Batch 3: Volatility-managed equity (Moreira & Muir)

Volatility-managed equity (Moreira & Muir) on SPY. Claim tested (declared before the run): Scaling equity exposure by the inverse of last month's realised variance raises the Sharpe ratio (and produces alpha) relative to the unmanaged market. Benchmark: buy-and-hold SPY. Part of Indicator Audit Batch 3, 'Strongest published evidence' (6 audits, sealed, Holm across the batch).

Executive summary

Scorecard

How the grade is set (Holdout Labs Fragility Rubric v1.2, sealed before this report was produced). PASS = 2 points, CAUTION = 1, FAIL = 0; N/A checks are excluded. Share of available points: A ≥ 85%, B ≥ 70%, C ≥ 55%, D ≥ 40%, otherwise F. Hard caps: Objective cap (beat the benchmark): if the strategy does not beat its benchmark (SPA check FAIL, or annualised mean excess return <= 0), the grade is at most C. Objective cap (reduce drawdown): if the drawdown reduction is not robust (drawdown check FAIL) or the declared return-cost tolerance is breached (tolerance check FAIL), the grade is at most C. Objective cap (improve Sharpe): if the Sharpe-ratio improvement is not robust (Sharpe check FAIL), the grade is at most C. Selection cap: if the deflated check FAILS, the grade is at most C. Overfitting cap: if the PBO check FAILS, the grade is at most D. Disclosure cap: if the number of variants tried was not declared and no variant matrix was supplied, the grade is at most B.

What was audited

Results in detail

1. Sharpe ratio and its uncertainty

Rubric checks use the excess series (strategy minus buy-and-hold of the traded asset); the absolute series is shown for reference.

2. The variant-count effect (Deflated Sharpe)

Dispersion of trial Sharpe ratios used: 0.215 (annualised standard deviation), estimated from the variant matrix.

3. Probability of backtest overfitting (CSCV)

Histogram of logit ranks: values left of the dashed line are splits where the in-sample winner finished at or below the out-of-sample median (logit 0).

4. Multiple-testing haircut

5. Reality Check and SPA

6. Holdout degradation

7. Period and regime stability

8. Transaction-cost sensitivity

Turnover 2.8 units per year; break-even cost 475.1 bps.

9. Parameter sensitivity

Chosen variant window=1m|target=0.16|cap=1.5 ranks 1 of 12. Grid median Sharpe 0.76, best 0.79; 100% of cells positive; neighbour ratio 0.97.

10. Drawdown distribution

What is fragile, and how to test the fix

These are suggestions for further statistical testing, not suggestions to trade. Any change to the rules creates a new variant: count it in N and confirm it on data it was not designed on.

What held up

• Transaction-cost headroom (absolute): The gross mean return falls to zero at 475.1 bps per unit traded.
• Parameter plateau (dSR): The chosen variant ranks 1 of 12 by Sharpe improvement; its one-step neighbours average 85% of its improvement.

Methods appendix

Sharpe ratio and its standard error

Per-period mean over standard deviation of the audited return series (risk-free rate taken as zero), annualised by the square root of periods per year. The standard error uses the IID-normal formula of Lo (2002) and the non-normal correction of Mertens (2002), which widens the error for negative skew and fat tails. Lo's autocorrelation-adjusted annualisation is reported alongside. [1], [2]

Probabilistic Sharpe ratio (PSR) and minimum track record

The probability that the true Sharpe ratio exceeds a benchmark (here zero), given the sample length, skewness and kurtosis; and the minimum sample length for 95% confidence. [3]

Deflated Sharpe ratio (DSR)

The PSR measured against the Sharpe ratio one would expect from the best of N skill-less trials, where N is the number of variants tried and the dispersion of trial Sharpe ratios is estimated from the variant matrix (or, without one, set to the null sampling variance 1/T). It corrects for selection and non-normality at once. [4]

Probability of backtest overfitting (PBO) via CSCV

The variant matrix is cut into 16 time blocks; for each of the 12,870 ways of picking half of them as in-sample, the in-sample best variant is ranked out of sample. PBO is the share of splits where it falls to or below the median. It assumes blocks long enough to preserve serial dependence and a variant set that represents the real search. [5]

Multiple-testing haircut

The Sharpe ratio is turned into a t-statistic and p-value, the p-value is adjusted for the number of tests (Bonferroni always; Holm and BHY when every variant's returns are supplied), and the adjusted p-value is mapped back to a haircut Sharpe ratio. [6], [7]

White's Reality Check and Hansen's SPA test

Tests whether the best of all variants beats the benchmark in mean return once the search over variants is accounted for. Uses the stationary bootstrap with mean block length T^(1/3) (at least 5) to keep short-range dependence. SPA studentises and recentres, so poor variants do not dilute the power. [8], [9], [10]

Holdout degradation

In-sample versus holdout Sharpe ratio at a fixed split date (by default the last 30% of the sample). The consistency p-value asks how surprising the holdout Sharpe would be if the in-sample Sharpe were the truth. A holdout only counts if it was not used to design the rule. [11]

Period and regime stability

Returns and Sharpe ratio by calendar year and by tercile of the underlying market's trailing 21-day volatility (lagged one day). Descriptive: the tercile cut points use the full sample. [12]

Parameter-sensitivity surface

Sharpe ratio across the supplied parameter grid. The neighbour ratio compares the chosen cell with its one-step neighbours: a plateau (ratio near 1) is less fragile than an isolated peak. [13]

Transaction-cost sensitivity

Net return = gross return minus turnover times a one-way cost per unit traded, over a grid of costs; the break-even cost is where the mean net return reaches zero. Market impact beyond a flat cost is not modelled. [12]

Drawdown distribution by bootstrap

The maximum drawdown is recomputed on 1,000 stationary-bootstrap resamples of the return series, showing how much deeper (or shallower) the worst loss could plausibly have been with the same return distribution in a different order. [10]

References

1. Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal 58(4), 36-52.
2. Mertens, E. (2002). Comments on Variance of the IID Estimator in Lo (2002). Working paper, University of Basel.
3. Bailey, D. H. & Lopez de Prado, M. (2012). The Sharpe Ratio Efficient Frontier. Journal of Risk 15(2), 3-44.
4. Bailey, D. H. & Lopez de Prado, M. (2014). The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality. Journal of Portfolio Management 40(5), 94-107.
5. Bailey, D. H., Borwein, J. M., Lopez de Prado, M. & Zhu, Q. J. (2016). The Probability of Backtest Overfitting. Journal of Computational Finance 20(4), 39-69.
6. Harvey, C. R. & Liu, Y. (2015). Backtesting. Journal of Portfolio Management 42(1), 13-28.
7. Harvey, C. R., Liu, Y. & Zhu, H. (2016). ... and the Cross-Section of Expected Returns. Review of Financial Studies 29(1), 5-68.
8. White, H. (2000). A Reality Check for Data Snooping. Econometrica 68(5), 1097-1126.
9. Hansen, P. R. (2005). A Test for Superior Predictive Ability. Journal of Business & Economic Statistics 23(4), 365-380.
10. Politis, D. N. & Romano, J. P. (1994). The Stationary Bootstrap. Journal of the American Statistical Association 89(428), 1303-1313.
11. Bailey, D. H., Borwein, J. M., Lopez de Prado, M. & Zhu, Q. J. (2014). Pseudo-Mathematics and Financial Charlatanism: The Effects of Backtest Overfitting on Out-of-Sample Performance. Notices of the AMS 61(5), 458-471.
12. Lopez de Prado, M. (2018). Advances in Financial Machine Learning. Wiley.
13. Pardo, R. (2008). The Evaluation and Optimization of Trading Strategies, 2nd ed. Wiley.

Data and hash appendix

• Prices from Yahoo Finance's public chart endpoint; we publish derived statistics only, never raw prices; no redistribution. Signals use the split-adjusted close; total returns use the split- and dividend-adjusted close.
• yahoo_spy.csv: 8471 rows ending 2026-09-24, SHA-256 8f10432b608927a6a88b67e6e98d72153ba70eb0254c2b5dce1fe3548fb3ceec (held before this batch; prices previously seen)
• Audit period 1994-01-03 to 2026-09-24; holdout from 2018-01-02; benchmark: buy-and-hold SPY (all fixed in the sealed batch preregistration).
• Cash earns 0% and borrowing above 100% exposure costs 0%; shorting has no borrow cost (stated in the preregistration).
• All of this audit's data files were held before the declaration was sealed (disclosed in the preregistration); the declaration-date check is therefore not applied.

Anyone with the same inputs and package version can re-run the audit and must obtain the same audited-series hash and the same statistics.