Sharpe Ratio: SHASHo Vs. Normal, Part 4/4

Sharpe Ratio: SHASHo vs. Normal, Part 4/4 | Dr Krzysztof Ozimek

Series results · 4 of 4

Posts 1–3 made the case that $\widehat{SR}$ fits SHASHo better than Normal for this dataset — and not just cosmetically: the exact R transform lets four of López de Prado and coauthors' (2026) measures (p-value, critical value, test power, MinTRL) be re-derived cleanly under SHASHo, with mean, variance, skewness, and kurtosis as explicit functions of its four parameters $(\mu, \sigma, \nu, \tau)$.

Throughout, SHASHo's four parameters were fit unconditionally: one fixed set of values for the whole sample. That's consistent with how I've approached this before — this wasn't the first time I tested SHASHo's fit against $\widehat{SR}$ on different data, and it has held up exploratorily each time. But it's worth being explicit that "unconditional" is a real limitation.

López de Prado et al. assume a Normal distribution for $\widehat{SR}$, with variance dependent on the returns' characteristics: $\hat\rho$, $\hat\gamma_3$, $\hat\gamma_4$ (regime-descriptive estimates), and the hypothesized SR level. That conditioning of the variance is the crux of their argument: a variance that responds to sample properties gives more trustworthy inference, which in turn means more effective portfolio management or strategy selection.

But in their approach, $\widehat{SR}$'s distribution is still Normal — only its variance adapts. What if that shape assumption itself doesn't hold under some market conditions? A better-behaved variance may not help if the underlying shape is wrong. That's where a conditioned SHASHo could matter — GAMLSS supports modeling any of its parameters as a function of covariates (Stasinopoulos et al., 2017), letting not just variance but mean, skewness, and kurtosis adapt to the regime too.

That leaves an open question this series doesn't resolve: if SHASHo's parameters were conditioned on the same variables López de Prado et al. use to condition $\sigma$, or some other way, would a conditional SHASHo deliver the same payoff they claim for conditional variance: more reliable inference on $\widehat{SR}$, and by extension, better strategy selection, portfolio management, and ultimately better trading results?

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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