Returns are iid with mean mu, variance sigma^2 and kurtosis k. Apply the delta method to the pair (mu_hat, sigma_hat^2) to derive the asymptotic variance of the estimated Sharpe ratio mu_hat/sigma_hat, then evaluate it for a true Sharpe of 0.5 under normality.

Returns are iid with mean mu, variance sigma^2 and kurtosis k. Apply the delta method to the pair (mu_hat, sigma_hat^2) to derive the asymptotic variance of the estimated Sharpe ratio mu_hat/sigma_hat, then evaluate it for a true Sharpe of 0.5 under normality.

Approach: Write the Sharpe as a smooth function of the first two moments, take its gradient, and sandwich the joint asymptotic covariance of the moment estimates between two copies of it.

1.125/T. Write SR = g(mu, v) = mu/sqrt(v) with v = sigma^2. The gradient is (1/sqrt(v), -mu/(2*v^{3/2})). Under iid sampling with zero third central moment, sqrt(T)*(mu_hat - mu, v_hat - v) is asymptotically normal with covariance diag(v, (k - 1)*v^2). The delta method gives asymptotic variance g'*Sigma*g = v*(1/v) + (mu^2/(4*v^3))*(k - 1)v^2 = 1 + SR^2(k - 1)/4. Under normality the kurtosis k = 3, so this is 1 + SR^2/2, and at SR = 0.5 the variance of the estimate is 1.125/T with standard error sqrt(1.125/T). Fat tails raise k and inflate it directly, and non-zero skewness adds a cross term proportional to the third moment times SR, so a strategy that sells options has a Sharpe standard error the normal formula understates in both terms at once.

Follow-up: What is the asymptotic variance of the log of the Sharpe ratio, and why does an interval built on that scale behave better in small samples?

Key concepts: delta method, asymptotic variance, Sharpe ratio, kurtosis.