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We've seen that survivorship bias might be one source of the equity premium puzzle. It turns out that survivorship bias also can affect our measurement of persistence in stock market returns, an issue that is crucial for tests of market efficiency. For a demonstration of the potential impact of survivorship bias, imagine that a new group of mutual funds is set up. Half the funds are managed aggressively and the other conservatively; however, none of the managers are able to beat the market in expectation. The probability distribution of alpha values is given by

406 PART III Equilibrium in Capital Markets

Probability | ||

Alpha Vàlue |
Conservative |
Aggressive |

(%) |
Manager |
Manager |

3 |
0 |
0.5 |

1 |
0.5 |
0 |

-1 |
0.5 |
0 |

-3 |
0 |
0.5 |

Because there are an equal number of aggressive and conservative managers, the frequency distribution of alphas in a given period is:

Relative Frequency of Funds | ||

Alpha Value |
Conservative |
Aggressive |

(%) |
Manager |
Manager |

3 |
0 |
.25 |

1 |
.25 |
0 |

- |
.25 |
0 |

- |
0 |
.25 |

Total |
.5 |
.5 |

Define a "winner" fund as one in the top-half of the distribution of returns in a given period; a "loser" is one in the bottom half of the sample. Manager alphas are assumed to be serially uncorrelated. Therefore, the probability of being a winner or a loser in the second quarter is the same regardless of first-quarter performance. A 2 X 2 tabulation of performance in two consecutive periods, such as in the following table, will show absence of any persistence in performance.

Second Period | ||

First Period |
Winners |
Losers |

Winners |
.25 |
.25 |

Losers |
.25 |
.25 |

But now assume that in each quarter funds are ranked by returns and the bottom 5% are closed down. A researcher obtains a sample of four quarters of fund returns and ranks the semiannual performance of funds that survived the entire sample. The following table, based only on surviving funds, seems to show that first-period winners are far more likely to be second period winners as well. Despite lack of true persistence in performance, the loss of just a few funds each period induces an appearance of significant persistence.

CHAPTER 13 Empirical Evidence on Security Returns 407

Second Period | |||

First Period |
Winners |
Losers |
Row Total |

Winners |
.3893 |
.1401 |
.5294 |

Losers |
.1107 |
.4706 |
.4706 |

Column total |
.5001 |
.4999 |
1.0000 |

The degree of survivorship bias depends first and foremost on how aggressively poorly performing funds are shut down. (In this example, the worst 5% of performers were shut down.) The bias increases enormously with the cut-off rate. Other factors affecting bias are correlation across manager portfolios, serial correlation of returns, the dispersion of style across managers, and the strategic response of managers to the possibility of cut-off.

To assess the potential effect of actual survivorship bias, Brown, Goetzmann, Ibbotson and Ross25 conducted a simulation using observed characteristics of mutual fund returns. Their results demonstrate that actual survivorship bias could be strong enough to create apparent persistence in the performance of portfolio managers. They simulate annual returns over a four-year period for 600 mutual fund managers, drawing from distributions that are constructed to mimic observed equity returns in the United States over the period 1926-1989,26 and mutual fund returns reported in a performance study by Goetzmann and Ibbotson.27 Four annual returns for each manager are generated independently so that relative performance over the first two-year period does not persist in the following two-year period. The simulated returns of the funds and the market index are used to compute four risk-adjusted annual returns (alphas) for each of the 600 managers. Winners (losers) are identified by positive (negative) alphas.

Two-by-two tabulations of the frequency of first-period/second-period winners and losers are shown in Table 13.6. When all 600 managers are included in the four-year sample, no persistence in performance can be detected. But when the poor performers in each year are eliminated from the sample, performance persistence shows up. Elimination of even a small number of poor performers can generate a significant level of apparent persistence.

The results for a 5% cut-off rate in Table 13.6 are not as strong as in the "clean" example we presented above; apparently, other factors mitigate the effect somewhat. Still, survivorship bias is sufficient to create an appearance of significant performance persistence even when actual returns are consistent with efficient markets.

SUMMARY 1. Although the single-factor expected return-beta relationship has not yet been confirmed by scientific standards, its use is already commonplace in economic life.

25 Stephen J. Brown, William Goetzmann, Roger G. Ibbotson, and Stephen A. Ross, "Survivorship Bias in Performance Studies," Review of Financial Studies 5, no. 4 (1992).

26 They draw market risk premiums from a normal distribution with mean 8.6% and standard deviation 20.8%. The 600 manager betas are drawn from a normal distribution with mean .95 and standard deviation .25. The residual standard deviation for each manager is estimated from actual data.

27 William Goetzmann and Roger Ibbotson, "Do Winners Repeat? Predicting Mutual Fund Performance," Journal of Portfolio Management 20 (1994).

Second-Period |
Second-Period | |||

Winners |
Losers | |||

No cut-off (n = |
600) | |||

First-period winners |
150.09 |
149.51 | ||

First-period losers |
149.51 |
150.09 | ||

= |
= -.004 | |||

Average annual excess return |
= 0.0% | |||

Average p |
= 0.950 | |||

5% cut-off (n = |
494) | |||

First-period winners |
127.49 |
1 19.51 | ||

First-period losers |
119.51 |
127.49 | ||

Average cross-section f-value |
= | |||

= |
= 0.44% | |||

Average p |
= | |||

= |
398) | |||

First-period winners |
106.58 |
92.42 | ||

First-period losers |
92.42 |
106.58 | ||

Average cross-section f-value |
= | |||

= |
= 0.61% | |||

p |
= | |||

= |
249) | |||

First-period winners |
71.69 |
53.31 | ||

First-period losers |
53.31 |
70.69 | ||

Average cross-section f-value |
= | |||

= |
= 0.80% | |||

p |
= |

In each of the four'years, managers who experience returns in the lowest percentile indicated by the cut-off value are excluded from the sample, and this experiment is repeated 20,000 times. Thus, the numbers in the first 2 x 2 fable give the average freguency with which the GOO managers fall into the respective classifications. The second panel shows the average frequencies forthe 494 managers who survive the performance cut, while the third and fourth panels give corresponding results for 398 and 249 managers. For each simulation, the winners are defined as those managers whose average two-year Jensen's a measure was greaferfhan or equal to that of the median manager in that sample.

In each of the four'years, managers who experience returns in the lowest percentile indicated by the cut-off value are excluded from the sample, and this experiment is repeated 20,000 times. Thus, the numbers in the first 2 x 2 fable give the average freguency with which the GOO managers fall into the respective classifications. The second panel shows the average frequencies forthe 494 managers who survive the performance cut, while the third and fourth panels give corresponding results for 398 and 249 managers. For each simulation, the winners are defined as those managers whose average two-year Jensen's a measure was greaferfhan or equal to that of the median manager in that sample.

2. Early tests of the single-factor CAPM rejected the SML, finding that nonsystematic risk did explain average security returns.

3. Later tests controlling for the measurement error in beta found that nonsystematic risk does not explain portfolio returns but also that the estimated SML is too flat compared with what the CAPM would predict.

4. Roll's critique implied that the usual CAPM test is a test only of the mean-variance efficiency of a prespecified market proxy and therefore that tests of the linearity of the expected return-beta relationship do not bear on the validity of the model.

5. Tests of the mean-variance efficiency of professionally managed portfolios against the benchmark of a prespecified market index conform with Roll's critique in that they provide evidence on the efficiency of the prespecific market index.

6. Empirical evidence suggests that most professionally managed portfolios are outperformed by market indexes, which lends weight to acceptance of the efficiency of those indexes and hence the CAPM.

7. Work with economic factors suggests that factors such as unanticipated inflation do play a role in the expected return-beta relationship of security returns.

8. Recent tests of the single-index model, accounting for human capital and cyclical variations in asset betas, are far more consistent with the single-index CAPM and APT. These tests suggest that macroeconomic variables are not necessary to explain expected returns. Moreover, anomalies such as effects of size and book-to-market ratios disappear once these variables are accounted for.

9. Volatility of stock returns is constantly changing. Empirical evidence on stock returns must account for this phenomenon. Contemporary researchers use the variations of the ARCH algorithm to estimate the level of volatility and its effect on mean returns.

10. The equity premium puzzle originates from the observation that equity returns exceeded the risk-free rate to an extent that is inconsistent with reasonable levels of risk aversion—at least when average rates of return are taken to represent expectations. Fama and French show that the puzzle emerges from excess returns over the last 50 years. Alternative estimates of expected returns using the dividend growth model instead of average returns suggest that excess returns on stocks were high because of unexpected large capital gains. The study implies that future excess returns will be lower than realized in recent decades.

KEY TERMS first-pass regression second-pass regression benchmark error

The website listed below gives you access to the Efficient Frontier: An Online Journal of Practical Asset Allocation. The journal has articles related to performance and efficiency.

http://www.efficientfrontier.com

The website listed below contains an article that explores market efficiency for the top Fortune 500 firms. The article entitled "Are High-Quality Firms Also High-Quality Investments?" investigates the relationship between stock performance and firm reputation.

http://www.ny.frb.org/rmaghome/curr isc/ci6-1.pdf

Longer-term performance on mutual funds can be found at the site listed below. Rankings for all types of mutual funds and their performance measures are available.

WEBSITES

PROBLEMS The following annual excess rates of return were obtained for nine individual stocks and a market index:

Stock Excess Returns (%)

Year Index

10 11 12

73 65 68 14

18 15

67 61

79 47

75 42

43 76 38 21

69 26

96 101

83 98

31 15

1. Perform the first-pass regressions and tabulate the summary statistics.

2. Specify the hypotheses for a test of the second-pass regression for the SML.

3. Perform the second-pass SML regression by regressing the average excess return of each portfolio on its beta.

4. Summarize your test results and compare them to the reported results in the text.

5. Group the nine stocks into three portfolios, maximizing the dispersion of the betas of the three resultant portfolios. Repeat the test and explain any changes in the results.

6. Explain Roll's critique as it applies to the tests performed in problems 1 to 5.

7. Plot the capital market line (CML), the nine stocks, and the three portfolios on a graph of average returns versus standard deviation. Compare the mean-variance efficiency of the three portfolios and the market index. Does the comparison support the CAPM?

Suppose that, in addition to the market factor that has been considered in problems 1 to 7, a second factor is considered. The values of this factor for years 1 to 12 were as follows:

% Chango in

FactorValue

% Chango in

FactorValue

I |
- |

2 |
6.46 |

3 |
16.12 |

4 |
- |

5 |
17.82 |

6 |
- |

7 |
- |

8 |
8.43 |

9 |
8.23 |

10 |
7.06 |

11 |
- |

12 |
2.03 |

Bodie-Kane-Marcus: Investments, Fifth Edition

8. Perform the first-pass regressions as did Chen, Roll, and Ross and tabulate the relevant summary statistics. (Hint: Use a multiple regression as in a standard spreadsheet package. Estimate the betas of the 12 stocks on the two factors.)

9. Specify the hypothesis for a test of a second-pass regression for the two-factor SML.

10. Do the data suggest a two-factor economy?

11. Can you identify a factor portfolio for the second factor?

12. Identify and briefly discuss three criticisms of beta as used in the capital asset pricing model.

13. Richard Roll, in an article on using the capital asset pricing model (CAPM) to evaluate portfolio performance, indicated that it may not be possible to evaluate portfolio management ability if there is an error in the benchmark used.

a. In evaluating portfolio performance, describe the general procedure, with emphasis on the benchmark employed.

b. Explain what Roll meant by the benchmark error and identify the specific problem with this benchmark.

c. Draw a graph that shows how a portfolio that has been judged as superior relative to a "measured" security market line (SML) can be inferior relative to the "true" SML.

d. Assume that you are informed that a given portfolio manager has been evaluated as superior when compared to the Dow Jones Industrial Average, the S&P 500, and the NYSE Composite Index. Explain whether this consensus would make you feel more comfortable regarding the portfolio manager's true ability.

e. Although conceding the possible problem with benchmark errors as set forth by Roll, some contend this does not mean the CAPM is incorrect, but only that there is a measurement problem when implementing the theory. Others contend that because of benchmark errors the whole technique should be scrapped. Take and defend one of these positions.

1. The SCL is estimated for each stock; hence we need to estimate 100 equations. Our sample consists of 60 monthly rates of return for each of the 100 stocks and for the market index. Thus each regression is estimated with 60 observations. Equation 13.1 in the text shows that when stated in excess return form, the SCL should pass through the origin, that is, have a zero intercept.

2. When the SML has a positive intercept and its slope is less than the mean excess return on the market portfolio, it is flatter than predicted by the CAPM. Low-beta stocks therefore have yielded returns that, on average, were higher than they should have been on the basis of their beta. Conversely, high-beta stocks were found to have yielded, on average, lower returns than they should have on the basis of their betas. The positive coefficient on -y2 implies that stocks with higher values of firm-specific risk had on average higher returns. This pattern, of course, violates the predictions of the CAPM.

3. According to equation 13.5, y0 is the average return earned on a stock with zero beta and zero firm-specific risk. According to the CAPM, this should be the risk-free rate, which for the 1946-1955 period was 9 basis points, or .09% per month (see Table 13.1). According to the CAPM, should equal the average market risk premium, which for the 1946-1955 period was 103 basis points, or 1.03% per month. Finally, the CAPM predicts that y3, the coefficient on firm-specific risk, should be zero.

SOLUTIONS TO CONCEPT CHECKS

PART III Equilibrium in Capital Markets

SOLUTIONS TO CONCEPT CHECKS

E-INVESTMENTS: FUNDS

4. A positive coefficient on beta-squared would indicate that the relationship between risk and return is nonlinear. High-beta securities would provide expected returns more than proportional to risk. A positive coefficient on o-(e) would indicate that firm-specific risk affects expected return, a direct contradiction of the CAPM and APT.

Go to http://www.morningstar.com and select the Funds tab. The index page for Funds contains a pull-down menu that should show "Find a Fund" when you enter the site. From the pull-down menu, select Long-Term Winners. A list of the long-term winners will appear. You can click on the name of the fund, and a more detailed report will appear. Another option on the report will allow you to view ratings details. Select that information for each of the top three long-term winners.

For each of the funds identify its beta, alpha, and R-sqr. Which if any of the funds outperformed the market for its level of risk? Which fund had the highest level of risk-adjusted performance?

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