15.4 Applications in Performance Measurement

Sharpe Ratio

Investors want a measure of how good a professional money manager is before they entrust their hard-earned funds to that professional for investing. Suppose that you see an advertisement in which McKinley Investment Management claims that the portfolios of its clients have an average return of 20% per year. You know that this average annual return is meaningless without also knowing something about the riskiness of the firm’s strategy. In this section, we consider some ways to evaluate the riskiness of an investment strategy.

A basic measure of investment performance that includes an adjustment for risk is the Sharpe ratio. The Sharpe ratio is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return, using the formula

The portfolio risk premium is the portfolio return R_P minus the risk-free return R_f; this is the basic reward for bearing risk. If the risk-free return is 3%, McKinley Investment Management’s clients who are earning 20% on their portfolios have an excess return of 17%.

The standard deviation of the portfolio’s return, ${\sigma }_{\mathrm{P}}$, is a measure of risk. Although you see that McKinley’s clients earn a nice 20% return on average, you find that the returns are highly volatile. In some years, the clients earn much more than 20%, and in other years, the return is much lower, even negative. That volatility leads to a standard deviation of returns of 26%. The Sharpe ratio would be $\frac{17\%}{26\%}$, or 0.65.

Thus, the Sharpe ratio can be thought of as a reward-to-risk ratio. The standard deviation in the denominator can be thought of as the units of risk the investor has. The numerator is the reward the investor is receiving for taking on that risk.

Treynor Measurement of Performance

Another reward-to-risk ratio measurement of investment performance is the Treynor ratio. The Treynor ratio is calculated as

Just as with the Sharpe ratio, the numerator of the Treynor ratio is a portfolio’s risk premium; the difference is that the Treynor ratio focus focuses on systematic risk, using the beta of the portfolio in the denominator, while the Shape ratio focuses on total risk, using the standard deviation of the portfolio’s returns in the denominator.

If McKinley Investment Management has a portfolio with a 20% return over the past five years, with a beta of 1.2 and a risk-free rate of 3%, the Treynor ratio would be $\frac{(0.20 - 0.03)}{1.2}=\mathrm{0.14.}$

Both the Sharpe and Treynor ratios are relative measures of investment performance, meaning that there is not an absolute number that indicates whether an investment performance is good or bad. An investment manager’s performance must be considered in relation to that of other managers or to a benchmark index.

Jensen’s Alpha

Jensen’s alpha is another common measure of investment performance. It is computed as the raw portfolio return minus the expected portfolio return predicted by the CAPM:

Suppose that the average market return has been 12%. What would Jensen’s alpha be for McKinley Investment Management’s portfolio with a 20% average return and a beta of 1.2?

Unlike the Sharpe and Treynor ratios, which are meaningful in a relative sense, Jensen’s alpha is meaningful in an absolute sense. An alpha of 0.062 indicates that the McKinley Investment Management portfolio provided a return that was 6.2% higher than would be expected given the riskiness of the portfolio. A positive alpha indicates that the portfolio had an abnormal return. If Jensen’s alpha equals zero, the portfolio return was exactly what was expected given the riskiness of the portfolio as measured by beta.