Statistical Analysis of the CAPM II. Black CAPM

Transcription

1 Statistical Analysis of the CAPM II. Black CAPM Brief Review of the Black CAPM The Black CAPM assumes that (i) all investors act according to the µ σ rule, (ii) face no short selling constraints, and (iii) exhibit perfect agreement with respect to the probability distribution of asset returns. It is not assumed that they can lend and borrow at a common risk free rate. Under these assumptions, the market portfolio is a mean variance efficient portfolio. 1

2 Thus, there is a portfolio Z, i.e., the zero beta portfolio with respect to the market portfolio, such that for each risky asset or portfolio of risky assets i, we have µ i = µ z + β i (µ m µ z ), (1) where µ m portfolio. is the expected return of the market 2

3 Framework for Estimation and Testing The CAPM relationship (1) is expressed in terms of expected values, which are not observable. To obtain a model with observable quantities, we describe returns using the following market model: r it = α i + β i r m,t + ϵ it i = 1,..., N (2) E(ϵ it ) = 0, i = 1,..., N (3) { σ ij if t = t E(ϵ it ϵ jt ) = 0 if t t i, j = 1,..., N (4) E(r m,t ϵ i,t ) = 0, i = 1,..., N. (5) Here r i,t is the return of asset i in period t, and r m,t is the return of the market portfolio in period t. This is very similar to the framework employed for testing the Sharpe Lintner CAPM. 3

4 However, in contrast to the market model considered last week, the relation (2) is not stated in terms of excess returns. According to equation (4), the asset specific error terms may be correlated. Thus, we allow for a non-diagonal covariance matrix, Σ, of the vector ϵ t = [ϵ 1t,..., ϵ Nt ], COV (ϵ t ) = Σ = σ 2 1 σ 12 σ 1N σ 12 σ 2 2 σ 2N σ 1N σ 2N σ 2 N Conditional on the excess return of the market, we then also have COV (r t ) = Σ, (6) where r t = [r 1t,..., r 2t ]. 4

5 We will also assume that the error terms follow a multivariate normal distribution, i.e., ϵ t iid N(0, Σ). (7) The Black CAPM implies a restriction on the intercept terms in (2), namely, α i = (1 β i )µ z, i = 1,..., N, (8) or, using vector notation, α = (1 N β)µ z. (9) Equation (9) imposes a nonlinear restriction on the parameters, because µ z is not known and has to be estimated, along with the further unknown parameters of the (restricted) model, i.e., β and Σ. 5

6 Estimation of the Parameters Write the market model as r t = α + βr m,t + ϵ t, t = 1,..., T, ϵ t iid N(0, Σ), where α = [α 1,..., α N ], and β = [β 1,..., β N ]. The maximum likelihood estimator (MLE) for the unconstrained model has been derived last week, and is given by α = r β r m, (10) β = T t=1 (r t r)(r m,t r m ) T t=1 (r m,t r m ) 2 (11) = = T t=1 (r t r)(r m,t r m ) T σ m 2 T t=1 (r m,t r m )r t T σ m 2, 6

7 and Σ = 1 T = 1 T T ϵ t ϵ t (12) t=1 T (r t α βr m,t )(r t α βr m,t ). t=1 where r = 1 T σ 2 m = 1 T T r t, r m = 1 T t=1 T (r m,t r m ) 2. t=1 T r m,t, t=1 7

8 Estimation of the Restricted Model Recall that the Black CAPM imposes α = (1 N β)µ z. (13) The parameters to estimate are µ z, β and Σ, and the log likelihood function is log L(µ z, β, Σ) = NT 2 log(2π) T 2 1 T ϵ 2 tσ 1 ϵ t, t=1 log Σ where ϵ t = r t (1 N β) µ z βr m,t. (14) 8

9 Note that, by the same arguments as last week, whatever the estimators of β and µ z will be, the estimator of Σ is Σ( µ z, β) = 1 T = 1 T T ϵ t ϵ t (15) t=1 T (r t (1 N β) µ z βr m,t ) t=1 (r t (1 N β) µ z βr m,t ). Moreover, for any given µ z, β will be the equation by equation OLS estimator of the regression through the origin (r t 1 N µ z ) = β(r m,t µ z ), t = 1,..., T. 9

10 It follows that β( µ z ) = T (r t 1 N µ z )(r m,t µ z ) t=1. (16) T (r m,t µ z ) 2 t=1 From last week s analysis, we also know that the log likelihood function, evaluated at the MLE, is log L = NT 2 [log(2π) + 1] T 2 log Σ. Thus, we have to find β and µ z such that 1 log Σ = log T T (r t µ z (1 N β) βr m,t ) t=1 (r t µ z (1 N β) βr m,t ) (17) is minimized. 10

11 But, as we have seen in (16), β can be written as a function of µ z, namely β( µ z ) = T (r t 1 N µ z )(r m,t µ z ) t=1. (18) T (r m,t µ z ) 2 t=1 Thus, (17) can be written as a function of just a single variable, µ z. Hence, we can find the MLE of the restricted model by first identifying µ z. This can be done, for example, by conducting a simple grid search. Then compute β via (18) and finally evaluate Σ using equation (15). 11

12 Likelihood Ratio (LR) Test Having estimated the parameters of both the unrestricted as well as those of the restricted market model, we can conduct a likelihood ratio (LR) test. If Σ 1 denotes the estimated error term covariance matrix under the unrestricted model, and Σ 0 is the estimated error term covariance matrix under the null hypothesis, then, by the same arguments as last week, the LR test statistic is LR = T [ ] asy log Σ 0 log Σ 1 χ 2 (N 1). Note that the degrees of freedom of the null distribution is N 1. Relative to the Sharpe Lintner model, we lose one degree of freedom because µ z is a free parameter. 12

13 5 4 3 portfolio mean portfolio standard deviation MVS with short sales DAX individual assets 13

14 5 4 3 portfolio mean MVS with short sales MVS without short sales DAX individual assets portfolio standard deviation 14

15 log likelihood as function of µ z µ z 15

16 sample mean return estimated betas We have also seen that the asymptotic likelihood ratio test may exhibit poor performance in finite samples. 16

17 To mitigate these effects, the adjusted statistic LR = ( T N ) [ ] 2 2 log Σ 0 log Σ 1 asy χ 2 (N 1) has been shown to more closely match the χ 2 distribution in finite samples. There also exists a further device that provides a useful check. 1 This also gives rise to a closed from estimator for the zero-beta rate, µ Z. 1 Cf. Shanken, J. (1986) Testing Portfolio Efficiency when the Zero Beta Rate in Unknown: A Note. Journal of Finance 41,

18 Lower Bound for the Exact Distribution Suppose for the moment that µ z is known. Then, we can proceed as last week when testing the Sharpe Linter model, i.e., we can consider the excess return market model r t µ z 1 N = α + β(r m,t µ z ) + ϵ t. (19) The zero beta CAPM is true if α = 0. The estimates of the unrestricted model are α 1 = r 1 N µ z β( r m µ z ) t β 1 = (r t r)(r m,t r m,t ) t (r m,t r m ) 2 Σ 1 = 1 [r t r T β 1 (r m,t r m )] t [r t r β 1 (r m,t r m )]. 18

19 Note that β 1 and Σ 1 do not depend on µ z, and, thus, the value of the log likelihood function does also not depend on µ z, as, at the maximum, log L 1 = NT 2 [log(2π) + 1] T 2 log Σ 1. The MLE under the restriction that α = 0 is β 0 (µ z ) = Σ 0 (µ z ) = 1 T t (r t 1 N µ z )(r m,t µ z ) t (r m,t µ z ) 2 (20) (r t µ z (1 N β 0 ) β 0 r m,t ) t (r t µ z (1 N β 0 ) β 0 r m,t ). 19

20 The value of the constrained log likelihood is log L 0 (µ z ) = NT 2 [log(2π) + 1] T 2 log Σ 0 (µ z ), which can be viewed as a function of only one variable, i.e., µ z. Consequently, the likelihood ratio test statistic, LR(µ z ) = T [ ] log Σ 0 (µ z ) log Σ 1, (21) can be viwed as a function of only µ z. Obviously, the value of µ z which minimizes the likelihood ratio statistic will be the MLE of µ z. (Recall that Σ 1 does not depend on µ Z.) 20

21 In the last week, we have developed a relation between the LR test and the F test for the Sharpe Linter CAPM, which used a formula expressing Σ 0 in terms of Σ 1 and α. Repeating the same line of arguments shows that (21) can be written as LR(µ z ) = T log where [ α Σ 1 1 α σ 2 m ( r m µ z ) 2 + σ 2 m ] + 1, (22) α = ( r β 1 r m ) (1 N β 1 )µ z. (23) is a function of µ z 21

22 Thus, the MLE of µ z is the value which minimizes where g(µ z ) = α Σ 1 1 α σ 2 m ( r m µ z ) 2 + σ 2 m = [µ2 za 2bµ z + c] σ 2 m σ 2 m + ( r m µ z ) 2, a = (1 N β 1 ) Σ 1 1 (1 N β 1 ), b = (1 N β 1 ) Σ 1 1 ( r β 1 r m ), c = ( r β 1 r m ) Σ 1 1 ( r β 1 r m ). 22

23 It follows that we can find the MLE of µ z by solving dg(µ z ) = (2aµ z 2b)[( r m µ z ) 2 + σ m] 2 dµ z [( r m µ z ) 2 + σ m] 2 2 (µ2 za 2bµ z + c)2(µ z r m ) [( r m µ z ) 2 + σ 2 m] 2 = 0, that is, µ z is a root of the quadratic where Aµ 2 z + Bµ z + C = 0, (24) A = b a r m, B = a( σ 2 m + r 2 m) c, C = b( σ 2 m + r 2 m) + c r m. This is a closed form solution for µ z. 23

24 It can be shown that the roots of (24) are real. The maximum likelihood estimator, then, is the root which corresponds to the smaller value of log(det( Σ(µ z ))). Once µ z is determined, we can apply the closed form expressions (20) to determine the MLE for the other parameters. 24

25 Now let us return to our objective of finding a test which does not rely on asymptotic arguments. Recall that, with µ z known, we could use the statistic J(µ z ) = T N 1 N [1 + ( r m µ z ) 2 σ 2 m to conduct an exact finite sample test. ] 1 α Σ 1 1 α (25) This uses the result that, in this case, (25) has an F distribution with N degrees of freedom in the numerator and T N 1 degrees of freedom in the denominator. As µ z is not known, this cannot be done. The MLE of µ z, µ z, is the value which minimizes the LR test statistic, LR(µ z ). As we know that J(µ z ) is a monotonic transformation of LR(µ z ), µ z also minimizes J(µ z ). 25

26 Thus, J( µ z ) J(µ z ), (26) where µ z is the true value of the zero beta portfolio s mean. That is, the F test based on µ z and using the exact F distribution will accept too often. But we know that, if it rejects, it will reject for any value of µ z, and we need not resort to asymptotic approximations in this case. This is a useful check because we have seen that the asymptotic likelihood ratio test rejects too often. 26