Empirical Methods in Finance Section 2 - The CAPM Model

Transcription

1 Empirical Methods in Finance Section 2 - The CAPM Model René Garcia Professor of finance EDHEC Nice, March 2010

2 Estimation and Testing The CAPM model is a one-period model E [Z i ] = β im E [Z m] with Z being excess returns with respect to a risk-free rate (Sharpe-Lintner). Z i = R i R f Z m = R m R j and: β im = Cov (Z i,z m) Var(Z m) For econometric analysis of the model, it is necessary to add an assumption about the time series behavior of returns. Returns IID though time and jointly multivariate normal Strong assumption but consistent theoretically with the CAPM model holding period by period Good empirical approximation for a monthly observation interval. Later on, we will remove this assumption.

3 Estimation and Testing Z t, a N 1 vector of excess returns on N assets or portfolios The excess returns can be described by the model: Z t = α + βz mt + ε t E [ε t ] = 0 E [ ε t ε ] t = Σ [ E [Z mt ] = µ m E (Z mt µ m) 2] = σ 2 m Cov [Z mt,ε t ] = 0 β = N 1 vector of betas Z mt = Return on the market portfolio Model implication: α = 0

4 Maximum Likelihood Approach Estimate the unconstrained model by ML. Conditional density function of excess returns Z t given Z mt. f (Z t Z mt ) = (2π) N/2 Σ 1 2 [ ] exp 2 1 (Z t α βz mt ) Σ 1 (Z t α βz mt ) Since iid excess returns, the joint probability density given T observations: T f (Z 1 Z 2 Z T Z m1 Z mt ) = f (Z t Z mt ) t=1 T = (2π) N 2 Σ 2 1 t=1 [ exp 1 ] 2 (Z t α βz mt ) Σ 1 (Z t α βz mt ) ML estimator: consistent, asymptotically efficient and asymptotically normal.

5 Maximum Likelihood Approach Log-likelihood function l(α,β,ε) = NT 2 log(2π) T 2 log Σ 1 T 2 (Z t α βz mt ) Σ 1 (Z t α βz mt ) t=1 Estimators (same as OLS) ˆα = ˆµ ˆβˆµ m ˆβ = Σ = T (Z t ˆµ)(Z mt ˆµ m) t=1 T (Z mt ˆµ m) 2 t=1 1 T T (Z t ˆα ˆβZ mt )(Z t ˆα ˆβZ ) mt t=1 with: ˆµ = 1 T T Z t t=1 ˆµ m = 1 T T Z mt t=1

6 Maximum Likelihood Approach Conditional distributions of the estimators ( [ ] ) ˆα N α, T ˆµ2 m ˆσ 2 Σ m ( [ ] ) ˆβ N β, T 1 1 ˆσ 2 Σ m T W N (T 2,Σ) Wishart with T 2 degrees of freedom and covariance matrix (multivariate generalization of the Chi-square distribution). ˆσ 2 m= T 1 T (Z mt ˆµ m) 2 t=1 [ [ ] ( Cov ˆα, β ) = T 1 ˆµ m ˆσ 2 Σ m ] Σ independent of ˆα,ˆβ

7 Maximum Likelihood Approach Null hypothesis H o : α = 0 H 1 = α 0 Wald test: J 0 = ˆα [Var(ˆα)] 1 ˆα = [ ] 1 T 1 + ˆµ2 m ˆσ 2 ˆα Σ 1 ˆα m Under H 0,J 0 has a χ 2 distribution with N degrees of freedom, we replace Σ 1 by Σ 1, a consistent estimator. We can choose the ML estimator of Σ.

8 With the joint normality assumption, we can use an exact test: Maximum Likelihood Approach [ ] 1 J 1 = T N ˆµ2 m N ˆσ 2 ˆα 1 Σ ˆα m Under H 0, J 1 is unconditionally distributed central F with N, (T N 1) degrees of freedom. Likelihood Ratio Test. To construct this test, we also need the estimators of the constrained model (α constrained to be zero). The constrained estimators are: ˆβ = T Z t Z mt t=1 T Z mt 2 t=1 Σ = 1 T T (Z t ˆβ Z mt )(Z t ˆβ ) Z mt t=1 Their distributions are: ( [ ] ) β N β, 1 1 T ˆµ 2 m + Σ ˆσ2 m T Σ W N (T 1,Σ)

9 Maximum Likelihood Approach LR, the log-likelihood ratio, is given by: LR = L L = T [ log 2 Σ ] log Σ [The summation in the 2nd part of the likelihood estimated at the ML estimators is NT for both the constrained and the unconstrained model]. Under the null: J 2 = 2LR [ = T log Σ ] log Σ a XN 2 Jobson, Korkie (1982), correction to J 2 which gives better finite sample properties. ( ) T N J 3 = 2 2 J T 2

10 Size of tests Size of tests. Since J 1 is an exact test and all other test statistics are monotonic transformation of J 1. J 1 = J 1 = J 1 = ( ) T N 1 J NT 0 ( )( [ T N 1 J2 exp N T ( ) ( T N 1 exp[ N ] ) 1 J 3 T N ]) We can compute the exact size of the tests based on large-sample statistics and their asymptotic 5% critical value. Example: J 0, 10 portfolios, 60 months of data J 0 χ 2 (10), a 5% asymptotic test will have a critical value of 18.31; this implies a value of for J 1. Since J 1 is F 10,49, the size will be 17%. The test overrejects.

11 Size of tests

12 Power of tests When drawing inferences using a test statistic it is important to evaluate its power. The power is the probability that the null hypothesis will be rejected given that an alternative hypothesis is true. Low power indicates that the test is not useful to discriminate between the alternative and the null. For a fixed value of N, considerable increases in power are possible with larger values of T. The power gain is substantial when N is reduced for a fixed alternative.

13 Non-Normal and non iid Returns - GMM-based Tests GMM-based tests since evidence of heteroskedasticity and temporal dependence in stock returns. Define: ε t = Z t α βz mt θ = [ α β ] h t = [1 Z mt ] f t (θ) = h t ε t = [ ] Z t α βz mt (Z t α βz mt )Z mt Given the model: E [f t (θ 0 )] = 0

14 GMM-based Tests For the sample average: GMM estimator Min g T (θ) = 1 T T f t (θ) t=1 Q T (θ) = g T (θ) W g T (θ) W = 2N 2N weighting matrix. Exactly identified system since 2N equations for 2N unknown parameters. The GMM estimators are in this case equivalent to the ML estimators (and OLS). ˆα = ˆµ ˆβˆµ m ˆβ = T t=1 (Z t ˆµ)(Z mt ˆµ m) T (Z mt ˆµ m) 2 t=1

15 GMM-based Tests The importance of the GMM approach is that a robust covariance matrix of the estimators can be formed. Variance-covariance of GMM ˆθ is given by: [ ] V = D o S D 0 where: D 0 = E [ ] gt (θ) θ S 0 = + E [ f t (θ)f t l (θ) ] l= ( θ a N θ, 1 [ ] ) D 1 T 0 S 1 0 D 0

16 GMM-based Tests D 0 for the CAPM is given by: [ 1 µm D 0 = µ m (σ 2 m + µ 2 m) ] I N. A consistent D T can be constructed with ˆµ m and ˆσ 2 m (ML). Construct S T by e.g. Newey-West method. V T = 1 T [ D T S 1 T D T ] 1 k ( k j S T = k j= k ) 1 T T ( ut u ) t k t=1 Since ˆα = R ˆθ, with R = [1 0] I N a robust estimator of Var(ˆα) is : A chi-square test of the model is: Under H 0 : α = 0 1 [ T R D T [ J 7 = T ˆα [R S 1 T D T S 1 T J 7 a X 2 N D T ] 1 R D T ] ] 1 1 R ˆα GMM allows to verify if model rejection is not due for example to the presence of heteroskedasticity in the data.

17 Empirical Evidence Black, Jensen and Scholes (1972) and Fama and McBeth (1973) find that there is a positive simple relation between average stock returns and β during the pre-1969 period. In general rejection of the Sharpe-Lintner-Black model after Anomalies: firm characteristics provide explanatory power for the cross-section of sample mean returns beyond the beta of the CAPM Size, Fama and French, 1992 Book-market ratio, Fama and French, 1992 Losers-winers portfolios Price-earnings ratio But little theoretical justification for the firm characteristics, so possibility of data snooping or survivorship bias (book-market ratios).

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19 Unobservability of the Market Portfolio The market portfolio (containing all assets) is unobserved and a proxy is used in the tests (most often value- or equal-weighted basket of NYSE and AMEX stocks). Roll (1977): The tests of the CAPM really only reject the mean-variance efficiency of the proxy. Are inferences sensitive to the use of a proxy? Test with broader proxies (include bonds, real estate) - Stambaugh (1982); inferences do not change. Compute an upper bound on the correlation between the market proxy return and the true market return to overturn a rejection of the CAPM [Kandel and Stambaugh (1987) and Shanken (1987)]; If correlation greater than 70% a rejection with a market proxy would also imply a rejection with the true market portfolio.

20 Cross-sectional Approach So far we have focused on the mean-variance efficiency of the market portfolio. Another way of viewing CAPM is that it links linearly expected returns and market betas: E (Z i ) = β i λ i = 1,2,,n The betas completely explain the cross-section of expected returns A natural idea is therefore to run a cross-sectional regression of the E (Z i ) on the β i. The λ is the regression coefficient and the residuals α i are the pricing errors. The problem is that we do not observe the β i. We need to estimate them. We do that by running N time-series regressions: Z i t = a i + β i Z m t + ε i t t = 1,2,,T for each i. Since the residuals in the cross-sectional regression are correlated with each other, a GLS regression should improve efficiency, but OLS might be more robust. The betas are not fixed and need to be estimated. This introduces an errors-in-variables problem. This problem can be solved by adjusting the standard errors of the estimators (see GMM approach by Cochrane). Grouping the assets into portfolios also minimizes the problem (by averaging the errors).

21 Cross-sectional Approach The cross-sectional approach is useful since it can be modified to accommodate additional risk measures beyond the CAPM beta. Say firm size S i is a candidate explanatory variable for the cross-section of expected returns. We can run the cross-sectional regression of E(Z i ) on β i and S i. One can test if size matters by testing if the coefficient of size is zero. Warning: The fact that we do not observe the true betas could make size appear significant in the CS regression.

22 Cross-sectional Approach - Fama-MacBeth Procedure Fama and MacBeth (1973) suggest an alternative procedure for running cross-sectional regressions and for producing standard errors and test statistics. 1. Find beta estimates with a time series regression. Fama and MacBeth use rolling 5-year regression. 2. Instead of estimating a single cross-sectional regression with the sample averages, run a cross-sectional regression at each time period. Z i t = λ t β i + α it i = 1,2,,N for each t 3. Fama and MacBeth suggest to estimate λ and α i as the average of the cross-sectional regression estimates 1 T ˆλ = T ˆλt ˆα i = 1 T T α it t=1 t=1

23 Cross-sectional Approach 4. Most importantly, they suggest to use the standard deviations of the cross-sectional regression estimates to generate the sampling errors for these estimates. σ 2 (ˆλ) = σ 2 ( α i ) = 1 T ( λt 2 ˆλ) T 2 t=1 1 T ( αit T 2 ) 2 α i t=1 This is similar to deducing the sampling variance of the sample mean of a series x t variation of x t through time in the sample with: by looking at the σ 2 ( x) = σ2 (x) T = 1 T T 2 (x t x) 2 t=1 The formula for σ 2 (ˆλ) could be adapted to account for autocorrelation of the λ t.

24 Cross-sectional Approach One can use naturally this sampling theory to test whether all the pricing errors are jointly zero. α vector of pricing errors across assets. ˆα = Cov(ˆα) = 1 T T α t t=1 1 T ( )( αt T 2 α αt α) t=1 and use the test: ˆα cov(ˆα)ˆα X 2 N 1 Analysis of the Fama-MacBeth procedure. Consider regression: y it = β x it + ε it, i = 1,2,,N;t = 1,2,,T We could: 1) Stack i and t observations together and estimate β by OLS. Pooled time-series cross-section estimate. However the errors are likely to be cross-sectionally correlated at any given time. So we have to correct the standard errors. 2) Take time series averages and run a pure cross-sectional regression of: E T (y it ) = β E T (x it ) + u i, i = 1,2,,N This procedure would lose any information due to variation of the x it over time, but easier to figure out a variance-covariance matrix for the u i and correct for residual correlation.

25 Cross-sectional Approach 3) Fama-MacBeth procedure: run a cross-sectional regression at each point in time; average the β t estimates to get ˆβ, and the use the time series standard deviation of β t to estimate the standard error of β. Proposition: If the x it do not vary over time and if the errors are cross-sectionally correlated but not correlated over time: The Fama-MacBeth estimate, the pure cross-sectional OLS estimate and the pooled time-series cross-sectional OLS estimates are identical. The standard errors for the Fama-MacBeth estimate are identical to the ones from the two other methods, corrected for residual correlation.

26 Cross-sectional Approach 1. Pooled OLS: Stack time series and cross sections y 1 x ε 1 Y = y 2. ; X = x. ; ε = ε 2. y T x ε T then: Y = Xβ + ε with: E(εε ) = Ω = Σ. Σ The estimate and its standard error are: βols = (X X) 1 X Y Cov( β OLS ) = (X X) 1 X ΩX(X X) 1 Since: X X = Tx x βols = (x x) 1 x E T (y t ) Cov( β OLS ) = 1 T (x x) 1 x Σx(x x) 1

27 Cross-sectional Approach This sampling variance is estimated with: 2. Pure Cross-Section One cross-section of the time-series averages: Σ= E T ( ε t ε t ); ε t y t x β OLS E T (y t ) = xβ + E t (ε t ) since E T (x t ) is a constant. Assuming i.i.d. errors over time: E[E T (ε t )E T (ε t )] = 1 T Σ The cross-sectional estimate and corrected standard errors are: βcs = (x x) 1 x E T (y t ) σ 2 ( β cs) = 1 T (x x) 1 x Σx(x x) 1 This is the same as the pooled OLS estimates and standard errors.

28 Cross-sectional Approach 3. Fama-Mac Beth βt = (x x) 1 x y t The estimate is the average of the cross-sectional regression estimates: βfm = E T ( β t ) = (x x) 1 x E T (y t ) It is the same as the pooled estimator (OLS in each sample). Cov( β FM ) = 1 T Cov( β t ) = 1 T (x x) 1 x Cov T (y t )x(x x) 1 with: Then: y t = xβ FM + ε t Cov T (y t ) = E T ( ε t ε t ) = Σ Finally: Cov( β FM ) = 1 T (x x) 1 x Σx(x x) 1 which is the same numerically as the OLS corrected standard errors. NOTE: If the x it vary over time none of the procedures are equal anymore.

29 New views about the Old Beta Fama and French (1992) put forward the size and value anomalies. Campbell and Vuolteenaho (2004) explain the size and value anomalies by an economically-motivated two-beta model. They break the beta of a stock with the market portfolio into two components, one reflecting news about the market s future cash flows (bad beta) and one reflecting news about the market s discount rates (good beta). An investor may demand a higher premium to hold assets that covary with the market s cash-flow news than to hold assets that covary with news about the market s discount rates, for poor returns driven by increases in discount rates are partially compensated by improved prospects for future returns. Required return on a stock determined by its bad beta (covariation with market cash-flow shocks), that earns a high premium, and its good beta (covariation with market discount rates), that earns a low premium. The high average return on value stocks is explained by the two-beta model: value and small stocks have a high bad beta, growth and large stocks have a high good beta.

30 Bad Beta, Good Beta Campbell and Mei (1993): another beta decomposition. The market beta of a portfolio is the sum of a cash-flow beta (covariation with market return of portfolio cash-flow news) and a discount rate beta (covariation with market return of portfolio discount rate news). Campbell, Polk, Vuolteenaho (2009): a four-way decomposition. β i,m = β CFi,CFM + β DRi,CFM + β CFi,DRM + β DRi,DRM Question: Are the high bad betas of value stocks and the high good betas of growth stocks attributable to their cash flows or their discount rates? Background: Barberis, Shleifer and Wurgler (2005), Market sentiment story, value stocks out of favor, growth stocks glamor stocks, favored by investors.

31 Bad Beta, Good Beta Campbell and Shiller (1988): loglinear approximation of r t+1 log(p t+1 + D t+1 ) log(p t ) around the mean log dividend-price ratio. r t+1 k + ρp t+1 + (1 ρ)d t+1 p t where ρ and k are linearization parameters. Solving forward iteratively: p t d t = 1 ρ k + E t j=0 ρj [ d t+1+j r t+1+j ] Campbell (1991): Decomposition of returns r t+1 E t r t+1 = (E t+1 E t ) ρ j d t+1+j (E t+1 E t ) ρ j r t+1+j j=0 j=0 = N CF,t+1 N DR,t+1

32 Bad Beta, Good Beta News Extraction VAR model: z t+1 = a + Γz t + u t+1 For the aggregate: excess returns, term yield spread, smoothed (10-year)log price-earnings ratio, small-stock value spread. N DR,t+1 = e1 λu t+1 N CF,t+1 = (e1 + e1 λ)u t+1 The cash-flow news is backed out as the difference between the total unexpected return and the DR news. Potential caveats with the methodology. 1 Cash flow news being a residual may capture noise. The quality of the return forecasting equation (choice of state variables) is crucial. Chen and Zhao (2006) apply the methodology to Treasury bond returns (with no actual cash flow news) and find that CF betas are larger than DR betas. 2 Statistical issues due to persistence of state variables. Main one is smoothed PE ratio. Results are sensitive to small changes in definition of this variable. 3 Results are sensitive to the sample period and to the type of portfolios (reverse results for industry portfolios). 4 Standard errors need to be adjusted (both for persistence and estimation error in extracting news). Weaker statistical evidence for CF betas. 5 Reduced-form results difficult to interpret economically.

33 Chen and Zhao Suggestions for Improvement First, while it is important to search for predictive variables well motivated by economic reasons, it is far from enough. The reason is that most predictive variables are related to the macroeconomy and thus are well motivated. Extensive robustness check with respect to model sensitivity is necessary before one can put faith into such conclusions. Second, a natural approach is to model both DR and CF in VAR. With such an approach, the unexpected return will be decomposed into DR news, directly modeled CF news, and residual news. Though we still do not know the nature of the residual news, we can at least compare the DR news and CF news on equal footing. Third, to mitigate model uncertainty, one can use the Bayesian model averaging approach (Avramov (2002) and Cremer (2002)). Another approach is to first recover the principal components of a large set of predictive variables, and then use them to predict stock returns (e.g., Ludvigson and Ng (2007)).

34 Cross-sectional forecasts of the equity premium Polk, Thompson and Vuolteenaho (2006) forecast the equity premium from the cross-sectional price of risk. Based on the Gordon model: From the CAPM: D i P i R rf + E(g i ) = E(R i ) R rf D i,t P i,t 1 β i E t 1 [R M,t R rf ] E(g i R rf ) To obtain the equity premium, regress the cross-section of dividend yields on betas and expected dividend growth: D i,t P i,t 1 λ 0,t 1 + λ 1,t 1 β i + λ 2,t 1 E(g i ) In the post-1965 subsample, the predictive ability of the cross-sectional beta-premium measures is less strong than in the pre-1965 subsample. Confirms empirical results about CAPM.

35 A long-run view about the CAPM Cohen, Polk and Vuoltaneeho (2009) adopt the perspective of a buy-and-hold investor to test the CAPM but use the price level instead of the short-horizon expected return. They choose return on equity (ROE) as the cash-flow measure and define the cash flow beta as the regression coefficient of a firm s or a portfolio s discounted log ROE on the market portfolio s discounted log ROE: N 1 ρ j log(1 + ROE k,t+j,j+1 ) = β CF N 1 k,0 + βcf k,1 ρ j log(1 + ROE M,t+j ) + ε k,t+n 1 j=0 j=0 ROE denotes the ratio of clean surplus earnings ( X t = BE t BE t 1 + D gross t )to beginning-of-the-period book equity ( BE t 1 ), with subscript k corresponding to the firm or portfolio under scrutiny and subscript M to the market portfolio. The second subscript refers to the year of observation and the third to the number of years from the sort. They find that value stocks betas sharply increase and growth stocks betas sharply decrease after portfolio formation. Within five years from portfolio formation, value stocks (three lowest price-to-book deciles) betas have increased to approximately 1.11 and growth stocks (three highest price-to-book deciles) betas have declined to approximately Results from long-run returns betas are consistent with the cash-flow betas for the period

36 A Long-Horizon Perspective on the Cross-Section of Expected Returns The existence of a value premium, whereby stocks with high book-to-market ratios have higher average returns than stocks with low book-to-market ratios, is widely documented over the post-1963 period in U.S. stock returns (Fama and French (1992)). However, whether the Capital Asset Pricing Model (CAPM) can explain this value premium is a source of lively debate. Fama and French (1993) find that the post-1963 value premium is left unexplained by the CAPM, while Ang and Chen (2007) show that the CAPM captures the value premium of the period. Moreover, the latter authors test a conditional CAPM with time-varying market betas and conclude that it can explain the value premium even in the post-1963 period. This is contested by Fama and French (2006), who conclude that it is size and book-to-market factors, or risks related to them, and not market beta risk that are rewarded in average returns. Garcia and Lioui (2009)contribute to the debate by providing evidence on the long-run relationship between expected returns and market beta risk in the post-1963 period.

37 A Long-Horizon Perspective on the Cross-Section of Expected Returns They show that in cross-sectional regressions returns averaged over longer periods (three to five years) are related positively and significantly to the market risk betas also computed over similar periods. Moreover, for the Fama-French 25 size and book-to-market portfolios, the market risk premium over five-year periods is estimated at a reasonable value close to 6% per annum. The explanation for this result rests in the gradual alignment between the betas of the portfolios and their average returns as the horizon increases.

38 A Long-Horizon Perspective on the Cross-Section of Expected Returns Excess Returns β L H L H 1m S B m S B

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40 The Conditional CAPM In the presence of a risk-free rate asset, the conditional CAPM can be stated as follows: E [r i (t) Ω t ] = β it E [r M (t) Ω t ] (1) where r i (t) is the one-period return on portfolio i in excess of the risk-free asset return, r M (t) the excess return on the market portfolio, and β it is given by the following expression: β it = Cov [r i (t),r M (t) Ω t ] Var [r M (t) Ω t ] (2) In this version of the CAPM, all moments are made conditional to the information available at time t represented by the information set Ω t. Many asset pricing studies on the US stock markets (Ferson and Harvey (1991), for example) have shown that allowing the moments to vary with time is essential, since there is evidence that both the beta, the ratio of the covariance to the variance, and the price of risk E [r M (t) Ω t ] are time-varying.

41 The Conditional CAPM In order to specify model (1) for estimation, we decompose the returns into a forecastable part and an unforecastable part, namely: r i (t) = E [r i (t) Ω t ] + u i (t) (3) r M (t) = E [r M (t) Ω t ] + u M (t) (4) where u i (t) and u m(t) are forecast errors orthogonal to the information in Ω t. Equation (1) can therefore be rewritten as follows: E [r i (t) Ω t ] = E [u i (t)u M (t) Ω t ] ] E [u M (t) 2 E [r M (t) Ω t ] (5) Ω t To obtain a set of moment conditions suitable for GMM estimation, we need to specify parametric models for the expectations on the right hand side of (5).

42 The Conditional CAPM - Time Series Approach Bodurtha and Mark (1991) choose to specify autoregressive processes for each of the expectations: ] kσ M E [u M (t) 2 Ω t = δ 0M + δ jm u M (t j) 2 (6) j=1 k i E [u i (t)u M (t) Ω t ] = δ 0i + δ ji u i (t j)u M (t j) (7) j=1 k M E [r M (t) Ω t ] = α 0M + α jm r M (t j) (8) j=1 The number of lags k σm,k i,k M to be included in each of the equations above remains an empirical issue, given the constraint imposed by the number of available observations. The final form of the moment conditions that will be used for GMM estimation can therefore be written as follows: r M (t) = α 0M + k M j=1 α jm r M (t j) + u M (t) u M (t) 2 = δ 0M + kσ M j=1 δ j M u M (t j) 2 + v M (t) u i (t)u M (t) = δ 0i + k i j=1 δ ji u i (t j)u M (t j) + v im (t),i = 1,...,N r i (t) = δ 0i + k i j=1 δ ji u i (t j)u M (t j) k δ 0M + kσ [α 0M + M j=1 α jm r M (t j)] + u i (t) M j=1 δ j u M M (t j) 2 where v M (t) and v im (t) are the conditional forecast errors corresponding to the second-moment conditions. Application to Brazil, Bonomo and Garcia (JIMF, 2001) (9)

43 The Conditional CAPM - Time-Varying Risk and Price of Risk Jagannathan and Wang (1996) derive the unconditional version of the conditional version of the CAPM. They show that when the conditional version of the CAPM holds (i.e., when betas and expected returns are allowed to vary over the business cycle), a two-factor model obtains unconditionally. Average returns are jointly linear in the average beta and in a measure of "beta instability". The zero-beta version of the conditional CAPM can be written as: E[R it I t 1 ] = γ 0t 1 + γ 1t 1 β it 1 Taking the unconditional expectation of this equation yields: E[R it ] = γ 0 + γ 1 βi + Cov(γ 1t 1,β it 1 ) where Define: ϑ i = Cov(γ 1t 1,β it 1 ) Var(γ 1t 1 ) Then: γ 0 = E[γ 0t 1 ] γ 1 = E[γ 1t 1 ] β i = E[β it 1 ] E[R it ] = γ 0 + γ 1 βi + Var(γ 1t 1 )ϑ i Cross-sectionally, the unconditional expected return on any asset i is a linear function of its expected beta and its beta-prem sensitivity.

44 The Conditional CAPM - Time-Varying Risk and Price of Risk Jagannathan and Wang (1996) choose the yield spread between BAA- and AAA-rated bonds, denoted by R prem t 1 as a proxy for the market risk premium. γ 1t 1 = κ 0 + κ 1 R prem t 1 They also make an assumption about the return on the market portfolio (consistent with Roll (1977): R mt = φ 0 + φ vwrt vw + φ labor Rt labor Using the value-weighted index from CRSP as the market portfolio, they find that the unconditional model implied by the conditional CAPM explains nearly 30 percent of the cross-sectional variation in average returns of 100 stock portfolios similar to those used in Fama and French (1992). This is a substantial improvement when compared to the 1 percent explained by the static CAPM. When human capital is also included in measuring wealth, the unconditional model implied by the conditional CAPM is able to explain over 50 percent of the cross-sectional variation in average returns, and the data fail to reject the model. More importantly, size and book-to-market variables have little ability to explain what is left unexplained.

45 The Conditional CAPM - Time-Varying Risk and Price of Risk Lettau and Ludvigson (2001): log-consumption wealth ratio as a conditioning variable. Petkova and Zhang (2005): find that time-varying risk goes in the right direction in explaining the value premium. Value betas tend to covary positively, and growth betas tend to covary negatively with the expected market risk premium. and r mt+1 = δ 0 + δ 1 DIV t + δ 2 DEF t + δ 3 TERM t + δ 4 TB t + e mt+1 γ t = ˆδ 0 +ˆδ 1 DIV t +ˆδ 2 DEF t +ˆδ 3 TERM t +ˆδ 4 TB t