Sometimes you want to estimate a Sharpe Ratio, but you don’t have a clean return time series i.e., the return data is available at heterogenous intervals of time. For example, perhaps you compute returns every day during month 1, then compute weekly returns during month 2, then revert to computing daily returns in month 3.

This typically happens when you a) don’t mark to market on a regular basis due to resource constraints (or laziness!) or b) you’re in a market where benchmark prices aren’t available at regular intervals.

Let’s imagine a case where we have m observations. For example, observation 1 might be the total return over 5 days, observation 2 might be the total return over 2 days, etc. Each observation i consists of nᵢ periods (e.g., 5 days).

where x is the return in each period. Note that each x is not observable directly. We only know the sum or mean of the x’s in each observation. The mean of x is distributed as follows.