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. This guarantees consistency of the HAC variance estimator and validity of the moving-block bootstrap.

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 (): (one trading year).
• Monthly data (): (three years), the value used in the original paper's empirical analysis.
• Other frequencies: defaults to (one calendar year of observations).

When the available history is too short to construct at least 60 rolling Sharpe observations at the chosen window, SSR is reported as unavailable rather than estimated from a shrunken 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.

Long-run variance and HAC estimation

The long-run variance of the rolling performance process accounts for all autocovariances:

The naive sample variance ignores the autocovariance terms and is severely anti-conservative under the overlap-induced dependence. We therefore use the Newey and West (1987) HAC estimator with a Bartlett kernel:

The Bartlett kernel guarantees and applies linearly declining weights to higher-order autocovariances. The bandwidth is selected by the Andrews (1991) data-driven procedure, which fits an AR(1) approximation to obtain

where is the first-order autocorrelation of . The plug-in bandwidth grows as in the textbook regime, slow enough to ensure consistency yet flexible enough to absorb both the mechanical overlap and any genuine serial dependence in the underlying returns.

Finite-sample cap. Because adjacent rolling windows share observations, the first-order autocorrelation of is mechanically close to one ( for daily data with ). The plug-in expression

then explodes — purely as an artifact of overlap, not of true persistence — and Andrews' AR(1) formula can return bandwidths in the hundreds. We therefore enforce a finite-sample cap

keeping the selected lag in the consistency regime while preventing hundreds of noisy high-order autocovariances from dominating finite-sample SSR estimates. The cap is conservative — it permits more autocovariance absorption than Andrews' optimal rate, which is appropriate given the strong overlap-induced dependence we are trying to absorb.

Sharpe Stability Ratio

Let denote the population mean of the rolling performance process and its long-run 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 rolling Sharpe ratios. Two strategies with identical ex-post can display markedly different SSR values: a stable strategy attains high SSR; an episodic strategy whose rolling Sharpe oscillates sharply attains low SSR despite comparable aggregate performance.

Asymptotic distribution

Under stationarity and mixing, Theorem 3.1 of the original paper establishes consistency of the HAC variance estimator,

and Theorem 3.2 establishes a Central Limit Theorem for the rolling mean,

Combining the two yields the inferential statistic

Under the boundary null , Theorem 3.3 shows ; under the fixed alternative , , so the test is consistent. The benchmark enters only through the null hypothesis; it does not affect the asymptotic variance, which is governed solely by the long-run variance of .

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

Our current implementation reports the asymptotic-Normal p-value derived from . The moving-block bootstrap is on the roadmap; the original paper recommends it for finite-sample inference under strong overlap-induced autocorrelation.

Three complementary one-sided tests

The original framework formalizes three distinct hypothesis tests, all implementable via either asymptotic or bootstrap p-values:

1. Mean rolling performance vs. benchmark. against . Test statistic (asymptotic) or (bootstrap). 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 rolling 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 long-run 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 rolling Sharpe to maintain constant SSR. This is the SSR counterpart of the mean–variance trade-off encoded in the classical SR.