Tests of linear beta pricing models (I) FINA790C Spring 2006 HKUST

Outline Linear beta pricing models Conditional and unconditional MV efficiency Base case: Maximum likelihood methods with conditional homoscedasticity –Non-traded factors –Traded factors –Example: riskfree rate, traded factors Extensions: –GMM –Cross-sectional regression methodology

SDF once again Suppose there is a stochastic discount factor m t+1 such that E t [m t+1 R it+1 ] = 1 Consider the portfolio p with maximal correlation with m m t+1 =  0t +  1t R pt+1 +  t+1 The asset z whose return is uncorrelated with m and p has conditional expected return 1/E t [m t+1 ] (riskfree rate if it exists)

Linear beta pricing This gives pricing relation in terms of z and p E t [R it+1 ] = E t [R zt+1 ] + β ipt+1 (E t [R pt+1 ]-E t [R zt+1 ]) From efficient set mathematics this is equivalent to saying that portfolio p is on the minimum variance frontier OR we can just assume that m t+1 is a linear function of pre-specified factors

Identifying the factors Theoretical arguments or economic intuition –Sharpe-Lintner-Black CAPM –Intertemporal CAPM –Consumption CAPM Statistical factors Empirical anomalies –Size, book-to-market, momentum

The role of conditioning information Theory is usually stated in terms of conditional expectations but we don’t know what is in investors’ information sets What is the relationship between unconditional minimum variance and conditional minimum variance? If a portfolio is unconditionally minimum variance then it must be conditionally minimum variance, but if it is conditionally minimum variance that does not imply that it is unconditionally minimum variance

Linear beta pricing models Suppose the return generating process for each of N assets follows a K-factor linear model: R it = α i + f t ’β i + u it E[u it |f t ] = 0, i=1,…,N where β i is the Kx1 vector of betas for asset i β i =  f -1 E[(R it – E(R it ))(f t – E(f t ))] and  f = E[(f t -E(f t ))(f t -E(f t ))’]

Linear beta pricing models Then the linear beta pricing model says E(R it ) = λ 0 + λ’β i where λ 0, λ are constants λ j is the risk premium for factor j (the expected return on a portfolio of the N assets which has beta of 1 on factor j and beta of 0 for all other factors)

Estimating & testing linear beta pricing model We can estimate and test the model using maximum likelihood methods if we make distributional assumptions for returns and factors Collect the N returns and K factors R t = α + Bf t + u t where α = μ R - Bμ F and suppose that u t is iid multivariate normal N(0,  u ) conditional on contemporaneous and past values of the factors

Maximum likelihood estimation The log likelihood of the unconstrained model is L = -½NTln(2π) -½Tln(|  u |)-½  (R t -α-Bf t )’  u -1 (R t -α-Bf t ) So the estimators are α* = μ R * – B*μ f * B* = [(1/T)  (R t –μ R *)(f t –μ f *)’]  f * -1 where μ R * = (1/T)  R t ; μ f * = (1/T)  f t ;  f * = [(1/T)  (f t –μ f *)(f t –μ f *)’] ;  u * = [(1/T)  (R t -α*-B*f t )(R t -α*-B*f t )’]

Maximum likelihood estimation For the constrained model α = λ 0 1+B(λ- μ f ) The constrained ML estimators are: let ξ= λ- μ f B c * = [(1/T)  (R t –λ 0 *1)(f t +ξ*)’][(1/T)  (f t +ξ*)((f t +ξ*)’] -1  uc * = [(1/T)  (R t - λ 0 *1-B c *(f t +ξ*)) (R t - λ 0 *1-B c *(f t +ξ*))’]  * = (λ 0 * ξ*’)’ = [Z’  uc * -1 Z] -1 [Z’  uc * -1 (μ R * – B c *μ f *)] where Z = [ 1 B c * ] The estimators can be computed by successive iteration

Testing The asymptotic variance of  * is (1/T)[1+(ξ*+μ f *)’  f * -1 (ξ*+μ f *)][Z’  uc * -1 Z] -1 The LRT statistic is J LR = -T{ ln(|  u *|) – ln(|  uc *|) } We can recover the factor risk premiums λ from λ* = ξ* + μ f which has asymptotic variance (1/T)  f * + var(ξ*)

Traded factors When the factors are traded then: –Factors f t are returns on benchmark portfolios. Estimators for the unconstrained model are exactly the same (under this new interpretation of the factors) –λ is the vector of expected returns on the benchmark portfolios in excess of the zero- beta rate So the restriction on α is λ 0 1-λ 0 B1 = λ 0 (1-B1)

Traded factors - constrained model Estimators for the constrained model are: B c *=[(1/T)  (R t –λ 0 *1)(f t –λ 0 *1)’][(1/T)  (f t –λ 0 *1)(f t –λ 0 *1)’] -1  uc * = [(1/T)  (R t - λ 0 *1-B c *(f t –λ 0 *1))(R t -λ 0 *1-B c *(f t –λ 0 *1))’] λ 0 *= [(1- B c *1)’  uc * -1 (1- B c *1)] -1 [(1- B c *1)’  uc * -1 (μ R *–B c *μ f *)] The asymptotic variance of λ 0 *= (1/T)[1+(μ f *-λ 0* 1)’  f * -1 (μ f *-λ 0 1)][(1-B c *1)’  uc * -1 (1-B c *1)] -1 We can recover the factor risk premiums λ from λ* = μ f *-λ 0 *1 which has asymptotic variance (1/T)  f * + var( λ 0 *)11’

Simple case Suppose there is a riskfree asset with rate r f, and there is only one factor r M (the Sharpe-Lintner CAPM)