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Risk-adjusted performance measures
Sharpe, Treynor, Jensen are the most common
The M2 measure
Risk-Adjusted Performance Measures
How do we assess the performance of any given investment portfolio?
There are alternative approaches depending on th purpose of the exercise.
Considering return alone is inadequate; there must be a focus on risk level, and also benchmarks for comparison By measuring both risk and return, we can calculate riskadjusted returns We initially examine three of the most popular measures,
which differ according to the risk measure used Sharpe, Treynor, Jensen
Uses the Capital Market Line
Sharpe Ratio (S) is the ratio of excess returns to volatility:
where the terms are average return on asset or portfolio i, risk free rate of return and standard deviation of return. Therefore, it is necessary to define a time period and data frequency in order to generate the data observations .
which average and standard deviation are calculated.
A potential issue arising with the Sharpe ratio is that its economic significance cannot be directly observed.
As an example, we want to compare two portfolios A and B. We calculate SA = 0.75 and SB = 0.8.
We find that portfolio A has under-performed portfolio B but is this difference of 0.05 sufficiently large to be economically relevant?
One approach is to consider the M2 measure (see later). Another approach is to use multiple performance measures to compare A and B.
Consider two fund managers who both achieved a return over one year of 10% on their portfolios.
Fund manager As portfolio had a returns standard deviation of 8%,
while fund manager Bs portfolio had a returns standard deviation of 9%.
The risk-free rate of interest over this period was 4%.
(a) What are the Sharpe ratios for the portfolios?
(b) Which portfolio performed better?
Sharpe versus Treynor
Sharpe ratio considers total risk, whereas Treynor measure considers only systematic risk
For this reason, Treynor measure may be preferred by investors who have a well-diversified portfolio If the portfolio is not well diversified, the Sharpe ratio may be preferred
Consider again the two managers from question 1.
The portfolio of manager A had a beta value of 0.9, and the beta of manager Bs portfolio was 1.5.
(a) Which portfolio had lesser/greater systematic risk than the market?
(b) What are the Treynor measures for the two portfolios?
(c) Which portfolio represented better performance?
Suppose that the trustees of a pension fund decide that they wish their equity portfolio to have a CAPM beta of 0.75 and instruct the fund manager of their equity portfolio to at least match the performance of an equity portfolio with a CAPM beta of 0.75 over the next year.
At the end of the period, the evaluation is undertaken:
The risk free rate of interest was 4% and the return on the market was 16%. The fund manager achieved a return of 15% with a portfolio that had the required CAPM beta.
What was the required return of the benchmark portfolio?
What was the Jensen measure for the fund managers portfolio?
Consider the same data as in Q3.
Suppose that the total risk of the managers portfolio (p) was 0.30 and that the standard deviation of return on the market (m) was 0.29.
(a) What is the of a well-diversified portfolio with the same total risk as the managers portfolio?
Hint: Well-diversified portfolio has correlation of 1 with the market.
(b) What is the Jensen measure of performance if this alternative value of is used?
Over the past three years, using monthly data, the return on a fund has been 14.5% per annum.
The benchmark return over this period was 16% and the standard deviation of the surplus was 12%.
(a) What is the information ratio for this fund?
(b) What do you conclude from the result?
A fund managers portfolio has achieved a return of 12% over the last year.
For the same period, the risk-free rate was 4% and the return on the market index was 11%.
The standard deviation of returns on the portfolio was 25% and the standard deviation of returns on the market index was 20%.
According to the Sharpe ratio, did the portfolio outperform the market?
According to the M2 measure, did the portfolio outperform the market
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