Calculate and Interpret the Sharpe Ratio, Treynor Ratio, M2, and Jensen’s Alpha

There are four ratios commonly used in performance evaluation.

Sharpe Ratio

The Sharpe Ratio is defined as the portfolio risk premium divided by the portfolio risk:

  • Sharpe ratio = (Rp – Rf) / σp

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 by itself is not informative. In order to rank portfolios, the Sharpe ratio for each portfolio must be computed. A 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 that instead of using total risk uses beta or systematic risk in the denominator.

  • Treynor ratio = (Rp – Rf) / βp

As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results and, the Treynor ratio does not work for negative beta assets. Also, while both the Sharpe and the Treynor ratio can rank portfolios, they do not provide information on whether the portfolios are better than the market portfolio or 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:

  • wp = σm / σp with the balance (1 – wp) invested in the risk-free asset

The return for the mimicking portfolio P’ is as follows:

  • Rp’ = wpRp + (1 – wp)Rf which we can reformulate as Rp’ = (σm / σ) x R + (1 – {(σm / σ)} x Rf
  • Rp’ = R + σm([R– Rf]/ σ)

The difference in return between the mimicking portfolio and the market return is M² which is expressed as:

  • M² = (R – R) x (σm / σ) – (R – R)

A portfolio that matches the return of the market will have a 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 to 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 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 measure of performance relative to the market.

  • Jensen’s alpha = αp = R – [Rf + βp(Rm– R)]

If αp is positive, the portfolio has outperformed the market whereas as negative value indicates underperformance. The values of alpha can also be used to rank portfolios or the managers of those portfolios with the alpha being a representation of the maximum an investor should pay for the active management of that portfolio.

Question

A client has three portfolio choices, each with the following characteristics:

  Expected return Volatility Beta
Portfolio A 15% 12% 10%
Portfolio B 18% 14% 11%
Portfolio C 12% 9% 5%

The efficient portfolio has an expected return of 20% and a volatility of 12%. The risk free rate of interest is 5%.

Based on the Sharpe ratio for each portfolio, the client should choose:

  1. Portfolio A
  2. Portfolio B
  3. Portfolio C

The correct answer is B.

Sharpe Ratio Formula = (Expected Return – Risk-Free rate of return) / Standard Deviation (Volatility)

The portfolio with the highest Sharpe ratio has the best performance.

  Sharpe measure
Portfolio A 83%
Portfolio B 93%
Portfolio C 78%

Note: The Sharpe ratio makes use of total risk not just the systematic risk of a portfolio (as represented by beta). Furthermore, information about the efficient portfolio has no use in this case.

Reading 41 LOS 41i:

calculate and interpret the Sharpe ratio, Treynor ratio, M2, and Jensen’s alpha


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