# Sharp Ratio Bias

Compounding the monthly returns but calculating the standard deviation from the ( not compounded) monthly returns will upward bias sharp ratio. Why is that? How are the compounded return different from simple return?

The issue with Sharpe Ratio bias is that the way that the SD grows over time (with the square root of time) is different than the way that the return grows over time. Letâ€™s assume for the moment that the (expected annual return) = 12 months * (expected monthly return). The challenge is that the expected annual SD = SqRT(12 months) * (expected monthly SD) Now, if you compare sharpe ratios, what you see is [annual Sharpe Ratio] = 12 * [monthly return] / [sqrt(12) * monthly SD] = SqRt(12) * [Monthly S.R.] So the main point is that the Sharpe Ratio for longer periods will look higher than the sharpe ratio for shorter periods. This is one of the reasons why most Sharpe Ratios need to be standardized to an annual period - to make them more comparable. Now, if you compound the returns, the compounded returns are going to be even higher than the uncompounded returns, which pushes up the sharpe ratio even more.

The annualized ST.DEV. of daily returns is generally higher than the weekly, which is in turn higher than th monthly.

why? because:

daily stdev* sqrt (252)

weekly stdev* sqrt (52)

monthly stdev* sqrt (12)

Lenghtenting the measurement interval will result in a lower estimate of volatility. (does this mean, i rather take volatiity for a month which i only multiply with sqrt of 12 which in turn gives me a lower STDV and in turn a higher sharpe ratio?

thanks for confirming / elaborating