Sharpe’s Ratio, Treynor’s Ratio and Jensen’s Alpha, CFP

Risk and Return for CFP-6

Risk Adjusted Performance Measurement (Important for Examination)

The Sharpe measure

Sharpe (1966) defined this ratio as the reward-to-variability ratio, but it was soon called the Sharpe ratio in articles that mentioned it. It is defined by

This ratio measures the excess return, or risk premium, of a portfolio compared with the risk-free rate, compared, this time, with the total risk of the portfolio, measured by its standard deviation.

If the portfolio is well diversified, then its Sharpe ratio will be close to that of the market. With this ratio the manager can check whether the expected return on the portfolio is sufficient to compensate for the additional share of total risk that he is taking. Since this measure is based on the total risk, it enables the relative performance of portfolios that are not very diversified to be evaluated, because the unsystematic risk taken by the manager is included in this measure.

The Treynor measure

The Treynor (1965) ratio is defined by

The term on the left is the Treynor ratio for the portfolio, and the term on the right can be seen as the Treynor ratio for the market portfolio, since the beta of the market portfolio is 1 by definition. Comparing the Treynor ratio for the portfolio with the Treynor ratio for the market portfolio enables us to check whether the portfolio risk is sufficiently rewarded.

The Treynor ratio is particularly appropriate for appreciating the performance of a well diversified portfolio, since it only takes the systematic risk of the portfolio into account.

The Jensen measure

Jensen’s alpha (Jensen, 1968) is defined as the differential between the return on the portfolio in excess of the risk-free rate and the return explained by the market model, or

Illustration:

You are evaluating the rankings based on Sharpe and Treynor Ratio of three funds A, B and C . The average returns obtained from funds A, B and C have been 16%, 19% and 14%, respectively against the market return of 13%. The standard deviations of fund returns have been 17, 22 and 16, respectively versus the market return standard deviation of 15. If the beta reported of these funds is 1.2, 1.4 and 1.1, respectively and the risk-free rate of return is 5.5%, what are your rankings in the order of best to worst?

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