# Historical Covariance Forecasts Underestimate Stress Risk by a Factor of 1.6

> Walk-forward test of four portfolio constructions on 49 US industries, 1973 to 2026. In the 33 most volatile months realised volatility ran 1.6 to 1.7 times the forecast, and the textbook optimiser underestimated risk in all 33. Rising volatility explains four fifths of the miss, correlation one fifth. 14 pages, 14 direct sources.

Published: 2026-09-29
Publisher: BlackRidge (https://blckridge.com/)
Canonical: https://blckridge.com/research/covariance-estimation-error-20260929/
PDF: https://blckridge.com/research/covariance-estimation-error-20260929/covariance-estimation-error-20260929-en.pdf

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# Historical Covariance Forecasts Underestimate Stress Risk by a Factor of 1.6

We tested four ways to build a portfolio using 54 years of daily US industry data. Every month we check the risk forecast against the volatility that actually arrives in the month that follows. The models consistently underestimate risk during market stress.

## In the 33 most volatile months since 1973 every covariance model’s median risk forecast was too low, and the textbook optimiser underestimated risk in all 33

The covariance matrix is fitted on a past that is much calmer than the moment it is needed. Pure estimation error makes the optimiser's forecast optimistic even in quiet markets, while during shocks all tested allocation methods miss reality by a similar magnitude. Closing the gap requires a faster volatility estimate paired with a stressed correlation [01] .

## In a typical year only 2 of 49 eigenvalues rise above the noise band

With 49 assets and one year of days, a random matrix already produces eigenvalues up to 2.08 [06] . The market factor carries more than half the variance. The rest cannot be mathematically distinguished from noise, yet the optimiser inverts all of it.

Largest to smallest eigenvalues of the correlation matrix for 49 US industries over 252 trading days (September 2025 to August 2026) against a shuffled control sample. Sources: Kenneth French Data Library, Laloux et al. [01] [06] Rank of eigenvalue (largest first)

## With one year of data the optimiser understates its own risk by a quarter even when nothing changes

This bias needs no crisis. A mean-variance optimiser simply loads on the directions that look least risky by pure chance [03] . Ledoit–Wolf shrinkage [04] roughly halves this bias, to 1.13 in our simulation, but does not remove it [01] .

True / forecast volatility of the minimum-variance portfolio. Simulated with 400 normal samples per window. The true covariance is the 49 industries over the latest year [01] [04] . Estimation window (trading days)

## Every risk model proves too optimistic in a crisis, and the naive allocations too cautious in calm markets

Equal-weighted and hierarchical risk parity [10] portfolios routinely overstate risk in quiet markets. Their realised volatility runs at roughly 0.75 of the forecast. Both models then violently understate risk in a crisis, hitting a median ratio near 1.65. The textbook optimiser fares worse: it understates risk even in calm months (1.10). Shrinkage is accurate in calm months (0.99) but still misses by 1.63 in stress.

12-month rolling median of realised/forecast volatility (1 = accurate), monthly, 1974 to August 2026 [01] .

- Portfolio
- Calm months
- All months
- Stress months
- Stress months underestimated
- Textbook optimiser
- 1.10
- 1.20
- 1.68
- 100.0%
- Shrinkage optimiser
- 0.99
- 1.08
- 1.63
- 93.9%
- Hierarchical risk parity
- 0.75
- 0.86
- 1.65
- 84.8%
- Equal weights
- 0.74
- 0.85
- 1.63
- 87.9%

## In the worst tenth of months industry correlation reaches 0.62, against 0.45 in ordinary up months

Risk models build their forecasts with the correlations of the trailing year. During the stress episodes shown below, the month itself was far more correlated than the recent past [01] . The overall curve is asymmetric: the best months also push correlation higher, just not as violently [07] .

Average pairwise correlation of daily returns within the month, 49 US industries, 643 months [01] . Deciles of monthly market return, worst to best

- Episode
- Correlation, trailing year
- Correlation, that month
- Realised / forecast risk (equal weight)
- Market, month
- October 1987
- 0.56
- 0.86
- 6.12
- −22.6%
- August 1998
- 0.51
- 0.72
- 2.00
- −15.7%
- October 2008
- 0.62
- 0.83
- 3.22
- −17.2%
- August 2011
- 0.65
- 0.91
- 3.26
- −5.9%
- March 2020
- 0.52
- 0.83
- 6.70
- −13.2%
- April 2025
- 0.37
- 0.75
- 3.23
- −0.4%

## Beyond two standard deviations down, correlation is 0.75 where a normal model expects 0.27

A normal model says extreme days should be less correlated. The data say the opposite. A covariance matrix, which describes the middle of the distribution, cannot carry this [07] [08] .

Exceedance correlation, average of 49 industries with the US market, daily 1972 to August 2026 [01] [07] . Threshold, standard deviations

## Rising correlation explains a fifth of the stress miss; rising volatility the rest

The prevailing narrative claims that correlations go to one in a crisis. Our decomposition of the equal-weight portfolio shows something else: most of the variance miss comes directly from each industry becoming more volatile [01] . Rising correlation multiplies the miss by a further 1.26, about a fifth of the total in log terms. Yet that specific fraction happens to be the cushion diversification was supposed to provide when you need it most.

## The 99% VaR broke 2.25% of days, and ten of 52 years ended in the Basel red zone

Normal models expect failure to be rare and random. In the data, breaks cluster violently in stress years. When a breach happens, the resulting loss towers over what a normal distribution allows [01] .

Days per calendar year on which the equal-weight portfolio lost more than its one-day 99% normal VaR from the trailing-year covariance, full calendar years 1974 to 2025 [01] [13] .

## Faster volatility plus a stressed correlation brought the stress-month forecast to 0.99, at a cost in calm months

Speed matters more than the correlation assumption. A stressed correlation alone helps less than an exponentially weighted covariance with the RiskMetrics decay of 0.94 [11] . Combined, they match the median stress month, but they still break Value at Risk on 6.3% of stress days. Overstating risk in calm months is the inevitable price.

- Model
- Stress, realised / forecast
- Stress months underestimated
- Stress days breaking VaR
- Calm, realised / forecast
- A · Trailing one-year covariance
- 1.65
- 87.5%
- 11.98%
- 0.74
- B · Exponentially weighted covariance
- 1.21
- 62.5%
- 8.08%
- 0.83
- C · Trailing volatility, stressed correlation
- 1.41
- 78.1%
- 9.13%
- 0.61
- D · Exponential volatility, stressed correlation
- 0.99
- 50.0%
- 6.29%
- 0.67

## The stock-bond correlation that protected portfolios from 2000 to 2021 has turned positive in 62.5% of months since 2022.

Two decades of negative correlation provided the window almost every multi-asset covariance matrix in use was fitted on. The 1970-1999 record was positive. On the worst stock days since 2022, Treasuries no longer rallied reliably [01] [02] .

252-day correlation of daily US market returns with minus the daily change in the 10-year yield, monthly points, 1963 to August 2026 [01] [02] .

## Industry correlation is at 0.17, the third lowest reading since 1972, which is when forecasts are most optimistic

A covariance matrix estimated today carries one of the calmest correlation structures on record [01] . April 2025 proved how fast it can double. History shows these extreme lows reliably give way to much higher readings.

63-day average pairwise correlation of daily returns, 49 US industries, month ends, 2000 to August 2026 [01] .

## Historical covariance models consistently underestimate risk during market stress

We tested standard portfolio construction methods over decades of data. In the 33 most volatile months, every historical covariance model’s median forecast was too low, and the textbook optimiser underestimated risk in all 33. The underlying math breaks down exactly when investors need protection.

- Measure
- What it shows
- Value
- Optimiser stress misses
- Share of stress periods where the textbook model underestimated risk
- 33 / 33
- Optimiser stress ratio
- Realised volatility divided by forecast in the most volatile months
- 1.68
- Correlation miss share
- Portion of the variance forecast error driven by changing correlations
- 20.3%
- VaR break rate
- Frequency of daily portfolio losses exceeding the modelled risk limit
- 2.25%
- Model D stress ratio
- Realised against forecast volatility using stressed correlation and exponential weights
- 0.99
- August 2026 industry correlation
- Average trailing correlation of industries at the end of August
- 0.17

Before taking a risk number to an investment committee, pin down three inputs. Ask for the estimation window, the speed of the volatility estimate, and the exact correlation assumed in stress.

## Selected sources

This report provides research material rather than investment advice. We calculated all results on historical US equity industries before transaction costs. Past estimation error does not establish a hard limit on future model failure.

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## Sources

- [01] [Kenneth French Data Library](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). We extracted daily returns for 49 industry portfolios, the US market and regional markets, matching the CRSP and Bloomberg August 2026 builds.
- [02] [FRED, 10-year Treasury constant maturity rate (DGS10)](https://fred.stlouisfed.org/series/DGS10). We used this series for daily bond yields to track stock-bond correlation.
- [03] [Michaud, The Markowitz Optimization Enigma: Is 'Optimized' Optimal?, Financial Analysts Journal 1989](https://www.newfrontieradvisors.com/media/t04a3oct/the-markowitz-optimization-enigma.pdf). Michaud recognised early on that mean-variance optimisers act as estimation-error maximisers.
- [04] [Ledoit and Wolf, A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis 2004](https://econpapers.repec.org/article/eeejmvana/v_3a88_3ay_3a2004_3ai_3a2_3ap_3a365-411.htm). We implemented their shrinkage estimator to test whether dampening sample noise improves risk forecasts.
- [05] [DeMiguel, Garlappi and Uppal, Optimal Versus Naive Diversification, Review of Financial Studies 2009](https://econpapers.repec.org/article/ouprfinst/v_3a22_3ay_3a2009_3ai_3a5_3ap_3a1915-1953.htm). The authors showed that an optimising model needs 6,000 months of data to beat naive diversification for 50 assets.
- [06] [Laloux, Cizeau, Potters and Bouchaud, Random matrix theory and financial correlations, 2000](https://www.cfm.com/wp-content/uploads/2022/12/234-1999-random-matrix-theory-and-financial-correlations.pdf). Their work defines the noise band we used to measure the random share of eigenvalues.
- [07] [Longin and Solnik, Extreme Correlation of International Equity Markets, Journal of Finance 2001](https://econpapers.repec.org/article/blajfinan/v_3a56_3ay_3a2001_3ai_3a2_3ap_3a649-676.htm). Exceedance correlation lets us track dependence deep in the tails.
- [08] [Ang and Chen, Asymmetric correlations of equity portfolios, Journal of Financial Economics 2002](https://econpapers.repec.org/article/eeejfinec/v_3a63_3ay_3a2002_3ai_3a3_3ap_3a443-494.htm). The data confirm downside correlations run 11.6% above normal distribution estimates.
- [09] [Forbes and Rigobon, No Contagion, Only Interdependence, Journal of Finance 2002](https://econpapers.repec.org/article/blajfinan/v_3a57_3ay_3a2002_3ai_3a5_3ap_3a2223-2261.htm). They proved that higher volatility mechanically biases correlation upward.
- [10] [López de Prado, Building Diversified Portfolios that Outperform Out of Sample, Journal of Portfolio Management 2016](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2708678). We ran hierarchical risk parity to see if bypassing full covariance inversion helps out of sample.
- [11] [J.P. Morgan and Reuters, RiskMetrics Technical Document, 1996](https://www.msci.com/research-and-insights/paper/1996-riskmetrics-technical-document). We built an exponentially weighted covariance benchmark with a decay factor of 0.94.
- [12] [Basel Committee, MAR33 Internal models approach: capital requirements calculation](https://www.bis.org/committees/bcbs/basel-framework/standard/mar/33/inforce/2023-01-01/published/2020-06-05). This standard shifts trading-book capital requirements to a 97.5% expected shortfall.
- [13] [Basel Committee, MAR99 Guidance on use of the internal models approach](https://www.bis.org/committees/bcbs/basel-framework/81331/chapter.pdf). The backtesting rules put a bank in the yellow zone from 5 exceptions and the red zone from 10, per 250 trading days.
- [14] [European Commission, press release of 12 June 2025](https://finance.ec.europa.eu/news/commission-proposes-postpone-one-additional-year-market-risk-prudential-requirements-under-basel-iii-2025-06-12_en). The Commission delayed the EU application of these market-risk rules to 1 January 2027.

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## Citation context

In 33 ex-post selected stress months (the top 5% by realised US-market volatility), the textbook sample minimum-variance portfolio's median next-month realised-to-forecast volatility ratio was 1.676 (1.68× rounded); its forecast was below realised volatility in all 33.

Sample and method: 49 daily value-weighted US industry portfolios, Ken French CRSP 202608 build, 3 Jan 1972–31 Aug 2026 (13,780 daily observations); 643 walk-forward monthly observations, Feb 1973–Aug 2026. Each forecast used the trailing 252 trading days and was compared with the following calendar month.

Limits: this measures volatility, not return or loss. The 33/33 under-forecast applies only to the textbook sample minimum-variance portfolio, not all four constructions. Portfolios were fully invested with shorts allowed, no long-only constraint and no leverage cap. Stress selection is ex-post and mechanically favors a trailing-forecast miss; this is not prospective, live or predictive evidence. The public page does not publish the archived raw extract or calculation script.

Primary input: [Kenneth R. French Data Library, CRSP 202608 build](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html).

Stable permalink: [https://blckridge.com/research/covariance-estimation-error-20260929/#citation-context](https://blckridge.com/research/covariance-estimation-error-20260929/#citation-context).

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