The Sharpe Stability Ratio: Evaluating the Sharpe Ratio Temporal Consistency

The Sharpe Ratio1, one of the most commonly used measure of risk-adjusted performance2, is usually reported as a point estimate (Morningstar, Quantalys, etc.). Thanks to the work of Lo3, Opdyke4 and more recently5 de Prado et al.6, it is nevertheless well understood that such a point estimate […] does not convey information about statistical significance6, so that a more meaningful way to measure and compare investment efficiency6 is rather to report the Probabilistic Sharpe Ratio (PSR)7, which expresses observed Sharpe Ratios in probability space, adjusting for skewness, kurtosis, serial correlation, and sample length6. However, these two measures cannot distinguish persistent skill from episodic outperformance8 because they both evaluate performance at a single temporal aggregation without quantifying how risk-adjusted returns evolve within the sample period8. In order to overcome this limitation, Traver and Rodriguez Dominguez8 introduces the Sharpe Stability Ratio8, a measure of the temporal consistency of the Sharpe Ratio across overlapping subperiods. In this blog post, I will first describe the Sharpe Stability Ratio in details and then, as examples of usage, I will: Show that it provides additional information compared to the Sharpe Ratio Show that it also provides additional information compared to another measure of risk-adjusted performances called the Ulcer Performance Index9 Experiment with it as a portfolio optimization objective and construct its associated efficient frontier Reminders The Sharpe Ratio Let be: $r_1,…,r_T$ the observed arithmetic returns of an investment (asset/fund/portfolio/trading strategy/etc.) over $T$ periods of time $r_{f,1},…,r_{f,T}$, the observed risk-free arithmetic returns10 over the same $T$ periods of time $r_1 - r_{f,1},…,r_T - r_{f,T}$, the associated investment observed excess arithmetic returns over the risk-free rate Then, the observed Sharpe Ratio1 $\widehat{SR}$ of the investment over the period $1..T$ is defined as: [\widehat{SR} = \frac{\widehat{\mu}{r - r_f}}{\widehat{\sigma}{r - r_f}}] , where: $\widehat{\mu}_{r - r_f}$ is the sample mean of the excess arithmetic returns $\widehat{\sigma}_{r - r_f}^2$ is the sample11 variance of the excess arithmetic returns The Sharpe Ratio provides a scale-free summary of expected excess return per unit of risk8 and can be interpreted as a measure of […] signal over noise, or skill over luck6. Prado et al.6 notes that the Sharpe ratio satisfies several important properties6, among which: It is rooted in Markowitz’s mean-variance analysis model. For example, as illustrated in Figure 1, the tangency portfolio is the mean-variance efficient portfolio with maximum Sharpe Ratio and all efficient portfolios are linear combinations of this portfolio and the risk-free asset12. Figure 1. The Sharpe Ratio in Mean-Variance Space. Source: de Prado. As another example, ranking portfolios by their Sharpe Ratios yields the same ordering as expected-utility maximization6 under three (sufficient) conditions: When portfolio returns follow a normal, or more generally an elliptical distribution When portfolio returns satisfy the LS property13, that is, when portfolio returns differ from one another only by location and scale parameters13 When investor preferences admit a mean–variance representation, either exactly or as a second-order approximation6 It is actually well defined and usable for any return process with finite mean and variance, regardless of higher-order moments or temporal dependence6. This property is of practical importance because a common misunderstanding regarding [the Sharpe Ratio] is that […] [its users] must implicitly assume that returns are i.i.d. Normal6. As explained in de Prado et al.6, this is incorrect: When returns are non-normal or serially correlated, variance may no longer fully characterize downside or tail risk, but it continues to measure dispersion, and the Sharpe Ratio retains its economic interpretation as expected excess return per unit of standard deviation. Moreover, while several alternative risk-adjusted performance measures have been proposed over the years - whether partial moments-based (Sortino Ratio, Omega Ratio…), drawdown-based (Calmar Ratio…) or Value-at-Risk-based (Rachev Ratio…) - empirical studies have regularly shown that portfolio rankings based on most14 of these measures are nearly identical to those determined by the Sharpe ratio13. This behavior is partially explained in Schuhmacher and Elingby13, which demonstrates that if portfolio returns satisfy the LS property15, then any “reasonable” risk-adjusted performance measure must be a strictly increasing function in the Sharpe ratio13. The Probabilistic Sharpe Ratio The usage of hats in the previous subsection for the definition of the Sharpe Ratio emphasises that $\widehat{\mu}_{r - r_f}$ , $\widehat{\sigma}_{r - r_f}^2$ and $\widehat{SR}$ are quantities computed from the observed excess returns $r_1 - r_{f,1}$,…,$r_T - r_{f,T}$ while the true mean excess return $\mu_{r - r_f}$, the true variance $\sigma_{r - r_f}^2$ and the true Sharpe ratio $SR = \frac{\mu_{r - r_f}}{\sigma_{r - r_f}}$ are quantities associated with the unobservable excess return generating process. It follows that the observed Sharpe Ratio $\widehat{SR}$ is a statistical estimator of its true counterpart $SR$, subject to an estimation error quantified by Lo3, Opdyke4 and de Prado et al.6 under various assumptions on the excess return generating process16. In order for practitioners to evaluate the credibility of [an] observed Sharpe Ratio8 without explicitely referring to the underlying statistical machinery17, Bailey and de Prado7 introduces the Probabilistic Sharpe Ratio $\widehat{PSR}(SR_0)$, a risk-adjusted performance measure defined as the probability that the Sharpe Ratio estimator $\widehat{SR}$ is greater than a benchmark Sharpe Ratio $SR_0$18 over the period $1..T$: [\widehat{PSR}(SR_0) = \Phi\left( \frac{\widehat{SR} - SR_0}{ \sqrt{\frac{1 - \widehat{\kappa}{r - r_f} \widehat{SR} + (\widehat{\gamma}{r - r_f} - 1) \frac{\widehat{SR}^2}{4}}{T}} } \right )] , where: $\kappa_{r - r_f}$ and $\gamma_{r - r_f}$ are the true skewness and the true kurtosis of the unobservable excess return generating process and $\widehat{\kappa}_{r - r_f}$ and $\widehat{\gamma}_{r - r_f}$ their empirical counterparts $\Phi$ is the cumulative distribution function of the standard normal distribution This makes the Probabilistic Sharpe Ratio one of the very few19 alternative risk-adjusted performance measures that explicitly penalizes uncertainty due to short samples, downside, tail and drawdown-related risks6 while still being grounded in modern portfolio theory6. As a side note, de Prado et al.6 further improves the Probabilistic Sharpe Ratio by taking into account the return generating process’s autocorrelation, which enables Sharpe Ratio inference under substantially more general assumptions than prior closed-form treatments6, including the original treatment in Bailey and de Prado7. The Sharpe Stability Ratio Definition Traver and Rodriguez Dominguez8 notes that neither the Sharpe Ratio nor the Probabilistic Sharpe Ratio enables statistical inference on temporal stability of risk-adjusted performance8, because they are both risk-adjusted performance measures computed over the full period $1..T$. Naturally, practitioners routinely compute rolling Sharpe Ratios to visualize [risk-adjusted performance] dynamics8, but no formal framework exists for statistical inference on temporal stability when these rolling estimates are constructed from overlapping windows that induce autocorrelation8. To resolve this shortcoming, Traver and Rodriguez Dominguez8 introduces the Sharpe Stability Ratio $\widehat{SSR}(SR_0)$, defined as the mean rolling Sharpe Ratio minus a benchmak Sharpe Ratio $SR_0$18 divided by its long-run (or asymptotic20) standard deviation: [\widehat{SSR}(SR_0) = \frac{\widehat{\mu}{\hat{z}} - SR_0}{\widehat{\sigma}{\infty, \hat{z}}}] , where: $\hat{z}_1 = \widehat{SR}_{1:w}$, $\hat{z}_2 = \widehat{SR}_{2:w+1}$, …, $\hat{z}_{T-w+1} = \widehat{SR}_{T-w+1:T}$ are the Sharpe Ratios over the time periods $1..w$, $2..w+1$, …, $T-w+1…T$ Here, because the time periods $1..w$, $2..w+1$, …, $T-w+1…T$ are overlapping, the associated Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$ are usually called rolling Sharpe Ratios. $\widehat{\mu}_{\hat{z}} = \frac{1}{T-w+1} \sum_{k=1}^{T-w+1} \hat{z}_k$ is the sample mean of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$ $\widehat{\sigma}_{\infty, \hat{z}}^2$ is an estimator of the long-run variance of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$ Contrary to the standard variance $\sigma^2$, the long-run variance $\sigma^2_{\infty}$ is a measure of uncertainty in the mean that accounts for serial correlation. In the specific case of the Sharpe Stability Ratio, using the long-run variance instead of the variance is made necessary by8: The usage of overlapping windows in the computation of the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$ Indeed, successive windows share $w-1$ observations, implying first-order autocorrelation close to one when $w$ is large relative to the sampling frequency8, which would lead the naive variance estimator [to understate] the long-run variance by a factor of order $w$8. The (potential) serial dependence in the excess returns $r_1 - r_{f,1},…,r_T - r_{f,T}$ As a practical estimator for $\sigma^2_{\infty}$, Traver and Rodriguez Dominguez8 chooses the Newey–West heteroskedasticity and autocorrelation consistent (HAC) estimator20 with the triangular kernel, also known as the Bartlett kernel21: [\widehat{\sigma}{\infty, \hat{z}}^2 = \widehat{\gamma}{0, \hat{z}} + 2 \sum_{k=1}^L \left( 1 - \frac{k}{L + 1} \right) \widehat{\gamma}_{k, \hat{z}}] , where: $\widehat{\gamma}_{0, \hat{z}}$ is equal to the sample variance of the rolling Sharpe Ratios, that is, $\frac{1}{T-w+1} \sum_{k=1}^{T-w+1} \left( \hat{z}_k - \widehat{\mu}_{\hat{z}} \right)^2$ $\widehat{\gamma}_{k, SRs}, k \geq 1$ is equal to the sample autocovariance of the rolling Sharpe Ratios at lag $k$, that is, $\frac{1}{T-w+1} \sum_{t=k+1}^{T-w+1} \left( \hat{z}_t - \widehat{\mu}_{\hat{z}} \right) \left( \hat{z}_{t-k} - \widehat{\mu}_{\hat{z}} \right)$ $L \geq 0$ is a lag truncation parameter, estimated by the nonparametric data-dependent procedure of Newey and West22 based on both theoretical asymptotic and empirical Monte Carlo results22 Interpretation Similar to the Sharpe Ratio that is a signal-to-noise ratio for the excess returns $r_1 - r_{f,1},…,r_T - r_{f,T}$, the Sharpe Stability Ratio is by definition a signal-to-noise ratio for the rolling Sharpe Ratios $\hat{z}_1$, $\hat{z}_2$, …, $\hat{z}_{T-w+1}$. In terms of values: A strongly positive Sharpe Stability Ratio is desirable because it indicates stable superior performances v.s. the benchmark Conversely, a strongly negative Sharpe Stability Ratio is usually not desirable because it indicates stable inferior performances v.s. the benchmark Last, a zero-ish Sharpe Stability Ratio is usually not desirable either because it typically indicates instability of performances23 - whether positive of negative - v.s. the benchmark A handful of reference values for 17 Barclay Hedge Fund Indices, representing real-world strategies with pronounced non-normality, serial dependence and regime-switching behavior8, are provided in Traver and Rodriguez Dominguez8. For convenience, they are reproduced in Figure 2 (full sample analysis) and Figure 3 (subperiods analysis). Figure 2. Misc. statistics inc. SSR(0) over the full period, 17 Barclay Hedge Hedge Fund strategy indices, January 1997 - December 2025. Source: Traver and Rodriguez Dominguez. Figure 3. Analysis of SSR(0) over subperiods, 17 Barclay Hedge Hedge Fund strategy indices, January 1997 - December 2025. Source: Traver and Rodriguez Dominguez. Two comments from these figures: It seems that reaching a Sharpe Stability Ratio of 1 is challenging but achievable24, so that this value could be considered as a (high) reference value to have in mind when interpreting Sharpe Stability Ratios. By extension, an interesting question is how high a Sharpe Stability Ratio could be or could not be? To answer that question, I propose to use Bernard Madoff’s Fairfield Sentry fund. Because it is now established that this fund was a fraud, studying it generally helps to find implausible risk-adjusted performance metrics. Figure 4 compares the performance of the S&P500 to that of the Fairfield Sentry fund25 over a little less than 20 years. Figure 4. S&P500 v.s. Fairfield Sentry, November 1990 - October 2008. Using 36 months26 as a rolling window length for the Sharpe Stability Ratio computation, figures are provided below. Instrument Monthly Sharpe Ratio (0) Probabilistic Sharpe Ratio (0) Sharpe Stability Ratio (0) S&P500 (Total Return) 0.19 0.99 0.37 Fairfield Sentry fund 1.19 1 1.55 From this table, an investment exhibiting a Sharpe Stability Ratio of about 1.5 over a 20-year period should then be considered as highly suspicious. Now, it is possible to go a little further and compute the Sharpe Stability Ratio of the Fairfield Sentry fund on an expanding window basis27, which will provide reference values for an implausible Sharpe Stability Ratio over all standard evaluation periods for an investment. Results are provided in Figure 5, on which the apparently huge Sharpe Stability Ratio of 4.31 reported for the Global Macro Hedge Fund Strategy Indice (Figure 3) is not so huge anymore when compared to the Sharpe Stability Ratio of 12 computed for the airfield Sentry fund over an equivalent period of 5 years! Figure 5. Fairfield Sentry Sharpe Stability Ratio, expanding window from November 1994 to October 2008. Temporal stability, as measured by [the Sharpe Stability Ratio], is inherently conditional on the prevailing market regime rather than an intrinsic, regime-invariant attribute of each strategy8. As a consequence, the Sharpe Stability Ratio also captures how consistently a strategy performs within a given macro-financial environment and how its stability profile shifts across environments8. Asymptotic distribution and statistical inference Traver and Rodriguez Dominguez8 establishes that under stationarity and $\alpha$-mixing assumptions, the Sharpe Stability Ratio satisfies [\sqrt{T-w+1} \widehat{SSR}(SR_0) \overset{a}{\sim} N \left( 0, 1 \right)] That asymptotic normality provides a theoretical basis for making inference about the Sharpe Stability Ratio, although Traver and Rodriguez Dominguez8 emphasises that in practice, due to the strong autocorrelation induced by overlapping windows8, bootstrap inference should be prefered to asymptotic approximations. A couple more words on asymptotic results, though. If we introduce the “normalized” Sharpe Stability Ratio $\widehat{nSSR}(SR_0)$ defined by [\widehat{nSSR}(SR_0) = \sqrt{T-w+1} \widehat{SSR}(SR_0)] , it then becomes possible to interpret the associated (normalized) Sharpe Stability Ratio values in statistical terms, which might be useful to become familiar with this new risk-adjusted performance measure. For example: A normalized Sharpe Stability Ratio value of about 1.96 would imply that $SR_0$ lies inside the 95% confidence interval around the mean rolling Sharpe Ratio A normalized Sharpe Stability Ratio value of about 2.576 would imply that $SR_0$ lies inside the 99% confidence interval around the mean rolling Sharpe Ratio As other examples: A strongly positive normalized Sharpe Stability Ratio value would imply that the mean rolling Sharpe Ratio is substantially greater than $SR_0$ A strongly negative normalized Sharpe Stability Ratio value would imply that the mean rolling Sharpe Ratio is substantially lower than $SR_0$ A normalized Sharpe Stability Ratio value close to zero would imply that the mean rolling Sharpe Ratio is close to $SR_0$ Comparison with other Sharpe Ratio-based measures With the introduction of the Sharpe Stability Ratio, there are now at least28 3 Sharpe Ratio-based risk-adjusted performance measures to choose from when analyzing an investment: The Sharpe Ratio The Probabilistic Sharpe Ratio The Sharpe Stability Ratio Figure 6 summarizes the main question answered by each of these measures. Figure 6. Different Sharpe Ratio Dimensions: SR, PSR, SSR. Source: Traver and Rodriguez Dominguez. As can be seen in Figure 6, each of these measures capture a different dimension of performance evaluation8. Among them, the Sharpe Stability Ratio focuses on the persistence of risk-adjusted performances over time, which allows it to distinguish between investments with a similar Sharpe Ratio and Probabilistic Sharpe Ratio. As an example, Figure 7 represents 3 different synthetic strategies with the same29 Sharpe Ratios and Probabilistic Sharpe Ratios but different Sharpe Stability Ratios. Figure 7. Three synthetic strategies with identical Sharpe Ratios and Probabilistic Sharpe Ratios (0, 0.5 and 1.0). Source: Traver and Rodriguez Dominguez. From a visual inspection of Figure 7, Strategy A would be the preferred strategy in terms of temporal stability, which is confirmed by its higher Sharpe Stability Ratio30. As another example31, Figure 8 represents 2 different synthetic strategies with again the same Sharpe Ratios but this time very different Sharpe Stability Ratios. Figure 8. Two synthetic strategies (consistent v.s. episodic) with identical Sharpe Ratios. On Figure 8, the episodic strategy can be considered as a “crisis alpha” strategy while the consistent strategy can be considered as a “structural alpha” strategy. In this case, relying on the Sharpe Ratio alone would not allow to distinguish between these two strategies and would actually favor the episodic strategy that exhibits a slightly higher Sharpe Ratio. On the contrary, the Sharpe Stability Ratio properly favors32 the consistent strategy that generates persistent alpha8. Practical considerations How to choose the rolling window length? Traver and Rodriguez Dominguez8 notes that the rolling window length $w$ is intrinsically a smoothing decision, analogous to selecting a bandwidth in nonparametric estimation: shorter windows produce noisy, high‑variance Sharpe estimates, whereas longer windows yield smoother but more “averaged‑out” dynamics8. Thus, there is no universally applicable rolling window length. That being said, standard rolling window sizes typically used in the litterature are the following: 252 days33, for daily returns 52 weeks34, for weekly returns 12 months33, 36 months35 or 60 months33, for monthly returns How to choose the benchmark Sharpe Ratio $SR_0$? The benchmark Sharpe Ratio that appears in the Sharpe Stability Ratio formula plays a similar role to that of the benchmark Sharpe Ratio that appears in the Probabilistic Sharpe Ratio formula. Nevertheless, the Sharpe Stability Ratio is somewhat unstable near the benchmark boundary8, which leads Traver and Rodriguez Dominguez8 to recommend using $SR_0 = 0$ as a base benchmark in empirical applications8. Implementation in Portfolio Optimizer The Portfolio Optimizer Web API allows to compute both the Sharpe Stability Ratio and the normalized Sharpe Stability Ratio as described in this blog post, c.f. the documentation. Examples of usage The Sharpe Stability Ratio v.s. the Sharpe Ratio In the previous section, it has been empirically shown that the Sharpe Stability Ratio is complementary to the Sharpe Ratio, c.f. for example Figure 7 and Figure 8. To illustrate the extent of that difference on real-world strategies with pronounced non-normality, serial dependence and regime-switching behavior8, Figure 9 plots the Sharpe Ratio v.s. the Sharpe Stability Ratio of the same 17 Barclay Hedge Fund Indices as used in Traver and Rodriguez Dominguez8 over the period