CAPM (1) where λ = E[r e m ], re i = r i r f and r e m = r m r f are the stock i and market excess returns.

Transcription

1 II.3 Time Series, Cross-Section, and GMM/DF Approaches to CAPM Beta representation CAPM (1) E[r e i ] = β iλ, where λ = E[r e m ], re i = r i r f and r e m = r m r f are the stock i and market excess returns. (a) Time-series estimation of λ If CAPM is true, equation (1) should work for all assets and because the market expected excess return or risk premium λ m = E[r e m ] is the same for all assets (thus being independent of assets), estimation of it should also be independent of individual assets. As a consequence, the time series estimate of the market risk premium simply is (2) ˆλ m = 1 T T rm,t. e t=1 1

2 The beta for the market portfolio is one and for the riskfree return zero. Thus, the line in the (beta, excess return) simply goes through the origin (0, 0) and (1, E[r e m ]). To produce the line, we estimate λ = E[r e m ] simply by (3) 2

3 Example 3.1: Consider the earlier (US) stock data example. The average excess return for the SP500 from the sample period is ˆλ = 0.090% (per week) (about 4.7% p.a). The graph below depicts the (estimated) expected excess-return beta relationships. The line is through the origin (0, 0) and (1, rm e ). Return-Beta Relationship Simple Time Series Estimation [Aug 15, 1988 to Jan ] Dell Excess return (%, p.a) IBM Apple MS Mkt Motorola 0 Ford GM Beta 3

4 Dell and Microsoft are kind of outliers with exceptionally high excess returns. IBM and Apple fit pretty well into the picture. Ford and GM have negative excess returns. The overall result is that CAPM does not tell much about the mean return behavior of these individual stocks. 4

5 (b) Cross-sectional estimation of λ The cross-sectional approach is simply to estimate λ with regression (3) E[r i ] = λβ i in two steps by first estimating βs and average excess returns from time series data, and in the second step estimate (3) by regressing the estimated average excess returns on the estimated betas. 5

6 Example 3.2: Let us next look at the same analysis as in Ex 3.1 using Fama-French size/bm sorted portfolios [monthly returns, ] Average excess returns vs. market beta for 25 stock porfolios sorted by size and book/market ratio Excess return er = lambda_ols x beta 0.2 er = aver(er_m) x beta Beta Source : French data library The blue line is again r e = ˆλ β, where ˆλ = r e m and the pink line is OLS cross-sectional regression of r e i on betas with ˆλ OLS (t-val 3.72). Here the Shanken corrected standard errors must be used to calculate the t- value (Shanken 1992 On estimating beta pricing models, Review of Financial Studies 5, 1 34). 6

7 The sample average method yields ˆλ = r e m (t-val 3.64). The differences of these two approaches is not big. 7

8 Remark 3.1: Because the returns and hence estimated betas are cross-sectionally correlated, we should have used in the Example 3.2 GLS instead of OLS. Nevertheless, the end result in this case would not be much different. The GLS estimator is (4) ˆλ = (β Σ 1 β) 1 β Σ 1 r e, where Σ = Cov(ˆβ) and r e is the vector of average excess returns of the stocks (portfolios). 8

9 Example 3.4: Allowing pricing error in CAPM by using intercept in (1), (5) E[r e i ] = α + λβ i we get by OLS estimated regression as (t-values in the parenthese) (6) ˆr e = β (1.56) (0.15) [Figure] That is the regression is completely flat. Thus, the bottom line seems to be that the CAPM does not at all capture the variability in the crosssectional (portfolio) excess returns. 9

10 GMM Discount Factor Approach Because m = 1 br e m (7) E[r e i ] = be[re i re m]. Thus the beta-return scatter plot in the discount factor representation is (E[r e r e m ], E[re ]) coordinate system. If m = 1 b f, the general factor model, then (8) E[r e i ] = E[re i f ]b and can check the graphically the fit of the model by plotting actual excess returns against the model predicted returns. If the model works, the observed the returns should plot approximately on the 45 degree line. 10

11 Example 3.5: Size/BM data. Equation (12) implies moment conditons (9) r e i bre i re m = 0. i = 1,... N (= number of shares). Estimating b with OLS gives ˆb OLS = (stderr = ) and with GMM (using HAC) (10) ˆb = (stderr = ). Average excess returns vs. discont factor predictor for 25 stock porfolios sorted by size and book/market ratio Excess return er_i = b_gmm x er_m x er_i E[er_m x er_i] Source : French data library Average excess returns and (CAPM) discount factor plot yields [Monthly returns ] The fit is of course similar as in the beta-representation. 12

12 Using (1.48) (i.e., λ = E[ff ]b) we get for the market risk premium (11) λ = E[(r e m )2 ]b. Thus, the discount factor estimate for the market price of risk is (12)ˆλ = (r e m )2 ˆb = , where (13) (rm) e 2 = 1 T T (rm,t) e 2. t=1 Again the estimates are quite close to the time series average and cross-sectional OLS. 13

13 Remrk 3.2: From the discount factor model m = 1 br e m, (14) b = E[re m ] E [ (r e m) 2]. Using the market data we have and r e m = 0.646% s r e m = %. Thus, the market data estimate of b is simply (15) ˆb = 0.646/( ) which is slightly larger than the GMM estimate given in (10). 14

14 Pricing errors The CAPM-testing was considered in Chapter 2. Here we shortly demonstrate the SDF priing error testing. The pricing errors are (16) u i = r e i bre i re m. We can test whether these are jointly zero using the J-statistic, which gives a χ 2 -statistic (17) χ 2 = T J χ 2 N 1 if the null hypothesis is true. 15

15 Example 3.5: (Continued) The GMM estimation yields J = and T = 950 months, we get (18) χ 2 = 950 J With N 1 = 25 1 = 24 degrees of freedom the p- value is , which indicates that market factor does price the portfolio returns adequately. 16

16 Cochrane Ch 15 works out additional comparisons of the three approaches. The end result is that differences are small. In one hand this shows reliability of the GMM/DF approach. What is gained through it? 17