Sharpe ratiosPosted by Yuling Yao on Aug 01, 2019.
This is a jitt question in Andrew’s class: to (by hand) approximate the standard deviation of X/Y if X is N(5,1) and Y is N(1,3).
OK it is a hook to bite: the answer is that it is ill-defined as the ratio of two independent normal random variables are Cauchy tailed so it makes very little sense to talk about the standard deviation… unless if I worked in finance and I compute the Sharpe ratio (arithmetic return average/ average sd), which in the best case scenario with all nice assumptions on time stationarity, the normality of return, and the independent starting/stopping time, I end up with a heavy-tailed student-t distribution and any direct comparison of any realization seems prohibitively noisy.
A slightly better remedy, is to shift from the deviation from the difference, say I use the metric average return - average variance. The logic is that log-return of portfolio = arithmetic return of stock −- variance of return/ 2 under the some normal assumption or approximately second order expansion – which however can be questioned in the first place. Nevertheless it is a more robust metric – both the empirical mean and variance has root n convergence rate. It is of course a different metric as we do not weight the return per risk, but still maybe useful in terms of finite sample computational stability.