that the average manager under-performed the benchmark, in proportion to fund expenses, and that performance did not persist from period to period. Some recent studies have shown that the average manager matches the benchmark net of fees, that top managers do have statistically significant skill, and that positive performance may persist. Other studies have found no evidence for persistence of performance. The conclusion of all these conflicting studies is that even if performance does
Page Figure 17.2 Top quartile performance (IR = 0.5).
Page 483 persist, it certainly doesn't persist at any impressively high rate. Do 52 percent or 57 percent of winners repeat, and is this statistically significant? Defining Returns We begin our in-depth discussion of performance analysis by defining returns—this may seem obvious, but there are several definitions. Should we use compound returns or average returns, arithmetic returns or logarithmic returns? Compound returns have the benefit of providing an accurate measure of the value of the ending portfolio.3 Arithmetic returns provide the benefit of using a linear model of returns across periods. We can see these points with an example. Let RP(t) be the portfolio's total return in period t, and let RB(t) and RF(t) be the total return on the benchmark and the risk-free asset. The compound total return on portfolio P over periods 1 through T, RP(1,T), is the product The geometric average return for portfolio P, gP, is the rate of return per period that would give the same cumulative return: The average log return zP is 3This is true unless the portfolio has experienced cash inflows and outflows. Even in that case, however, the industry standard approach to performance analysis is to equally weight each period's return, without accounting for differing portfolio values in different periods.
Page 484 The geometric average return is compounded annually, while the average log return is compounded continuously. Finally, the (arithmetic) average return aP is It is always4 true that zP ≤ gP ≤ aP. This does not necessarily say that one measure is better to use than the others. It does indicate that consistency is important, to make sure we are not comparing apples and oranges. These issues become even more important when we attribute each period's return to different sources, and then aggregate all the sources over time. To cumulate returns, we need to account for cross products. We discuss one approach to this in the technical appendix. Cross-Sectional Comparisons The simplest type of performance analysis is a table that ranks active managers by the total performance of their fund over some period. Table 17.1 illustrates a typical table, showing median performance, key percentiles, and the performance of a diversified and widely followed index (the S&P 500), for a universe of institutional equity portfolios covered by the Plan Sponsor Network (PSN) over the period January 1988 through December 1992. These cross-sectional comparisons can provide a useful feel for the range of 4First, zP = ln{1 + gP} < gP by the convexity of the logarithm function. We have a useful approximation . Again, by the convexity of the logarithm function, so gP ≤ aP. Finally, we have an approximation (exact for log normal) that , where is the variance of ln{RP(t)}. This reduces to .
Page 485 TABLE 17.1 Percentile Annualized Return, 1988–1992 5th 23.57% 25th 18.97% Median 16.31% 75th 14.50% 95th 10.92% S&P 500 15.80% performance numbers over a period; however, they have four drawbacks: • They typically do not represent the complete population of institutional investment managers. Table 17.1 includes only those institutional equity portfolios that began no later than 1983, still existed in 1993, and are covered in the PSN database. • These cross-sectional comparisons usually contain survivorship bias, which is increasingly severe the longer the horizon. Table 17.1 does not include firms that went out of business between 1983 and 1993. • These cross-sectional comparisons ignore the fact that some of the reporting managers are managing $150 million portfolios, while others are managing $15 billion portfolios. The rule is one man, one vote—not one dollar, one vote. • Cross-sectional comparisons do not adjust for risk. The top performer may have taken large risks and been lucky. We cannot untangle luck and skill in this comparison. Figure 17.3 shows the impact of using a cross-sectional snap-shot. Compare two managers, A and B. Over a 5-year period, Manager A has achieved a cumulative return 16 percent above the benchmark, while Manager B has outperformed by almost 20 percent. Based on this rather limited set of information, most people would prefer B to A, since B has clearly done better over the 5-year period.
Page 4 Figure 17.3 Cumulative return comparison.
Page 487 Figure 17.3, however, shows the time paths that A and B followed over the 5-year period. After seeing Fig. 17.3, most observers prefer A to B, since A obviously incurred much less risk than B in getting to the current position.5 If you had stopped the clock at most earlier times in the 5-year period, A would have been ahead. Performance analysis must account for both return and risk. Returns-based Performance Analysis: Basic The development of the CAPM and the notion of market efficiency in the 1960s encouraged academics to consider the problems of performance analysis. The CAPM maintained that consistent exceptional returns by one manager were unlikely. The academics devised tests to see if their theories were true, and the by-products of these tests are performance analysis techniques. These techniques analyze time series of returns. One approach, first proposed by Jensen (1968), separates returns into systematic and residual components, and then analyzes the statistical significance of the residual component. According to the CAPM, the residual return should be zero, and positive deviations from zero signify positive performance. The CAPM also states that the market portfolio has the highest Sharpe ratio (ratio of excess return to risk), and Sharpe (1970) proposed performance analysis based on comparing Sharpe ratios. We will discuss the Jensen approach first, and then the Sharpe approach. Returns Regression Basic returns-based performance analysis according to Jensen involves regressing the time series of portfolio excess returns against benchmark excess returns, as discussed in Chap. 12. 5Manager A has realized an information ratio of 1.0 over this period, while Manager B has realized an information ratio of 0.7.
Page 488 Figure 17.4 Returns regression. Figure 17.4 shows the scatter diagram of excess returns to the Major Market Index portfolio and the S&P 500, together with a regression line, over the period from January 1988 through December 1992. The estimated coefficients in the regression are the portfolio's realized alpha and beta: Alpha appears in the diagram as the intercept of the regression line with the vertical axis. Beta is the slope of the regression line. For the above example, αP = 0.03 percent per month and βP = 0.92. The regression divides the portfolio's excess return into the benchmark component, βP · γB(t), and the residual component, θP(t) = αP + ∈P(t). Note that in this example, the residual return is quite different from the active return, because the active beta is –0.08. While the alpha is 3 basis points per month, the average active return is –4 basis points per month. The CAPM suggests that alpha should be zero. The regression analysis provides us with confidence intervals for our estimates of alpha and beta. The t statistic for the alpha provides a rough test
Page 489 of the alpha's statistical significance. A rule of thumb is that a t statistic of 2 or more indicates that the performance of the portfolio is due to skill rather than luck. The probability of observing such a large alpha by chance is only 5 percent, assuming normal distributions. The t statistic for the alpha is approximately where αP and ωP are not annualized and T is the number of observations (periods). The t statistic measures whether αP differs significantly from zero, and a significant t statistic requires a large αP relative to its standard deviation, as well as many observations. For the example above, the t statistic for the estimated αP is only 0.36, not statistically distinct from zero.