4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required. Ross (1976) introduced the Arbitrage Pricing Theory (APT) as an alternative to the CAPM. The basic assumption is that there are a number of, say K, common risk factors generating the returns so that with R i = a i + b i f + i, E[ i f] =0 E[ 2 i ]=σ2 i σ2 <, Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13, 341{ 360. 58
and E[ i j ]=0, whenever i = j, where i =1,...,N, the number of assets, a i is the intercept term of the factor model, b i is a K 1 coe±cient vector of factor sensitivities (loadings) for asset i, f is a K 1 vector of common factors, and i is the disturbance term. Without loss of generality we may assume that the common factors have zero mean, i.e., E[f] =0, which implies that a i =E[R i ]= µ i are the mean returns. In the matrix form the return generating model is (17) R = µ + Bf + and E[ f] =0 E[ f] =, 59
where R =(R 1,R 2,...,R N ), µ =(µ 1,...,µ N ), B =(b 1,...,b N ) is an N K factor loading matrix, =( 1,..., N ),and is a N N matrix (assumed diagonal in the original Ross model). Furthermore, it is assumed that K N. The derivation of the APM relies on the no arbitrage assumption. Let w =(w 1,...,w N )' be an arbitrage strategy. Then (18) w ι = N i=1 w i =0, and the implied portfolio should be riskfree (or more precisely its starting value should be zero, non-negative in the meantime with probability one, and have a strictly positive expected end value). The return of the portfolio is R p = w R = w a + w Bf + w. 60
In order to make the portfolio riskfree, the market risk w Bf and unsystematic risk w must be eliminated. The unsystematic risk can be eliminated by letting N be large, so that Var(w ) = w w = N i=1 wi 2σ2 i. The weights w i are of order 1/N,so N i=1 wi 2σ2 i 0 as N. Next in order to eliminate the market risk the weights must be selected such that (19) w B = 0 In the language of linear algebra the columns of the expanded matrix ~B =(ι, B) spans a K + 1-dimensional linear subspace, call it V, in IR N. Because N>K+1 all vectors lying in the orthogonal complement, V,ofV are valid candidates for w to satisfy conditions (18) and (19). More precisely V = {y IR N : y = ~Bx, x IR K+1 } and V = {z IR N : z y =0 y V }. Note further that IR N = V V,actuallyIR N = V V, the direct sum of V and V. 61
Given an arbitrage strategy w that satis es (19), we get with large N approximately (because w 0) (20) R p = N i=1 which is a riskfree return. w i µ i = w µ The absence of arbitrage implies that any arbitrage portfolio must have a zero return. In other words (21) R p = w µ =0, which implies that the expected return vector µ is orthogonal to w. But then it is a vector in the linear space V and hence of the form (22) µ = ~Bλ, where λ =(λ 0, λ 1,...,λ K ) IR K+1. In other words the expected returns of the single assets are of the form (23) µ i = λ 0 + b i1 λ 1 +...+ b ik λ K 62
If there is a riskfree asset with return R f = µ 0, it has by de nition, zero exposure on the common market risk factors. That is b 0j =0, j =1,...,K. Then from (23) with b 0j =0, we get (24) R f = λ 0, and we can rewrite (23) as (25) µ i = R f + λ 1 b i1 + + λ K b ik. This is the APT equilibrium model of the expected asset returns. Because b ij is a sensitivity to the jth common risk factors it is natural to interpret that λ j represents the risk premium (the price of risk) for factor j in the equilibrium. Note. Generally if there is no riskfree return λ 0 can be interpreted the zero-beta return. In the matrix form the APM for the expected equilibrium returns is (26) E[R t ]=µ = ιλ 0 + Bλ where λ 0 = R f is the riskfree return if it exists, and λ =(λ 1,...,λ K ). 63
Estimation and Testing of APT Assumption: Returns are normally and temporally independently distributed. APT does neither specify the factors nor the number of factors. We consider four versions: Factors are (1) portfolios of traded assets and a riskfree asset exists; (2) portfolios of traded assets and a riskfree asset does not exist; (3) not portfolios of traded assets; (4) portfolios of traded assets and the factor portfolios span the mean-variance frontier of risky assets. The derivation of the test statistics is analogous to the CAPM case. Relying on normality the LR test statistic is of the form (27) J = T N 2 K 1 log ^ log ^, where ^ and ^ are the unconstrained and constrained ML-estimators, respectively. Again as before the asymptotic null distribution of J is chi-square with degrees of freedom equal to the number of restrictions imposed by the null hypothesis. 64
(1) Portfolios as Factors with a Riskfree Asset Denote the unconstrained form of the factor model (17) in this case as (28) Z t = a + BZ Kt + t with E[ t ]=0, E[ t t]=, E[Z Kt ]=µ K, E[Z Kt µ K )(Z Kt µ K ) ]= K and Cov[Z Kt, t ]=0, where B is the N K matrix of factor sensitivities, Z Kt is the K 1 vector of factor portfolio excess returns, and a and t are N 1 vectors of intercepts and error terms, respectively. The APM implies that a = 0. In order to test this with the LR test wee need to estimate the unconstrained and constrained model. 65
Model (28) is a seemingly unrelated regression (SUR) case, but because each regression equation has the same explanatory variables the ML-estimators are just the OLS estimators ^a =^µ ^B^µ K, ^B = T (Z t ˆµ)(Z Kt ˆµ K ) T 1 (Z Kt ^µ K )(Z Kt ^µ K ), ^ = 1 T T (Z t ^a ^BZ Kt )(Z t ^a ^BZ Kt ), T T ^µ = 1 Z t, ^µ K = 1 T T The constrained estimators are (29) ^B = and T Z t Z Kt T Z Kt Z Kt Z Kt. 1 ^ = 1 T T (Z t ^B Z Kt )(Z t ^B Z Kt ). 66
The exact multivariate F -test becomes then (30) where (31) J 1 = T N K N ^ K = 1 T 1+^µ K ^ 1 K ^µ K 1 ^a ^ 1^a, T (Z K ^µ K )(Z K ^µ K ). Under the null hypothesis J 1 F (N, T N K). (2) Portfolios as Factors without a Riskfree Asset Let R Kt =(R 1t,...,R Kt ) be portfolios that are factors of the APT model, and denote the related factor model (17) as (32) R t = a + BR Kt + t. If there does not exist a riskfree asset, then as in the CAPM there exist a portfolio which is uncorrelated with the portfolios in R Kt. Let γ 0 denote the expected return of this portfolio. 67
Then in the APM λ 0 = γ 0 and the constrained factor model is R t = ιγ 0 + B(R Kt ιγ 0 )+ t so that = (ι Bι)γ 0 + BR Kt + t, a =(ι Bι)γ 0. The constrained ML-estimators are ^B = ^ = 1 T and T T (R t ι^γ 0 )(R Kt ι^γ 0 ) (R Kt ι^γ 0 )(R Kt ι^γ 0 ) 1 T Rt ι^γ 0 ^B (R Kt ι^γ 0 ) R t ι^γ 0 ^B (R Kt ι^γ 0 ), ^γ 0 = (ι ^B ι) ^ 1 (ι ^B ι) 1 (ι ^B ι) ^ 1 (ι ^B ^µ). Again a suitable iterative procedure must be implemented in the estimation., The null hypothesis H 0 : a =(ι Bι)γ 0 can be tested again with the likelihood ratio test J of the form (27), which is asymptotically chi-squared distributed with N 1 degrees of freedom under the null hypothesis. 68
(3) Macroeconomic Variables as Factors Let f K be macroeconomic factors of the APM, with E[f Kt ]=µ fk. Then (17) becomes (33) R t = a + Bf Kt + t with parameter structure and unconstrained ML estimators similar as derived in the case of (28). To formulate the constrained parameters consider the expected value of (33) (34) µ = a + Bµ fk If the APM holds then this should be equal to(26). Soequatingthesewegetfora a = ιλ 0 + B(λ µ fk ). Denoting γ 0 = λ 0 and γ 1 = λ µ fk,ank 1 vector, the restricted model is (35) R t = ιλ 0 + Bγ 1 + Bf Kt + t. 69
Again estimating the parameters, the APM implied null hypothesis (36) H 0 : a = ιλ 0 + Bγ 1 canbetestedwiththelikelihoodratiotestof the form (27) which in this case is under the null hypothesis asymptotically chi squared distributed with degrees of freedom N K 1. (4) Factor Portfolios Spanning the Mean-Variance Frontier Consider again the factor portfolio model (32). If the factor portfolios in R Kt span the meanvariance frontier then in (26) λ 0 is zero (c.f. the zero-beta case considered earlier). This form of APM imposes the restriction (37) on (32). H 0 : a = 0 and Bι = ι We say that a set of vectors S = {x 1,...,x n } spans the linear space V,ifeachy V can be represented as a linear combination of elements of S, i.e.,ify V then y = a 1 x 1 + +a n x n for some a =(a 1,...,a m ) IR n. 70
Example. Zero-beta version of CAPM (a two-factor model). Let R mt denote the market portfolio (a MVP) and R 0t the associated zero-beta portfolio. Then K =2, B =(β 0m, β m ): (N 2), and R Kt =(R 0t,R mt ) : (2 1), sothat R t = a + β 0m R mt + β m R mt + t. As found earlier a = 0, andβ 0m + β m = ι. Again estimating the constrained and unconstrained model, we can use the LR-statistic (27) to test the null hypothesis (37). The degrees of freedom are 2N. These come from N restrictions in a = 0 and N restrictions in B to satisfy Bι = ι. Again, assuming that the multivariate normality holds, there exist an exact test J 2 = T N K ^ (38) N ^ 1, which is F [2N, 2(T N K)]-distributed under the null hypothesis. 71
Estimation of Risk Premia and Expected Returns As given in (26) the expected return of the assetsareundertheapt µ = λ 0 ι + Bλ. To make the model operational one needs to estimate the riskfree or zero beta return λ 0, factor sensitivities B, and the factor risk premia λ. The appropriate estimation procedure varies across the four cases considered above. The principle is that we use the appropriate restricted estimators in each case to estimate the parameters of (26). 72
For example, in the excess return case of (28), B is estimated by (29), and (39) ^λ =^µ K = 1 T T Z Kt, which is the average excess return of the market factors. Aninterestingquestionthenmightbeifthe factors are jointly priced. That is to test the null hypothesis (40) H 0 : λ = 0 An appropriate test procedure is the multivariate mean test (known as Hotelling T 2 test) (41) where (42) J 3 = T K TK Var[^λ] = 1 T ^ K = 1 T 2 ^λ Var[^λ] 1 ^λ, T (Z Kt ^µ Kt )(Z Kt ^µ Kt ). Asymptotically J 3 F (K, T K) under the null hypothesis (40). 73
Signi cance of individual factors, i.e., null hypotheses (43) H 0 : λ j =0 canbetestedwiththet-test (44) J 4 = ^λ j vjj, which is asymptotically N(0, 1)-distributed under the null hypothesis, and where v jj is the jth diagonal element of Var[^λ], j =1,...,K. Note. Test of individual factors is sensible only if the factors are theoretically speci ed. If they are empirically speci ed they do not have clear-cut economic interpretations. Note. Another way to estimate factor risk premia is to use a two-pass cross-sectional regression approach. In the rst pass the factor sensitivities (B) are estimated and in the second pass the premia parameters of the regression (45) Z t = λ 0t ι + ^Bλ + η t can be estimated time-period-by-time-period. Here ^B is identical to the unconstrained estimator of B. 74
Selection of Factors The factors of APM must be speci ed. This can be on statistical or theoretical basis. Statistical Approaches Linear factor model as given in (17), in general form is (46) R t = a + Bf t + t with (47) E[ t t f t ]=. In order to nd the factors there are two primary statistical approaches factor analysis and principal component analysis 75
Factor Analysis Using statistical factor analysis the assumption is that there are K common latent (not directly observable) common factors that affect the stock prices. Especially it is assumed thatthecommonfactorscapturethecrosssectional covariances between the asset returns. The factor model is of the form (46), which with the above additional assumption implies the following structure to the covariance matrix of the returns (48) = B B + D where =Cov(f t )isak K covariance matrix of the common factors, and D =Cov( ) is an N N diagonal matrix of the residuals (unique factors). 76
All the parameter matrices on the right hand side of the decomposition (48) (altogether (NK + K(K +1)/2 + N parameters) are unknown. Furthermore the decomposition is not unique, because (given that is positive de nite) we can always write =GG so that rede ning B as BG (49) = BB + D. Even in this case B is not generally unique, because again de ning C = BT where T is an arbitrary K K matrix such that TT = I, we get = CC + D. Matrix T is called a rotation matrix. The APT does not give the number of common factors K. Thus the rst task is to determine the number of factors. Empirically this can be done with various statistical criteria using factor analysis packages. 77
A popular method is to select as many factors as there are eigenvectors larger than one computed from the return correlation matrix. Modern statistical packages provide also explicit statistical tests as well as various criterion functions for the purpose. An example of FA approach is in Lehmann, B. and D. Modest (1988). The empirical foundations of the Arbitrage Pricing Theory, Journal of Financial Economics, 21, 213{254. 78
Principal Component Principal component analysis is another tool for deriving common factors. This however is a more technical approach. Furthermore usually di erent components are obtained from correlation matrix than from covariance matrix. Nevertheless, there is no clear-cut research results which one, PCA or FA, should be a better choice in APT analysis. An example of PCA application is in Connor, G. and R. Korajczyk (1988). Risk an return in an equilibrium APT: Application of a new test methodology.journal of Financial Economics, 21, 255{290. 79
Theoretical Approaches One approach is to macroeconomic and - nancial market variables that are thought to capture systematic risk of the economy. Chen, Roll and Ross (1986) Journal of Business, 31, 1067{1083, use ve factors: (1) long and short government bond yield spread (maturity premium), expected in ation, (2) unexpected in ation, (3) industrial production growth, yield spread between high- and low-grade bonds (default premium), (4) aggregate consumption growth and (5) oil prices. Another approach is to specify di erent characteristics of rms and form a portfolio of these. For example: market value of equity, PE-ratio and book value to market value. 80
Some Empirical Results The evidence supporting the exact factor pricing, i.e., model (26), are mixed. Especially di±culties are to explain the "size" e ect and the "book to market" e ect. Nevertheless APM seem to provide an attractive alternative to the single-factor CAPM. 81