# Sharpe Ratio, Treynor Ratio, M2, and Jensen’s Alpha

The following are the four ratios commonly used in performance evaluation.

### Sharpe Ratio

The Sharpe Ratio is the portfolio risk premium divided by the portfolio risk. $$\text{Sharpe ratio} = \frac{ R_p – R_f } { \sigma_p }$$ The Sharpe ratio, or reward-to-variability ratio, is the slope of the capital allocation line (CAL). The greater the slope (higher number), the better the asset. Note that the risk being used is the total risk of the portfolio, not its systematic risk, which is a limitation of the measure. The portfolio with the highest Sharpe ratio has the best performance, but the Sharpe ratio is not informative. In order to rank portfolios, the Sharpe ratio for each portfolio must be computed. Further limitation occurs when the numerators are negative. In this instance, the Sharpe ratio will be less negative for a riskier portfolio, resulting in incorrect rankings.

### Treynor Ratio

The Treynor ratio is an extension of the Sharpe ratio. Instead of using total risk, Treynor uses beta or systematic risk in the denominator. $$\text{Treynor ratio} = \frac{ R_p – R_f } { \beta _p }$$ As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results. Apart from this, the Treynor ratio does not work for negative beta assets. Also, while both the Sharpe and Treynor ratios can rank portfolios, they do not provide information on whether the portfolios are better than the market portfolio. Similarly, they do not offer information about the degree of superiority of a higher ratio portfolio over a lower ratio portfolio.

### M-Squared (M²) Ratio

The concept behind the M² ratio is to create a portfolio P’ that mimics the risk of the market portfolio by altering the weights of the actual portfolio P and the risk-free asset until portfolio P’ has the same total risk as the market. The return on the mimicking portfolio P’ is determined and compared with the market return. The weight in portfolio P (wp), which sets the portfolio risk equal to the market risk, can be written as: $$w_p = \frac{ \sigma_m } { \sigma_p }$$ With the balance (1 – wp) invested in the risk-free asset. The return for the mimicking portfolio P’ is as follows: $$R_{ p’} = w_p R_p + (1 – w_p ) R_f$$ Which we can reformulate as: $$R_{ p’} = \frac{ \sigma_m } { \sigma_p } × R_p + (1 – \frac{ \sigma_m } { \sigma_p } )×R_f$$ Therefore, $$R_{ p’} = R_f + \sigma_m \frac{ [R_p – R_f ] } { \sigma_p }$$ The difference in return between the mimicking portfolio and the market return is M² which is expressed as: $$M^2 = \left[R_p – R_f \right] \frac{ \sigma_m } { \sigma_p }+ R_f=SR\times\sigma_m+R_f$$ A portfolio that matches the market’s return will have an M² value equal to zero, while a portfolio that outperforms will have a positive value. By using the M² measure, it is possible to rank portfolios and also determine which portfolios beat the market on a risk-adjusted basis.

### Jensen’s Alpha

Jensen’s alpha is based on systematic risk. The daily returns of the portfolio are regressed against the daily returns of the market. Essentially, this is done in order to compute a measure of this systematic risk in the same manner as the CAPM. The difference between the actual return of the portfolio and the calculated or modeled risk-adjusted return is a gauge of performance relative to the market. $$\text{Jensen’s alpha} = \alpha_p = R_p – [R_f + \beta_p (R_m– R_f)]$$ If \alphap is positive, the portfolio has outperformed the market, while a negative value indicates underperformance. The alpha values can also be used to rank portfolios or the managers of those portfolios, with the alpha being a representation of the maximum amount an investor should pay for the active management of that portfolio.

### Question

A client has three portfolio choices, each with the following characteristics: $$\begin{array}{l|r|r|r} \textbf{} & \textbf{Expected Return} & \textbf{Volatility} & \textbf{Beta} \\ \hline \text{Portfolio A} & 15\% & 12\% & 10\% \\ \text{Portfolio B} & 18\% & 14\% & 11\% \\ \text{Portfolio C} & 12\% & 9\% & 5\% \\ \end{array}$$ The efficient market portfolio has an expected return of 20%, a standard deviation of 12%, and a risk-free interest rate of 5%. Based on the Sharpe ratio for each portfolio, the client should choose:
1. Portfolio A.
2. Portfolio B.
3. Portfolio C.
Solution The correct answer is portfolio B. $$\text{Sharpe ratio} = \frac{ R_p – R_f } { \sigma_p }$$ The portfolio with the highest Sharpe ratio has the best performance. $$\begin{array}{l|r|r} \textbf{} & \textbf{Calculation} & \textbf{Sharpe Measure} \\ \hline \text{Portfolio A} & (15\%-5\%)/12\% & 0.83 \\ \text{Portfolio B} & (18\%-5\%)/14\% & 0.93 \\ \text{Portfolio C} & (12\%-5\%)/9\% & 0.77 \\ \end{array}$$ Note: The Sharpe ratio uses total risk, not just the systematic risk of a portfolio (as represented by beta). Further, note that the information about the efficient market portfolio is useless in this case.
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