Sharpe Ratio
We use the Ex-Post (Historic) Sharpe Ratio to compare the "effectiveness" of portfolios. The Sharpe Ratio indicates how much of an additional return a portfolio gets for an additional measure of variability (risk). The higher the Sharpe Ratio of a portfolio, the higher the additional return per unit of variability, and therefore, the more "efficient" the portfolio. It is a standard practice in financial literature to compare the Sharpe ratios of different portfolios: the higher the Sharpe Ratio, the "better" the portfolio.
Let Di be the difference between the portfolio's rate of return and the risk-free rate of return in month i:
Di = Xi - Xif
| where: | |
| | Xi is the portfolio's monthly rate of return in month i; |
| | Xif is the risk-free monthly rate of return in month i; |
| | i = 1, 2,...T; |
| | T - number of months in the sample. |
To simplify our calculations, instead of using Xif
- the risk-free rate for every month in the sample
- we use an average of (X1 + XT) / 2,
that is, the average of the first-month risk-free rate of return
and the last-month risk-free rate of return.
This substitution practically doesn't affect the results.
As above, the standard deviation  equals:
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