Sharpe Ratio

1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility

1.3.7 Sharpe Ratio

All calculation results are provided in Table 1.14. As can be seen in cell D22, the bull market beta stands at0:9726, i.e., if the benchmark increases by 1%, the portfolio, on average, increases only by0:9726%. The bear market beta of0:9191indicates that when the benchmark decreases by 1%, the fund decreases only by0:9191%. Such a portfolio would behave nicely during up- and downward movements of the market: the bull beta should be much higher than the bear beta which is here the case. Ideally, the bull beta should be higher than1and the bear beta should be close to0 or even negative, but the latter is difficult to achieve with long-only portfolios.

End of Example 9

1.3.6.2 Conclusion

The bull and bear beta are plausible concepts that build on the general beta concept.

Investors can more thoroughly analyze a portfolio’s behavior compared to the market development. Therefore, it makes sense to not only look at the overall beta but also to consider the bull and bear market beta.

SRPf D SR^p:a:_Pf D rPf^p:a:

Pf^p:a:

(1.74) or

SRPf D SR^p:a:_Pf D rPf^p:a:rrf^p:a:

Pf^p:a:

: (1.75)

Hereby:

rPf^p:a: D annualized return of the portfolio, rrf^p:a: D annual risk-free interest rate, Pf^p:a: D annualized volatility of the portfolio.

1.3.7.1 Notes

• The Sharpe ratio is a risk-adjusted return measure in the absolute world, i.e., no benchmark is considered.⁴⁶It was developed by William Sharpe⁴⁷in 1966.⁴⁸

• The definition using the risk premium is a key part of modern portfolio theory when deriving the efficient frontier and the CAPM as will be done in Chap.2. But this definition of the Sharpe ratio requires the specification of the risk-free rate.

In order to avoid this, the first definition is often used in risk and performance measurement practice.

• Also for portfolios which are actively managed against a benchmark it makes sense to calculate the Sharpe ratio. In this situation, Eqs. (1.74) and (1.75) are applied and the two Sharpe ratios are compared. In active portfolio management one would expect the Sharpe ratio of the portfolio to be higher than the benchmark’s Sharpe ratio over the same time period using the same subintervals.

1.3.7.2 Interpretation

If we analyze a portfolio that is managed against a benchmark, it does not make sense to only look at the portfolio’s Sharpe ratio. In this situation we have to compare the Sharpe ratios of the portfolio and the benchmark. For passively managed

46If we include a benchmark, the risk-adjusted return measure is a relative measure versus a benchmark calledinformation ratio. This concept is explained in Sect.1.3.8.

47William F. Sharpe was born on June 16, 1934, in Boston, MA/USA. He is the STANCO 25 Professor of Finance, Emeritus at Stanford University’s Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences. Sharpe was one of the originators of the capital asset pricing model (CAPM) and created the Sharpe ratio for risk-adjusted investment performance analysis. He contributed to the development of the binomial method for the valuation of options, the gradient method for asset allocation optimization, and returns-based style analysis for evaluating the style and performance of investment funds.

48Sharpe (1966, p. 123).

Table 1.15 Example 10: Calculation of the Sharpe ratio for portfolio and benchmark

A B C

Monthly portfolio Monthly benchmark

1 Month performance performance

2 07/2012 6.10 % 6.01 %

3 08/2012 5.50 % 5.45 %

4 09/2012 4.70 % 4.63 %

5 10/2012 5.00 % 6.99 %

6 11/2012 5.10 % 4.16 %

7 12/2012 6.70 % 7.07 %

8 01/2013 6.03 % 5.97 %

9 02/2013 3.23 % 2.95 %

10 03/2013 5.12 % 4.66 %

11 04/2013 5.21 % 4.91 %

12 05/2013 4.10 % 4.01 %

13 06/2013 4.50 % 3.87 %

14 07/2013 1.75 % 2.95 %

15 08/2013 3.71 % 4.52 %

16 09/2013 4.20 % 3.93 %

17 10/2013 4.26 % 4.99 %

18 11/2013 4.00 % 3.84 %

19 12/2013 5.10 % 4.99 %

20 Annual. return 15.79 % 13.02 %

21 Annual. volatility 16.49 % 17.02 %

22 Sharpe ratio 0.96 0.76

Source: Own, for illustrative purposes only

portfolios we expect these two to be (almost) identical. For actively managed portfolios we expect the portfolio’s Sharpe ratio to be higher.

If we look at an absolute return portfolio like a hedge fund, the Sharpe ratio has explanatory power on a stand-alone basis: the higher its value, the better.

Example 10

This example, displayed in Table 1.15, shows how the Sharpe ratio can be calculated for a portfolio which is managed against a benchmark. Column A lists the month, columnBthe monthly portfolio return, columnC the monthly benchmark return. To calculate the Sharpe ratio, we choose the formula without the risk-free rate to avoid additional calculations which are unnecessary for the interpretation. Further,T D1:5years andN D18months.

The lower part of the table shows the performance, risk and risk-adjusted performance of the portfolio and benchmark. The Excel^r calculations behind these results are as follows:

• Annualized portfolio returnrPf^p:a:in cellB20:

15:79% D fPRODUCT.1C.B2WB19//^{^}.12=18/1g

• Annualized benchmark returnrBm^p:a:in cellC 20:

13:02% D fPRODUCT.1C.C 2WC19//^{^}.12=18/1g

• Annualized portfolio volatilityPf^p:a:in cellB21: 16:49% D SQRT.12/STDEV:S.B2WB19/

• Annualized benchmark volatilityBm^p:a:in cellC 21: 17:02% D SQRT.12/STDEV:S.C 2WC19/

• Sharpe ratio of portfolioSRPf in cellB22: 0:96 D B20=B21

• Sharpe ratio of benchmarkSRBmin cellC 22: 0:76 D C 20=C 21

The portfolio has a higher Sharpe ratio than the benchmark, i.e., for one unit of risk (volatility) the investor in the portfolio receives a higher additional return than when passively investing in the index.

End of Example 10 1.3.7.3 Conclusion

The Sharpe ratio is a key ratio when evaluating a portfolio. It is an absolute risk- adjusted return measure as it does not look at portfolio returns versus a benchmark, but rather at the portfolio returns by themselves. In order to calculate the Sharpe ratio, either the portfolio’s percentage return or the return premium (return minus risk-free rate) can be used. The associated risk is always volatility. In practice, the risk-free rate should be omitted to avoid questions on the choice of the risk-free rate or its computation. For benchmark-oriented portfolios, the Sharpe ratio should be calculated both for the portfolio and the benchmark in order to find out if the portfolio or the benchmark delivers a better risk-adjusted return in absolute terms.

However, what risk-adjusted return measure would we need to analyze a portfolio versus the benchmark performance? In principle, a relative risk-adjusted return measureis very similar to theabsolute risk-adjusted return measureSharpe ratio.

We only need to replace the absolute return by the relative return (alpha) and the absolute risk measure (volatility) by the relative risk measure (tracking error). This new risk-adjusted return measure is calledinformation ratio.