**Sharpe Ratio **

The most popular measure of risk-adjusted performance for investments is the Sharpe ratio (see “Measuring sticks,” below).

When using annual returns and an annual standard deviation of returns, the Sharpe ratio may be interpreted as the annual risk premium that the investment earned per percentage point in annual standard deviation. In this case, the investment’s return exceeded the riskless rate by 35 basis points for each percentage point in standard deviation. In an analysis of past data, the mean return of the portfolio is used as an estimate of its expected return, and the historical standard deviation of the sample is used as an estimate of the asset’s true risk. *Throughout the remainder of this analysis of performance measures, the analysis may be viewed as interchangeable between using historical estimates and expectations.*

Obviously, both the numerator and denominator of the Sharpe ratio should be measured in the same unit of time, such as quarterly or annual values. But the resulting Sharpe ratio is sensitive to the length of the time period used to compute the numerator and the denominator. Note that the numerator is proportional to the unit of time, ignoring compounding. Thus the excess return expressed as an annual rate will be two times larger than a semiannual rate and four times larger than a quarterly rate, ignoring compounding. However, the denominator is linearly related to the square root of time, assuming that returns are statistically independent through time:

σT = σ1 √ T where σT is the standard deviation over T periods, σ1 is the standard deviation over one time period, such as one year, and T is the number of time periods.

This formula assumes that the returns through time are statistically independent. Thus, a one-year standard deviation is only √2 times a semiannual standard deviation, and a one-year standard deviation is only twice (√4) the quarterly standard deviation. Thus switching from quarterly returns to annualized returns roughly increases the numerator fourfold but increases the denominator only twofold, resulting in a twofold higher ratio.