Problem Description

A financial return series consists of 2n observations. Exactly n of these observations are zero returns. The remaining n observations are non-zero returns, denoted x1, x2, ..., xn, which have a known mean mu and standard deviation sigma.

Assume the risk-free rate is zero, so the Sharpe ratio is defined as: Sharpe = mean(return) / std(return)

Questions

1. Calculate and compare the Sharpe ratio computed on all 2n observations with the Sharpe ratio computed after removing the n zero-return observations (i.e., computed only on x1, ..., xn).
2. Determine if the Sharpe ratio of the filtered series (with zeros removed) is higher or lower than the original series.
3. If your conclusion from point 2 depends on the sign of the mean return (mu), explain the underlying reasons.
4. Briefly discuss whether the conclusion from point 2 would still hold if the n removed observations were not necessarily the zero-return ones, but rather any n arbitrary observations.