Sharpe Ratio Distribution: When Normal’s a No-Go, Why Not SHASHo? (3/4)
Post 2/4 (Block 3) landed on the exact, unconditional result: $P(\widehat{SR}\le\widehat{SR}^*)=\Phi(R^*)$ — SHASHo's probability that the SR estimator falls at or below any value collapses exactly to a standard Normal one.
Five slides attached, being the main content of this post, follow: the first sets up the test's conditional probability, the remaining four present the test-related measures built on it.
1.Test setup
2. 1️⃣ p-value ($p$)
3. 2️⃣ Critical value ($SR_c$)
4. 3️⃣ Test power ($1-\beta$)
5. 4️⃣ Minimum Track Record Length (MinTRL)
One notation note before that: López de Prado et al. (2026) define the upper-tail probability as $P(X\ge a)$, not through the more standard $\le$-based CDF form. Slides 1–5 mirror that $\ge$ convention to keep the comparison to their formulas direct — it doesn't replace the $\le$-based CDF identity from Post 2/4, which the $\ge$ form is simply the complement of: $P(X\ge a)=1-P(X\le a)$ for continuous distributions.
Each slide 2–5 follows the same three-part structure. General is the definition of the measure itself — true regardless of which distribution $\widehat{SR}$ follows. Normal is a specific closed-form formula, derived in López de Prado et al. (2026) from the asymptotic Normal distribution shown there for the Sharpe ratio estimator. SHASHo is the corresponding formula under the SHASHo distribution, derived in this series (the derivation itself isn't repeated here).
Throughout slides 2–5, $\sigma[SR_0]$ and $\sigma[SR_1]$ denote the standard error of the Sharpe ratio estimator (López de Prado et al., 2026) evaluated under the null and alternative hypothesis respectively — each conditioned on the number of observations $T$, the estimated return autocorrelation $\hat\rho$, skewness $\hat\gamma_3$, and kurtosis $\hat\gamma_4$, and on $SR_0$ or $SR_1$ itself.
Quick definition: $u_{1-\alpha}\equiv\Phi^{-1}(1-\alpha)$, the standard Normal quantile at the $1-\alpha$ level — used throughout slides 3–5.
An additional note: $T_{obs}$ and Normal's $T$ are the same actual sample size — the different label only signals that SHASHo's scale is anchored at that one observed value, whereas Normal's scale formula holds at any $T$, which matters once MinTRL (slide 5) solves for a different one.
One interesting detail worth a closer look: comparing the monotonicity of all four measures — in $\widehat{SR}^*$, $SR_0$, and $SR_1$ — between Normal and its SHASHo counterpart. For those eager to dig deeper, that comparison is left to uncover.
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
