bkmsol_ch13

# However with this sample size the power of this test

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to the average excess return and the difference is less than one standard error. However, with this sample size, the power of this test is extremely low. 11. When we use the actual factor, we implicitly assume that investors can perfectly replicate it, that is, they can invest in a portfolio that is perfectly correlated with the factor. When this is not possible, one cannot expect the CAPM equation (the second pass regression) to hold. Investors can use a replicating portfolio (a proxy for the factor) that maximizes the correlation with the factor. The CAPM equation is then expected to hold with respect to the proxy portfolio. Using the bordered covariance matrix of the nine stocks and the Excel Solver we produce a proxy portfolio for factor F, denoted PF. To preserve the scale, we include constraints that require the nine weights to be in the range of [-1,1] and that the mean equal the factor mean of 0.60%. The resultant weights for the proxy and period returns are:

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13-7 Proxy Portfolio for Factor F (PF) Weights on Universe Stocks Year PF Holding Period Returns A -0.14 1 -33.51 B 1.00 2 62.78 C 0.95 3 9.87 D -0.35 4 -153.56 E 0.16 5 200.76 F -1.00 6 -36.62 G 0.13 7 -74.34 H 0.19 8 -10.84 I 0.06 9 28.11 10 59.51 11 -59.15 12 14.22 Average 0.60 This proxy (PF) has an R-square with the actual factor of 0.80. We next perform the first pass regressions for the two factor model using PF instead of P: A B C D E F G H I R-square 0.08 0.55 0.20 0.43 0.33 0.88 0.16 0.71 0.72 Observations 12 12 12 12 12 12 12 12 12 Intercept 9.28 -2.53 -1.35 -4.45 -0.23 -3.20 4.99 -2.92 5.54 Beta M -0.50 0.80 0.49 1.32 1.00 1.64 0.76 1.97 2.12 Beta PF -0.06 0.42 0.16 -0.13 0.21 -0.29 0.21 0.11 0.08 t- Intercept 0.72 -0.21 -0.12 -0.36 -0.02 -0.55 0.27 -0.33 0.58 t-Beta M -0.83 1.43 0.94 2.29 1.66 6.00 0.90 4.67 4.77 t-Beta PF -0.44 3.16 1.25 -0.97 1.47 -4.52 1.03 1.13 0.78 Note that the betas of the nine stocks on M and the proxy (PF) are different from those in the first pass when we use the actual proxy.
13-8 The first-pass regression for the two-factor model with the proxy yields: Average Excess Return Beta M Beta PF A 5.18 -0.50 -0.06 B 4.19 0.80 0.42 C 2.75 0.49 0.16 D 6.15 1.32 -0.13 E 8.05 1.00 0.21 F 9.90 1.64 -0.29 G 11.32 0.76 0.21 H 13.11 1.97 0.11 I 22.83 2.12 0.08 M 8.12 PF 0.6 The second-pass regression yields: Regression Statistics Multiple R 0.71 R Square 0.51 Adjusted R Square 0.35 Standard Error 4.95 Observations 9 Coefficients Standard Error t Statistic for β =0 t Statistic for β =8.12 t Statistic for β =0.6 Intercept 3.50 2.99 1.17 Beta M 5.39 2.18 2.48 -1.25 Beta PF 0.26 8.36 0.03 -0.04 We can see that the results are similar to, but slightly inferior to, those with the actual factor, since the intercept is larger and the slope coefficient smaller. Note also that we use here an in-sample test rather than tests with future returns, which is more forgiving than an out-of-sample test. 12. (i) Betas are estimated with respect to market indexes that are proxies for the true market portfolio, which is inherently unobservable. (ii) Empirical tests of the CAPM show that average returns are not related to beta in the manner predicted by the theory. The empirical SML is flatter than the theoretical one. (iii) Multi-factor models of security returns show that beta, which is a one- dimensional measure of risk, may not capture the true risk of the stock of portfolio.

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