ and in Gourieroux et al.⁷, which gives a slightly more explicit formula for marginal VaRs:

\[\text{MVaR}_{\alpha,i}(w) = - \Bbb{E} \lbrack X_i \vert X = - \text{VaR}_{\alpha}(w) \rbrack\]

, where $X_i$ is a random variable representing the $i$-th asset return over the time horizon $T$.

Multivariate Value-at-Risk sensitivities estimation through empirical quantile estimators

In a scenario-based approach to VaR estimation, marginal VaRs are typically estimated from either⁸:

One scenario⁹, in case the portfolio VaR is estimated through a single order statistic.
A linear combination of two subsequent scenarios, in case the portfolio VaR is estimated through a linear combination of two subsequent order statistics.

In more details, let $r^i_1,…,r^i_n$, $i=1..m$, be samples of the $m$ asset returns over the horizon $T$.

Then, from the formulas of the previous sub-sections:

In case the empirical portfolio VaR estimator is $\text{VaR}_{\alpha} = - r_{\left( \lfloor n (1 - \alpha) \rfloor + 1 \right)}$, the associated marginal VaRs are
\[\text{MVaR}_{\alpha,i} = - r^i_{\left[ \lfloor n (1 - \alpha) \rfloor + 1 \right)]}, i=1..m\]
, with $r^i_{[1]}, …, r^i_{[n]}$ the concomitants of the order statistics $r_{(1)}, …, r_{(n)}$.

For future reference, that approach to computing marginal VaRs is called scenario extraction in Epperlein and Smillie¹⁰.
In case the empirical portfolio VaR estimator is $\text{VaR}_{\alpha} = - \left( 1 - \gamma \right) r_{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor \right)} - \gamma r_{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1 \right)}$, the associated marginal VaRs are
\[\text{MVaR}_{\alpha,i} = - \left( 1 - \gamma \right) r^i_{\left[ \lfloor (n+1) \left( 1 - \alpha \right) \rfloor \right]} - \gamma r^i_{\left[ \lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1 \right]}, i=1..m\]

Value-at-Risk estimation through the Harrell-Davis quantile estimator

As mentioned in the introduction, the main issue with the traditional empirical quantile estimators introduced in the previous section is that they experience a substantial lack of efficiency, caused by the variability of individual order statistics¹¹.

Additionally, Mausser³ notes that while the accurary of these estimators improves with increasing sample size, this is not necessarily true of the estimated quantile sensitivities³.

Such a puzzling behaviour¹² is illustrated in Figure 1 and is explained in Epperlein and Smillie¹⁰ as follows:

The reason for this is quite subtle: recall that [marginal Value-at-Risk] is defined as the [opposite of the] expectation of the component P&L, given that the parent P&L is equal to the [opposite of the] VaR.

Using [traditional empirical quantile estimators], one computes $-X_i \vert X = - \text{VaR}_{\alpha}(w)$, not $- \Bbb{E} \lbrack X_i \vert X = - \text{VaR}_{\alpha}(w) \rbrack$.

Thus, by using scenario extraction, we simply draw a single sample from a (conditional) probability distribution.

The distribution will have some variance, which we cannot expect to reduce simply by increasing the number of Monte Carlo runs.

Figure 1. Noise in marginal Value-at-Risk estimators as the number of Monte Carlo scenarios is increased, empirical quantile-based estimator (scenario extraction) v.s. kernel-based estimator. Source: Epperlein and Smillie.

In all cases, that behaviour is a huge limitation in terms of scenario-based marginal Value-at-Risk estimation…

L-estimators

Sheather and Marron¹¹ argues that an obvious way of improving the efficiency of sample quantiles is to reduce [their] variability by forming a weighted average of all of the order statistics, using an appropriate weight function¹¹.

Such a weighted average of order statistics, called an L-estimator, has a general form given by³:

\[Q_p(x) = \sum_{j=1}^n w_{p, n, j} x_{(j)}\]

, with:

$x_1, …, x_n$ the series of interest and $x_{(1)} \leq … \leq x_{(n)}$ the associated order statistics
$p \in [0,1]$ the $p$-th quantile of interest
$\sum_{j=1}^n w_{p, n, j} = 1$

To be noted that the empirical quantile estimators are examples of L-estimators, albeit ones that place the entire weights on at most two order statistics, so that the problem [with L-estimators] then becomes one of choosing¹³ the weight function¹¹ $w_{p, n, j}$.

The Harrell-Davis quantile estimator

The Harrell–Davis quantile estimator $HD_p$, introduced in Harrell and Davis⁴, is an L-estimator whose weights¹⁴ are related to the beta function:

\[HD_p(x) = \sum_{j=1}^n \left( I_{\frac{j}{n}}(a, b) - I_{\frac{j-1}{n}}(a, b) \right) x_{(j)}\]

, where:

$a = (n+1)p$
$b = (n+1)(1-p)$
$I_x(\alpha, \beta)$ is the regularized incomplete beta function

The typical shape of these weights is represented in Figure 2, which plots the weights for estimating the 0.95 quantile from a sample of 100 observations³.

Figure 2. Harrell–Davis quantile estimator weights, p=0.95, n=100 observations. Source: Hausser.

On Figure 2, it is visible that the Harrell–Davis weights are distributed around a range of order statistics instead of being concentrated on at most two order statistics, which allows it to offer a significant gain in efficiency over the traditional [quantile estimators]⁴.

That gain in efficiency is empirically studied in Harrell and Davis⁴, which shows that - for small samples up to $n=60$ and over a wide variety of light- and heavy-tailed symmetric and asymmetric distributions⁴ - the Harrell–Davis quantile estimator is generally more than 10% as efficient as the traditional quantile estimator based on two order statistics.

Under a different experimental setup, Dielman et al.¹⁵ empirically establishes - for small samples up to $n=30$ and over misc. distributions -, that the Harrell–Davis quantile estimator is the overall best quantile estimator among 10 candidates, although it still has deficiencies concerning extreme percentile¹⁵.

That limitation¹⁶ is not surprising, though, with Harrell and Davis⁴ already mentioning that for some distributions like the exponential distribution, much higher sample sizes - $n \approx 80-100$ - may be required for $p = 0.9$ or above⁴.

Under yet another experimental setup - this time for sample sizes ranging from $n=100$ to $n=10000$ and focused on fat-tailed Student-t distributions ,- Inui and Kijima¹⁷ concludes that the Harrell–Davis estimates are more efficient [than that of the ordinary empirical quantile] except in the case of t-distributions with $n \approx 200-300$ and [a degree of freedom] lower than 4¹⁷.

The exact gains in efficiency reported in Inui and Kijima¹⁷ are detailed in Figure 3.

Figure 3. Efficiency of the Harrell–Davis estimates v.s. the ordinary empirical quantile estimates for 99% confidence level, with varying sample size (n) and Student-t degree of freedom (DF). Source: Inui and Kijima.

Value-at-Risk and Value-at-Risk sensitivities estimation

In order to compute more robust risk analytics³ than with the empirical quantile estimators, Mausser³ proposes to use the Harrell–Davis quantile estimator for non-parametric Value-at-Risk and Value-at-Risk sensitivities estimation.

For this, Mausser³ defines:

The opposite of the Harrell–Davis $(1 - \alpha)$-th quantile estimator of the sample of portfolio returns $r_1,…,r_n$ as the Harrell–Davis portfolio VaR estimator
\[\text{VaR}_{\alpha} = - HD_{1 - \alpha}(r)\]
The opposite of the Harrell–Davis $(1 - \alpha)$-th concomitant estimator of the sample of asset returns $r^i_1,…,r^i_n$ as the Harrell–Davis $i$-th marginal VaR estimator, $i=1..m$
\[\text{MVaR}_{\alpha,i} = - \sum_{j=1}^n \left( I_{\frac{j}{n}}(a,b ) - I_{\frac{j-1}{n}}(a, b) \right) r^i_{[j]}\]
, where $a = (n+1)(1 - \alpha)$ and $b = (n+1)\alpha$.

That formula is derived in Mausser³, with the following comments in Bonollo et al.¹⁸:

The [marginal VaR] of each position is given by a weighted average of its P&Ls; the weights are given by the Harrell-Davis formula […], where the scenario with higher weight will be the VaR scenario of the portfolio.

For both estimators, the fact that VaR is calculated as a weighted combination of the losses across multiple scenarios has an inherent smoothing effect³.

This can be seen for example in Figure 4, which compares the Value-at-Risk trade risk profile³ obtained using the Harrell–Davis quantile estimator and the single-order statistic empirical quantile estimator.

Figure 4. Harrell–Davis quantile estimator v.s. single order statistics empirical quantile estimator, 99% VaR trade risk profile for Mitsubushi EC 6mo 860 call. Source: Hausser.

That smoothness explains why the Harrell–Davis VaR estimator is more efficient than the single-order statistic based VaR estimator, with for example Bonollo and Mastrototaro¹⁹ reporting error estimates down by as much as 32 percentage points²⁰ in the context of default risk charge²¹ estimation, which is a Value-at-Risk at a 99.9% confidence level estimated over a 1 year horizon.

In the specific case of VaR sensitivies estimation, that smoothness also provides the Harrell–Davis marginal VaR estimator with the desirable property that, as the sample size increases, the standard deviation of the estimate decreases³, contrary to its empirical counterparts whose estimates variability not only exceeds that of the [Harrell–Davis] estimates, but actually gets larger as the sample size increases³, c.f. Figure 1.

That phenomenom is also reported in Bonollo et al.¹⁸, which observes that the [marginal] VaR obtained with Harrell-Davis weights is way more stable compared to the [empirical quantile-based] one¹⁸.

Comparison with other L-estimators

Several other Value-at-Risk estimators based on linear combinations of order statistics have been proposed in the litterature, among which:

The rectangular VaR estimator

That estimator, introduced in Hallerbach⁵ in the context of marginal Value-at-Risk estimation, is an equally-weighted average of all the order statistics in a neighborhood of the single-order statistic based empirical quantile estimator.
The triangular VaR estimator

That estimator, introduced in Epperlein and Smillie¹⁰ in the context of marginal Value-at-Risk estimation, is an L-estimator whose maximum weight is located at the index of the single-order statistic based empirical quantile estimator and whose weights are symmetrically decreasing to 0 at a selected rate from that maximum weight.
General kernel-smoothed VaR estimators

Those estimators, which effectively smooth the empirical distribution function by spreading each sample observation over an interval with a so-called kernel³ and of which the previous estimators are examples, are described in details in a previous blog post.

Compared to all these semi-parametric VaR estimators, one practical advantage of the Harrell-Davis VaR estimator is that it is fully non-parametric; that is, there are no parameters to choose (window size, kernel type, bandwidth…).

Another practical avantage, and maybe the most important one, is that its empirical performances are comparable to general kernel-smoothed VaR estimators:

Bonollo and Mastrototaro¹⁹ compares the Harrell-Davis estimator to the first two estimators and concludes that the best result was obtained with the Harrell-Davis estimator¹⁹.
Sheather and Marron¹¹ compares the Harrell-Davis estimator to an optimal Gaussian kernel-smoothed estimator and concludes that the former performs surprisingly well¹¹.

Incidentally, Sheather and Marron¹¹ demonstrates that the Harrell-Davis quantile estimator is asymptotically a Gaussian kernel estimator, although one whose bandwidth is in fact suboptimal¹¹ but often not too far from the optimum¹¹.

That suboptimality is an important feature of the Harrell-Davis quantile estimator, because general kernel-smoothed quantile estimators require the selection of a bandwidth parameter, which is typically done in a data-driven way.

Unfortunately, the choice of an optimal bandwidth is a difficult task, especially when smoothing the tails of underlying distributions with possible data scarcity²², so that a guaranteed suboptimality is probably more important in practice than a non-guaranteed optimality…

Caveats

Despite its very good performances, the Harrell-Davis quantile estimator also has some limitations to be aware of.

Bias

Inui and Kijima¹⁷ empirically shows that the Harrell–Davis estimator tends to have a more significant positive bias than the ordinary quantile estimator when used for fat-tail distributions¹⁷.

More precisely, and under a Monte Carlo simulation setting that uses Student-t distributions to model portfolio returns, Inui and Kijima¹⁷ demonstrates that:

The bias of both the Harrell–Davis VaR estimator and the ordinary quantile VaR estimator increases with higher confidence level $\alpha$, heavier tails for the portfolio returns and smaller sample size $n$.
The small sample bias ($n \approx 200$) of the Harrell–Davis VaR estimator is greater than that of the ordinary quantile VaR estimator for fat-tailed distributions with a degree of freedom lower than $\approx 6$.

Here, interestingly, the reverse is true for $n \approx 100$ and Inui and Kijima¹⁷ more generally states that the bias of the Harrell–Davis estimates of VaR with $n \approx 100$ is always smaller than that with $n \approx 200$, whatever the degree of freedom¹⁷.

That slightly higher²³ bias is not necessarily an issue, but because some empirical researches suggest to use [Student-t] distributions with [a degree of freedom] not greater than 4¹⁷ to model asset returns, it is a limitation of the Harrell–Davis VaR estimator to keep in mind.

As a side note, Kim and Hardy²⁴ argues that bias correction may increase the bias and substantially increase the rMSE²⁴ of bias-corrected VaR estimators, so that it is probably useless to try to improve the Harrell–Davis VaR estimator on this aspect.

Statistical robustness

Akinshin¹ highlights that the improved efficiency of the Harrell–Davis quantile estimator has a price¹ in that it becomes non robust in the statistical sense.

Indeed¹:

Since the estimation is a weighted sum of all order statistics with positive weights, a single corrupted element may spoil all the quantile estimations, including the median.

It may become a severe drawback in the case of heavy-tailed distributions in which it’s a typical situation when we have a few extremely large outliers.

For practical situations in which this non-robustness is an issue, Akinshin¹ proposes to use a trimmed modification of the Harrell-Davis quantile estimator¹ that is both more robust than the Harrell–Davis estimator […] and typically more statistically efficient than [the single-order statistic empirical estimator]¹.

The core idea of Akinshin’s¹ trimmed Harrell-Davis quantile estimator is to discard elements outside of the highest density interval of the underlying beta distribution¹ under the assumption that they might be contaminated.

This is done through the introduction of a parameter that needs to be carefully¹ selected - the width of the beta distribution highest density interval¹.

Nevertheless, in practice, determining that width parameter in the context of VaR estimation might prove to be difficult due to the extreme nature of the associated quantiles²⁵, so that the interested reader will probably need to perform some more analysis here.

Tail extrapolation

Like all L-estimators, the Harrell-Davis quantile estimator is limited by the fact that the estimated quantile cannot lie outside the range of the order statistics, which may be a concern when estimating extreme quantiles based on small samples³.

Hopefully, this is a well-known issue in VaR estimation with several alternative approaches readily available like extreme value theory-based VaR estimation.

Implementation in Portfolio Optimizer

Portfolio Optimizer implements the Harrell–Davis quantile estimator for both Value-at-Risk and Value-at-Risk contributions estimation, c.f. the documentation.

Conclusion

As Dielman et al.¹⁵ points out, there is not one [quantile] estimator that is always best; however, the results presented here do strongly argue in favor of [using the Harrell-Davis quantile estimator], in a wide range of cases¹⁵.

In the financial industry, this seems to be the choice made for example by Allianz France for their internal Solvency Capital Requirement (SCR) model, at least until April 2021²⁶.

So, why not giving it a try?

As usual, feel free to connect with me on LinkedIn or to follow me on Twitter.

–

See Andrey Akinshin (2022): Trimmed Harrell-Davis quantile estimator based on the highest density interval of the given width, Communications in Statistics - Simulation and Computation. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵
See Leszek Gadomski, Vasile Glavan, Expected shortfall and Harell-Davis estimators of value-at-risk, QUANTITATIVE METHODS IN ECONOMICS, Vol. XI, No. 1, 2010, pp. 81-89. ↩
See Mausser, H., 2001. Calculating quantile-based risk analytics with L-estimators. Algo Res. Quart. 4, 33–47. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹
See Harrell, F. E., and C. E. Davis. 1982. A new distribution-free quantile estimator. Biometrika 69 (3):635–40. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸
See Winfried G. Hallerbach, Decomposing portfolio value-at-risk: a general analysis, Journal of Risk, volume 5, number 2 (winter 2002), pages: 1-18. ↩ ↩² ↩³
See Hyndman, R. J., & Fan, Y. (1996). Sample Quantiles in Statistical Packages. The American Statistician, 50(4), 361–365. ↩
See Gourieroux, C., Scaillet, O. and Laurent, J.P. (2000). Sensitivity analysis of Values at Risk. Journal of Empirical Finance, 7, 225-245. ↩ ↩²
Both of these approaches are non-parametric in nature. ↩
As noted in Mausser³, the marginal VaR for an instrument equals its per-unit loss in the particular scenario for which the portfolio loss equals the VaR³. ↩
See Epperlein, E., Smillie, A. Cracking VaR with kernels. RISK, 2006, vol. 9, pages 70-74. ↩ ↩² ↩³
See Sheather, Simon J., and J. S. Marron. “Kernel Quantile Estimators.” Journal of the American Statistical Association, vol. 85, no. 410, 1990, pp. 410–416. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹
Mausser³ also notes that such a weird behaviour tends to occur whenever an instrument’s losses do not relate with those of the portfolio in a consistent manner³. ↩
When the weight function is properly chosen, L-estimators exhibit certain desirable asymptotic properties such as asymptotic normality (under very general conditions)¹⁵ - which facilitates the construction of confidence intervals for the quantile³ - and compete favorably with M estimators¹⁵. ↩
For practical applications, it is worth noting that the Harrell–Davis weights depend only on the sample size $n$ and on the quantile $q$. ↩
See Dielman, T., Lowry, C., Pfaffenberger, R., 1994. A comparison of quantile estimators. Comm. Statist.: Simulation Comput. 23, 203–228. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶
Another limitation is that since the Harrell–Davis quantile estimator requires evaluation of the incomplete beta function, this estimator cannot be computed over ranges of values for which the incomplete beta function is undefined¹⁵, which is the case for small $n$ and very small $p$. ↩
See Koji Inui, Masaaki Kijima, Atsushi Kitano, VaR is subject to a significant positive bias, Statistics & Probability Letters, Volume 72, Issue 4, 2005, Pages 299-311. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹
See Michele Bonollo, Giacomo Colombo, Giacomo Giannoni and Antonio Menegon, Risk Attribution, iason, Research Paper Series, N.48, 05/07/2022. ↩ ↩² ↩³
See Michele Bonollo and Leonardo Mastrototaro, A Comparison of Advanced Methods for the Quantile Estimation in the Risk Management Field, iason, Research Paper Series, N.80, 07/10/2025. ↩ ↩² ↩³
From 73.2% to 41.2% (3rd case of $n = 1000$ in Table 1), c.f. Bonollo and Mastrototaro¹⁹. ↩
The default risk charge (DRC) is a regulatory measure designed to capture default risk within the trading portfolio, as required by Basel standards¹⁹. ↩
See Keming Yu & Abdallah K. Ally & Shanchao Yang & David J. Hand, Kernel quantile based estimation of expected shortfall, Journal of Risk, Journal of Risk. ↩
The bias of the Harrell–Davis VaR estimator seems to be greater by about 5 percentage point (19% v.s. 14%) for a Student-t distribution with a degree of freedom equal to 3, c.f. Figure 6 of Inui and Kijima¹⁷. ↩
See Kim JHT, Hardy MR. Quantifying and Correcting the Bias in Estimated Risk Measures. ASTIN Bulletin. 2007;37(2):365-386. ↩ ↩²
Especially at very high confidence levels $\alpha=99$%, $\alpha=99.5$%, … ↩
See IBAZIZEN Rachid, Creation d’un cadre prudentiel pour le pilotage d’un FRPS de grande taille, Memoire. ↩

The NAAIM Exposure Index: Incorporating Active Investment Managers Sentiment into Asset Allocation

2026-05-07T00:00:00-05:00

The NAAIM Exposure Index represents the average exposure to U.S. equity markets as reported by members of the National Association of Active Investment Managers (NAAIM) in a weekly survey.

That index, like any other sentiment indicator, is a useful gauge of the possible future direction of a market¹ that can be incorporated into one’s asset allocation process.

In this blog post, I will analyze an example of such incorporation that allows advisors to move to a modified buy-and-hold strategy while dipping their toes into the waters of active management².

The NAAIM Exposure Index

Definition

The NAAIM Exposure Index is the result of a weekly data collection process organized by the NAAIM since July 2006³:

Each Wednesday, NAAIM members are asked to report their exposure to the U.S. equity markets at that Wednesday’s close.

The range of possible responses is quite wide, going from -200% (leveraged short) to 0% (cash or market neutral) and to 200% (leveraged long).
Each Thursday, responses are collected and processed to compute the average exposure of NAAIM members to the U.S. equity markets, known as the NAAIM Exposure Index.

Here, for example:
- A reading of 93.79 on 29th April 2026 indicates that NAAIM members are on average 93.79% invested in U.S. stocks and 6.21% invested in other assets (cash, bonds…)
- A reading of 106.56 on 26th August 2020 indicates that NAAIM members are on average leveraged long 106.56% in U.S. stocks
- A reading of -3.56 on 5th October 2011 indicates that NAAIM members are on average short 3.56% in U.S. stocks and 96.44% invested in other assets (cash, bonds…)
In addition to the average exposure, the NAAIM also provides other distributional statistics about their members’ U.S. equities exposure (median exposure, standard deviation of exposures…).

These other statistics help contextualizing the raw value of the NAAIM Exposure Index.

For example, compare:
- A reading of 92.43 on 22th November 2006, with a standard deviation of 20.82
- A reading of 92.93 on 31th December 2025, with a standard deviation of 48.97
Both values of the NAAIM Exposure Index are nearly identical, but the later has about twice the uncertainty of the former, highlighting stronger exposure disagreement between NAAIM members.

An interesting remark made by Hepburn⁴ is that prior to the NAAIM weekly survey of investment managers sentiment, there were no surveys of actual money managers⁴, most probably explained by the fact that data on what active investment managers do within their portfolios is often a closely guarded secret⁴.

Limitations

Due to the nature of NAAIM members, the NAAIM Exposure Index presents some limitations that must be kept in mind.

Most are detailled on the NAAIM website, like the fact that reporting a single number for each adviser may not accurately represent the market view of a manager who has short term and long term strategies that are providing conflicting signals⁵.

Some additional caveats are highlighted by the current⁶ NAAIM president Ryan Redfern in an interview by StockCharts, like the fact that at Thursday’s open, NAAIM members can wildly change their exposure to U.S. equities compared to their reported position at Wednesday’s close.

Usage as a sentiment indicator

At heart, the NAAIM Exposure Index quantifies how active investment managers are adjusting exposure in real time, managing risk, aligning strategy, and responding to market dynamics⁷.

As such, the NAAIM Exposure Index is not predictive in nature and is of little value in attempting to determine what the stock market will do in the future⁵.

Nevertheless, the NAAIM Exposure Index is a sentiment indicator and, like any other sentiment indicator, can be tentatively used for market timing.

For example, Rob Hanna from QuantifiableEdges notes that⁸:

Extreme highs or lows can signal overbought or oversold conditions and may precede reversals in the market.

For example, if [the NAAIM Exposure Index] falls sharply while markets are relatively stable, it may reflect growing concern among professionals—an early warning signal. Conversely, a surge in exposure may suggest confidence in market upside.

Implementation in Portfolio Optimizer

Portfolio Optimizer historically allowed to easily retrieve the NAAIM Exposure Index, as well as its median and its standard deviation, through the endpoint /markets/indicators/naaim-exposure-index.

Unfortunately, since 1st August 2026, access to the live NAAIM Exposure Index has been paywalled by the NAAIM, with a 3-month delay v.s. live data for non-paying users.

In addition, in a subsequent update, the NAAIM explicited its redistribution policy, with only NAAIM Program Partners allowed to redistribute the NAAIM Exposure Index data.

As a consequence, this endpoint has been suspended until Portfolio Optimizer (hopefully) obtains that status.

Incorporating the NAAIM Exposure Index into asset allocation - Hepburn’s adaptive rebalancing strategy

Hepburn⁴ proposes to use the NAAIM Exposure Index to bridge the gap between strategic asset allocation [(SAA)] and tactical asset allocation [(TAA)]⁴ by making periodic adjustments in a [60/40 U.S.] stock and bond allocations to match current market conditions⁴.

In more details, Hepburn’s adaptive rebalancing⁴ strategy works as follows:

At the end of each quarter, compute the 13-week simple moving average of the NAAIM Exposure Index $\overline{NAAIM}_{13w}$
Allocate $\overline{NAAIM}_{13w}$% of the portfolio to U.S. stocks⁹ and $\left( 100 - \overline{NAAIM}_{13w} \right)$% of the portfolio to U.S. bonds¹⁰
Hold the portfolio for the following quarter

A little bit less⁶ than 20 years after its initial publication by William T. Hepburn, how did that strategy fare?

Using the SPY ETF as a proxy for U.S. stocks and the AGG ETF as a proxy for U.S. bonds, Figure 1 compares the evolution of Hepburn’s strategy to a 60/40 portfolio¹¹.

Figure 1. Hepburn's adaptive rebalancing strategy v.s. 60/40 rebalancing, September 2006 - February 2026.

Figures:

Portfolio Management Strategy	Average Stocks Exposure	CAGR	Annualized Volatility	Annualized Sharpe Ratio	Maximum (Weekly) Drawdown
60/40 portfolio	60%	7.9%	11.2%	0.73	34%
Hepburn’s adaptive rebalancing	66%	9.1%	11.5%	0.82	25%

From Figure 1 and associated figures¹², it seems Hepburn’s conclusion in 2026 is still valid in 2009⁴, that is:

Tactical Asset Allocation using the NAAIM Survey equity allocations proved itself to be superior to the fixed ratio Strategic Asset Allocation model

As a side note for people familiar with tactical asset allocation and for whom an annualized Sharpe Ratio of ~0.82 might not be particularly attractive, Hepburn⁴ makes it clear that the proposed adaptive rebalancing strategy is a first (baby) step for investors and advisers wishing to actively adjust their various investment allocations⁴.

In other words, that strategy is aimed at traditional buy and hold oriented investors and their advisers⁴.

For those investors, by the way, pairing Hepburn’s adaptive rebalancing strategy together with Meb Faber’s Tactical Yield strategy - recently⁶ analyzed by Allocate Smartly - might be something to look at.

Less traditional investors are encouraged to add their own twist on that baseline strategy, for example by adding a layer of risk management over the stocks and bonds allocation (moving average, momentum, whatever).

Conclusion

This concludes this short post on the NAAIM Exposure Index.

As usual, feel free to connect with me on LinkedIn or to follow me on Twitter.

–

See Proactive Advisor Magazine, Industry insights, Will Hepburn, The backward nature of sentiment indicators. ↩
See Forbes, A System For Market Timing. ↩
The first reading of the NAAIM Exposure Index dates back to 5th July 2006. ↩
See William T. Hepburn, Hepburn Capital Management, Using Adaptive Rebalancing to Bridge the Gap between Strategic Asset Allocation and Tactical Asset Allocation, October 2009. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹
See NAAIM Website, NAAIM Exposure Index. ↩ ↩²
At the date of publication of this blog post. ↩ ↩² ↩³
See NAAIM Facebook. ↩
See Rob Hanna, Modeling with the NAAIM Exposure Index. ↩
Represented by the S&P 500 Index. ↩
Represented by the Barclay’s Aggregate Bond Index. ↩
Quarterly rebalanced. ↩
To be noted that Hepburn’s figures⁴ are based on quarterly data v.s. weekly data for my reproduction, which naturally degrades drawdown and volatility statistics; in addition, beyond the figures displayed, the Ulcer Index of Hepburn’s adaptive rebalancing is about 20% less than that of the 60/40 portfolio! ↩

Mean-Variance Optimization in Practice: Reverse Optimization and Implied Expected Returns

2026-04-19T00:00:00-05:00

The fact that mean-variance optimizers are highly sensitive to changes in expected returns […] is well known in investment practice¹, with a couple of practical solutions already described in this blog, for example using near efficient portfolios or subset resampling-based efficient portfolios.

In this blog post, I will introduce another approach originally described in Sharpe² and known as reverse optimization³, which consists in trying to improve the robustness of expected returns estimates by turning the mean-variance optimization problem around.

In more detail, instead of optimizing portfolio weights using a set of expected returns and a covariance matrix⁴, reverse optimization starts with the weights of a given portfolio⁵ and solves for the corresponding set of [implied] expected returns³ that makes it mean–variance efficient.

That approach, illustrated in Figure 1 taken from Two Sigma⁶, has been popularized by the Black-Litterman methodology and has empirically been shown to improve out-of-sample performance of portfolios⁷, c.f. for example Ardia and Boudt⁴ or Ni et al.⁷.

Figure 1. Differences between traditional and reverse mean-variance optimization. Source: Two Sigma.

As a main example of usage, I will show how reverse optimization allows to analyze the hidden assumption made by index investors in the MSCI World w.r.t. future U.S. over ex-U.S. equities performance.

Reverse mean-variance optimization and implied expected returns

Let be:

$n$, the number of assets in a universe of assets
$\mu \in \mathbb{R}^{n}$, the vector of expected asset arithmetic returns at a given holding horizon (one day, one month…)
$\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$, the covariance matrix of the asset arithmetic returns at the holding horizon
$\lambda$, $0 < \lambda < +\infty$, a risk tolerance⁸ parameter

Under the Markowitz’s mean-variance framework, a mean–variance efficient portfolio is defined as a vector of portfolio weights $w^* \in \mathbb{R}^{n}$ that maximizes the mean–variance utility function

\[w^* = \operatorname{argmax} \mu {}^t w - \frac{\lambda}{2} w {}^t \Sigma w \newline \textrm{s.t. } w \in C\]

, where $C \subset \mathbb{R}^{n}$ is the set of investment constraints on the portfolio weights (budget constraint, no short-sale constraint, sector exposure constraints…).

Reverse unconstrained mean-variance optimization

When no constraints are imposed on the mean-variance optimization problem or when no linear constraint is binding at optimality¹², the first-order optimality condition associated to the mean–variance utility function maximisation problem above is

\[\frac{\partial \mathcal{L}}{\partial w} = \mu - \lambda \Sigma w = 0\]

, where $\mathcal{L}(w) = \mu {}^t w - \frac{\lambda}{2} w {}^t \Sigma w$ is the Lagrangian of the mean–variance utility function maximisation problem.

Now, assuming that the solution to this problem $w^* \in \mathbb{R}^{n}$ is known but $\mu$ is not, it is then possible to isolate $\mu$, which leads to the classical¹ definition of the vector of implied expected returns $\mu_{impl} \in \mathbb{R}^{n}$ as

\[\mu_{impl} = \lambda \Sigma w^*\]

Ardia and Boudt⁴ comments that definition as follows:

We thus see that the implied expected return for [a] stock is proportional to the covariance between the stock’s return and the return on the mean–variance efficient portfolio.

Ceteris paribus, this covariance will tend to be higher for stocks that have: (i) a relatively larger weight in the efficient portfolio proxy, (ii) a high volatility, and (iii) a high correlation with the stocks in the efficient portfolio.

The propagation of the weights bets to implied returns bets thus interacts with the variance and correlation properties of the individual stock returns.

Reverse budget-constrained mean-variance optimization

Herold¹ notes that in the literature, implied returns are regularly computed according to [the relationship established in the previous sub-section, which] will often result in unreasonable values¹.

Indeed, the assumption that no constraints are imposed on the mean-variance optimization problem or that no constraints are binding at optimality is not realistic, because in practice the budget constraint is relevant to the investor¹.

So, let be $C = \{ w \in \mathbb{R}^{n}, \sum_{i=1}^n w_i = 1 \}$ representing the budget constraint.

The Lagrangian of the mean–variance utility function maximisation problem then becomes¹⁹

\[\mathcal{L}(w) = \mu {}^t w - \frac{\lambda}{2} w {}^t \Sigma w - \gamma \left( w {}^t 1_n - 1 \right)\]

, where:

$\gamma \in \mathbb{R}$ is the Lagrange multiplier associated to the budget constraint $\sum_{i=1}^n w_i = 1$
$1_n$ is a $n$-dimentional vector of 1s

and the first-order optimality condition becomes¹⁹

\[\frac{\partial \mathcal{L}}{\partial w} = \mu - \lambda \Sigma w - \gamma 1_n = 0\]

Consequently, and again assuming that the solution to this problem $w^*$ is known but $\mu$ is not, the vector of implied expected returns $\mu_{impl}$ turns into a two-parameter family that depends on a […] multiplicative factor and an additive component¹:

\[\mu_{impl} = \lambda \Sigma w^* + \gamma 1_n\]

Herold¹ argues¹⁰ that this definition - although involving an additional parameter $\gamma$ - results in sensible and more realistic values for the implied returns¹ than the common procedure of determining the multiplier only and implicitly setting the additive constant to zero¹.

As a side note, under the additional assumption that the marginal investor’s objective is to maximize risk-adjusted returns (or Sharpe Ratio)⁶, it can be shown¹¹ that $\gamma$ can be set to 0, which justifies the widespread use of the definition of $\mu_{impl}$ established in the previous sub-section.

Reverse linearly-constrained mean-variance optimization

Beyond the budget constraint analyzed in the previous sub-section, the standard mean-variance portfolio optimization problem originally introduced in Markowitz¹² involves general linear constraints on the portfolio weights $w$ modelled through the set $C = \{ w \in \mathbb{R}^{n}, Ax = b, Cx \leq d \}$, where:

$A \in \mathcal{M}(\mathbb{R}^{n_e \times n})$ and $b \in \mathbb{R}^{n_e}$ represent linear equality constraints, like the budget constraint $\sum_{i=1}^n w_i = 1$
$C \in \mathcal{M}(\mathbb{R}^{n_i \times n})$ and $d \in \mathbb{R}^{n_i}$ represent linear inequality constraints like nonnegativity constraints¹³ $w_i \geq 0, i=1..n$ or sector constraints

While that formulation is apparently much more complex than that of the previous sub-section, Sharpe² actually notes that the solution, once determined, can he characterized as the solution to a [linear equality-constrained] problem², so that only the case $C = \{ w \in \mathbb{R}^{n}, Ax = b \}$ needs to be analyzed.

As Sharpe² puts it:

Inequality constraints not binding in the solution are simply omitted from the expression of the problem; those that were binding are included as equalities.

Thus, assuming that the solution to this problem $w^*$ is known¹⁴, the Lagrangian of the mean–variance utility function maximisation problem becomes²

\[\mathcal{L}(w) = \mu {}^t w - \frac{\lambda}{2} w {}^t \Sigma w - \langle {\Gamma, A' w - b'} \rangle\]

, where:

$n_b \geq 0$ represents the number of binding inequality and bound constraints at optimality turned into linear equality constraints
$A’ \in \mathcal{M}(\mathbb{R}^{(n_e + n_b) \times n})$ and $b \in \mathbb{R}^{n_e + n_b}$ represent the original linear equality constraints $(A,b)$ plus the binding inequality and bound constraints at optimality turned into linear equality constraints
$\Gamma \in \mathbb{R}^{n_e + n_b}$ are the Lagrange multipliers associated to the $n_e + n_b$ equality constraints

and the first-order optimality condition becomes²

\[\frac{\partial \mathcal{L}}{\partial w} = \mu - \lambda \Sigma w - A' {}^t \Gamma = 0\]

This time, the vector of implied expected returns $\mu_{impl}$ turns into a $n_e + n_b + 1$-parameter family that depends on a multiplicative factor and several additive components:

\[\mu_{impl} = \lambda \Sigma w^* + A' {}^t \Gamma\]

One important consequence of that result is that for a budget and nonnegativity-constrained mean-variance optimization problem:

If there are no asset with a null weight in the solution $w^*$, then the definition of $\mu_{impl}$ established in the previous sub-section remains valid¹⁵.
Otherwise, the definition of $\mu_{impl}$ established in the previous sub-section provides an upper bound for the implied expected return of any asset whose weight is null².

Reverse non-linearly-constrained mean-variance optimization

Non-linear constraints imposed on the portfolio weights (tracking error constraints, turnover constraints, cardinality constraints…) are out of scope of this blog post.

Nevertheles, the interested reader is refered to Bertsimas et al.¹⁶, which shows that reverse optimization can be characterized as an inverse optimization problem, with applications to the Black-Litterman model.

Reverse optimization in practice

From the previous section, using reverse optimization within a universe of assets requires:

The vector of a reference mean–variance efficient portfolio weights $w^*$
An estimate of the covariance matrix $\Sigma$ of the asset arithmetic returns at a given holding horizon
An estimate of the risk tolerance $\lambda > 0$
An estimate of the Lagrange multiplier $\gamma$ associated to the budget constraint or an estimate of the Lagrange multipliers $\Gamma$ associated to the equality constraints binding at optimality for $w^*$

Unfortunately, most of these quantities cannot be directly observed and so [it is needed to] resort to estimating them instead¹⁷.

Choosing a reference mean–variance efficient portfolio $w^*$

Ardia and Boudt⁴ notes that traditionally, the standard choice [for a reference mean-variance efficient portfolio] has been to use the market capitalization weighted portfolio⁴ because of the result that under the capital asset pricing model (CAPM) the market capitalization portfolio is mean–variance efficient⁴.

Nevertheless, for almost all [investors], the investable universe does not correspond to all tradable assets and hence, the CAPM conditions [under which that portfolio is mean–variance efficient] are by construction not verified⁴.

As a consequence, Ardia and Boudt⁴ argues that the market capitalization weighted portfolio is only one possible proxy for a mean–variance efficient portfolio and that other proxies may lead to more accurate [implied] expected returns².

Al-Thani et al.³ even goes further and states that the most neutral and economically defensible starting point for reverse optimization and implied-return extraction […] should be chosen from [a risk-based portfolio]³.

Examples of such risk-based portfolios - some empirically demonstrated in Ardia and Boudt⁴ to provide a small forecast accuracy gain […] compared with the market capitalization weighted portfolio⁴ - are the following:

The equally weighted portfolio
The inverse volatility weighted portfolio
The equal-risk-contribution portfolio
The maximum diversification portfolio
The minimum variance portfolio
The conviction-parity portfolio

One practical remark about these alternative reference mean–variance efficient portfolios is that their asset weights $w^*$ might not be easily observable (unlike the market capitalization portfolio) nor computable (unlike the risk-based portfolios above).

In such a case, and provided historical portfolio returns are available, it should be possible to infer $w^*$ through an index tracking optimization procedure.

Estimating the asset covariance matrix $\Sigma$

As noted in Shumway et al.¹⁷, the covariance matrix of returns can be reasonably estimated using historical return data¹⁷ because risk tends to be strongly persistent through time⁶.

This is what is done for example in Herold¹, in Ardia and Boudt⁴ or in Bevan and Winkelmann¹⁸.

Depending on the choice of the mean–variance efficient portfolio, though, the primary goal when estimating the asset covariance matrix might not be to find the best estimate of the true covariance matrix¹⁷ but simply to find the best estimate of [whatever} estimate of the covariance matrix¹⁷ was originally used to produce that portfolio.

In other words, in the specific context of reverse optimization, the problem of estimating the asset covariance matrix is slightly different from the problem of forecasting the asset covariance matrix for the next holding period¹⁹.

Estimating the risk tolerance parameter $\lambda$

The risk tolerance parameter $\lambda$ represents the magnitude of the trade-off between expected return and variance² and acts as a scaling factor for the reverse optimization estimate of [implied expected] returns²⁰.

Two Sigma⁶ explains this as follows:

Crucially, the implied returns that come from reverse optimization are only relative return levels - not absolute return expectations.

That is, reverse optimization may tell us that investors expect twice the return from [asset] A as from [asset] B, but provides no guidance on whether those expected returns are 2% versus 1% annually or 20% versus 10%.

In practice, $\lambda$ can be estimated by several different methods:

Using a sensible fixed value, or a range of sensible fixed values, like in Ardia and Boudt⁴ which uses $\lambda \in \{1, 2.4, 5 \}$.
Using the risk premium of the chosen reference mean–variance efficient portfolio, through the formula¹
\[\lambda = \frac{\mu_p}{\sigma_p^2}\]
, where:
- $\mu_p$ is the risk premium of the chosen reference mean–variance efficient portfolio, that is, the expected portfolio return over the risk free rate²¹
- $\sigma_p$ is the standard deviation of the chosen reference mean–variance efficient portfolio returns over the risk free rate
A couple of remarks:
- The portfolio risk premium $\mu_p$ and the portfolio standard deviation $\sigma_p$ might both be estimated from a set of historical portfolio returns.
  
  In that case, those historical returns must be arithmetic returns so as to make $\lambda$ invariant w.r.t. time, since both numerator and denominator [are then] linear in time²².
- The portfolio risk premium $\mu_p$ might be estimated from a set of historical portfolio returns and the portfolio standard deviation might be estimated through the formula $\sigma_p = \sqrt{ w^* {}^t \Sigma w^*}$.
  
  Same remark as above, since $\Sigma$ is supposed to be the covariance matrix of the asset arithmetic returns.
  
  In addition, for internal consistency, $\Sigma$ should then be the covariance matrix of the asset arithmetic excess returns over the risk free rate²³.
- In both cases, due to the constraint $\lambda > 0$, the estimated portfolio risk premium $\mu_p$ must be stritly positive for the estimation of the risk tolerance parameter to make sense.
  
  This might not always be possible, depending on the exact set of historical portfolio returns used.
- In both cases also, it is important to keep in mind that one concern when inferring the risk tolerance parameter from market data is the extent of the sampling variation²⁴.
Using the Shape Ratio of the chosen reference mean–variance efficient portfolio, through the formula¹⁸
\[\lambda = \frac{ \text{SR}_p }{\sigma_p}\]
, where:
- $\text{SR}_p$ is the Sharpe Ratio of the chosen reference mean–variance efficient portfolio
- $\sigma_p = \sqrt{ w^* {}^t \Sigma w^*} $ is the standard deviation of the chosen reference mean–variance efficient portfolio returns over the risk free rate
Three remarks:
- The periodicity of the Sharpe Ratio must be consistent with the holding horizon of the asset covariance matrix $\Sigma$.
  
  For example, using an annualized Sharpe Ratio and an asset covariance matrix at a daily horizon would not be consistent…
- For internal consistency, $\Sigma$ should be the covariance matrix of the asset arithmetic excess returns over the risk free rate²³.
- Examples of annualized Sharpe Ratios used in the litterature are ranging from 0.20-0.30¹²⁵ to 1.0¹⁸.
  
  Ultimately, this depends on the chosen reference mean–variance efficient portfolio as well as on the holding horizon.
Using the target expected return of a reference asset.

It is not uncommon⁶²⁶, especially when computing long-term capital market assumptions (CMAs), to set the risk tolerance parameter to a value that makes the implied expected return of an asset equal to a prespecified value¹.

When there is no Lagrange multiplier to consider, this is simply done through the formula
\[\lambda = \frac{ \left( \Sigma w^* \right)_i}{ \mu_{target,i} }\]
, where $1 \leq i \leq n$ is the index of the reference asset whose target expected return $\mu_{target,i}$ is known.

A couple of examples:
- When the universe of assets contains an equity index, it is possible to use a²⁷ forecast of the equity risk premium - to which the risk free rate must be added - as the target expected return.
  
  Such a forecast is readily available on Aswath Damodaran’s website, who maintains estimates of the historical implied equity risk premiums for different countries with plenty of details in his yearly-updated paper Equity Risk Premiums (ERP): Determinants, Estimation and Implications²⁸.
  
  Such a forecast can also be directly extracted from the equity index historical returns data, c.f. the three procedures in Merton²⁹.
- When the universe of assets contains government bonds, it is possible to use the²⁷ forecast produced by the Bogle model for bonds as the target expected return.
- When the universe of assets contains a riskless asset with a non-null weight in $w^*$, it is possible to use the risk free rate as the target expected return, c.f. Sharpe².
One general remark:
- The periodicity of the target implied expected return $\mu_{target,i}$ must be consistent with the holding horizon of the asset covariance matrix $\Sigma$.

Estimating the Lagrange multiplier $\gamma$

Like the risk tolerance parameter, the Lagrange multiplier $\gamma$ associated to the budget constraint can be estimated by several different methods³⁰:

If $\lambda$ is already known
- Using the risk premium $\mu_p$ of the chosen reference mean–variance efficient portfolio, through the formula
  \[\gamma = \mu_p - \lambda \sigma_p^2\]
- Using the Shape Ratio $\text{SR}_p$ of the chosen reference mean–variance efficient portfolio, through the formula
  \[\gamma = \text{SR}_p \sigma_p - \lambda \sigma_p^2\]
To be noted that if $\lambda$ has been estimated using the same information as for $\gamma$, the formulas above will lead to $\gamma = 0$.

In other words, two independent pieces of information are required in order to properly estimate $\gamma$.
If $\lambda$ is not already known, using the target expected returns or risk premiums of two reference assets to simultaneously determine the two parameters $\lambda$ and $\gamma$.

That method is proposed in Herold¹ and consists in solving the following system of linear equations
\[\begin{cases} \mu_{target,i} = \lambda \left( \Sigma w^* \right)_i + \gamma \\ \mu_{target,j} = \lambda \left( \Sigma w^* \right)_j + \gamma \end{cases}\]
, where where $1 \leq i \leq n$ and $1 \leq j \leq n$, $j \ne i$, are the indexes of the two reference assets whose target expected returns $\mu_{target,i}$ $\mu_{target,j}$ are known.

Estimating the Lagrange multipliers $\Gamma$

In case of general linear constraints, the estimation of several Lagrange multipliers $\Gamma$ will require yet additional independent pieces of information.

Unfortunately, assuming that these pieces of information are available might partially defeat the purpose of finding implied returns³¹, which explains why practical guidelines for general linear constraints are difficult to find in the litterature.

What can be said, though, is that Herold’s method¹ is particularly suited to this case.

Caveats

In the previous sub-sections, several different procedures to estimate the risk tolerance parameter $\lambda$ and the Lagrange multiplier $\gamma$ have been described.

Best and Grauer⁹ shows that these procedures lead to differences and inconsistencies⁹ in the resulting implied expected returns and concludes that simply stated, it is impossible to satisfy simultaneously all […] ways of restricting [$\lambda$ and $\gamma$]⁹.

From a practical perspective, this means that the way $\lambda$ and $\gamma$ are estimated matters.

Implementation in Portfolio Optimizer

Portfolio Optimizer uses reverse optimization to compute implied expected through the endpoint /assets/returns/expected/estimation/implied.

Examples of usage

Black-Litterman portfolio allocation model

The Black-Litterman asset allocation model, created by Fischer Black and Robert Litterman, is a sophisticated portfolio construction method that […] uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of [implied] expected returns (the prior distribution) to form a new, mixed estimate of expected returns²⁰.

That equilibrium vector of implied expected returns is classically defined through the formula

\[\Pi = \lambda \Sigma w_{mkt}\]

, where:

$\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ is the covariance matrix of the asset arithmetic excess returns over the risk free rate at a given holding horizon
$\lambda$, $0 < \lambda < +\infty$ is the risk tolerance parameter
$w_{mkt} \in \mathbb{R}^{n}$ is the vector of the asset market capitalization weights
$\Pi \in \mathbb{R}^{n}$ is the vector of the implied arithmetic excess returns over the risk free rate over the holding horizon “at equilibrium”

and which comes from the usage of reverse unconstrained mean-variance optimization as detailled in the first section, with:

$\mu_{impl} = \Pi$
$w^* = w_{mkt}$, that is, the reference mean–variance efficient portfolio is chosen to be the market capitalization weighted portfolio
Arithmetic excess returns over the risk free rate instead of arithmetic returns

Another example of usage of reverse optimization in the Black-Litterman model can be found in Allaj³², which proposes to use reverse optimization to allow investors to express views on portfolio weights rather than directly on the parameters of the distribution of returns³².

Sharpe² and Herold¹ suggest a way to use reverse optimization for improving a portfolio in an iterative manner², with a case study presented in Herold¹.

In both cases, the underlying idea is to start from a portfolio an investor is confident with (the market portfolio, a core/satellite portfolio, etc.) and iteratively:

Compute its implied expected returns
Compare them with the investor’s absolute or relative return expectations
Revise the portfolio accordingly

In the case of an investment organization like a mutual fund, Sharpe² explains that procedure as follows:

Given a set of holdings, the implicit expected returns could be computed, then shown to the organization’s analysts.

Discrepancies could be used to suggest modifications in relative holdings.

The revised portfolio could then be subjected to the same analysis, leading to a new set of implicit expected returns with the process repeated as many times as appears useful.

In the case of an individual investor, Herold¹ explains that procedure as follows:

This motivates the investor to start with a portfolio which is intuitively reasonable, calculate several statistics such as portfolio risk, risk decomposition and implied returns, and then change this portfolio in an iterative way to align it to one’s expectations.

Prediction of future stock returns from mutual funds’ portfolio holdings

Shumway et al.¹⁷ applies reverse optimization³³ to extract implied expected returns from U.S. mutual funds’ portfolio holdings and constructs a measure of fund managers’ stock picking ability by correlating each manager’s revealed beliefs about stock returns with the subsequently realized returns¹⁷.

From that measure, Shumway et al.¹⁷ separates mutual funds’ managers into two categories - “informed” and “uninformed” managers - and proceeds to show that the difference in beliefs between these two groups of managers - [called the stock belief difference index (BDI)] - reveal information not embedded in the stock price¹⁷, especially for mid-cap stocks.

In more details, in the words of Shumway et al.¹⁷:

We sort stocks into deciles according to BDI and examine the subsequent three-month performance across the decile portfolios.

The results show that, on average, stocks with higher BDI statistics outperform stocks with lower BDI statistics, indicating that revealed beliefs contain valuable information about future stock returns.

We find the annualized performance spread between the top and bottom decile funds is about two to five percent, which is significant, both economically and statistically.

These performance differences are not explained by variations in risk or style factors.

We also sort stocks into three groups: small, medium, and big according to their size at the end of each quarter and redo our analysis.

We find that the significant performance difference between the top and bottom BDI deciles comes from the medium size group.

This result suggests that the stock picking skills of fund managers are reflected mostly in medium size stocks.

Shumway et al.¹⁷ finally concludes that the information provided in portfolio holdings on investor belief about expected future returns may improve existing estimation techniques, and hence have important implications for empirical asset pricing¹⁷.

Shrinkage estimator for asset sample mean returns

With Herold’s method¹ to compute implied expected returns, what would happen if the target expected returns were known for all assets instead of only two?

Mathematically, the associated system of linear equations would then be overdetermined and it would therefore seem natural⁹ to determine the two parameters $\lambda$ and $\gamma$ as solutions of the constrained linear least squares problem

\[\operatorname{argmin_{(\lambda, \gamma)}} \frac{1}{2} \lVert \mu_{impl} - \mu_{target} \rVert_2^2 \newline \textrm{s.t. } \lambda > 0\]

, where $\mu_{target} \mathbb{R}^{n}$ is the vector of the asset target expected returns.

It turns out that this is exactly what is proposed in Best and Grauer⁹, in which:

The vector of the asset target expected returns is taken equal to the vector of the asset sample mean returns.
The chosen reference mean–variance efficient portfolio is the market capitalization weighted portfolio.

The resulting implied expected returns can be viewed as a shrinkage of the asset sample mean returns towards expected returns compatible with mean-variance optimality for the market capitalization weighted portfolio.

Such a shrinkage estimator of the asset expected returns is complementary to the other well-known shrinkage estimators like the Bayes-Stein shrinkage estimator introduced in Jorion³⁴ or the quadratic shrinkage estimator introduced in Wang et al.³⁵.

Inspired by Best and Grauer⁹, Levy and Roll³⁶ proposes to move beyond the shrinkage of the asset sample mean returns and to additionally incorporate the asset sample standard deviations in the shrinkage procedure.

Analyzing the hidden assumptions embedded in an asset allocation

Do you believe US stocks will outperform the rest of the world by 2.5% per year going forward?³⁷

This is the punchline used by Felix Goltz in a recent LinkedIn post³⁷ to show how reverse optimization can be used to identify the hidden assumption made by index investors in the MSCI World w.r.t. future U.S. over ex-U.S. equities performance.

Figure 2, taken from that post³⁷, summarizes Goltz’s findings.

Figure 2. Implied expected returns for U.S. and ex-U.S. equities. 1985 - 2025. Source: Goltz.

In this sub-section, I will try to reproduce the underlying methodology.

From Figure 2, Goltz³⁷ uses reverse unconstrained mean-variance optimization to compute implied expected returns for U.S. and ex-U.S. equities.

For this, the following pieces of information are needed:

The universe of assets³⁸

The MSCI World index will be used as the reference (market capitalization weighted) mean-variance efficient portfolio³⁸ and:
- The MSCI U.S. index will be used to represent its U.S. equities component
- The MSCI World ex-USA index will be used to represent its ex-U.S. equities component
The expected risk free rate

In the context at hand, the aim is to analyze the difference of implied expected returns between U.S. equities and ex-U.S. equities.

Hence, there is no need to use an expected risk free rate and it will be set to 0%.
The vector of the reference mean-variance efficient portfolio weights $w^* \in \mathbb{R}^{2}$

Because the MSCI World index is used as the reference mean-variance efficient portfolio, the vector of portfolio weights is represented by the vector $w^* = \left( w_{mkt,US}, w_{mkt,ex-US} \right) {}^t$.

From the MSCI World Index factsheet, the weight of U.S. equities in the MSCI World index is 71.27% on 31st March 2026.

The corresponding vector of portfolio weights is thus $w^* = \left( 0.7127, 0.2873 \right) {}^t$.
An estimate of $\Sigma \in \mathcal{M}(\mathbb{R}^{2 \times 2})$, the covariance matrix of the arithmetic returns of the MSCI U.S. index and of the MSCI World ex-USA index, at a one year holding horizon

Using monthly arithmetic returns³⁹ over the period January 1979 - March 2026 and a simple annualization rule⁴⁰, the empirical covariance matrix $\Sigma$ of these two assets at a one year holding horizon is equal to
\[\Sigma = \begin{pmatrix} 0.0276 & 0.0179 \\ 0.0179 & 0.0228 \end{pmatrix}\]
An estimate of the risk tolerance parameter $\lambda$

It is possible to estimate $\lambda$ thanks to the historical risk premium⁴¹ of the MSCI U.S. index over the period January 1979 - March 2026.

As illustrated in Figure 3, the resulting value of the risk tolerance parameter seems to be around 3.01⁴² on 31st March 2026, even if one needs to be extremely cautious with this kind of historical estimation due to sampling variation²⁴.

Figure 3. Historical risk tolerance parameter, MSCI World index, expanding window over the period January 1979 - March 2026.

Thanks to the different estimates above, it is possible to compute the vector of implied expected returns $\mu_{impl} \in \mathbb{R}^{2}$, with $\mu_{impl} = \left( \mu_{impl,US}, \mu_{impl,ex-US} \right) {}^t$, through the formula $ \mu_{impl} = \lambda \Sigma w^* $.

This results in $ \mu_{impl} \approx \left( 7.48\%, 5.82\% \right) {}^t$, which means that investing in the MSCI World index in March 2026 comes with the hidden assumption of an outperformance of U.S. equities over ex-U.S. equities of about 1.7% annualy going forward.

In order to validate the robustness of that conclusion - keeping in mind the remarks in Best and Grauer⁹ - it is also possible to estimate the risk tolerance parameter $\lambda$ using a target annualized expected return of 9.09% for U.S. equities, as reported by Aswath Damodaran at the end of March 2026.

This results in $ \mu_{impl} \approx \left( 9.09\%, 7.07\% \right) {}^t$ and an implied outperformance of U.S. equities over ex-U.S. equities of about 2% annualy going forward.

That figure is closer to Goltz’s³⁷ 2.5%, and since 1.7-2% and 2.5% are in the same ballpark, this validates the methodology detailled above.

How is an investor now supposed to act when presented with that information?

Three possibilities:

Either he thinks that U.S. equities are indeed likely to outperform ex-U.S. equities by such an amount, in which case he has nothing to do
Or he thinks that the performance of U.S. equities v.s. ex-U.S. equities might revert to its historical mean⁴³, in which case he might want to decrease his exposure to U.S. equities⁴⁴

In that case, he can use Figure 4 as a decision support, for it depicts different possible ranges⁴⁵ of expected annualized outperformance of U.S. equities over ex-U.S. equities as implied by U.S. equities allocation weights.

Figure 4. Relative outperformance of U.S. equities over ex‑U.S. equities as implied by U.S. equities allocation weights, annualized expected range.

Incidentally, and maybe contrary to intuition⁴, Figure 4 highlights that neutrality towards U.S. equities outperformance is achieved with an allocation of about only⁴⁶ 35% to U.S. equities…
Or, for example because he his a long-time reader of the blog, he thinks that that information is incorrect because the budget constraint has not properly been taken into account in the previous exercice

In that case, instead of using unconstrained reverse mean-variance optimization with either the risk aversion parameter $\lambda$ or the target annualized expected return for U.S. equities, he can use budget-constrained reverse mean-variance optimization with both estimates and obtain the corresponding implied expected returns for ex-U.S. equities.

This is what is done in Battistella and McLoughlin⁴⁷, which applies budget-constrained reverse mean-variance optimization to 6 major equity components⁴⁸ of the MSCI All Country World index and which ultimately concludes that the problem [of implied U.S. equities outperformance] appears less severe⁴⁷ than what is usually reported!

Conclusion

To sum up this blog post, reverse optimization is a useful method that can help “harness the wisdom of crowds”⁶ and allows to derive the implicit return assumptions of market participants⁶ by exploiting the high predictability of risk, and relying on the plain math of mean-variance optimization⁶.

As usual, feel free to connect with me on LinkedIn or to follow me on Twitter.

–

See Herold, U. Computing implied returns in a meaningful way. J Asset Manag 6, 53–64 (2005). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹ ↩²⁰ ↩²¹ ↩²²
See Sharpe, William F., Imputing Expected Security Returns from Portfolio Composition., The Journal of Financial and Quantitative Analysis 9, no. 3 (1974): 463–72. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴
See Al-Thani, Khalifa and Mignacca, Domenico and Steyn, Andries Christiaan and Fusai, Gianluca and Caccioli, Fabio and Germano, G., The Quest for Neutrality in Asset Allocation (November 24, 2025). ↩ ↩² ↩³ ↩⁴
See David Ardia, Kris Boudt, Implied Expected Returns and the Choice of a Mean–Variance Efficient Portfolio Proxy, The Journal of Portfolio Management Summer 2015, 41 (4) 68 - 81. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³
Represented by a given list of asset weights and a given covariance matrix. ↩
See Two Sigma Client Solutions Team, Estimating Global Investor Views with Reverse Optimization. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸
See Ni, X., Malevergne, Y., Sornette, D., Woehrmann, P., 2011. Robust reverse engineering of cross-sectional returns and improved portfolio allocation performance using the CAPM. Journal of Portfolio Management 37 (4), 76–85. ↩ ↩²
Sometimes also called the risk aversion parameter, although strictly speaking, the risk aversion is defined as the inverse of the risk tolerance. ↩
See Best, M. J., & Grauer, R. R. (1985). Capital Asset Pricing Compatible with Observed Market Value Weights. The Journal of Finance, 40(1), 85–103. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹
The reasoning can be summarized as follows: an hypothetical stock/bond investor needs to increase his stock weight when the risk tolerance $\lambda$ increases due to the budget constraint becoming binding and not because he is more bullish on stocks; in that case, implied expected returns should not vary with $\lambda$. ↩
Because the Sharpe Ratio is homogeneous of degree 0, the budget constraint only fixes the scale, so that it is possible to first solve the unconstrained Sharpe Ratio maximization problem and then normalize the obtained optimal portfolio weights. ↩
See Markowitz, H.M. (March 1952). Portfolio Selection. The Journal of Finance. 7 (1): 77–91. ↩
Bound constraints are sometimes included in inequality constraints for theoretical analysis, but numerically, they are better handled separately. ↩
So that binding inequality constraints at optimality are known as well. ↩
Because nonnegativity constraints are not binding at optimality. ↩
See Dimitris Bertsimas, Vishal Gupta, Ioannis Ch. Paschalidis, (2012) Inverse Optimization: A New Perspective on the Black-Litterman Model. Operations Research 60(6):1389-1403. ↩
See Shumway, Tyler and Szefler, Maciej and Yuan, Kathy Zhichao, The Information Content of Revealed Beliefs in Portfolio Holdings (March, 18 2009). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹²
See Bevan, A., & Winkelmann, K. (1998). Using the Black-Litterman Global Asset Allocation Model: Three Years of Practical Experience. Fixed Income Research. ↩ ↩² ↩³
Which can for example be solved by using simple and exponentially weighted moving average models. ↩
See Thomas Idzorek, A step-by-step guide to the Black-Litterman model: Incorporating user-specified confidence levels, Forecasting Expected Returns in the Financial Markets, Academic Press, 2007, Pages 17-38. ↩ ↩²
For example, in case the reference mean-variance efficient portfolio is the market portfolio, the portfolio risk premium is by definition equal to the market risk premium. ↩
See Satchell, S., Scowcroft, A. A demystification of the Black–Litterman model: Managing quantitative and traditional portfolio construction. J Asset Manag 1, 138–150 (2000). ↩
In case of a constant risk free rate, the covariance matrix of the asset arithmetic returns is strictly equal to the covariance matrix of the excess asset arithmetic returns over the risk free rate; in case of a varying risk free rate, though, which is more realistic, the two asset covariance matrices are slighty different from each other but typically not different enough to materially alter any result. ↩ ↩²
See Grant, A., Kwon, O.K. & Satchell, S. Properties of risk aversion estimated from portfolio weights. J Asset Manag 25, 427–444 (2024). ↩ ↩²
See Partridge, L. and Croce, R., 2012. Risk Parity for the Long Run. ↩
See for example Envestnet Capital Markets Assumptions Methodology, 2023. ↩
Or, more generally, to use a/the forecast range as the target upper, lower, and central⁶ implied expected return for that asset. ↩ ↩²
See Damodaran, Aswath, Equity Risk Premiums (ERP): Determinants, Estimation and Implications - The 2023 Edition. ↩
See Robert C. Merton. On Estimating the Expected Return on the Market: An Exploratory Investigation. Journal of Financial Economics 8 (December 1980), 323-61. ↩
Unlike the risk tolerance parameter, there exists no sensible fixed value³⁰ for the Lagrange multiplier $\gamma$; attempts to circumvene that problem have been made, c.f. for example Xin and Ding³¹, but they are still relatively impractical. ↩
See Xin, L., Ding, S. Expected returns with leverage constraints and target returns. J Asset Manag 22, 200–208 (2021). ↩ ↩²
See Allaj, E. (2020). The Black–Litterman model and views from a reverse optimization procedure: An out-of-sample performance evaluation. Computational Management Science, 17(3), 465-492. ↩ ↩²
Called the dual problem of mean-variance optimization in Shumway et al.¹⁷. ↩
See P. Jorion, Bayes-Stein estimation for portfolio analysis, Journal of Financial and Quantitative Analysis 21 (3) (1986) 279–292. ↩
See C. Wang, T. Tong, L. Cao, B. Miao, Non-parametric shrinkage mean estimation for quadratic loss functions with unknown covariance matrices, Journal of Multivariate Analysis 125 (2014) 222–232. ↩
See Levy, M., and Roll, R. The Market Portfolio may be Mean-Variance Efficient After All. Review of Financial Studies, 23 (2010), pp. 2464-2461. ↩
See Felix Goltz, LinkedIn post. ↩ ↩² ↩³ ↩⁴ ↩⁵
All indexes are Gross Total Return and denominated in USD. ↩ ↩²
To be noted that, like for the expected risk free rate set to 0%, there is no need either to use asset historical excess arithmetic returns for the computation of $\Sigma$ - raw historical arithmetic returns are perfectly fine since we are not computing implied excess expected returns but implied expected returns. ↩
Multiplication of the monthly covariance matrix by 12. ↩
The U.S. historical monthly risk free rate has been taken from the Fama and French data library. ↩
The exact value used 3.015222148. ↩
See Marin Lolic, Robert Panariello, Som Priestley, Validating Your U.S. and Non-U.S. Equity Exposure, T. ROWE PRICE INSIGHTS, February 2023. ↩
An investor in a MSCI World ETF cannot directly decrease his exposure to U.S. equities, but there are at least two solutions: either keep the MSCI World ETF and buy an additional MSCI World ex-US ETF or fully replace the MSCI World ETF by a U.S. equities ETF and a MSCI World ex-US ETF. ↩
The ranges have been computed using the two procedures to estimate the risk tolerance parameter $\lambda$ already used to compute the 1.7-2% range in case the U.S. equities allocation weight was fixed to 71.27%; another possibility would be to use different risk tolerance parameters, for example 2 and 4, as suggested by Figure 3. ↩
And not 50% for example. ↩
See Battistella, Arnaud and McLoughlin, Nicholas, How much is too much? Part 1: Why 60% in US equities isn’t as crazy as it might sound (June 12, 2025). ↩ ↩²
U.S. equities, Japan equities, Europe ex-U.K. equities, U.K. equities, Pacific ex-Japan equities, Emerging Markets equities. ↩

The Market Rank Indicator: Measuring Financial Risk, Part 3

2026-03-04T00:00:00-06:00

In the previous post of this series on measuring financial risk, I described the absorption ratio, a measure of financial market fragility based on principal components analysis, introduced in Kritzman et al.¹.

In this new blog post, I will describe another measure of financial distress called the market rank indicator (MRI), this time related to the notion of condition number² of a matrix, introduced in Figini et al.².

As an example of usage, I will show how to use the market rank indicator to dynamically scale the market exposure of a portfolio of U.S. equities.

The market rank indicator

Definition

Let be:

$n$, the number of assets
$T$, the number of time periods, with $T > n$
$X \in \mathbb{R}^{T \times n}$, the matrix of the asset arithmetic or logarithmic returns for each of the $T$ time periods
$\sigma_1,…,\sigma_n$, the singular values of $X$ ordered such that $0 < \sigma_1 \leq … \leq \sigma_n$
$1\leq k \leq n$, the number of singular values $\lambda_1,…,\lambda_k$ to retain in the computation of the market rank indicator

The market rank indicator $\text{MRI}$ of the assets is defined² as the ratio between the largest singular value and the geometric mean of the $k$ smallest singular values of the matrix $X$², that is

\[\text{MRI}_k = \frac{ \sigma_n }{ \left( \prod_{i=1}^k \sigma_i \right)^{1/k} }\]

Alternative definition

Let be:

$n$, the number of assets
$T$, the number of time periods, with $T > n$
$X \in \mathbb{R}^{T \times n}$, the matrix of the asset arithmetic or logarithmic returns for each of the $T$ time periods
$\Sigma = \frac{1}{T} X {}^t X \in \mathcal{M}(\mathbb{R}^{n \times n})$, the asset returns covariance matrix
$\lambda_1,…,\lambda_n$, the eigenvalues of $\Sigma$ ordered such that $0 < \lambda_1 \leq … \leq \lambda_n$

Thanks to the relationship $\lambda_i = \frac{\sigma_i^2}{T}$, $i=1..n$, between the eigenvalues of $\Sigma$ and the singular values of $X$, the market rank indicator can alternatively be defined as the square root of the ratio between the largest eigenvalue and the geometric mean of the $k$ smallest eigenvalues of the matrix $\Sigma$, that is

\[\text{MRI}_k = \sqrt{ \frac{ \lambda_n }{ \left( \prod_{i=1}^k \lambda_i \right)^{1/k} } }\]

Generalized definition

In the two previous sub-sections, it is assumed that $T > n$.

This is to guarantee that the singular values of the matrix $X$ or that the eigenvalues of the matrix $\Sigma$ are not null.

In case $T \leq n$, the definition of the market rank indicator needs to be adapted, which is simply done by redefining [it] on the first $T$ [non zero] singular values².

Rationale

Figini et al.² highlights that what contains valuable information on market synchronization in a principal components analysis on asset returns is not only the strength of the first principal component, but also the weakness of the last components².

This observation leads to the proposal of the market rank indicator as a generalization of the condition number $\kappa(X)$ of the matrix $X$², with $\text{MRI}_1 = \kappa(X)$ as a special case.

Interpretation

Figini et al.² proposes both a mathematical and a financial interpretation of the market rank indicator:

Mathematically, the market rank indicator measures the difficulty to span $\mathbb{R}^n$ with the $n$ columns of $X$².

Indeed, for a given $1\leq k \leq n$, a large value of $\text{MRI}_k$ indicates that […] the columns of $X$ are a basis for an $(n − k)$-dimensional space².
Financially, the market rank indicator may be interpreted as a measure of distance of the asset return series from a market where the number of independent assets is reduced².

Indeed, again for a given $1\leq k \leq n$, a large value of $\text{MRI}_k$ indicates that $k$ dimensions/assets out of $n$ are not very useful in diversifying a portfolio².

Theoretical and empirical properties

Figini et al.² establishes several theoretical properties of the market rank indicator, among which:

Property 1: For any $1 \leq k \leq n$, $ 1 \leq \text{MRI}_k \leq \text{MRI}_1 $.

Property 2: For a given matrix $X$, $\text{MRI}_k$ is non-increasing in $1 \leq k \leq n$.

In addition, Figini et al.² empirically demonstrates that the market rank indicator is a suitable early warning system for future turbulence on the market² by showing³ that a large value of [that indicator] signals that the market will experience worse performances in the near future and that the probability of a large and negative return increases².

How to determine the number of singular values to retain?

When computing the market rank indicator, Figini et al.² suggests to use a number of singular values of about 1/3th the number of assets²:

The value for $k$ is set after a careful sensitivity analysis: an extensive comparison of the results obtained under different parameter settings showed that the best choice for $k$ is around one third of $n$.

To be noted, though, that depending on the specific context at hand, the number of singular values to retain could also be chosen using the numerical optimization method, assuming for example $k$ stochastic².

Comparison with other measures of financial risk

Comparison with the absorption ratio

Figini et al.² notes that critical market conditions² are characterized by two connected but not perfectly equivalent² phenomena:

An increase in the weight of the first principal components²
A reduction in the weight of the last components²

The absorption ration focuses on the quantity of variability explained by the largest components², that is, it focuses on the first phenomenon.

The market rank indicator, on the other hand, focuses on the second phenomenon², which theoretically makes it a perfect complement to the absorption ratio as a measure of financial risk.

Comparison with the covariance matrix effective rank

Familiar readers might remember that in a previous blog post, I discussed the effective rank⁴ of a covariance matrix, which is a real-valued extension of its rank allowing to measure the dimensionality of the associated universe of assets.

It turns out - as its name suggests - that the market rank indicator is very similar to the effective rank.

Indeed, when there is an increase in the co-movement of the asset returns², the vectors of the return time series become closer², which leads to a reduction in the diversification opportunities or, from a numerical point of view, a reduction in the market “dimension”² which is also captured by the effective rank.

As an illustration, Figure 1 compares both indicators when applied to the 11 Vanguard U.S. sector ETFs⁵ over the period 31th October 2005 - 30th January 2026, using the same methodology as described in the Exemple of usage section.

Figure 1. Market rank indicator v.s. effective rank, 11 Vanguard U.S. sector ETFs, 31th October 2005 - 30th January 2026.

On Figure 1, it is clearly visible that the market rank indicator and the effective rank move in a near perfect opposite direction to each other, which is confirmed numerically by a -84% correlation between these two indicators.

Still, the inverse relationship between the market rank indicator and the effective rank is not perfect⁶, so that they might be considered as two similar-but-different indicators.

Horse race for Value-at-Risk forecasting

Figini et al.² compares the empirical performances of the market rank indicator to those of three other measures of financial risk (the absorption ratio, the turbulence index and the average correlation) when forecasting the level of the 20-day ahead Value-at-Risk, and shows that it is the indicator which displays the best performances².

In more details, Figini et al.²:

First notes that in general, a good systemic risk indicator would allow to well discriminate between the regular behavior of the market and periods of distress² because intuitively, the VaR (or other risk indicators) should be low when the indicator is low and large when the indicator is large².
Then proceeds to empirically test this assumption in the case of the S&P 500 index, the STOXX Europe 600 index and the DAX index and concludes that:

Although each indicator presents a degree of forecasting power, the MRI is the indicator which displays the best performances for the three markets.

In fact, [as can be seen in Figure 2] not only the VaR associated to the highest percentile interval is the largest, but also the difference between the VaR associated to the highest interval and the one associated to the lowest interval present the largest outcomes as well […].

This shows that the MRI can best discriminate between regular and turbulent periods, with respect to other systemic risk indicators.

Figure 2. Future 20-day S&P 500 Value-at-Risk at 5% level v.s. quintiles of various systemic risk indicators, 02nd January 1992 - 01st July 2011. Source: Figini et al.

Implementation in Portfolio Optimizer

Portfolio Optimizer implements the computation of the market rank indicator through the endpoint /assets/indicators/market-rank, with the default number of singular values to retain suggested in Figini et al.².

Example of usage - Scaling the market exposure of a portfolio of U.S. equities

I will now illustrate the performances of the market rank indicator in a trading strategy desgined to capture the idea that exposure to stocks should be decreased when the market rank indicator is increasing.

For this:

I will work within the universe of U.S. equities represented by the SPY ETF and the 11 Vanguard U.S. sector ETFs⁵.
I will rely on daily return data⁷ over the period 31th October 2005 - 30th January 2026.
I will use the following trading strategy at the end of each month:
- Compute the sample covariance matrix of the 11 Vanguard U.S. sector ETFs over the past month, using the ETF daily arithmetic returns.
- Compute the market rank indicator associated to that covariance matrix, retaining 3 eigenvalues.
- Determine how elevated is this market rank indicator relative to its past 12-month history on a scale $s$ from 0% to 100%, thanks to a percentile rank.
- If $s$ is greater than 75% (i.e., relatively elevated recent market rank indicator), allocate the portfolio to cash (with 0% interest) else allocate the portfolio to U.S. equities (SPY ETF); in both cases, hold that portfolio for the next month.

The equity curve associated to that market rank indicator-based trading stategy is depicted in Figure 3.

Figure 3. MRI-based trading strategy v.s. buy and hold, SPY ETF, 31th October 2005 - 30th January 2026.

From Figure 3, the market rank indicator-based trading stategy does a good job in avoiding some serious drawdowns when compared to the buy and hold strategy, but unfortunately lags in terms of total return.

Figures:

Portfolio Management Strategy	Average Exposure	CAGR	Annualized Sharpe Ratio	Maximum (Monthly) Drawdown
Buy and hold	100%	10.9%	0.78	51%
MRI-based	68%	8.2%	0.79	33%

For comparison, Figure 4 additionally depicts the equity curve associated to a similar strategy, this time based on the absorption ratio of the 11 Vanguard U.S. sector ETFs⁸.

Figure 4. MRI-based and AR-based trading strategies v.s. buy and hold, SPY ETF, 31th October 2005 - 30th January 2026.

With figures:

Portfolio Management Strategy	Average Exposure	CAGR	Annualized Sharpe Ratio	Maximum (Monthly) Drawdown
Buy and hold	100%	10.9%	0.78	51%
MRI-based	68%	8.2%	0.79	33%
AR-based	69%	10.6%	1.03	21%

From Figure 4, it seems that that the absorption ratio is superior⁹ to the market rank indicator in terms of predictive performances for the specific universe of assets and over the specific period considered here.

Incidentally, that result is at odds with the findings of Figini et al.², so that the interested reader might want to dig deeper…

Conclusion

Together with the turbulence index and the absorption ratio¹⁰, the market rank indicator is a simple early warning indicator² for dangerous periods on financial markets.

Although it did not particularly shine v.s. the absorption ratio in the specific example studied in this blog post, the market rank indicator is definitely an additional tool to integrate² so as to improve [one’s] warning system².

Waiting for the analysis of the other measures of financial risk, feel free to connect with me on LinkedIn or to follow me on Twitter.

–

See Mark Kritzman, Yuanzhen Li, Sebastien Page and Roberto Rigobon, Principal Components as a Measure of Systemic Risk, The Journal of Portfolio Management Summer 2011, 37 (4) 112-126. ↩
See Silvia Figini, Mario Maggi, Pierpaolo Uberti, The market rank indicator to detect financial distress, Econometrics and Statistics, Volume 14, 2020, Pages 63-73. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹ ↩²⁰ ↩²¹ ↩²² ↩²³ ↩²⁴ ↩²⁵ ↩²⁶ ↩²⁷ ↩²⁸ ↩²⁹ ↩³⁰ ↩³¹ ↩³² ↩³³ ↩³⁴ ↩³⁵ ↩³⁶ ↩³⁷ ↩³⁸ ↩³⁹ ↩⁴⁰
Using returns of the 10 sectors composing the S&P 500 index. ↩
See Olivier Roy and Martin Vetterli, The effective rank: A measure of effective dimensionality, 15th European Signal Processing Conference, 2007. ↩
These are “VOX”, “VCR”, “VDC”, “VDE”, “VFH”, “VHT”, “VIS”, “VGT”, “VAW”, “VNQ”, “VPU”, c.f. https://investor.vanguard.com/investment-products/etfs/sector-etfs. ↩ ↩²
Most probably due to the fact that the number of eigenvalues used by each indicator is different. ↩
(Adjusted) prices of the ETFs have have been retrieved using Tiingo. ↩
The absorption ratio-based trading strategy is exactly the same as the market rank indicator-based trading strategy, except that the absorption ratio is computed instead of the market rank indicator; that’s the only change. ↩
The two indicators are agreeing most of the time - their correlation is 85%; as a side note, it could be interesting to understand what happens when they are disagreeing… ↩
As well as the effective rank; I cannot insist too much on that one, though, because - to this date - I did not analyze its usefulness as a measure of financial risk on the blog. ↩

More Bootstrap Simulations with Portfolio Optimizer: the Autoregressive Online Bootstrap

2026-02-01T00:00:00-06:00

In a previous article, I described several classical bootstrap techniques — i.i.d. bootstrap, circular block bootstrap, and stationary block bootstrap — and showed how the stationary block bootstrap could be used to simulate future price paths for financial assets by following the methodology of Anarkulova et al.¹.

In this blog post, I will detail another bootstrap technique called the autoregressive online bootstrap² and introduced in Palm and Nagler², that is best described as a multiplier bootstrap³ coupled with an autoregressive sequence of weights specifically chosen to make it useable with streaming time series data.

As an example of usage, I will simulate alternative price histories for the SPY and TLT ETFs, which are ETFs representative of the US stock market and of the long-term US Treasury bonds.

Mathematical preliminaries

Let $X_1, …, X_n$⁴, $n \geq 1$ be a sample of data observed from a population.

Limitations of classical bootstrap techniques in an online setting

In a typical⁵ online setting, the length $n$ of the sample of data $X_1, …, X_n$ is growing infinitely, with new events [that] are observed at the moment they occur².

In such a setting, the classical bootstrap techniques described in the previous blog post on bootstrap simulations require all data observed so far² at any given point in time.

Indeed:

The i.i.d. bootstrap, by definition, requires keeping track of the entire observed sample $ {X_1, …, X_n} $².
The circular and the stationary block bootstrap require all blocks […] to increase in size with $n$², so that to compute the bootstrap in practice, the entire data set $ {X_1, …, X_n} $ needs to be kept in memory and processed fully, every time the block size changes².

Depending on the domain of application, this can be a serious computational limitation when:

n is large
n is moderately large but the underlying $n$ observations require a lot of computer memory to be stored
n is moderately large but the computation time is limited (e.g. real-time applications)

The multiplier bootstrap

The multiplier bootstrap³ is a general class of bootstrapping schemes based on perturbations of the original observations with suitable weights².

In other words, compared to classical bootstrap techniques, the multiplier bootstrap replaces the idea of “randomly resampling observations” with the idea of “randomly reweighting observations”, which enables it to be applied in an online setting.

For i.i.d. data $X_1, …, X_n$, it is defined as follows:

Let $V_1, …, V_n$ be $n$ i.i.d. random variables of unit mean/variance.
Let $ \bar V_i = \frac{1}{i} \sum_{j=1}^{i} V_j $ the running mean of these random variables, which can be computed recursively through the formula
\[\bar V_i = \frac{(i-1) \bar V_{i-1} + V_i }{i}\]
The multiplier bootstrap samples $ X_1^*,…,X_n^* $ are then defined as
\[X_i^* = \frac{V_i}{\bar V_i} X_i\]

The autoregressive online bootstrap

Methodology

The autoregressive online bootstrap² is a specific instance of the multiplier bootstrap that generates a sequence of random weights evolving according to an autoregressive process centered around 1:

\[V_i = 1 + \rho_i \left( V_{i-1} - 1 \right) + \sqrt{1 - \rho_i^2} \zeta_i, i=1..n\]

, with:

$V_0 = 0$
$\rho_i = 1 - i^{-\beta}$, $ 0 < \beta < \frac{1}{2}$
$\zeta_i \sim \mathcal N(0,1), i=1..n$

The associated autoregressive online bootstrap samples $ X_1^*,…,X_n^* $ are then defined as

\[X_i^* = \frac{V_i}{\bar V_i} X_i\]

Rationale

Palm and Nagler² notes that for the multiplier bootstrap to remain valid for time series:

The dependencies between weights $V_i$ and $V_j$ must increase with the sample size $n$, but at the same time remain almost independent when the time gap $|i-j|$ is sufficiently large compared to $n$².
A scaling of the weights by their arithmetic mean is also necessary²

This is what they propose with the autoregressive online bootstrap technique.

Properties

The main result of Palm and Nagler² is that under mild conditions on the original data $X_1, …, X_n$, the autoregressive online bootstrap is a consistent resampling scheme for the mean and for any continously differentiable transformation of the mean⁶ of an univariate or a multivariate time series.

How to select the parameter $\beta$?

The parameter $\beta$ controls the behaviour of the autoregressive online bootstrap samples in a way similar to how the block length controls the behaviour of the block bootstrap samples in classical block bootstrap techniques.

Intuitively:

A small value of $\beta$ leads to slowly changing weights and thus to long stretches of bootstrap samples that are consistent with the original observations, although up- or down-weighted.

This is similar in spirit to having a long block size in a block bootstrap technique.
On the contrary, a large value of $\beta$ leads to quickly changing weights and thus to rapidly varying bootstrap samples.

This is similar in spirit to having a short block size in a block bootstrap technique.

Palm and Nagler² demonstrates that $\beta_{opt} = \sqrt{2} - 1$ allows for an optimal bias-variance trade-off², so that unless there is a specific need to play with this parameter, using that value should be the default choice.

Practical performances

Palm and Nagler² compares the practical performances of the autoregressive online bootstrap to an i.i.d. multiplier bootstrap and to a block bootstrap technique called the moving average block bootstrap⁷.

In particular, Palm and Nagler² shows that:

The two time series bootstraps achieve approximately correct coverage in all [studied] scenarios, even in the presence of nonlinear [AR(2)-GARCH(1,1)] dependencies² corresponding to a stochastic volatility process.
The autoregressive online bootstrap allows for cheap online updates - in constant time, as illustrated in Figure 1 - contrary to the moving average block bootstrap.

Figure 1. Computation time per online update of 200 bootstrap samples as the bootstrap techniques progress through a stream of 2000 samples. Source: Palm and Nagler.
The autoregressive online bootstrap has a slightly higher variance compared to the moving average block bootstrap, which is the cost [to] pay for its computational advantage².

Caveats

One important limitation of the autoregressive online bootstrap is that the weights $V_i$,$i=1..n$ can be negative.

For this reason, in finance, logarithmic asset returns should then be prefered to either asset prices (since they could become negative) or arithmetic asset returns (singe they could become < -100%).

Implementations

Implementation in Portfolio Optimizer

Portfolio Optimizer implements the autoregressive online bootstrap through the endpoint /assets/returns/simulation/bootstrap/online.

Implementation elsewhere

An implementation of the autoregressive online bootstrap in Python is available at https://github.com/nicolaipalm/online-bootstrap-implementation by its authors.

Example of usage - Simulation of alternative price histories for ETFs

As an example of usage, I propose to use the autoregressive online bootstrap to generate alternative price histories for the SPY and TLT ETFs and compute a couple of associated descriptive statistics⁸.

Alternative price histories for the SPY ETF

Figure 2 illustrates 10 synthetic price histories generated by applying the autoregressive online bootstrap to the logarithmic returns of the SPY ETF over the period 1st January 2025 - 31th December 2025⁹.

Figure 2. Alternative price histories for the SPY ETF, autoregressive online bootstrap, 1st January 2025 - 31th December 2025.

On Figure 2, it is visible that the autoregressive online bootstrap allows for a wide variety of scenarios to be generated.

This is confirmed in Figure 3, which depicts the distributions of the first four moments of the logarithmic returns of 1000 synthetic price histories generated by applying the autoregressive online bootstrap to the logarithmic returns of the SPY ETF over the period 1st January 2025 - 31th December 2025⁹.

Figure 3. Distributions of the first four moments of 1000 alternative price histories for the SPY ETF, autoregressive online bootstrap, 1st January 2025 - 31th December 2025.

Figure 3 empirically demonstrates two points:

It confirms the mean-preservation property of the autoregressive online bootstrap.

As a side note, in case this is a problem for the usage at hand, it is always possible to alter the mean of the generated scenarios in a perfectly controlled way, c.f. a previous blog post.
It shows that the autoregressive online bootstrap can generate scenarios with a wildly different standard deviation/skewness/kurtosis compared to the original one.

Here, to be noted that some scenarios might seem implausible, with a kurtosis > 100 for example.

Nevertheless, implausible does not equal impossible¹⁰, so that depending on the usage at hand, such scenarios could be eliminated or put aside for specific stress-testing.

Figure 4 illustrates the variety of scenarios in another way, by looking at the correlation between the logarithmic returns of the SPY ETF and of its 1000 alternatives, which approximatively ranges from -0.4 to 1.

Figure 4. Distribution of the correlation between the SPY ETF price history and 1000 alternative price histories, autoregressive online bootstrap, 1st January 2025 - 31th December 2025.

Alternative price histories for the SPY and TLT ETFs

Figure 5 illustrates the behaviour of the autoregressive online boostrap in terms of bivariate correlation when applied to the logarithmic returns of both the SPY and TLT ETFs over the period 1st January 2025 - 31th December 2025⁹.

Figure 5. Distribution of the correlation between the two components of 1000 alternative price histories for the SPY and TLT ETFs, autoregressive online bootstrap, 1st January 2025 - 31th December 2025.

From Figure 5, it appears that the whole range of admissible correlations $[-1,1]$ is achievable through the autoregressive online bootstrap, with the majority of scenarios falling in the interval $[-0.25, 0.50]$.

Conclusion

The autoregressive online bootstrap introduced in Palm and Nagler² and detailled in this blog post provides an alternative to classical block bootstrap techniques for time series that is specifically taylored to streaming data.

Don’t hesitate to experiment with it and evaluate whether it could replace your current boostrap methodology!

As usual, feel also free to connect with me on LinkedIn or to follow me on Twitter.

–

See Anarkulova, Aizhan and Cederburg, Scott and O’Doherty, Michael S., The Long-Horizon Returns of Stocks, Bonds, and Bills: Evidence from a Broad Sample of Developed Markets (November 15, 2021). ↩
See Nicolai Palm, Thomas Nagler; An Online Bootstrap for Time Series; Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:190-198. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹ ↩²⁰
See Van Der Vaart, A. W. and Wellner, J. A. Weak convergence. In Weak convergence and empirical processes, pp. 16–28. Springer, 1996. ↩ ↩²
The observations $X_1, …, X_n$ can be observations of random variables or of random vectors. ↩
It is also possible that for computational reasons - like a huge $n$ - a complete data set is available from the start², but is processed sequentially or in batches². ↩
Like the variance, c.f. Palm and Nagler². ↩
See Buhlmann, P. L. (1993). The blockwise bootstrap in time series and empirical processes. PhD thesis, ETH Zurich. ↩
In practice, the generation of such alternative price histories would typically be integrated into a backtesting engine. ↩
(Adjusted) prices of the SPY and TLT ETFs have have been retrieved using Tiingo. ↩ ↩² ↩³
Who could have predicted the price action of gold and silver on 30th January 2026? ↩

Completing a Correlation Matrix: Another Problem from Finance

2026-01-06T00:00:00-06:00

The previous post of this series on mathematical problems related to correlation matrices introduced the nearest correlation matrix problem¹, which consists in determining the closest² valid correlation matrix to an approximate correlation matrix

In this blog post, I will now describe the correlation matrix completion problem, which consist in filling in the missing coefficients of a partially specified correlation matrix in order to produce a valid correlation matrix.

As noted in Dreyer et al.³, such matrices often arise in financial applications when the number of stochastic variables becomes large or when several smaller models are combined in a larger model³.

After a couple of reminders on correlation matrices, I will detail the mathematical formulation of the correlation matrix completion problem, discuss its exact and approximate solutions and I will finally illustrate how this problem naturally appears when working with capital market assumptions from financial institutions.

Mathematical preliminaries

Let $n$ be a number of assets.

Correlation matrices

Let $C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ be a square matrix.

$C$ is a correlation matrix if and only if

$C$ is symmetric: $C {}^t = C$
$C$ is unit diagonal: $C_{i,i} = 1$, $i=1..n$
$C$ is positive semi-definite, that is, $x {}^t C x \geqslant 0, \forall x \in \mathbb{R}^n $

The sets of correlation matrices

Let be the following convex sets:

$\mathcal{S}^n_{+} = \{ X \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $X {}^t = X$ and $X$ is positive semi-definite $\}$
$\mathcal{S}^n_{++} = \{ X \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $X {}^t = X$ and $X$ is positive definite $\}$
$\mathcal{E}^n = \{ X \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $X {}^t = X$ and $x_{ii} = 1, i = 1,…,n \}$

Then:

The set of correlation matrices is the convex compact set $\mathcal{S}^n_{+} \cap \mathcal{E}^n$.
The set of invertible correlation matrices is the open bounded convex set $\mathcal{S}^n_{++} \cap \mathcal{E}^n$.

As a geometrical side note, the set of vectorized correlation matrices defines a subset of the unit hypercube in $\mathbb{R}^n$ called the elliptope, c.f. for example the website Convex Optimization.

Partial correlation matrices completion

A partial correlation matrix $C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ is a symmetric unit diagonal matrix⁴ whose coefficients $c_{ij} = c_{ji}$ are not all specified.

A partial correlation matrix is said to be partial positive (semi-)definite when its specified principal minors are positive (semi-)definite.

A positive semi-definite completion - or simply a completion - of a partial correlation matrix $C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ is a correlation matrix $C^* \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $c_{ij}^{*} = c_{ij}$ whenever the coefficient $c_{ij}$ is specified in $C$.

A positive definite completion of a partial correlation matrix $C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ is a positive definite correlation matrix $C^* \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $c_{ij}^{*} = c_{ij}$ whenever the coefficient $c_{ij}$ is specified in $C$.

Undirected graph associated to a correlation matrix

Let $C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ be a partial correlation matrix.

It is possible⁵ to associate an undirected graph $G = \left( V, E \right)$ to $C$, defined as follows:

$V = \{ 1, 2, …, n \}$ is the set of vertices
$E$ is the set of edges, with $(i,j) \in E$ whenever the coefficient $c_{ij}$ is specified, with $i \ne j$.

The graph $G$ is then said to be chordal if every cycle of length greater than or equal to 4 has a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle⁶.

As a reminder, a cycle in G is a sequence of $k \geq 3$ pairwise distinct vertices $\left( v_1, …, v_s \right)$ such that $\left( v_1, v_2 \right)$, $\left( v_2, v_3 \right)$, …, $\left( v_{k-1} v_k \right)$, $\left( v_k v_1 \right)$ $\in E$, with $k$ called the length of the cycle.

The correlation matrix completion problem

Problem formulation

Let be:

$C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ a partial correlation matrix.
$\mathcal{N}$ the index set of the specified off-diagonal elements of $C$.
$\mathcal{U}^n = \{ X \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $x_{ij}=c_{ij}$ for $(i,j) \in \mathcal{N} \}$

Then:

The correlation matrix completion problem is the problem of finding a correlation matrix $C^* \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ that completes the matrix $C$, that is, finding $C^* \in \mathcal{S}^n_{+} \cap \mathcal{U}^n$.
The positive definite correlation matrix completion problem is the problem of finding a positive definite correlation matrix $C^* \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ that completes the matrix $C$, that is, finding $C^* \in \mathcal{S}^n_{++} \cap \mathcal{U}^n$.

Existence of solutions

Contrary to the nearest correlation matrix problem, the correlation matrix completion problem does not always admit a solution.

Indeed, an already necessary condition for a completion to exist is for the partial correlation matrix to be partial positive semi-definite⁵.

So, for example, the following partial correlation matrix does not admit any completion:

\[C_1 = \begin{pmatrix} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & . \\ -1 & -1 & . & 1 \end{pmatrix}\]

Beyond this necessary condition, Fiedler⁷, Grone et al.⁵ and Smith⁸ all establish the following result⁹:

A partial positive semi-definite correlation matrix is completable regardless of the values of the specified correlations if and only if the undirected graph associated to these specified correlations is chordal.

To be noted that when that graph is not chordal, nothing can be said in general because the existence of a completion then depends on the exact values of the specified correlations.

As an illustration:

Let $C_2 \in \mathcal{M} \left( \mathbb{R}^{4 \times 4} \right)$ be the following partial positive definite correlation matrix:
\[C_2 = \begin{pmatrix} 1 & 1 & . & 1 \\ 1 & 1 & 1 & . \\ . & 1 & 1 & . \\ 1 & . & . & 1 \end{pmatrix}\]
The undirected graph associated to that partial correlation matrix is the graph with the set of vertives $V = \{ 1, 2, 3, 4 \}$ and the set of edges $G = \{ (1,2), (1,4), (2,3) \}$, depicted in Figure 1.

Figure 1. Undirected graph associated to the partial correlation matrix C2.

From Figure 1, that graph is chordal, because there are no cycles of length >= 4.

Consequently, the partial correlation matrix $C_2$ admit a positive definite completion and the existence of that completion does actually even not depend on the values of the specified correlations (here, all 1s).
Let now $C_3 \in \mathcal{M} \left( \mathbb{R}^{4 \times 4} \right)$ be the following partial positive semi-definite correlation matrix:
\[C_3 = \begin{pmatrix} 1 & 1 & . & 0 \\ 1 & 1 & 1 & . \\ . & 1 & 1 & 1 \\ 0 & . & 1 & 1 \end{pmatrix}\]
The undirected graph associated to that partial correlation matrix is the graph with the set of vertives $V = \{ 1, 2, 3, 4 \}$ and the set of edges $G = \{ (1,2), (1,4), (2,3), (3,4) \}$, depicted in Figure 2.

Figure 2. Undirected graph associated to the partial correlation matrix C3.

From Figure 2, that graph is not chordal, because there is only one cycle of length 4 and that cycle does not have any chord.

Consequently, the partial correlation matrix $C_3$ might or might not admit a completion, nothing can be said at this stage.

For the interested reader, a couple of other theoretical results can be found in Fiedler⁷, which also notes that the general [completion] problem […] seems to be difficult⁷.

Unicity of solutions

Again contrary to the nearest correlation matrix problem, the correlation matrix completion problem does not generally admit a unique solution when one exists.

Indeed, Grone et al.⁵ establishes that the set of all positive semi-definite completions of a partial correlation matrix is in general a convex compact set and not a singleton.

This result can be illustrated with one and two missing correlations:

One missing correlation

The partial correlation matrix $C_4 = \begin{pmatrix} 1 & . \\ . & 1 \end{pmatrix} $ has one missing correlation.

By introducing the variable $x$ representing that missing correlation, a completion is a valid completion if and only if it has a positive or null determinant, that is, $\det(x) = 1 - x^2 \geq 0$.

That condition being equivalent to the condition $x \in [-1,1]$, all correlation matrices of the form $ C^*_4(x) = \begin{pmatrix} 1 & x \\ x & 1 \end{pmatrix} $ with $x \in [-1,1]$ are completions of $C_4$.
Two missing correlations

The partial correlation matrix $C_5 \in \mathcal{M} \left( \mathbb{R}^{5 \times 5} \right)$ represented in Figure 3 has two missing correlations.

Figure 3. Partial correlation matrix with 2 missing correlations. Source: Georgescu et al.

Because all the specified sub-matrices of $C_5$ are positive semi-definite, a completion is again a valid completion if and only if it has a positive or null determinant.

By introducing the variables $x$ and $y$ representing the two missing correlations, the determinant of $C_5$ can be factored as $ \det(x,y) $ $ \approx$ $ −0.18 \left( 0.67x^2 +0.75y^2 −xy −0.32x−0.24y+0.17 \right)$.

Non-negativity of that determinant is then equivalent to the condition
\[0.67x^2 +0.75y^2 −xy −0.32x−0.24y+0.17 \leq 0\]
, which defines the 2-dimensional ellipse (and its interior) depicted in Figure 4.

Figure 4. Feasible region for Georgescu et al.'s partial correlation matrix completions.

All correlation matrices $C_5^{*}(x,y) \in \mathcal{M} \left( \mathbb{R}^{5 \times 5} \right)$ with the same correlations as $C_4$, a correlation $c^{*}_{34}$ $=$ $x$ and a correlation $c^{*}_{35}$ $=$ $y$ - with $x,y$ satisfying the above relationship - are thus completions of $C_5$.

The existence of infinitely many completions¹⁰ naturally leads to trying to find a best-estimate completion in some sense⁶, which does exist and is called the maximum determinant completion⁶.

The maximum determinant correlation matrix completion problem

Fiedler⁷ originally demonstrated that if a positive definite partial correlation matrix admits a completion, then, there is a unique matrix in the (nonempty) class of all positive definite completions […] that has maximum determinant⁸.

More recently, by introducing a generalized determinant that gives the determinant of the nonsingular part of [a] matrix¹¹, Dreyer¹¹ established a similar result for positive semi-definite partial correlation matrices.

That completion, called the maximum determinant completion - or the Max-Det completion - has several intesting theoretical properties, which makes it an ideal candidate for guaranteeing the unicity of the completion of a correlation matrix.

Problem formulation

Let be:

$C \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ a partial correlation matrix.
$\mathcal{N}$ the index set of the specified off-diagonal elements of $C$.
$\mathcal{U}^n = \{ X \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ such that $x_{ij}=c_{ij}$ for $(i,j) \in \mathcal{N} \}$

The maximum determinant correlation matrix completion problem can be cast as a generalization of a semidefinite programming problem:

\[C^* = \operatorname{argmax} \log ( \det (X) ) \text{ s.t. } X \in \mathcal{S}^n_{+} \cap \mathcal{E}^n \cap \mathcal{U}^n\]

Assuming that a solution exist, c.f. the previous section, Grone et al.⁵ and Olvera Astivia¹² show using standard results from mathematical optimization theory that it is necessarily unique.

Mathematical properties of the solution

Georgescu et al.⁶ summarizes the main properties of the maximum determinant correlation matrix completion:

For the multivariate normal model, it maximizes the entropy of the distribution described by the matrix

van der Schans and Boer¹³ comments this property as follows:

Since the already specified [correlations] imply a dependence between variables between which no dependence is specified and dependence reduces the amount of uncertainty in a system, the intuitive interpretation of entropy maximization is not introducing more dependence than is already implied by the already specified [correlations].
Again for the multivariate normal model, it maximizes the likelihood of the correlation matrix
It corresponds to the analytic centre of the feasible region described by the positive semi-definiteness constraints⁶

In other words, for a given partial correlation matrix, its maximum determinant completion lies as “deep” as possible¹⁴ inside the set of all its positive definite completions.

Computation of the solution

The original algorithmic approach to solving determinant maximization problems with matrix constraints has been to use interior point methods¹⁵.

More recently, dual projected gradient methods have been developed that are better able to scale with the dimensionality of the problem¹⁶.

Nevertheless, the drawbacks of these algorithms, from an application point of view, are that they are difficult to implement […], do not always converge if inconsistent starting [correlations] are specified and, finally, global optimization is too slow and memory consuming for large matrices¹³.

More on “inconsistent starting correlations” later.

For these reasons, it is tempting to [simply] set the missing [correlations] to zero³, but this approach has several shortcomings³, among which the fact that forcing unspecified correlations to zero¹⁷ might not be ideal for financial applications where most assets are correlated to each other.

To illustrate this point, Figure 5 represents a partial correlation matrix with a block of missing correlations.

Figure 5. Partial correlation matrix with a block of missing correlations. Source: Georgescu et al.

The maximum determinant completion of the block of missing correlations is⁶

\[E_1 = \begin{pmatrix} 0.1000 & 0.1500 & 0.0500 & 0.0750 \\ 0.2400 & 0.3600 & 0.1200 & 0.1800 \\ 0.2200 & 0.3300 & 0.1100 & 0.1650 \\ 0.2600 & 0.3900 & 0.1300 & 0.1950 \\ 0 & 0 & 0 & 0 \end{pmatrix}\]

, while the completion of the same block of missing correlations obtained after setting them to zero and computing the nearest correlation matrix to the resulting full approximate correlation matrix is⁶

\[E_2 = \begin{pmatrix} 0.0022 & 0.0084 & 0.0004 & 0.0035 \\ 0.0003 & 0.0011 & 0.0001 & 0.0005 \\ 0.0025 & 0.0098 & 0.0005 & 0.0040 \\ 0.0042 & 0.0164 & 0.0008 & 0.0067 \\ 0 & 0 & 0 & 0 \end{pmatrix}\]

Comparing the two sub-matrices $E_1$ and $E_2$, it is clear that the completed correlations of $E_2$ are much closer to zero than those of $E_1$, thus imposing an additional soft constraint of “closeness to zero” to the correlation matrix completion problem.

Reverting to simple heuristics for the correlation matrix completion problem might thus be potentially dangerous, depending on the context.

Hopefully, van der Schans and Boer¹³ proposes an heuristic which does not suffer from these shortcomings and that:

Is fast, which makes it suitable for applications in which the computation time is important¹³
Tries to not introduce more dependence between variables than is implied by the initially specified [correlations]¹³, contrary to the “setting missing correlations to zero” heuristic
Additionally corrects for inconsistencies in the already specified [correlations]¹³, but again, more on this later
Empirically produces completions with an average correlation difference with the maxdet completion [that is] reasonable¹³

A drawback of this heuristic, though, is that it depends on the ordering of the rows and columns of the [correlation] matrix, i.e. the heuristic will yield a different result if first rows and columns of [the matrix] are interchanged before starting the procedure¹³, which might or might not be acceptable in practice.

As a side note, explicit solutions of the maximum determinant correlation matrix completion problem are known in specific cases, for example for correlation matrices that follow an L-shaped, block diagonal pattern which is a common structure in insurance problems¹², c.f. Georgescu et al.⁶

The infeasible maximum determinant correlation matrix completion problem

In the previous section, a solution to the maximum determinant correlation matrix completion problem was assumed to exist.

There are at least two practical problems with this assumption:

A partial correlation matrix might theoretically admit a completion, but numerical round-off errors might prevent an algorithm from finding it

An example of such an ill-behaved matrix can be found in Glunt et al.¹⁸, but it is a partial covariance matrix and not a partial correlation matrix…

Building a similar example using a partial correlation matrix is certainly doable, but I failed to do so quickly, so, no example here.
More typically, the initially specified [correlations] can be inconsistent in the sense that no valid completion exists¹³

Here, the partial correlation matrix $C_1$ perfectly illustrates this point, even though it is a rather extreme example.

In both cases, simply failing to compute a solution might be unacceptable (e.g. for downstream pipelines), so that something must be done.

Unfortunately, the currently existing literature does not focus on algorithms that both complete partially specified matrices and also correct for inconsistencies¹³.

In other words, there is no well-established formulation of what could be called the infeasible maximum determinant correlation matrix completion problem.

At minimum, what can be said is that any solution to this new problem must involve a trade-off between

The closeness to the initial partial correlation matrix, for example in terms of the Frobenius distance
The value of the determinant

Incidentally, this is exactly how the heuristic algorithm of van der Schans and Boer¹³ is working: the specified [correlations] are adjusted as little as possible¹³ and the introduced extra (conditional) dependence between variables is as little as possible¹³.

When applied to the partial correlation matrix $C_1$, van der Schans and Boer’s algorithm¹³ gives

\[C^*_1 = \begin{pmatrix} 1 & -0.99 & -0.07 & -0.07 \\ -0.99 & 1 & -0.07 & -0.07 \\ -0.07 & -0.07 & 1 & 0.98 \\ -0.07 & -0.07 & 0.98 & 1 \end{pmatrix}\]

As expected, the completed correlation matrix $C^*_1$ is very far from the partial correlation matrix $C_1$, with only one correlation close to its initial value of -1 (-0.99).

Again, this is an extreme infeasible example, but it empirically demonstrates that adjusting the initially specified correlations, even as little as possible¹³, can lead to a completed correlation matrix that does not ressemble at all the initial partial correlation matrix.

Implementation in Portfolio Optimizer

Portfolio Optimizer implements two methods to complete a partial correlation matrix through the endpoint /assets/correlation/matrix/completed:

One proprietary method that guarantees to find the maximum determinant completion if it exists or that guarantees to minimally adjust - in terms of Frobenius distance - the partially specified correlation matrix so that a maximum determinant completion exists
The heuristic method of van der Schans and Boer¹³, with additional tweaks to improve its numerical robustness

For comparison with van der Schans and Boer’s algorithm¹³, the proprietary method of Portfolio Optimizer applied to the partial correlation matrix $C_1$ gives:

\[C^{**}_1 = \begin{pmatrix} 1 & -0.30 & -0.59 & -0.59 \\ -0.30 & 1 & -0.59 & -0.59 \\ -0.59 & -0.59 & 1 & 1 \\ -0.59 & -0.59 & 1 & 1 \end{pmatrix}\]

Comparing the two completed matrices $C_1^{*}$ and $C_1^{**}$, it appears that the matrix $C_1^{**}$ is much¹⁹ closer to the partial correlation matrix $C_1$ than the matrix $C_1^{*}$, which is consistent with the expected behaviour of Portfolio Optimizer.

Example of usage - Completing partial correlation matrices from financial institutions

Major financial institutions regularly provide forecasts of future risk/return characteristics for broad asset classes over the next 5 to 20 years, called (Long Term) Capital Market Assumptions (LTCMA).

In addition to the future expected volatility, the risk forecasts sometimes also include future expected correlation matrices, which is for example the case with J.P. Morgan.

Unfortunately, these correlation matrices are typically partially specified.

In addition, even if they were fully specified, combining capital market assumptions from several financial institutions would render them partial due to all institutions not covering the same asset classes…

So, as an example of usage, I propose to compute the maximum determinant completion of the partial correlation matrix provided by Blackrock as part of their November 2025 5-year capital market assumptions and represented in Figure 6.

Figure 6. Five-year expected future partial correlations between major asset classes, NZD currency, 13th November 2025. Source: Blackrock.

The resulting maximum determinant completed correlation matrix is displayed in Figure 7.

Figure 7. Five-year expected future Max-Det completed correlations between major asset classes, NZD currency, 13th November 2025.

I would like to mention that such an example of usage was inspired by a LinkedIn post from Peter Urbani, who proposes to complete Blackrock’s partial correlation matrix into the correlation matrix displayed in Figure 8 thanks to a 2-factor model built from the fully specified global equities/global bonds correlations.

Figure 8. Five-year expected future 2-factor model completed correlations between major asset classes, NZD currency, 13th November 2025. Source: Peter Urbani.

Comparing the “Max-Det” completed correlation matrix with Peter Urbani’s “2-factor model” completed correlation matrix , both matrices are perfectly identical²⁰.

As a closing (fun) remark, Olvera Astivia¹² highlights that having half of the entries in a correlation matrix missing [can be] considered a rather extreme condition¹².

What could then be said about the initial Blackrock’s partial correlation matrix that has around 82% of missing entries?!

Conclusion

With this first blog post of 2026, my hope is that you have added to your quantitative toolbox a useful methodology to obtain, in a mathematically principled way, values implied in correlational structure of the data, even if said data has not (or cannot) be obtained¹².

In any cases, for more mathematics of correlation matrices, feel free to connect with me on LinkedIn or to follow me on Twitter.

–

Nicholas J. Higham, Computing the Nearest Correlation Matrix—A Problem from Finance, IMA J. Numer. Anal. 22, 329–343, 2002.. ↩
In terms of the Frobenius distance. ↩
Olaf Dreyer, Horst Kohler, Thomas Streuer, Completing correlation matrices, arXiv. ↩ ↩² ↩³ ↩⁴
That is, belonging to the set $\mathcal{E}^n$. ↩
R. Grone, C.R. Johnson, E. Sa, H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109–124. ↩ ↩² ↩³ ↩⁴ ↩⁵
Georgescu DI, Higham NJ, Peters GW. 2018 Explicit solutions to correlation matrix completion problems, with an application to risk management and insurance. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸
Fiedler, M. Matrix Inequalities. Numer. Math. 9, 109–119 (1966). ↩ ↩² ↩³ ↩⁴
Ronald L. Smith, The positive definite completion problem revisited, Linear Algebra and its Applications, Volume 429, Issue 7, 2008,Pages 1442-1452. ↩ ↩²
One interesting feature of that result is that the existence of a correlation matrix completion - which seems quite algebric in nature - is actually equivalent to a “visual” condition. ↩
When one exists. ↩
Olaf Dreyer, Matrix completion and semidefinite matrices, arXiv. ↩ ↩²
Oscar L. Olvera Astivia (2021) A Note on the General Solution to Completing Partially Specified Correlation Matrices, Measurement: Interdisciplinary Research and Perspectives, 19:2, 115-123. ↩ ↩² ↩³ ↩⁴ ↩⁵
See van der Schans, Martin and Boer, Alex, A Heuristic for Completing Covariance And Correlation Matrices (March 14, 2013). Technical Working Paper 2014-01 November 2014. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷
The maximum determinant correlation matrix completion maximizes the product of distances to the defining hyperplanes⁶. ↩
See Vandenberghe, Lieven and Boyd, Stephen and Wu, Shao-Po, Determinant Maximization with Linear Matrix Inequality Constraints, SIAM Journal on Matrix Analysis and Applications, Volume 19, Number 2, Pages 499-533. ↩
See Nakagaki, T., Fukuda, M., Kim, S. et al. A dual spectral projected gradient method for log-determinant semidefinite problems. Comput Optim Appl 76, 33–68 (2020). ↩
Or close to zero in case the nearest correlation matrix to the resulting approximate correlation matrix is computed. ↩
W. Glunt, T.L. Hayden, Charles R. Johnson, P. Tarazaga, Positive definite completions and determinant maximization, Linear Algebra and its Applications, Volume 288, 1999, Pages 1-10. ↩
The Frobenius distance between $C_1$ and $C^*_1$ is $\approx 2.630$, while the Frobenius distance between $C_1$ and $C^{**}_1$ is $\approx 1.575$. ↩
Incidentally, this allows to conclude - due to the entropy maximization property of the maximum determinant correlation matrix completion under a multivariate normal model - that the 2-factor model used by Peter Urbani is a vanilla multivariate normal model. ↩