The Sharpe ratio (Sharpe, 1966) computes the risk premium of the investment portfolio per unit of total risk of the portfolio. The risk premium, also known as excess return, is the return of the portfolio less the risk-free rate of interest as measured by the yield of a Treasury security. The total risk is the standard deviation of returns of the portfolio. The numerator captures the reward for investing in a risky portfolio of assets in excess of the risk-free rate of interest while the denominator is the variability of returns of the portfolio. In this sense, the Sharpe measure is also called the ”reward-to-variability” ratio. Equation (34.1) gives the Sharpe ratio:
where S is the Sharpe ratio, rp the return of the portfolio, rf the risk-free rate, and sp the standard deviation of returns of the portfolio.
The Sharpe ratio for an investment portfolio can be compared with the same for a benchmark portfolio such as the overall market portfolio. Suppose that a managed portfolio earned a return of 20 percent over a certain time period with a standard deviation of 32 percent. Also assume that during the same period the Treasury bill rate was 4 percent, and the overall stock market earned a return of 13 percent with a standard deviation of 20 percent. The managed portfolio’s risk premium is (20 percent — 4 percent) = 16 percent, while its Sharpe ratio, S, is equal to 16 percent/32 percent = 0.50. The market portfolio’s excess return is (13 percent — 4 percent) = 9 percent, while its S equals 9 percent/20 percent = 0.45. Accordingly, for each unit of standard deviation, the managed portfolio earned a risk premium of 0.50 percent, which is greater than that of the market portfolio of 0.45 percent, suggesting that the managed portfolio outperformed the market after adjusting for total risk.
The Treynor ratio (Treynor, 1965) computes the risk premium per unit of systematic risk. The risk premium is defined as in the Sharpe measure. The difference in this method is in that it uses the systematic risk of the portfolio as the risk parameter. The systematic risk is that part of the total risk of an asset which cannot be eliminated through diversification. It is measured by the parameter known as ‘beta’ that represents the slope of the regression of the returns of the managed portfolio on the returns to the market portfolio. The Treynor ratio is given by the following equation:
where T is the Treynor ratio, rp the return of the portfolio, rf the risk-free rate, and bp the beta of the portfolio.
Suppose that the beta of the managed portfolio in the previous example is 1.5. By definition, the beta of the market portfolio is equal to 1.0. This means the managed portfolio has one-and-half times more systematic risk than the market portfolio. We would expect the managed portfolio to earn more than the market because of its higher risk. In fact, in the above example, the portfolio earned an excess return of 16 percent whereas the market earned only 9 percent. These two numbers alone do not tell anything about the relative performance of the portfolio since the portfolio and the market have different levels of market risk. In this instance, the Treynor ratio for the managed portfolio equals (20 percent — 4 percent)/1.5 = 10.67, while that for the market equals (13 percent — 4 percent)/1.00 = 9.00. Thus, after adjusting for systematic risk, the managed portfolio earned an excess return of 10.67 percent for each unit of beta while the market portfolio earned an excess return of 9.00 percent for each unit of beta. Thus, the managed portfolio outperformed the market portfolio after adjusting for systematic risk.