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).
Let’s assume returns are normally distributed (and IID). Then we can write
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.
Now we can use maximum likelihood estimation (MLE) to estimate μ and σ.
From elementary Statistics, we know that the probability density function of the normal distribution is
The likelihood function for our case is
and the log-likelihood function is
To find μ that maximizes the likelihood function, we differentiate the log-likelihood function with respect to μ and set it to 0 i.e.,
To find σ² that maximizes the likelihood function, we differentiate the log-likelihood function with respect to σ² and set it to 0 i.e.,
Finally, we can write
