Risk-Adjusted Performance Measurement Issues In a Bear Market>

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Overview

Today’s bear market requires investment professionals to re-evaluate the meth-ods used to assess risk-adjusted return measures. Most risk-adjusted return measures are reward-to-risk ratios. These ratios generally consist of a reward that is an excess return in the numerator and a risk in the denominator. This excess return is the average portfolio return less the average risk-free return for many risk-adjusted measures. The metric for risk varies between measures, as shown in Table 1 below. Although these measures have proven effective in a bull market, there are computational and interpretation issues with these measures in a down market, when excess returns are negative.

In a bull market, average portfolio excess returns would be positive, and the risk-adjusted return measure would decrease as risk increased. However, in a bear market, average portfolio excess returns would generally be negative, and the risk-adjusted measure would become less negative (or increase) as risk in-creased. This anomaly occurs in a wide range of measures impacted by the bear market. It also occurs in a measure such as the information ratio, in which the reward is the excess return of the portfolio over the benchmark. The informa-tion ratio is impacted more by relative performance than by market direction. However, the information ratio has the same problem as other ratios when the average excess return of the portfolio versus the benchmark is negative.

Risk Adjusted MeasuresThe following is a list of some frequently used measures in which higher ratios are generated when the rewards are negative and the risk measure increases.

Measure Reward Risk CommentSharpe Ratio Rp-Rf Excess Return

Standard DeviationWidely used for relative ranking of portfolios or funds

Treynor Ratio Rp-Rf Portfolio Beta Well known tool for rank-ing portfolios or funds

Sortino Ratio Rp-Rf Downside Risk Measure receiving more interest since focus on downside

Calmar Ratio Rp-Rf MaximumDrawdown

Used by hedge funds and alternative investments

Information Ratio

Rp-Rb Tracking error of portfolio vs bench-mark

Widely used to show value added while tracking benchmark

where Rp=Portfolio Return, Rf=Risk Free Return, and Rb=Benchmark Return

Table 1

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Approaches in Practice

A range of approaches is used when dealing with negative rewards as numera-tors in risk-adjusted performance measures:

(1) Avoid using the measures since results are counterintuitive.

An argument can be made that clients would not be interested in accepting less than the risk-free rate (or less than the benchmark) for performance while taking on risk.

(2) Reinterpret results.

It has been argued that results are meaningful when negative rewards with high volatility have a higher probability of achieving positive returns. The benefit of high volatility is very questionable and would depend on the probability distri-bution of returns favoring positive performance. However, portfolio volatility is generally a drag on performance, and it is more likely that negative performance will outweigh positive performance.

(3) Measurement modification to always reward higher performance and lower risk.

Craig L. Israelsen has published a modification for the Sharpe ratio and the Information ratio that provides rankings more consistent with reward-volatility measures. This approach was published in Financial Planning magazine and the Journal of Asset Management. An adjustment is made to the denominator of the Sharpe ratio when the average excess return is negative. This adjustment uses the inverse of the annualized standard deviation of returns in the de-nominator. There would be no adjustment for portfolios if the average excess return were positive. We will be using this adjustment approach in the example with the Sharpe ratios below. The adjusted Sharpe ratios with negative excess returns vary significantly from the standard Sharpe measurement with positive excess returns.

Sharpe Ratio ExampleIn the following sections, we will focus on the development and evaluation of Sharpe ratios in a negative market environment. For proxy portfolios to be ranked we will consider style indices provided by PPCA. In the example, we will compute the Sharpe ratio for nine styled equity indices based on 24 months in 2007-2008. In this time frame, all of the nine styled indices averaged negative monthly excess returns. Table 2, on the following page, provides the monthly style index returns and the monthly risk-free rate (90-day T-bill) returns. The monthly excess returns are provided in Table 3 on the page 6.

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Table 2: Style Returns

Month LrgValue LrgCore LrgGrow MidValue MidCore MidGrow MinValue MinCore MinGrow T-Bill

200701 0.68 1.2 3.17 2.68 4.34 3.41 1.49 1.23 1.95 0.16

200702 (2.49) (1.89) (2.78) 0.19 0.25 0.4 (0.87) 0.76 (0.43) 0.164

200703 2.27 (0.53) 1.44 1.26 0.49 1.12 0.94 1.68 0.43 0.162

200704 4.88 4.82 3.6 4.12 3.61 3.27 2.28 2.68 2.2 0.16

200705 4.11 3.34 3.23 4.08 4.95 4.34 3.75 4.13 2.8 0.157

200706 (1.78) (1.71) (1) (2.25) (1.54) (0.27) (1.73) (0.29) (1.14) 0.158

200707 (3.69) (1.03) (1.2) (5.08) (4.55) (2.15) (6.58) (5.04) (6.54) 0.16

200708 1.23 1.89 2.17 (0.12) 0.4 0.16 0.89 1.75 (1.58) 0.144

200709 4.01 3.64 5.78 2.33 3.14 3.94 0.61 2.57 3.24 0.14

200710 0.84 4.47 4.47 0.85 1.36 4.46 0.95 3.71 3.53 0.142

200711 (4.73) (2.3) (5.13) (4.9) (5.02) (5.59) (6.87) (7.32) (10.92) 0.126

200712 (0.55) 0.19 (0.35) (1.36) (0.9) 1.05 (1.21) (0.47) (0.14) 0.131

200801 (4.05) (5.65) (10.58) (3.03) (5.84) (8.03) (2.4) (8.91) (9.53) 0.095

200802 (3.35) (0.67) (1.42) (3.62) (2) (0.84) (3.08) (2.29) (4.13) 0.091

200803 (0.22) 2.32 (2.11) (1.94) (0.63) (4.28) (0.03) (0.9) (3.62) 0.075

200804 4.19 3.97 7.31 5.15 7.13 6.37 3.83 4.68 2.37 0.077

200805 (0.83) 2.8 3.06 2.69 4.98 5.24 3.8 5 4.89 0.092

200806 (9.44) (6.53) (6.5) (10.15) (5.78) (6.44) (10.27) (7.89) (8.29) 0.093

200807 (0.15) (0.29) (3.51) 0.38 (2.04) (6.23) 3.44 0.57 0.69 0.086

200808 1.48 2.32 (0.69) 3.27 0.97 (0.48) 3.93 3.59 3 0.087

200809 (5.2) (6.41) (15.42) (8.01) (13.5) (16.83) (5.91) (11.28) (13.95) 0.056

200810 (15.98) (14.66) (17.56) (22.73) (21.37) (22.19) (20.79) (20.55) (27.22) 0.029

200811 (4.74) (8.53) (8.41) (10.17) (9.28) (10.69) (12.94) (9.94) (16.02) 0.001

200812 1.64 (0.32) 1.18 3.85 5.43 3.48 5.31 5.46 5.41 0.009

Ranking Performance With and Without Adjustments

We first compute the monthly returns for the portfolio and the risk-free rate. Then we compute the excess monthly returns as the difference between the portfolio return and the risk-free rate of 90-day T-bill returns. We then com-pute the monthly average and standard deviation of the excess returns. Then we annualize the excess-return standard deviation and the arithmetic average of the excess returns. We then compute the Sharpe ratio.

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Table 3: Excess Returns

Month LrgValue LrgCore LrgGrow MidValue MidCore MidGrow MinValue MinCore MinGrow

200701 0.522 1.037 3.003 2.52 4.179 3.251 1.328 1.066 1.789

200702 (2.653) (2.051) (2.944) 0.03 0.089 0.233 (1.033) 0.594 (0.594)

200703 2.104 (0.688) 1.279 1.101 0.332 0.961 0.776 1.521 0.265

200704 4.724 4.665 3.441 3.964 3.446 3.105 2.12 2.518 2.045

200705 3.954 3.188 3.072 3.925 4.794 4.181 3.595 3.975 2.639

200706 (1.941) (1.864) (1.154) (2.407) (1.698) (0.427) (1.883) (0.448) (1.296)

200707 (3.853) (1.193) (1.356) (5.24) (4.714) (2.306) (6.74) (5.199) (6.704)

200708 1.084 1.75 2.021 (0.262) 0.255 0.018 0.746 1.604 (1.723)

200709 3.868 3.496 5.639 2.186 2.998 3.797 0.469 2.434 3.099

200710 0.693 4.332 4.324 0.708 1.217 4.32 0.81 3.57 3.386

200711 (4.854) (2.43) (5.255) (5.03) (5.148) (5.714) (6.992) (7.444) (11.043)

200712 (0.681) 0.061 (0.481) (1.492) (1.034) 0.919 (1.345) (0.596) (0.266)

200801 (4.144) (5.748) (10.673) (3.128) (5.935) (8.125) (2.492) (9.002) (9.62)

200802 (3.443) (0.765) (1.512) (3.71) (2.087) (0.93) (3.174) (2.384) (4.226)

200803 (0.29) 2.245 (2.185) (2.016) (0.704) (4.351) (0.109) (0.976) (3.694)

200804 4.109 3.895 7.23 5.078 7.053 6.295 3.754 4.607 2.298

200805 (0.922) 2.707 2.969 2.597 4.885 5.147 3.704 4.911 4.799

200806 (9.531) (6.623) (6.594) (10.242) (5.876) (6.531) (10.366) (7.982) (8.382)

200807 (0.236) (0.38) (3.592) 0.299 (2.126) (6.312) 3.354 0.482 0.602

200808 1.391 2.235 (0.774) 3.179 0.88 (0.567) 3.845 3.503 2.918

200809 (5.259) (6.467) (15.471) (8.069) (13.561) (16.881) (5.963) (11.335) (14.007)

200810 (16.007) (14.692) (17.584) (22.757) (21.401) (22.221) (20.824) (20.575) (27.249)

200811 (4.74) (8.529) (10.173) (10.173) (9.276) (10.691) (12.936) (9.946) (16.022)

200812 1.61 (0.329) 3.841 3.841 5.421 3.471 5.301 5.451 5.401

We first compute the arithmetic average return for the portfolio and risk-free rate. Then we compute the excess monthly return as the difference between the portfolio return and the risk-free rate or 90-day T-bill returns. We then compute the monthly average and standard deviation of the excess returns. Next we compute the monthly portfolio-return standard deviation. Then we annualize the portfolio-return standard deviation and the arithmetic average of the excess returns. We then compute the Sharpe ratio.

The standard Sharpe ratio rankings illustrate the problem with negative excess returns. For example, compare the reward and volatility information of Large Cap Value versus Mid Cap Core. Large Cap Value has a higher average excess return and a lower volatility than Mid Cap Core, although the Sharpe ratio is higher for Mid Cap Core. In fact, Large Cap Value has a higher average excess return and lower volatility than all indices except Large Cap Core and was ranked number 8 out of 9 using the standard Sharpe ratios.

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With the Israelsen method, an adjustment can be made to the denominator of the Sharpe ratio when the average excess return is negative. This adjustment results in using the inverse of the annualized standard deviation of returns in the denominator. This is the equivalent of developing a ranking metric that is the product of the excess return multiplied by the standard deviation of the excess return. We can refer to this adjustment of the Sharpe ratio as a ranking metric and not an adjusted Sharpe ratio, since the dimensionality is in terms of the product of the average excess-return multiplied by the excess-return standard deviation rather than the amount of the excess return per unit of standard deviation. There would be no adjustment for ranking portfolios if the average excess return were positive.

Using the adjustment approach, the rankings make more intuitive sense. Note, that Large Cap Value was ranked 2 using this adjustment and number 8 without the adjustment. Mid Cap Growth moved in rankings from number 4 to number 8 after the adjustment. This makes sense since Mid Cap Growth had next to the lowest return and next to the highest standard deviation.

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LrgValue LrgCore LrgGrow MidValue MidCore MidGrow MinValue MinCore MinGrow

MONTHLY Excess Return (1.44) (0.92) (1.83) (1.88) (1.58) (2.06) (1.84) (1.65) (3.15)

Std Deviation 4.7 4.68 6.28 6.24 6.48 7.07 6.2 6.44 7.87

Annualized Excess Return (17.24) (11.07) (21.92) (22.55) (19) (24.68) (22.03) (19.82) (37.79)

(Standard) Std Deviation 16.28 16.2 21.74 21.63 22.45 24.48 21.47 22.31 27.27

Sharpe Ratio (1.05409) (0.67967) (1.00395) (1.03872) (0.84337) (1.00388) (1.02289) (0.88535) (1.3816)

Standard Ranking 8 1 5 7 2 4 6 3 9

Annualized Excess Return (17.24) (11.07) (21.92) (22.55) (19) (24.68) (22.03) (19.82) (37.79)

(Adjusted) Std Dev(Adjusted)

0/061 0.062 0.046 0.046 0.045 0.041 0.047 0.037 0.037

Adj Ranking Measure

(280.67) (179.37) (476.57) (487.69) (426.7) (604.23) (472.37) (1030.68) (1030.68)

Adjusted Ranking 2 1 6 7 3 8 5 4 9

Table 4: Rankings

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Conclusion

The computation and interpretation of risk-adjusted performance measures re-quire special consideration when the excess returns computed are negative. Several measures are impacted. These risk-adjusted performance measures provide a method of ranking portfolios based on one metric for each portfolio. In bear markets, several of these measures may provide questionable rankings. Initial reviews of technical publications have provided an adjustment approach only for the Sharpe ratio and the Information ratio.

Portfolio ranking approaches such as those developed by Craig Israelsen have potential in negative excess-return situations. Future research will expand the search for adjustment methods to support more-accurate portfolio rankings. Until validated methods for adjustment are available for handling negative excess returns, it still makes sense to focus on multiple metrics for risk-return measure-ment, such as returns, volatility, market capture, drawdowns, recovery periods, downside risk and return-distribution properties.

ReferencesIsraelsen, Craig L., “Modifications of Sharpe Ratio,” Financial Planning, January 2003.

Israelsen, Craig L., “Modifications of Information Ratio,” Financial Planning, May 2004.

Israelsen, Craig L., “A Refinement to the Sharpe Ratio and Information Ratio,” Journal of Asset Management, Vol 5, 6, 423-427.