Sharpe Ratio Distribution: When Normal’s a No-Go, Why Not SHASHo? (1/4)
I recently came across López de Prado et al.'s (2026) treatment of the Sharpe ratio estimator: $\widehat{SR}\sim N\big(SR,\ \sigma^2[SR]=f(T,\rho,\gamma_3,\gamma_4,SR)\big)$ — Normal, with the true $SR$ as its mean, and a variance $\sigma^2[SR]$ (see the Normal section of the image below) conditioned on sample size, the excess return series' autocorrelation, its skewness, its kurtosis, and the level of $SR$ itself.
It's an elegant correction — but stubbornly a bell curve underneath 🙃
That sent me down a research rabbit hole. I bootstrapped the Sharpe ratio estimator 5,000 times on two years of daily S&P 500 returns and let GAMLSS pick its own unconditional shape. SHASHo came out with all four parameters significant and a better GAIC than Normal — only one dataset, but a spur to dig deeper. It also seems rarely used in mainstream toolkits, and I found no prior work applying it to the Sharpe ratio.
SHASHo — the sinh-arcsinh distribution from Jones and Pewsey (2009) — is a four-parameter family $(\mu,\sigma,\nu,\tau)$: location, scale, skewness shape, kurtosis shape. For $\widehat{SR}\sim\text{SHASHo}(\mu,\sigma,\nu,\tau)$, the mean depends on all four parameters, variance on $\sigma,\nu,\tau$, and skewness and excess kurtosis on $\nu,\tau$ alone. Set $\nu=0,\tau=1$ and it collapses back to Normal; move away from that point and it models skewness and fat tails — which Normal can't.
That raised the question: if $\widehat{SR}$ can be shaped like SHASHo rather than Normal, can you still test $H_0: SR\le SR_0$ vs. $H_1: SR>SR_0$ like López de Prado et al. — with the same four picked measures they derive, listed below — but now for a different assumed distribution?
- p-value
- critical value
- test power
- Minimum Track Record Length
It turns out you can. The SHASHo variable has an exact $R$ transform that is itself standard Normal — and that fact alone makes all four measures re-derivable under this richer shape.
The SHASHo distributional properties and the $R$ transform of $\widehat{SR}$ — both used throughout this series — come from Rigby et al. (2020).
Over the next three posts, I'll cover:
- Structural assumptions behind SHASHo
- Four mentioned measures — López de Prado et al.'s original, and its SHASHo counterpart
- Summary and a look ahead
References
López de Prado, M., Lipton, A., & Zoonekynd, V. (2026). How to use the Sharpe ratio. SSRN.
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5520741
Jones, M. C., & Pewsey, A. (2009). Sinh-arcsinh distributions: A broad family giving rise to powerful
tests of normality and symmetry. Biometrika, 96(4), 761–780.
Rigby, R. A., Stasinopoulos, M. D., Heller, G. Z., & De Bastiani, F. (2020). Distributions for modeling
location, scale, and shape: Using GAMLSS in R. Chapman and Hall/CRC.
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