The Sharpe ratio is a widely used measure of risk-adjusted return in finance. It was introduced by William F. Sharpe in 1966, and it measures how much excess return an investor can obtain per unit of risk taken on an investment.
Mathematically, the Sharpe ratio is defined as the ratio of the excess return of a portfolio over the risk-free rate (typically the rate of return on a government bond) to the standard deviation of that excess return:
$$Sharpe\ Ratio = \frac{R_p - R_f}{\sigma_p}$$
where,
Rp is the portfolio’s average annual return over a certain period,
Rf is the risk-free rate of return over the same period,
σp is the portfolio’s standard deviation.
The Sharpe ratio calculation provides a way to measure the risk-adjusted performance of a portfolio. The higher the Sharpe ratio, the better the portfolio’s risk-adjusted return. A Sharpe ratio of 1 or more is generally considered good, while a ratio below 1 may indicate that the portfolio is not generating enough excess return to justify the risk taken on.
For example, suppose that a portfolio has an average annual return of 10% and a standard deviation of 12%, while the risk-free rate of return for the same period is 2%. The Sharpe ratio for this portfolio would be:
$$Sharpe\ Ratio = \frac{10\% - 2\%}{12\%} = 0.667$$
This means that for each unit of risk taken on by the portfolio, the excess return generated is equal to 0.667 units.
In practice, it is important to consider the drawbacks of the Sharpe ratio. For instance, it assumes that returns are normally distributed and that the investor’s utility function is represented by a quadratic function of the returns. In addition, the Sharpe ratio can be sensitive to outliers and to the choice of the risk-free rate. Nevertheless, it remains a valuable tool for comparing the risk-adjusted performance of different portfolios or investment strategies.