1. Chapter 5 Multifactor Pricing Models 5.1Multifactor Pricing Models Motivation: Empirical evidences show that the market β does not completely explain the cross section risk; More factors are needed to explain the cross-sectional risk. Commonly-used factors: — proxies of market portfolio; 149

2. 150 — Firm characteristics: “size e ff ect”, “PE e ff ect”, “book-market ratios”: *** size e ff ect: di ff erence of returns on a portfolio with small capi- talization and high capitalization; *** PE e ff ect: di ff erence of returns on a portfolio with small PE and high PE; — Macroeconomic and market variables: “maturity premium” (yield spread between long and short rates), expected inflation, unexpected inflation, industrial production growth, “default pre- mium” (yield spread between high and low grade bonds). Notation:

3. 151 — R = (R 1, · · ·, R N ) T is a vector of returns of N traded assets. — f is the measurements (returns) related to K common factors. The Models: The cross-sectional risk is modelled as R = a + Bf + ε, E(ε) = 0,Var(ε) = Σ. For example, the return of the ith portfolio is R it = a i + b i1 f 1t + · · · + b iK f Kt + ε it, t = 1, · · ·, T. Here, the factor sensitivity parameters (loadings) b ij are an extension of the market beta to the multi-factor model. Statistical point of view: With more factors, the market portfo-

4. 152 lio can be better approximated and pricing errors can be reduced. Constraints: Financial economics theory puts theoretical constraints on the intercepts a i. Here are some backgrounds. Theoretical Background: — derived by Ross (1976) using the Arbitrage Pricing Theory (APT). In the absence of arbitrage in large economies, Ross (1976) showed that µ = ER ≈ γ 0 1 + Bλ K. * γ 0 is the model zero-beta parameter, equals to riskfree return if such an asset exists. * λ K is a vector of factor risk premia, e.g. λ K = E(f − γ 0 1) if tradable. — derived by the Intertemporal CAPM (ICAPM) of Merton (1973), which shows µ = γ 0 1 + Bλ K. — a generalization of CAPM.

5. 153 Question: — How to select factors? — How to empirically test the above theory? — How to use the multiple factor model to forecast the expected return and volatility of a firm?

6. 154 5.2Estimation and Testing Situations: We will mainly focus on the following 3 settings: (a) Factors are portfolios of traded assets and γ 0 is known. (b) Factors are portfolios of traded assets and γ 0 is unknown. (c) Factors are not portfolios of traded assets (macroeconomic var). Hypothesis: H 0 : µ = γ 0 1 + Bλ K, namely the intercepts a = γ 0 1 + B(Ef − γ 0 1). (5.1) MLR test: As in CAPM, the MLR test can be shown to have form a

7. 155 — d = # of restrictions imposed by the null hypothesis. — the statistic has been scaled by (T − N/2 − K − 1) instead of T in an hope to improve the finite sample performance; — di ff erent situation gives di ff erent statistical models, which result in di ff erent d, Σ and Σ 0. 5.2.1Portfolios as factors with a riskfree asset Excess Returns: Y t = R t − r f,t 1, X t = f t − r f,t 1,

8. (Y t − Y¯ )(X t − X) T (X t − X)(X t − X) T a = Y¯ − BX, 156 where r f,t = risk-free rate at time t. Models: Y t = a + BX t + ε t. The risk premium is λ K = E(f − γ 0 1). Hence, under the APT and ICAPM, a = 0 by (5.1). Hypothesis: H 0 : a = 0. Note that Cov(Y t, X t ) = BCov(X t, X t ) = ⇒ B = Cov(Y t, X t )var(X t ) −1. and a = EY t − BEX t. Substituting them by the empirical moments, we have MLE: B = T t=1 ¯ T ¯ −1, ¯ ε t = Y t − a − BX t, Σ = T −1 T t=1 ε t ε Tt.

9. t X t ][ 157 Remark: The coe ffi cients a i and (b i1,... b ik ) of the ith row of B are just the regression coe ffi cients obtained from the marginal models: Fit the multiple linear model for each given asset: Y it = a i + b i1 X 1t + · · · + b iK X Kt + ε it, t = 1, · · ·, T. This yields the residual vector ε t = (ε 1t,..., ε N t ) at each given time period t, which captures the cross-section risk. MLE under H 0 : a = 0: B 0 = [ T t=1 Y T T X t X Tt ] −1, Σ 0 = T −1 T t=1 t ε ot ε oT,ε ot = Y t − BX t.

10. ¯ T −1 [1 + X Ω K X] −1 a T Σ −1 a ∼ 0 F N,T −N −K. 158 Remark: The coe ffi cients in B 0 and residuals in ε ot can be obtained via the marginal model: For each given i, fit without the intercept term of the following model: Y it = b i1 X 1t + · · · + b ik X Kt + ε it. a Exact test: Let Ω K = Var(X t ) be the covariance structure of K factors and Ω K be its sample covariance. Then, the MLR test is equivalent to the Wald test T 1 = T − N − K N H This null distribution is exact rather than the approximate one if ε ∼ N (0, Σ). This is an extension of the single-factor model.

11. 159 Example 1. Fama and French (1993) consider the following factors (K = 2, 3 and 5 factors): 1. di ff erence of returns between large and small capitalization; 2. di ff erence of returns between high and low book-to-market ratios; 3. CRSP value-weighted stock index; 4. a term structure factor (yield spread between long and short bonds); 5. a default risk (yield spread high and low grade bonds) They use 25 stock portfolios and 7 bond portfolios, namely N = 32. The stock portfolios are created using a two way sort based on the market capitalization and book-to-equity ratio. The bond portfolios

12. 160 include five US government and two corporate bound portfolios. The period of study is 1963/07–1991/12, namely T = 354. The P-values are summarized as follows based on the test statistic T 1. Table 5.1: Summary of testing results using the exact F-test K235 P-value 0.010 0.039 0.025 Fama and Fama find — some improvement going from two factors to five factors; — three factors are necessary when testing portfolio consisting only of stocks; — five factors when bond portfolio is included.

13. 161 Remarks: ∗ 1. In the multi-factor model Y t = a + BY Kt + ε t, when a linear combination of Y Kt forms the tangency portfolio, a = 0. Thus, multi-factor model is an e ff ort to expand the model so that the tangency portfolio is better approximated. 2. When K is too large and T is too small, it is easy to overfit the model. Often, K ≈ 5. 3. For a subset of traded assets, one can also define the tangency portfolio within this subset. The intercepts of regressing the returns of this subset of asset are 0. Thus, if the tested assets have high weights in one of factors, its intercept will be zero (artifact of factor selection). 4. The Sharpe ratio increases as the subset of trade assets increase. 5.2.2Portfolios as factors without a riskfree asset.

14. 162 This is an extension of the Black version of CAPM. Model: R t = a + BR Kt + ε. — R t is the return of N assets —R Kt is the returns of K factors, which are a traded portfolio. MLE: The same as the last section, as the models are identical. The marginal model technique continues to apply. Null hypothesis: ER t − 1γ 0 = B(ER Kt − 1γ 0 ). Constrained Model: Under the null hypothesis, R t = 1γ 0 + B(R Kt − 1γ 0 ) + ε t = (1 − B1)γ 0 + BR Kt + ε t. Constrained MLE: Given γ 0, the situation is the same as 5.2.1

15. T Σ −1 (1−B 0 1) −1 (1−B 0 1) T Σ −1 (R−B 0 R K ), 163 and B 0 and Σ 0 can be easily be obtained. Now, given the estimated B 0 and Σ 0, the coe ffi cient γ 0 is determined by γ 0 = (1−B 0 1) 0 ¯ Algorithm: Starting from unconstrained model, iterate the above equations once, and obtain a one-step estimate. This is as good as the fully iterative procedure, since B and Σ are root-T consistent. 5.2.3 Macroeconomic variables as factors ∗ Example: innovations in GNP, changes in bond yields, unanticipated inflation. These factors are observable, but not traded. So the risk premia is unknown. Model: R t = a + Bf Kt + ε t.

16. 164 Since the risk premia λ K are not the same as Ef K −γ 0 1, they are treated as unknown parameters. From APT and ICAMP, we have under the null hypothesis µ = a + BEf K = γ 0 1 + Bλ K. Constrained Model: Letting γ 1 = λ K − Ef K, we have a = γ 0 1 + Bγ 1. Comparison: K extra parameters under H 0. The same method as that in 4.2.2 can be used to obtain the MLE under H 0. MLE under the full model: The same as before, since f Kt is observable MLR: The maximum likelihood ratio continues to apply.

17. λ k = X = T −1 165 5.3Estimation of risk premia and expected returns ¯ Expected returns: µ = γ 0 1 + Bλ K. The risk premia for three situations can be estimated as follows. The estimated covariance matrix is also attached. Trade portfolios as factors with riskless rate r f : T Xt,Xt, Var(λ K ) = T −1 Ω K. λ K = T −1 t=1 Trade portfolios as factors without r f : T t=1 R Kt − γ 0 1, Var(λ K ) = T −1 Ω K + Var(γ 0 )11 T.

18. 166 λ K = T −1 Macroeconomic var. as factors ∗ : Since λ K = Ef K + γ 1, T t=1 f Kt + γ 1. 5.4Applications of multifactor models Two important applications: Forecasting the return of a firm and to estimate the covariance matrix of assets. For simplicity, we consider the model in §5.2.1. Forecast return: For any asset, its expected return is r¯ f + β 1 (¯r F,1 − r¯ f ) + · · · + β K (¯r F,K − r¯ f ), where r¯ f is the average riskfree rate, r¯ F,i is the average return of the

19. 167 factor, and β i is the associated beta with the i-th factor. Covariance matrix: Covariance matrix is important for asset allo- cation and portifolio and risk management. However, large covariance matrices are hard to be estimated with good accuracy. From the mul- tifactor model, var(Y t ) = var(R t ) = Bvar(X t )B T + var(ε). Idea: If we assume that the factors capture the cross-section risk, we may assume that var(ε) is a diagnonal matrix. Note that var(X t ) is a low-dimensional matrix (e.g. 3 × 3 if we use a three factor model). Estimation. Based on the multiple regression, we can obtain an estimate of B and residual variance and hence var(ε). The low-

20. 168 dimensional matrix var(X t ) can be estimated by the sample covari- ance matrix. 5.5Selection of Factors Two approaches: — Economics Theory: (a) Characteristics of Firms: “Size e ff ect”, “ PE”, “book-market ratio”, “ PS”. (b) macroeconomic and financial market variables: maturity pre- mium, expected unexpected inflation, industrial production growth, default premium.

21. = B B + Σ, B = Ω K B. 169 — Statistical Methods: Motivated by the APT, build factors from a comprehensive set of asset returns (much larger than the test sets), e.g. factor analysis, principal analysis 5.5.1 Factor analysis ∗ Factor analysis: Assume that K factors account for all covariance of asset returns and hence Σ is diagonal — strict factor structure, imposed also by Ross in his development of APT. Model: R t = a + Bf t + ε t, Var(ε t |f t ) = Σ. ⇒ Var(R t ) = B T Ω K B + Σ, ∗ T ∗ Ω K = Var(f t ), 1 ∗ 2 Identifiability: B can be identified only up to a K × K nonsingular matrix. B ∗ is unique up to an orthogonal transform. MLE: If ε t is normal, then B ∗ and Σ can be estimated from {R t, t = 1,..., T }.

22. we have a = ER t and a = R. By generalized least-squares, ¯ f t = (B Σ −1 B) −1 B Σ −1 (R t − R) ≡ W (R t − R). 170 Estimation of factors: WOLG, Ef t = 0. From R t = a + Bf t + ε t, ¯ T ¯ Renormalization: (optional) Factor j is taken as f jt = n i=1 w ji R it / n i=1 w ji. This di ff ers from the above f only by a linear scale. These factors can be used directly to fit a multi-factor model. Interpretations: The weights w j is the same as solving w j s.t.w Tj b k = 0, ∀ k = j, w Tj b j = 1, i.e. minimizing the residual variance, subject to orthogonality constraint.

23. 171 Proof: Let e j be the K-dimensional unit vector with one at the jth position. Then, the side condition can be written as T B w j = e j. By the Lagrange multiplier method, we minimize T Taking derivative and setting it to zero Σw j − Bλ = 0 or w j = Σ −1 Bλ. Now, using the constraint T The converse follow obvious from the above derivation: T B w j = e j.

24. 172 5.5.2 5.6 Principal component analysis ∗ Empirical results Lehmann and Modest (1988): (Statistical method) — daily data to build factors using factor analysis — weekly interval for testing Connor and Korajczyk (1988): — use principal component on the sample covariance of R t to obtain factors — account for January e ff ects (intercept of January is 0) and non Jan.

25. 173 Test intervals: — CK works with monthly data — CK and LM are based on tests from four five-year periods, aggre- gated together. Results: CK and LM find little sensitivity to increasing the number of factors beyond five.

26. 174 Table 5.2: Summary of results for tests of exact factor pricing using zero-intercept F-test Time period 64:01-83:12 63:01-82:12 Portfolio characteristic market value of equity dividend yield own variance Study CK LM N 10 5 20 5 20 5 20 K 5 10 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 p-value 0.002 ** 0.11 0.14 0.42 0.17 0.18 0.17 0.94 0.97 0.98 0.29 0.57 0.55 0.83 0.97 0.98 ** Less than 0.001.