4.5 Statistics of the Sharpe ratio

4.5.1Setup#

Let $r_1,\dots,r_T$ be per-period excess returns (over the risk-free rate, or over a benchmark), stationary with mean $\mu$, variance $\sigma^2$, third and fourth central moments $\mu_3,\mu_4$, skewness $\gamma_3=\mu_3/\sigma^3$ and (non-excess) kurtosis $\gamma_4=\mu_4/\sigma^4$; a normal distribution has $\gamma_3=0$, $\gamma_4=3$. The per-period Sharpe ratio is $S=\mu/\sigma$, estimated by $\hat S=\hat\mu/\hat\sigma$. Throughout, $S$ and $T$ are in the same period (daily Sharpe with $T$ days). Annualise only at the end.

4.5.2The sampling distribution#

Proof

Let $\theta=(\mu,\sigma^2)$ and $\hat\theta=(\hat\mu,\hat\sigma^2)$. Write $\hat\sigma^2=\frac1T\sum(r_t-\mu)^2-(\hat\mu-\mu)^2$; the second term is $O_p(1/T)$ and does not affect the $\sqrt T$ limit. By the multivariate CLT applied to the iid vectors $(r_t-\mu,(r_t-\mu)^2-\sigma^2)$,

$$\sqrt T(\hat\theta-\theta)\xrightarrow{d}N(0,\Sigma),\qquad \Sigma=\begin{pmatrix}\sigma^2&\mu_3\\ \mu_3&\mu_4-\sigma^4\end{pmatrix},$$

because $\operatorname{Cov}(r-\mu,(r-\mu)^2)=\mathbb E(r-\mu)^3=\mu_3$ and $\operatorname{Var}((r-\mu)^2)=\mu_4-\sigma^4$. The Sharpe ratio is $g(\theta)=\mu/\sqrt{\sigma^2}$ with gradient $\nabla g=\big(1/\sigma,\;-\mu/(2\sigma^3)\big)$. By the delta method the asymptotic variance is

$$\nabla g^\top\Sigma\nabla g=\frac{\sigma^2}{\sigma^2}-2\cdot\frac1\sigma\cdot\frac{\mu}{2\sigma^3}\mu_3+\frac{\mu^2}{4\sigma^6}(\mu_4-\sigma^4)=1-S\gamma_3+\frac{S^2}{4}(\gamma_4-1).$$

With $\gamma_3=0$, $\gamma_4=3$ this is $1+S^2/2$.∎

The two non-normal terms have intuitive signs. Negative skew ($\gamma_3<0$) with $S>0$ increases the variance: option-selling strategies (M09) show smooth gains and rare large losses, and their Sharpe ratios are less reliable than they look. Fat tails ($\gamma_4>3$) increase it further. For daily data $S$ is small (around 0.06 for an annual 1.0), so the correction terms are small and $\operatorname{SE}\approx1/\sqrt T$ to a good approximation. The code checks (4.12) against Monte Carlo for normal and Student-$t$ returns.

4.5.3Annualisation with autocorrelated returns#

Proof

The $q$-period sum has mean $q\mu$ and variance $\sum_{i,j=1}^{q}\operatorname{Cov}(r_i,r_j)=\sigma^2\big(q+2\sum_{k=1}^{q-1}(q-k)\rho_k\big)$, since lag $k$ appears in $q-k$ pairs on each side of the diagonal. Divide.∎

Positive autocorrelation, which smoothed or stale prices create (illiquid stocks, monthly NAVs, mark-to-model books), makes $\sqrt q$ scaling overstate the annual Sharpe ratio. With monthly data and $\rho_1=0.2$ only, $\eta(12)=12/\sqrt{12+2\cdot11\cdot0.2}=2.963$ against $\sqrt{12}=3.464$: the naive annualised Sharpe ratio is 17% too high.

4.5.4Probabilistic Sharpe ratio and minimum track record#

Minimum track record length Setting $\mathrm{PSR}(S^*)=1-\alpha$ and solving for $T$:

For an annual Sharpe ratio of 1.0 on daily data, $S^*=0$ and 95% confidence ($z=1.645$), MinTRL is 684 days (2.7 years) with normal returns, 730 days (2.9 years) with $\gamma_3=-1,\gamma_4=8$, and 778 days (3.1 years) with $\gamma_3=-2,\gamma_4=15$. Problem 4.9 asks you to derive (4.16).