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.
Follow me on LinkedIn and see the heated comments on this post.
Further learning
To see how SHASHo and Normal compare when the Sharpe ratio estimator's sampling distribution is conditioned on the return characteristics, on S&P 500 data at three frequencies, see: SHASHo vs. Normal: Conditioning the Sharpe Ratio Estimator's Sampling Distribution.
