## Sharpe ratio (‘historic’)

The Sharpe ratio is useful for assessing risk-adjusted returns of an investment.  Higher ratios are better outcomes.

## Calculation of the Sharpe ratio

Consider the historic series of returns from an investment portfolio where ri represents any single return during one interval of time.  The corresponding individual return from a benchmark security or portfolio is bi [the typical risk-free benchmark security is a U.S. Treasury bond of appropriate maturity date].  The differential return, di, reflects the difference between the investment portfolio’s return and the benchmark’s return at the same interval of time.

• di = ribi

D is the arithmetic average of all differential returns.

• D = sum of all di‘s divided by the number of di’s in the series

σdi is the standard deviation of all differential returns.

• single deviation = diD at one interval of time in the series
• single squared deviation = (diD)2
• N is the number of time intervals in the series
• σdi = (sum of single squared deviations/(N-1))½

The historic Sharpe ratio is the average difference between investment returns and benchmark returns relative to the variability of differences in returns.

• Sharpe = D/σdi.

The Sharpe ratio is useful in at least two ways:

1. A negative Sharpe ratio indicates that the benchmark yields a higher return than the investment portfolio.  For example, the benchmark Treasury bond would outperform the investment portfolio.
2. A comparatively high Sharpe ratio indicates that the investment returns are comparatively high in relationship to investment risk.  In this analysis, D is an index of investment returns and σdi is an index of investment risk.

## Reference

William F. Sharpe.  The Sharpe Ratio.  Stanford University.  Reprinted fromThe Journal of Portfolio Management, Fall 1994.  www.stanford.edu/~wfsharpe/art/sr/sr.htm