ASSET PRICING MODELS
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1 ASSE PRICING MODELS [1] CAPM (1) Some notation: R it = (gross) return on asset i at time t. R mt = (gross) return on the market portfolio at time t. R ft = return on risk-free asset at time t. X it = R it R ft = excess return on asset i. X mt = R mt R ft = excess return on the market portfolio. R1 t X1 t 1 R 2t X 2t 1 Rt = ; Xt = ; en = : : : R X 1 Nt N 1 Nt N For simplicity, we assume that (R t,r ft,r mt ) is iid over time. 1 (2) Sharpe-Lintner version of CAPM Lintner, 1965, Review of Economics and Statistics Sharpe, 1964, Journal of Finance Campbell, Lo and Mackinlay (CLM), 1997, book, chapter 5. ASSE PRICING-1
2 1) Basic idea: var( R mt ) = risk from the market portfolio of risky asset. risk price = p. cost of bearing the market risk = pvar( R mt ). At equilibrium, cost of risk = expected gain from risk. pvar( R ) = E( R ) R mt mt ft E( Rmt ) Rft p =. var( R ) Systematic risk of an individual asset i: mt he risk of asset i due to correlation between returns on asset i and the whole risky-asset market cov( R, R ). it mt Let β i be the systematic risk of an individual asset i relative to the market risk: cov( Rit, Rmt ) β i =. var( R ) mt his β i can be estimated by the time-series OLS on: R it = α i + β i R mt + ε it. Cost of bearing the (systematic) risk of asset i: β var( R ) p = β [ E( R ) R ]. i mt i mt ft ASSE PRICING-2
3 Equilibrium condition: ER ( ) β [ ER ( ) R] = R, for all i = 1,..., N it i mt ft ft ER ( ) = R + β [ ER ( ) R] or E( X ) = β E( X ). it ft i mt ft [Capital Asset Pricing Model] it i mt 2) Empirical Model Model: X 1t = α 1 + β 1 X mt + ε 1t ; X 2t = α 2 + β 2 X mt + ε 2t ; : X Nt = α N + β N X mt + ε 3t. X t = α + βx mt + ε t, where α = ( α1,..., α N ), β = ( β1,..., β N ) and εt = ( ε1 t,..., ε Nt). Assume that the ε t are iid over time with Cov( εt) = Σ= [ σ ii] N N. Comments. No heteroskedasticity over time. Reasonable if ε t is normal. If ε t is t-distributed, heteroskedasticity should exist. [MacKinlay and Richardson (1991, JF).] ASSE PRICING-3
4 Equilibrium condition: ER ( ) β [ ER ( ) R] = R it i mt ft ft, for all i = 1,..., N ER ( ) = R + β [ ER ( ) R] or E( X ) = β E( X ). it ft i mt ft [Capital Asset Pricing Model] it i mt 2) Empirical Model Model: X 1t = α 1 + β 1 X mt + ε 1t ; X 2t = α 2 + β 2 X mt + ε 2t ; : X Nt = α N + β N X mt + ε 3t. X t = α + βx mt + ε t, where α = ( α1,..., α ), β = ( β1,..., β ) and ε = ( ε1,..., ε ). N N t t Nt Assume that the ε t are iid over time with Cov( ε t) = Σ= [ σ ii] N N. Comments. No heteroskedasticity over time. Reasonable if ε t is normal. If ε t is t-distributed, heteroskedasticity should exist. [MacKinlay and Richardson (1991, JF).] ASSE PRICING-4
5 Estimation: SUR model with the same regressor: GLS = OLS. Use MLE or OLS. CAPM implies H o : α 1 =... = α N = 0. Can test these restrictions by Wald or LR. [See Ch. 4-5 of CLM.] If N is too large, the test result would be unreliable. [See Ahn and Gadarowski, 2004] wo-pass Regression Method (Fama- MacBeth, 1973, JPE) Suppose that β i s are known. hen, we can consider the following cross-sectional regression model for each t: (*) X = e γ 1 + βγ 2 + error, t N t t where β = ( β1,..., β ). N If the CAPM is correct, it should be the case that E(γ 1t ) = 0 and E(γ 2t ) 0. ASSE PRICING-5
6 Estimation procedure: SEP 1: SEP 2: For each i, do time-series OLS to estimate β i ( ˆi β ). For each t, do cross-section OLS to estimate γ 1t and γ 2t ( ˆ γ 1t and ˆ γ 2 t ). SEP 3: Compute: ˆ γ 1 ˆ ˆ 1 ˆ γ γ ( 1) γ γ ( ) 2 1 ;var( ) ˆ j = Σ t= jt j = Σt= 1 jt j SEP 4: Do t-tests to check whether γ 1 = 0 and γ Equivalent Procedure [Shanken, 1992, RFS] SEP 1: For each i, do time-series OLS to estimate β i ( ˆi β ). Let 1 t t Bˆ = ( e, ˆ N β ), γ = ( γ1, γ 2) and X = Σ = 1X SEP 2: Do OLS on X = Bˆ γ + error : ( BB ˆˆ) 1 ˆ γ = BX ˆ ; ( ) ( ˆˆ) ˆ 1 ˆ γ = Σ ˆ 2 1( )( ) ( ˆˆ t= t t ) 1 1 Cov B B B X X X X B B B he above covariance matrix is valid only if true β i s are used. Correct form of the covariance matrix will be discussed below. ASSE PRICING-6
7 (3) Black-version of CAPM: 1) Basic Model Model when there is no risk-free asset. R omt = return on the zero-beta portfolio associated with m. [portfolio that has the minimum variance of all portfolios uncorrelated with m] Let γ = E(R omt ). Black-Version of CAPM: E(R it ) = γ + β i [E(R mt )-γ] E(R it ) = γ(1-β i ) + β i E(R mt ). 2) Empirical estimation and testing. Empirical Model: R 1t = α 1 + β 1 R mt + ε 1t ; R 2t = α 2 + β 2 R mt + ε 2t ; : R mt = α n + β n R mt + ε 3t. R t = α + βr mt + ε t, Black-version of CAPM implies H o : (e N -β)γ = α. See CLM for how to test this hypothesis. ASSE PRICING-7
8 (3) When returns are heteroskedastic or autocorrelated over time. Estimate the parameters by GMM. Moment conditions: 1 E ( Xit αi βixmt = 0, i = 1,..., N X ) mt 1 E ( Xt α βxmt) = 0. X mt ASSE PRICING-8
9 [2] Multifactor Pricing Model (1) Arbitrage Pricing Model [Ross, JE, 1976] Assumption 1: R it = α i + β i1 f 1t + β i2 f 2t β ik f kt + ε it, where f 1t,... f kt are macroeconomic or portfolio factors. R t = α + β 1 f 1t β k f kt + ε t = α + Bf t + ε t, where B = (β 1,..., β k ) and f t = (f 1t,..., f kt ). Cov(ε t ) = Σ N N (ε it are cross-sectionally correlated). If there is no missing factor, Σ should be diagonal. R t and f t are covariance-stationary and ergodic. he factors in f t are strictly exogeous: E(f s ε it ) = 0 for all i = 1,..., N, and all t and s. E( f ε ) = 0. s t he beta matrix B is of full column. How could we test for rank(b)? What happens if B is not full column? [See below.] Assumption 2 (No Autocorrelation): Assumption 1 plus εε t s,..., = 0N N. E( f1 f ) ASSE PRICING-9
10 Assumption 3 (No Heteroskedasticity): Assumption 2 plus t 1 Cov( ε f,..., f ) =Σ, for all t. Assumption 4 (Autocorrelation in f t ): Let Σ ˆ F be the Newey-West estimator of 1 lim Var Σt= 1( ft E( ft))( ft E( ft )) Assumption 5 (No Autocorrelation in f t ): ˆ 1 Σ F = Σt= 1( ft f )( ft f ) 1 p lim Var Σ t= 1( ft E( ft))( ft E( ft)) Ross (1976) shows that the absence of arbitrage implies: γ 0 Ho: ER ( t) = enγ 0 + Bγ1 = ( en, B) Bcγ γ. Or equivalently, 1 α o : N o 1 H α = e λ + Bλ = B λ, where λ 1 = γ 1 E(f t ). c ASSE PRICING-10
11 (2) Estimation and esting [Ahn and Gadarowski, 2004] Let Z = (1, f ). t t 1 Let Ξ= lim Var Σt= 1Zt ε t. Let ˆΞ 1 be the Newey-West estimator of Ξ using OLS residuals instead of ε t. hen, it is consistent under Assumption 1. Under Assumption 2, ( ZZ εε ) ˆ 1 Ξ2 Σt= 1 t t t t p Ξ. Under Assumption 3, ˆ 1 1 Ξ ˆ ˆ 3 Σt 1ZZ = t t Σt= 1εε t t ZZ Σ p Ξ. ASSE PRICING-11
12 wo-pass Estimation of lambdas: Let Λ= ˆ ( ˆ α, B ˆ ) be the OLS estimator of α and B. Let A be any N N positive definite matrix. hen, a two-pass estimator of γ is given: ˆ γ ( B AB ) 1 = ˆ ˆ Bˆ Aˆ α. P c c c If A = I N, the P estimator is the Fama-MacBeth estimator. Cov( ˆ γ ) ˆ ˆ ˆ ( ) ˆ ˆ ˆ ˆ P B c ABc B c A ABc ( B c ABc ) where 1 1 = Ω, ˆ ˆ ˆ ˆ ˆ 1 1 Ω= ( λ* ZZ I N) Ξ( ZZλ* I N), * is any consistent estimator of λ 1. Asymptotically optimal choice of A = ( Ω ˆ ) 1. ˆ ˆ λ = (1, ˆ λ 1 ), and ˆ λ 1 λ be the optimal P estimator using ( ˆ ) 1 Let ˆOMD Ω. Estimation of gammas: ˆ λ 1 0, ˆ ( ˆ ) ˆ ˆ P γp = B c ABc B c AR = λp + Jf =, ˆ λ1, P + f where J 01 k = I. k ASSE PRICING-12
13 Under Assumption 4, ˆ 1 Cov( ˆ γ ) ( ) ˆ P = Cov λ P + JΣ J. F Under Assumption 5, ˆ 1 Cov( ˆ γ ) ( ) ˆ P = Cov λ P + JΣ F J. Model Specification test: 2 N + 1 ˆ 1 ˆ ˆ ( Q ( ˆ OMD Bc OMD Bc OMD) N k χ 1 ) α λ Ω α λ. N 1 k N 1 k ( ˆ ˆ ˆ ) Alternative test: Assume: α = e λ + Bλ + Sλ, N where S contains firsm specific variables such as firm sizes or book values. H o and H α o imply λ 2 = 0. For detailed test procedures, see Ahn and Gadarowski. ASSE PRICING-13
14 Empirical Suggestions from Ahn and Gadarowski Need to check persistency of factors. When factors follow unit root or near-unit-root processes, the P estimators are unreliable. Do not use too many assets (N). 25 or fewer would be appropriate. esting autocorrelation in time-series OLS residuals and factors are important. Heteroskedasticity-robust Q tests are more reliable than autocorrelation-robust Q tests. he Q tests generally have low power. he t-tests based on nonoptimal P estimators are more reliable than those based on the optimal P estimators. (3) What happens if the beta matrix B is not of full column? When? Some factors are in fact not the determinants of returns. Kan and Zhang (1999, JF) call such factors useless factors. he columns of B corresponding to useless factors are zero vectors. rank(b) < k. ASSE PRICING-14
15 Consequences? he P estimator is severely biased. he price of a useless factor would appear to be significant. How to find out correct factors? Conner and Korajczyk (1993, JF) What happens if some unimportant factors are used? he P estimators are severely biased. [Kan and Zhang, 1999, JF] Important to test how many factors to be used. How many factors to be used? When candidate factors are all observed: Connor and Korajczyk (1993, JF). Jagannathan and Wang (1996, JF). ASSE PRICING-15
16 [3] Estimating the Number of Factors Question: Suppose we do not observe factors. Wish to estimate factors. How may factors? Methods: MLE assuming factors are iid standard normal over time: Large and Small N. [See Campbell, Ch. 6.4]. Jones (2001, JFE): Large N and small. Bai (2003, ECON), and Bai and Ng: Both N and are large. Chi, Nardari and Shephard (2002, JEC): N small and large. ASSE PRICING-16
17 [4] Stochastic Discount Factor Model here are many asset pricing models (CAPM, AP, consumptionbased CAPM, intertemporal equilibrium model): See Ch. 8 of CLM. A particular asset pricing model typically implies: E[R t m t (f t δ)] = e N, where m(f t δ) is a scalar function of factors and a parameter vector δ, e N is the normalized price vector, and m(f t δ) is called stochastic discount factor (SDF). For linear factor models, m(f t δ) = δ 1 + f t δ 2, where δ = (δ 1,δ 2 ). he parameter vector δ and model specification can be tested by GMM. (1) Jagannathan and Wang (1996, JF): Let w t (δ) = R t m t (δ) e N ; then, E[w t (δ)] is the pricing error. If no pricing error, then, E[w t (δ)] = 0 N 1. HJ-distance (Hansen and Jagannathan, 1997): 1 HJ ( δ ) = E[ wt( δ)] G E( w t ( δ )), where G = E(R t R t ). Correct model has HJ(δ) = 0. For a given δ, this measure equals the maximum pricing error generated by a given asset pricing model. ASSE PRICING-17
18 Standard GMM test for no pricing error: 1 Let S be a consistent estimator of lim Cov Σt= 1wt( δ ). [by White (1980) or Newey and West (1987).] Let ˆ δ GMM be the optimal GMM estimator. hen, under the hypothesis of no pricing error, J = w ˆ S w ˆ N p, 1 2 ( δ ) GMM ( δgmm) d χ ( ) 1 where w ( δ ) = Σ t = 1w t ( δ ), and p is the # of parameters in δ. Jagannathan-Wang test: 1 HJ ( δ ) = w ( δ) G w ( δ). Let ˆHJ δ is the minimizer of HJ (δ). Under the hypothesis of no pricing error, SHJ [ HJ ( ˆ δ )] 2 = HJ weighted χ JW (1996) provides a simulation method to compute p-value for this statistic. JW conjecture that this HJ test would have better finite sample properties, because non-optimal GMM often has better finite sample properties than optimal GMM. 2. ASSE PRICING-18
19 Ahn and Gadarowski (2004, Journal of Empirical Finance) For linear factor models, the JW method has extremely poor finite sample properties, poorer than optimal GMM, especially when N is large. Better to use optimal GMM! ASSE PRICING-19
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