SHASHo vs. Normal: Conditioning the Sharpe Ratio Estimator’s Sampling Distribution

Disclosure: EODHD provides me with free access to its data, which is used in this article. The article also contains an affiliate link to EODHD: if you sign up through it, I may receive a commission at no extra cost to you. The analysis and conclusions presented are my own and are for educational purposes only, not investment advice.

Data & methodology

All price data used in this analysis comes from EODHD (eodhd.com): S&P 500 index (GSPC.INDX) history at three sampling frequencies — 5-minute, 1-hour, and daily — split into six analysis windows, one uptrend and one downtrend regime at each frequency.

For each window ("cell"), log returns \(r_t = \ln P_t - \ln P_{t-1}\) are computed, and four real, empirical statistics are estimated from them: the raw (non-annualized) Sharpe-ratio point estimate \(\widehat{SR} = \bar r / s_r\), where \(\bar r\) is the sample mean and \(s_r\) the sample standard deviation of the returns in that window — no risk-free rate is subtracted from the numerator, since the object of study is the estimator's own sampling variability, not an excess-return figure. Also estimated: the returns' lag-1 sample autocorrelation \(\hat\rho\), and their sample skewness \(\hat\gamma_3\) and excess kurtosis \(\hat\gamma_4\).

To study how much \(\widehat{SR}\) would vary run to run under the cell's own return-generating behavior, each cell's returns are used to fit an AR(1) process plus a kernel-density estimate of its residuals; that fitted process is then simulated 10,000 times at the cell's real sample length \(T\). Each simulated path yields its own \(\widehat{SR}\), together building the empirical sampling distribution shown as the gray histogram in each chart below.

Two probability models are then fit to that simulated distribution using GAMLSS distributional regression (Stasinopoulos et al., 2017), in which each distributional parameter — not just the mean — can be modeled as its own function of predictor variables. Here that function is nn(), a small neural-network-based smoother from the gamlss.add package: rather than assuming a linear or otherwise fixed form, it lets each parameter flex nonlinearly with the simulated \((\hat\rho,\hat\gamma_3,\hat\gamma_4)\). A third, formula-based distribution is overlaid alongside them for comparison. All three are evaluated at the cell's real, empirical tuple:

• Normal (NO) — mean fixed at the cell's real \(\widehat{SR}\); standard deviation \(\sigma\) conditioned on \((\hat\rho,\hat\gamma_3,\hat\gamma_4)\) via nn().
• SHASHo — a four-parameter sinh-arcsinh distribution (Jones & Pewsey, 2009); all four of \((\mu,\sigma,\nu,\tau)\) are conditioned on \((\hat\rho,\hat\gamma_3,\hat\gamma_4)\) via nn() the same way. Unlike the Normal, none of these four raw parameters is directly one of \(\widehat{SR}\)'s moments: its actual mean is a function of all four parameters, its variance a function of \((\sigma,\nu,\tau)\), and its skewness and excess kurtosis are each a function of \((\nu,\tau)\) alone — which is why the "Fitted distributions" panel in each image below reports back-calculated moments rather than the raw \((\mu,\sigma,\nu,\tau)\) values.
• Normal (Prado) — mean fixed at the cell's real \(\widehat{SR}\), same as NO, but standard deviation not fit via nn() at all: instead taken from a closed-form analytical formula (López de Prado, Lipton & Zoonekynd, 2026) for \(\widehat{SR}\)'s standard error under non-normal, autocorrelated returns, evaluated directly from the cell's real \(T\), \(\hat\rho\), \(\hat\gamma_3\), and \(\hat\gamma_4\) — an independent, formula-based benchmark against the two nn()-fitted models above, shown as the dashed "Normal (Prado)" curve.

Conclusions

Across all 6 cells, SHASHo's curve tracks the simulated \(\widehat{SR}\) histogram's shape more closely than the Normal's does — though how much of that gap is really about skewness and tail weight varies cell by cell: modest in some (skewness under 0.03 in absolute value in two of the six), more pronounced in others (up to roughly ±0.12).

The likelihood-ratio test confirms SHASHo's edge is statistically decisive everywhere — it beats the plain, \((\hat\rho,\hat\gamma_3,\hat\gamma_4)\)-conditioned Normal in 6 of 6 cells (p < 0.0001 throughout) — but the mechanism behind that isn't uniform: in cells with more pronounced skew/kurtosis, letting \((\nu,\tau)\) flex explains much of it; in others (1-Hour — Uptrend most starkly, where SHASHo's fitted skewness is just 0.02), a bigger driver is that Normal (NO)'s mean is fixed by design to the cell's real \(\widehat{SR}\), while SHASHo's mean is fit freely and can track the simulated distribution's actual center instead.

The Vuong tests complicate the practical verdict, though: Normal (Prado) — a closed-form formula fit to nothing — out-predicts both nn()-fitted models in 11 of 12 comparisons run; SHASHo avoids losing to it in only 1 of 6 cells.

So is it worth modeling with SHASHo? For understanding why \(\widehat{SR}\)'s distribution departs from Normal, yes — its extra parameters let where the distribution centers, how it skews, and how heavy its tails are all be estimated from the data instead of assumed. For predicting the distribution itself as accurately as possible, these 6 cells don't make a clean case for the added complexity over Prado's much simpler formula.

References

EODHD. (n.d.). EODHD Financial Data APIs [Data set]. Use my personal code DROZIMEK10 for 10% off your first purchase.

Jones, M. C., & Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96(4), 761–780. https://doi.org/10.1093/biomet/asp053

López de Prado, M., Lipton, A., & Zoonekynd, V. (2026). How to use the Sharpe ratio. http://dx.doi.org/10.2139/ssrn.5520741

Ozimek, K. (2026). Sharpe Ratio Distribution: When Normal's a No-Go, Why Not SHASHo? (1/4).

Stasinopoulos, M. D., Rigby, R. A., Heller, G. Z., Voudouris, V., & De Bastiani, F. (2017). Flexible regression and smoothing: Using GAMLSS in R. Chapman & Hall/CRC.