Sharpe Stability Ratio (SSR)

Excess returns

Let denote portfolio returns and the corresponding risk-free rates. Excess returns are

The process is assumed strictly stationary and -mixing - a regularity condition satisfied by standard return models including AR, GARCH, and Markov-switching processes. The original paper uses this condition for the consistency of its HAC variance estimator and for its moving-block bootstrap.

FolioLab subtracts one cash series for the whole run: is what cash earned over day 's interval at the run's annual risk-free rate, accrued by calendar day, so a Monday carries three days. The rate itself is one value for the run, not a separate rate for each day. The default window is one year of the run's own trading days, about 246 on an Indian run and 252 on a US run.

Full-sample Sharpe ratio

The classical (point-in-time) Sharpe ratio over is

The annualized Sharpe is , where is the number of observations per year ( for daily, for weekly, for monthly).

Rolling performance process

For a window length , define the rolling mean and rolling sample variance

The corresponding rolling Sharpe ratio and the rolling performance process are

The choice of is a smoothing decision analogous to bandwidth selection in nonparametric estimation: shorter windows produce noisy, high-variance Sharpe estimates, while longer windows yield smoother but more averaged-out dynamics. The window must be long enough for the Central Limit Theorem to justify approximate normality of local Sharpe-type estimators.

Our backend uses paper-aligned fixed defaults - no adaptive shrinkage:

• Daily data: is one trading year of the run's own sessions, on a US run and about 246 on an Indian run.
• Monthly data (): (three years), the value used in the original paper's empirical analysis.
• Other frequencies: defaults to (one calendar year of observations).

The rolling process above is the paper's definition. FolioLab splits the series into non-overlapping windows of the same length instead. With windows that end on the last observation, the window Sharpe ratios are

The earliest observations do not fill a whole window, so the calculation does not use them. A window with zero volatility is dropped.

When the history holds fewer than three whole windows, SSR is reported as unavailable rather than estimated from a shorter window. This is deliberate: a partial-window SSR is harder to interpret and easier to over-trust than an explicit "insufficient data" verdict.

Mechanical autocorrelation from overlapping windows

Consecutive rolling windows share observations, creating mechanical overlap that induces strong serial dependence even when the underlying returns are i.i.d. For rolling means, Proposition 3.1 in the original paper shows that the autocorrelation function of the rolling sequence satisfies

Because the rolling Sharpe is a smooth function of the rolling mean and rolling standard deviation, the delta-method linearization

transfers the same first-order autocorrelation structure to , with higher-order remainder terms vanishing as . For , this implies - adjacent rolling Sharpe observations are nearly perfectly correlated by construction.

Why FolioLab does not use the HAC long-run variance

The paper estimates the long-run variance of the rolling process, which accounts for all autocovariances,

with the Newey and West (1987) Bartlett-kernel estimator and an Andrews (1991) bandwidth :

FolioLab used this estimator before, with the bandwidth capped at . The cap is not conservative. Adjacent windows share observations, so the autocovariances of stay large out to lag . A cap of is far below on every practical history (35 lags on 1,260 rolling observations, against ), so it cuts off most of that dependence. The long-run variance was then too small, and in a simulation it was about 6.3 times too small. In a simulation of daily return series with a true Sharpe ratio of zero, the test rejected at the 5% level in 24% to 40% of cases.

A larger bandwidth does not repair the test. A series of rolling values holds only about independent blocks, and a HAC estimate from so few blocks is biased downward and noisy. A bandwidth of still rejected in 22% to 34% of cases.

Inference on non-overlapping windows

FolioLab therefore computes the SSR on the non-overlapping window Sharpe ratios defined in Setup & Notation. The windows do not overlap, so their Sharpe ratios are close to independent, and no HAC correction or bandwidth is necessary. With the mean and the sample standard deviation of the window Sharpe ratios:

The SSR is reported only when . With two windows the test has one degree of freedom, a one-sided 5% critical value of 6.31, and a standard deviation from a single difference.

Sharpe Stability Ratio

Let denote the population mean of the window Sharpe ratios and their standard deviation. The structural Sharpe Stability Ratio is

For hypothesis testing relative to a required Sharpe benchmark , the inferential (benchmark-centered) form is

The benchmark represents the required efficiency level. The choice serves as the natural baseline (and, as shown in the elasticity analysis below, eliminates divergences in economically relevant regions). High SSR requires both (i) superior average performance relative to the benchmark and (ii) low temporal dispersion of window Sharpe ratios. Two strategies with identical ex-post can display markedly different SSR values: a stable strategy attains high SSR; an episodic strategy whose window Sharpe oscillates sharply attains low SSR despite comparable aggregate performance.

The paper's asymptotic test

Under stationarity and mixing, Theorems 3.1 to 3.3 of the original paper show that the HAC variance is consistent and that converges to a standard Normal under the boundary null, as . On a practical history the rolling series holds only a few independent blocks, so that limit is far away, and the Mathematical Formulation section shows the size this gave.

The test FolioLab runs

On the non-overlapping window Sharpe ratios, the inferential statistic is

The one-sided p-value for against is

where is the Student t distribution with degrees of freedom. This is the one-sample t-test on the window Sharpe ratios.

What the p-value assumes. The Student t reference is exact if the window Sharpe ratios are independent and Normal with a common variance. Each window Sharpe ratio averages returns, so it is close to Normal. A Sharpe ratio does not scale with volatility, so a volatility regime that changes between windows leaves the window variances close to equal. For unequal variances, the t-test stays conservative at two-sided levels up to 8.3% (Bakirov and Székely, 2006; Ibragimov and Müller, 2010). That result does not cover this test exactly, because a one-sided 5% test uses the critical value of a two-sided 10% test.

Size in simulation. In seeded simulations of 2,000 to 4,000 series per case, the rejection rate at 5% under a true null stayed between 4.3% and 5.7%. The cases were daily returns with 3 to 20 windows from iid Normal, Student t(4), GARCH(1,1) and AR(1) processes and from nonzero benchmarks, and monthly returns with . These simulations are the evidence for the 5% level.

The p-value remains approximate. The test does not correct dependence that lasts longer than one window, or the small-sample bias of a Sharpe ratio. With few windows the test also has low power: against a true annualized Sharpe of 1.0, it rejected in about 32% of cases with 3 windows and about 90% with 10.

Moving-block bootstrap (finite-sample inference)

Asymptotic approximations may be unreliable under the strong autocorrelation induced by overlapping windows. The original paper therefore proposes a moving-block bootstrap (MBB) - Künsch (1989), Hall et al. (1995) - that resamples contiguous blocks of the original return series, reconstructs bootstrap return paths, and recomputes for each replicate . Block length is selected via the Politis and White (2004) automatic procedure, which adapts to the autocorrelation structure of each series. The bootstrap p-value for the one-sided test against is

FolioLab does not run this bootstrap. It reports the Student t p-value on non-overlapping windows that the previous section defines.

Three complementary one-sided tests

The original framework formalizes three distinct hypothesis tests, each with an asymptotic or a bootstrap p-value. FolioLab runs only the first, with the Student t-test on non-overlapping windows:

1. Mean window performance vs. benchmark. against . Test statistic . This is the test exposed by our API; the verdict on the SSR card reports its one-sided -value at .
2. SSR exceeds a stability threshold . against .
3. Pairwise stability comparison. against , with a joint block bootstrap that preserves cross-sectional dependence between the two return series.

Partial derivatives

Improvements in average performance translate into SSR gains proportionally to the inverse of temporal dispersion: when instability is low, temporal stability acts as a performance amplifier. Reducing raises SSR nonlinearly, so high-SSR strategies benefit more in absolute terms from stability improvements than low-SSR strategies.

Total differential

The total differential decomposes SSR changes into three economic channels:

The performance channel reflects improvements in average window Sharpe; the exigence channel captures changes in the required benchmark; the stability channel reflects variations in temporal dispersion.

Elasticities

The unit-elasticity result for is striking: a 1% increase in temporal dispersion produces a 1% decrease in SSR regardless of or . SSR penalizes instability in a scale-invariant manner. The performance and benchmark elasticities both diverge as , which motivates the choice in empirical applications: at zero benchmark, and , eliminating divergence in economically relevant regions.

Iso-SSR curves: the performance–stability trade-off

For a fixed benchmark , level sets of SSR satisfy

where is the SSR level. Iso-SSR curves are upward-sloping straight lines in space with slope : an increase in temporal instability of one unit must be compensated by additional units of average window Sharpe to maintain constant SSR. This is the SSR counterpart of the mean–variance trade-off encoded in the classical SR.