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.
September 2026
BlackRidge Research
BlackRidge Research: Covariance Error01
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Forecast Failure
02 / 14
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].
Textbook optimiser stress failures
100%
Share of volatile months where the model underestimated risk.
Optimiser stress forecast miss
1.68
Median ratio of realised to predicted volatility in market shocks.
Equal weight calm bias
0.74
Median ratio showing how the naive allocation overstates risk.
Stressed exponential model
0.99
Realised volatility ratio using fast decay and stressed correlation.
Textbook optimiser
1.68
Shrinkage optimiser
1.63
Hierarchical risk parity
1.65
Equal weights
1.63
Median realised volatility relative to forecast in the 33 stress months for four portfolio methods, tested on 49 US industries from February 1973 to August 2026 [01].
Observation
A mean-variance model requires an impossibly long estimation window to beat a naive equal-weight allocation out of sample [05]. Yet during market stress, the equal-weight and hierarchical portfolios underestimate risk just as severely as the optimisers [01]. The gap stems from the covariance inputs rather than the portfolio math.
Interpretation
Turbulence shreds the historical covariance structure exactly when you rely on it most. You can shrink the matrix or build a complex hierarchy, but neither fixes inputs that are too calm.
Implication
Risk teams must decouple the volatility estimate from the correlation estimate. Accelerate the volatility decay and hardcode a stressed correlation assumption to survive market shocks.
Caveat
We define stress months retrospectively by their realised volatility, mechanically selecting periods where any trailing forecast will look too low. The test also isolates US equity industries. This ignores the cross-asset dynamics that might cushion a real portfolio shock.
BlackRidge Research: Covariance Error02
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Random matrix theory
03 / 14
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.
Eigenvalue (log scale)
US industry returnsShuffled returns
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)
Eigenvalues above noise edge
2 / 49
Median count out of 49 across 643 rolling windows.
Variance in first eigenvalue
58.2%
Median share of total variance captured by the market factor.
Condition number
536
Median ratio of the largest to smallest eigenvalue across all windows.
Interpretation
Standard optimisers treat every matrix component as hard fact. They place massive bets on the smallest eigenvalues to force theoretical diversification. Matrix inversion mathematically amplifies this error, letting random static dictate the final portfolio weights.
Implication
Risk teams must strip the noise before optimisation. Stop feeding raw empirical covariance matrices into your models and apply spectral filtering or shrinkage instead.
Caveat
The theoretical noise band assumes independent and identically distributed returns. Weak but genuine economic relationships can easily hide inside it. Small eigenvalues do not necessarily lack information, they just lack the strength to survive matrix inversion.
BlackRidge Research: Covariance Error03
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The optimiser turns noise into leverage
04 / 14
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 volatility divided by forecast volatility (log scale)
Textbook optimiserWith shrinkage
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)
Understatement factor
1.24
With one year of data.
Gross position
367%
Median in the walk-forward test.
Short positions
133%
Median in the textbook model.
Observation
Testing 14 optimising models reveals none consistently beat a naive equal-weight allocation out of sample [05]. An unconstrained mean-variance model requires an estimation window of 6,000 months for 50 assets to reliably outperform equal weights [05].
Interpretation
The algorithm treats sample noise as structural truth. It finds asset combinations with low historical variance and builds massive offsetting positions. The resulting portfolio reflects the errors in your specific sample rather than the actual market.
Caveat
The simulation assumes normal returns and a stable covariance. This isolates pure estimation error and understates the real problem. Real optimisers usually add position limits, which cut both the extreme leverage and part of the bias.
BlackRidge Research: Covariance Error04
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The Walk-Forward Record
05 / 14
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%
Interpretation
Trailing correlations break down exactly when you need them to hold. You build a covariance matrix on a year of quiet data, but volatility and correlations both gap higher in a shock. The risk estimate fails mainly because volatility jumps; rising correlation adds the rest.
Caveat
These portfolios hold entirely different weights. Their absolute realised volatilities naturally diverge. We isolate the forecasting error here. The ratio compares a specific model's risk prediction only against its own eventual outcome.
BlackRidge Research: Covariance Error05
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CORRELATION DYNAMICS
06 / 14
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
Average correlation
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%
Observation
In stress months the trailing year averages a correlation of 0.58. The stress month itself jumps to 0.71. Calm months do the opposite and drop from 0.53 to 0.44 [01].
Interpretation
Diversification vanishes exactly when investors need it most. An equal-weight portfolio expects a diversification ratio of 1.38 heading into a stress month. The realised ratio falls to 1.20 [01].
Caveat
Correlation moves toward one, it does not reach it. The highest 63-day average on record is 0.86, set in August 2011. Correlation measured in a volatile month is partly inflated by the volatility itself [09].
BlackRidge Research: Covariance Error06
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Tails
07 / 14
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].
Correlation
ObservedNormal distribution
Exceedance correlation, average of 49 industries with the US market, daily 1972 to August 2026 [01][07]. Threshold, standard deviations
Observed downside correlation
0.75
Average across industries when both asset and market fall past two standard deviations.
Normal benchmark
0.27
Theoretical expectation for the exact same extreme downside moves.
Observed upside correlation
0.65
Average across industries when both asset and market rise past two standard deviations.
Interpretation
Covariance is a linear metric built on calm days. It assumes extreme moves will scatter. The data show they converge. A risk model relying on this metric is blind to the actual mechanics of a market crash.
Implication
Portfolios need stress tests built on empirical tail dependence. Diversification models fed by historical covariance will fail precisely when the market breaks.
Caveat
The downside to upside gap in our data is smaller than the gap to the normal model. Tail dependence drives this result, alongside asymmetry. Forbes and Rigobon show volatility regimes mechanically inflate measured correlation [09].
BlackRidge Research: Covariance Error07
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Volatility vs Correlation
08 / 14
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.
Total variance miss
3.09
Volatility contribution
2.45
Correlation contribution
1.26
Ratio of realised to forecast variance, geometric mean over 33 stress months, equal-weight portfolio of 49 industries [01].
Correlation share of miss
20.3%
Share of the log variance prediction error, leaving the rest to rising volatility.
Stress diversification ratio
1.20
Falls from 1.59 in calm months, indicating a severe loss of portfolio cushioning.
Realised stress correlation
0.71
Average pairwise coefficient during the shock, rising from 0.58 in the trailing forecast window.
Observation
Over 33 stress months the equal-weight portfolio generated a variance 3.09 times higher than predicted [01]. Re-pricing the forecast with the actual stress volatilities but the stale trailing correlations explains a factor of 2.45. The change in correlation accounts for the remaining multiple of 1.26, or 20.3% of the total log variance miss.
Interpretation
Volatility does the heavy lifting in breaking your risk budget. Individual industries gap down, and their standard deviations explode. The correlation breakdown simply removes the remaining diversification benefit exactly when the portfolio takes the brunt of the volatility shock.
Implication
Risk teams obsess over modelling complex correlation dynamics when they should first stress-test baseline volatility. If you cannot survive a pure volatility shock, a better correlation matrix will not save you.
Caveat
This exact split depends on the mathematical order of the decomposition. Computing the correlation effect first would yield slightly different attribution shares. The analysis also relies specifically on an equal-weight allocation, so a heavily concentrated portfolio might show a different balance between the two effects.
BlackRidge Research: Covariance Error08
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VaR Backtest
09 / 14
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].
Breaks
Breaks per yearExpected at 99%Basel red zone
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].
Daily break rate
2.25%
against the 1% failure rate expected under normality
Years in the red zone
10 / 52
of 52 full calendar years measured by Basel rules
Severity multiple
1.47
average loss on a break day as a multiple of VaR, compared to a normal 1.15
Observation
Measured in forecast standard deviations, the empirical 97.5% expected shortfall hits 3.31. A normal model expects 2.34. The extremes scale even worse: the worst break on record destroyed 9.0 times the estimated VaR in October 1987.
Interpretation
Trailing covariance maps the center of the distribution. Risk management happens in the tails. By the time a portfolio hits a stress event, the matrix has already stopped describing reality, leaving capital buffers sized for a calm market that no longer exists.
Implication
Risk teams rely on unadjusted normal VaR at their own peril. Internal limits require expected shortfall metrics calibrated directly to historical stress.
Caveat
This specific test applies a normal VaR to a static equal-weight portfolio. Banks typically rely on historical-simulation VaR.
BlackRidge Research: Covariance Error09
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Model Fixes
10 / 14
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.
A · Trailing one-year covariance
1.65
B · Exponentially weighted covariance
1.21
C · Trailing volatility, stressed correlation
1.41
D · Exponential volatility, stressed correlation
0.99
Median realised/forecast volatility of the equal-weight portfolio in the 32 of the 33 stress months that fall after this test’s January 1977 start, to August 2026 [01][11]
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
Interpretation
Reacting quickly to rising volatility prevents more damage than assuming extreme correlation. Models fail in stress primarily because variance explodes. Fixing the correlation matrix without speeding up the volatility estimate leaves the system too slow.
Implication
Prioritise short-memory volatility estimators over static correlation buffers. You must accept inflated risk estimates during calm periods to cover the actual risk of a crisis.
Caveat
Estimating correlations only on the worst tenth of market days is a failed variant. It lowered the forecast by 9.7% because truncated samples bias measured correlation downward [07]. Model D's stressed correlation is still taken from the past.
BlackRidge Research: Covariance Error10
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Cross-Asset Hedge
11 / 14
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
Stock-bond correlation
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].
Correlation 2000-2021
−0.37
The negative correlation era that trained modern risk models.
August 2026 correlation
+0.30
The highest reading since June 2024.
Regional equity correlation
0.76
Average across five regions since 2015, up from 0.46 in the 1990s.
Observation
On the worst 1% of stock days from 2000 to 2021, the 10-year yield fell 9.7 basis points on average and dropped on 83.6% of those days [01][02]. Since 2022, the yield rose 2.2 basis points on average and fell on only 41.7% of the worst equity days [01][02].
Interpretation
A positive correlation regime means stocks and bonds sell off together, stripping multi-asset portfolios of their built-in shock absorber. Risk models calibrated purely to the previous two decades blindly expect bonds to save the day.
Caveat
The post-2022 sample of worst equity days contains only 12 days. Yield changes provide a mechanical proxy for bond price direction, not a precise return series.
BlackRidge Research: Covariance Error11
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Current Environment
12 / 14
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
63-day average pairwise correlation of daily returns, 49 US industries, month ends, 2000 to August 2026 [01].
August 2026 correlation
0.17
The third lowest reading of 654 month ends since 1972.
Realised risk multiple
3.23
Equal-weight portfolio forecast mismatch in April 2025.
Historical reversion rate
77.8%
Share of lowest-tenth readings followed by a correlation jump above 0.50 within two years.
Interpretation
Models trained on current data generate massive position sizes because they perceive immense diversification benefits. During April 2025 the market dropped just 0.4%, yet correlation jumped to 0.75 and the textbook optimiser realised 4.06 times its predicted volatility.
Implication
A portfolio team does not need to wait for the rule. The European Commission postponed its application of the Basel 97.5% expected shortfall standard calibrated to stress [12] to 1 January 2027 [14]. Compute risk with stressed correlations today.
BlackRidge Research: Covariance Error12
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Conclusion
13 / 14
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.
01
A trailing year of data leaves just 2 of 49 eigenvalues above the noise band [06]. The optimiser understates true volatility by 1.24 times on this sample.
02
All four methods’ median stress ratio was 1.63–1.68.
03
The change in correlation explains only 20.3% of the variance miss, leaving rising volatility to account for the rest.
04
The standard normal Value at Risk model broke its limits enough to trigger the Basel red zone in 10 of 52 years [13].
05
Applying exponentially weighted volatilities to a stressed correlation matrix aligns the forecast with realised risk. This combination achieves a 0.99 ratio during market shocks. The model degrades during calm months, where realised volatility is only 0.67 of the forecast.
Summary of Forecast Errors
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.
The Commission delayed the EU application of these market-risk rules to 1 January 2027.
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.
BlackRidge Research: Covariance Error14
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.
When you use a result that another publication established, cite that original work; it is linked in Sources. Cite this report for our synthesis, explanation or an identified recalculation. No link is required in return.
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