Financial Econometrics Short Course Lecture 2 Portfolio Choice and CAPM

Transcription

1 Financial Econometrics Short Course Lecture 2 Portfolio Choice and CAPM Oliver Linton obl20@cam.ac.uk Renmin University Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 1 / 64

2 We have cross section of risky assets with return R i, i = 1,..., N, and market return R m, which is a portfolio of risky assets. We may have a risk free asset with return R f. We suppose that ER i = µ i var(r i ) = σ ii cov(r i, R j ) = σ ij Portfolio choice. Choose weights w i such that N i=1 w i = 1 to find the mean variance effi cient frontier (the achievable risk/return tradeoff), either Minimize portfolio variance for given portfolio mean, or Maximize portfolio mean for given portfolio variance. CAPM leads to restrictions on the risk/return tradeoff taking account of the market portfolio Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 2 / 64

3 Let R = (R 1,..., R N ) be the N 1 vector of returns with mean vector and covariance matrix ER = µ = µ 1. E [ (R µ) (R µ) ] = Σ = µ N σ σjk... σnn. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 3 / 64

4 We consider the problem of mean variance portfolio choice in this general setting. Let R w denote the random portfolio return R w = N w j R j = w R, j=1 where w = (w 1,..., w N ) are weights with N j=1 w j = 1. The mean and variance of the portfolio are µ w = w µ = N w j µ j ; σ 2 w = w Σw = j=1 N j=1 N k=1 w j w k σ jk. There is a trade-off between mean and variance, meaning that as we increase the portfolio mean, which is good, we end up increasing its variance, which is bad. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 4 / 64

5 To balance these two effects we choose a portfolio that minimizes the variance of the portfolio subject to the mean being a certain level. Assume that the matrix Σ is nonsingular, so that Σ 1 exists with Σ 1 Σ = ΣΣ 1 = I (otherwise, there exists w such that Σw = 0). We first consider the Global Minimum Variance (GMV) portfolio. Definition The Global Minimum Variance portfolio w is the solution to the following minimization problem where i = (1,..., 1). min w Σw subject to w i = 1, w R N Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 5 / 64

6 To solve this problem we form the Lagrangian, which is the objective function plus the constraint multiplied by the Lagrange multiplier λ This has first order condition L = 1 2 w Σw + λ(1 w i). L w = Σw λi = 0 = w = λσ 1 i. Then premultiplying by the vector i and using the constraint we have 1 = i w = λi Σ 1 i, so that λ = 1/i Σ 1 i and the optimal weights are w GMV = This portfolio has mean and variance Σ 1 i i Σ 1 i. µ GMV = i Σ 1 µ i Σ 1 i ; σ 2 GMV = 1 i Σ 1 i. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 6 / 64

7 The global minimum variance portfolio may sacrifice more mean return than you would like so we consider the more general problem where we ask for a minimum level m of the mean return. Definition The portfolio that minimizes variance for a given level of mean return solves min w R N w Σw subject to the constraints w i = 1 and w µ = m. The Lagrangian is L = 1 2 w Σw + λ(1 w i) + γ(m w µ). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 7 / 64

8 The first order condition is L w which yields = Σw λi γµ = 0, w opt = λσ 1 i + γσ 1 µ, where λ, µ R are the two Lagrange multipliers. Then imposing the two restrictions: 1 = i w opt = λi Σ 1 i + γi Σ 1 µ and m = µ w opt = λµ Σ 1 i + γµ Σ 1 µ, we obtain a system of two equations in λ, γ, which can be solved exactly to yield λ = C Bm ; γ = Am B A = i Σ 1 i, B = i Σ 1 µ, C = µ Σ 1 µ, = AC B 2, provided > 0. This portfolio has mean m and variance σ 2 opt(m) = Am2 2Bm + C, which is a quadratic function of m. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 8 / 64

9 We consider the practical problem of implementing portfolio choice. Suppose just risky assets vector R t R N, t = 1,..., T, where the population mean and covariance matix is denoted by µ, Σ. We let µ = 1 T T t=1 R t, Σ = 1 T T (R t µ) (R t µ), t=1 which are consistent estimates of the population quantities as T for fixed N. Let also  = i Σ 1 i, B = i Σ 1 µ, Ĉ = µ Σ 1 µ, = ÂĈ B 2, and ŵ opt (m) = λ Σ 1 i + γ Σ 1 µ, λ = Ĉ Bm ; γ = Âm B, which is the optimal sample weighting scheme. The corresponding estimated variance is ŵ opt (m) Σŵ opt (m). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renmin anduniversity CAPM 9 / 64

10 Oliver Linton Figure: () Financial Effi cient Econometrics Frontier Short Course of Dow LectureStocks 2 Portfolio Choice Renminand University CAPM 10 / 64

11 A necessary condition for Σ to be full rank (and hence invertible) is that T > N. In practice, there are many thousands of assets that are available for purchase. There is now an active area of research proposing new methods for estimating optimal portfolios when the number of assets is large. Essentially there should be some structure on Σ that reduces the number of unknown quantities from N(N + 1)/2 to some smaller quantity. The market model and factor models are well established ways of doing this and we will visit them shortly. Example The Ledoit and Wolf (2003) shrinkage method replaces Σ by Σ = α Σ + (1 α) D, where D is the diagonal matrix of Σ and α R is a tuning parameter. For α (0, 1) the matrix Σ is of full rank and invertible regardless of the relative sizes of N/T. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 11 / 64

12 The Capital Asset Pricing Model (Risk return tradeoff) Sharpe-Lintner version with a riskless asset (borrowing or lending) for all i. E [R i ] = R f + β i (E [R m ] R f ) = R f + π i = R f + β i π m Relates three quantities µ i = E [R i R f ] ; µ m = E [R m R f ] ; β i = cov(r i, R m ) var(r m ) all of which can be estimated from time series data Risk/return tradeoff - more risk, more return. The β i is the relevant measure of riskiness of stock i not var(r i ) Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 12 / 64

13 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 13 / 64

14 Fisher Black version without a riskless asset Find the (zero beta) portfolio return R 0 such that R 0 = arg min var(r w ) subject to cov(r w, R m ) = 0 Then for all i. E [R i ] = E [R 0 ] + β i (E [R m ] E [R 0 ]) Since R 0 is not observed, this creates some diffi culties Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 14 / 64

15 Testable versions embed within some class of alternatives. Sharpe-Lintner. Letting Z i = R i R f and Z m = R m R f E [Z i ] = α i + β i E [Z m ] and test H 0 : α i = 0 for all i Black. We have E [R i ] = α i + β i E [R m ] and test H 0 : α i = (1 β i )E [R 0 ] for all i. Here, R 0 is the return on the (unobserved) zero beta portfolio Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 15 / 64

16 Market Model We have a time series sample on each asset, the market portfolio, and the risk free rate {R it, R mt, R ft, i = 1,..., N, t = 1,..., T } Definition For Z it = R it R ft or Z it = R it and Z mt = R mt R ft or Z mt = R mt : Z it = α i + β i Z mt + ε it, E [ε it Z mt ] = 0 ; var [ε it Z mt ] = σ 2 εi ; cov(ε it, ε is ) = 0. Definition We may further assume that ε it are normally distributed ε it N(0, σ 2 εi ). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 16 / 64

17 Evidence for Normality Early proofs of the CAPM often assumed joint normality of returns, but later it was shown that it can hold under weaker distributional assumptions. Neverthelss, much of the literature uses exact tests based on assumption of normality. Measures of non-normality: skewness and excess kurtosis [ ] (r µ) 3 κ 3 E σ 3 [ (r µ) 4 κ 4 E σ 4 ] 3 For a normal distribution κ 3, κ 4 = 0. Daily stock returns typically have large negative skewness and large positive kurtosis. Fama for example argues that monthly returns are closer to normality Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 17 / 64

18 Theorem (Aggregation of (logarithmic) returns). Let A be the aggregation (e.g., weekly, monthly) level such that r A = r r A. Then under RW1 Er A = AEr var(r A ) = Avar(r) κ 3 (r A ) = 1 A κ 3 (r) κ 4 (r A ) = 1 A κ 4(r). This says that as you aggregate more (A ), returns become more normal, i.e., κ 3 (r A ) 0 and κ 4 (r A ) 0 as A. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 18 / 64

19 Theorem Suppose that returns follow a stationary martingale difference sequence after a constant mean adjustment and possess four moments. Then Er A = AEr, var(r A ) = Avar(r). Let r t = (r t E (r t ))/std(r t ). Then κ 3 (r A ) = 1 A κ 3 (r) + 3 A A 1 A 1 j=1 κ 4 (r A ) = 1 A κ 4(r) + 4 A + 6 A 1 ( A 1 j j=1 A + 6 A A 1 j=1 A 1 k=1 j =k j=1 ( 1 j ) E ( r 2 A t r t j ), ( 1 j ) E ( r 3 A t r t j ) ) [ E ( r 2 t r 2 ] t j ) 1 ( 1 j ) ( 1 k ) E ( r 2 A A t r t j r t k ). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 19 / 64

20 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 20 / 64

21 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 21 / 64

22 QQ plots Oliver Linton () Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 22 / 64

23 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 23 / 64

24 We next ask how the market model aggregates Suppose that MM holds for the highest frequency of data. Then for integer A we have t Z is = Aα i + β i s=t A t s=t A for t = A + 1,..., T, which can be written Z mt + t s=t A ε it Z A it = α A i + β A i Z A mt + ε A it. Suppose that ε it is independent of Z ms for all s, then this is a valid regression model for any A, with α A i = K α i, β A i = β i, var(ε A it) = Kvar(ε it ). If ε it are iid, then κ 3 (ε A it ) = κ 3(ε it )/ A and κ 4 (ε A it ) = κ 4(ε A it )/A, so that the aggregated error terms should be closer to normality than the high frequency returns. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 24 / 64

25 Market Model (S&P500-tbill) Daily estimates α se(α) β se(β) se W (β) R 2 Alcoa Inc AmEx Boeing Bank of America Caterpillar Cisco Systems Chevron du Pont Walt Disney General Electric Home Depot HP IBM Intel Johnson Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 25 / 64

26 α se(α) β se(β) se W (β) R 2 JP Morgan Coke McD MMM Merck MSFT Pfizer Proctor & Gamble AT&T Travelers United Health United Tech Verizon Wall Mart Exxon Mobil Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 26 / 64

27 Market Model (S&P500-tbill) Monthly estimates with standard errors α se(α) β se(β) se W (β) R 2 Alcoa Inc AmEx Boeing Bank of America Caterpillar Cisco Systems Chevron du Pont Walt Disney General Electric Home Depot HP IBM Intel Johnson Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 27 / 64

28 α se(α) β se(β) se W (β) R 2 JP Morgan Coke McD MMM Merck MSFT Pfizer Proctor & Gamble AT&T Travelers United Health United Tech Verizon Wall Mart Exxon Mobil Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 28 / 64

29 Portfolio weights Tangency portfolio has weights that are proportional to where i is the N vector of ones. w TP Σ 1 (ER R f i) Under the CAPM, these weights should be the weights of the market portfolio and hence should always be positive. Global Minimum Variance portfolio has weights that are proportional to w MV Σ 1 i Empirically, find many negative weights in both cases (short selling). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 29 / 64

30 Annualized returns, std, and portfolio weights µ σ w MV w TP Alcoa Inc AmEx Boeing Bank of America Caterpillar Cisco Systems Chevron du Pont Walt Disney General Electric Home Depot HP IBM Intel Johnson Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 30 / 64

31 µ σ w MV w TP JP Morgan Coke McD MMM Merck MSFT Pfizer Proctor & Gamble AT&T Travelers United Health United Tech Verizon Wall Mart Exxon Mobil Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 31 / 64

32 Chinese Data (Cambridge undergrad Rose Ng did this work in her thesis testing risk/return in Shanghai/HK). Two markets for same stocks. She provides tests of pricing relationships. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 32 / 64

33 Maximum Likelihood Estimation and Testing Suppose that Z t = α + βz mt + ε t, where ε t N(0, Ω ε ). Do not restrict Ω ε to be diagonal but require N << T. Essentially require to work with portfolios rather than individual stocks. Recent work on large matrices attempts to address this. The Gaussian log likelihood is l(α, β, Ω ε ) = c T 2 log det Ω 1 2 Define: µ m = 1 T T t=1 T t=1 Z mt ; σ 2 m = 1 T µ i = 1 T (Z t α βz mt ) Ω 1 ε (Z t α βz mt ) T Z it t=1 T t=1 (Z mt µ m ) 2 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 33 / 64

34 The maximum likelihood estimates β are the equation-by-equation time-series OLS estimates β i = T t=1(z mt µ m )(Z it µ i ) T t=1(z mt µ m ) 2 = 1 σ 2 m The maximum likelihood estimates of α are 1 T T t=1 (Z mt µ m )(Z it µ i ) α i = µ i β i µ m The maximum likelihood estimate of Ω ε is ( Ω ε = 1 T 1 T ε ε = T t=1 ε it ε jt ) ε it = Z it α i β i Z mt i,j Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 34 / 64

35 Maximum Likelihood Estimation and Testing Under the normality assumption we have, conditional on excess market returns Z m1,..., Z mt, the exact distributions: ( ( ) ) Ω ε α N β N α, 1 T ( β, 1 T 1 + µ2 m σ 2 m 1 σ 2 m Ω ε ) Without normality (but under iid) we have for large T ( ( ) ) T ( α α) = N 0, 1 + µ2 m σ 2 Ω ε m Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 35 / 64

36 Wald Test Statistic Wald test statistic for null hypothesis that α = 0 W = α [ var( α) 1 ] α = T Under null hypothesis for large T ( W = χ 2 (N) 1 + µ2 m σ 2 m ) 1 α Ω 1 ε α provided N < T. The asymptotic approximation is valid regardless of whether the errors are normally distributed or not. For the Dow stocks: Daily W = (p-value = ); Monthly W = (p-value = ) Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 36 / 64

37 Exact Finite-Sample Variant of the Wald Test Statistic Under normality, we can get an exact test statistic by using an F distribution and a degrees of freedom correction F = (T N 1) N ((1 + µ2 m σ 2 ) 1 ) α Ω 1 ε α F (N, T N 1) m This is superior to the Wald test (under the assumption of normality) Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 37 / 64

38 Likelihood Ratio Test The likelihood ratio test is a natural alternative to a Wald test: LR = 2(log l c log l u ) = T [log det Ω ε log det Ω ε ] = χ 2 (N) Ω ε = 1 T ε ε = ( 1 T T t=1 ε it ε jt where ε it are the constrained residuals ie the no intercept residuals These tests have an exact relationship which allows us to derive the exact distribution for the likelihood ratio test under normality. ) i,j Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 38 / 64

39 CLM Table 5.3. Works with Monthly returns on ten value weighted portfolios based on size. CRSP value weighted index, 1 month tbill rate. Results for full period show rejection at 5% but not at 1% level. Five year subperiods: some rejections at 5% some not. Aggregate assuming independence, ie make use of χ 2 (j) + χ 2 (i) = χ 2 (j + i) very strong rejections. Likewise for ten year subperiods. The aggregated results allow for different parameter values across the subperiods but hide whether the evidence is getting stronger agains the CAPM or not Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 39 / 64

40 Testing Black Version of the CAPM Tests for the Black version are more complicated to derive. Estimate the same unconstrained model as before using total returns instead of excess returns. The constrained model is R t = (i β)γ + βr mt + ε t for some scalar unknown γ, where i is the N vector of ones. There are N 1 "nonlinear cross-equation" restrictions α 1 1 β 1 = = α N 1 β N = γ The model to be estimated is nonlinear in the parameters. Estimating the constrained model requires numerical maximization of the nonlinear (in parameters) system of equations. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 40 / 64

41 Useful trick (profiling or concentration): assume that the expected return on the zero-beta portfolio is known exactly (use a noisy estimate as proxy) R t γi = β(r mt γ) + ε t so that conditionally on γ the model is linear in β. With the zero-beta return known, the Black model can be estimated using the same methodology as the Sharpe-Lintner model Then, relax the assumption that the zero-beta return is known. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 41 / 64

42 For θ = (γ, β 1,..., β N ) the (constrained) likelihood function is l(θ, Ω) = c T 2 log det Ω ε 1 2 T t=1 (R t γi β(r mt γ)) Ω 1 (R t γi β(r mt γ)) maximize with respect to θ. Profile/concentration method. Define β i (γ) = T t=1(r mt γ)(r it γ) T t=1(r mt γ) 2 Ω ε (γ) = 1 T ε (γ) ε (γ) = ( 1 T T t=1 ε ε it(γ) ε jt(γ) ) i,j Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 42 / 64

43 Then search the profiled likelihood over the scalar parameter γ l P (γ) = c T 2 log det Ω ε (γ) 1 2 T t=1 (R t γi β (γ)(r mt γ)) Ω ε (γ) 1 (R t γi β (γ)(r mt γ)) and let γ be the maximizing value and then let β i ( γ ) and Ω ε ( γ ) be the corresponding estimates of β i and Ω ε. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 43 / 64

44 Testing Black Version of the CAPM The LR statistic compares the relative fit of the constrained and unconstrained models. As T LR = T [log det Ω ε log det Ω ε ] = χ 2 (N 1) where Ω ε is the MLE of Ω ε in the constrained model. Note only N 1 degrees of freedom Exact theory much more tricksy, require simulation methods. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 44 / 64

45 Robustness of MLE and tests to Heteroskedasticity (RW2.5) Maximum likelihood estimation apparently assumes multivariate normal returns. CAPM can hold under weaker distributional assumptions (e.g., elliptical symmetry, which includes multivariate t-distributions with heavy tails). Actually, the Gaussian MLE of α, β is robust to heteroskedasticity, serial correlation, and non-normality since the estimates are just least squares. The asymptotic test statistics are robust to normality of the errors, but they are not robust to heteroskedasticity or serial correlation, and in that case we need to adjust the standard errors Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 45 / 64

46 Suppose that E (ε t Z mt ) = 0 E (ε t ε t Z mt ) = Ω t, where Ω t is a potentially random time varying covariance matrix. This is quite a general assumption, but as we shall see below it is quite natural to allow for dynamic heteroskedasticity for stock return data. In this case, it is not possible to perform an exact test and the tests we already defined are unfortunately not properly sized in this case. However, we can construct robust Wald tests based on large sample approximations. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 46 / 64

47 Let Ω T = 1 T T t=1 ε t ε t ; Ψ T = 1 T T (Z mt µ m ) 2 ε t ε t t=1 V = Ω T + µ2 m Ψ σ 4 T m J H = T α V 1 α Then, under the null hypothesis as T J H = χ 2 (N). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 47 / 64

48 Cross-Sectional Regression Tests The CAPM says that µ i = β i µ m, where µ i = E (R i R f ) and µ m = E (R m R f ). Fama and MacBeth (1973) say embed this in a richer cross-sectional relationship µ i = γ 0 + β i γ 1. We should find γ 0 = 0 and γ 1 > 0 with γ 1 = µ m = E (R m R f ). In fact we should find γ 0, γ 2, γ 3 = 0 and γ 1 > 0 in µ i = γ 0 + β i γ 1 + β 2 i γ 2 + σ2 ɛi γ 3 Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 48 / 64

49 Problem: We dont observe β i (or σ 2 ɛi ). Solution First estimate β i for each stock or portfolio using time series data. Then estimate the cross-sectional regression by OLS (or GLS) R i R f = γ 0 + β i γ 1 + u i Under the CAPM, γ 0 = 0 and γ 1 = E (R m R f ) (which can be separately estimated by the sample average R m R f ). Test γ 0 = 0 by t-test Empirically find that there is a positive and linear relationship between beta risk and return with a high R 2, but γ 0 > 0 and γ 1 significantly lower than market excess return. In fact, they include additional variables, e.g., R i R f = γ 0 + β i γ 1 + β 2 i γ 2 + σ2 εi γ 3 + u i where σ 2 εi is an estimate of the idiosyncratic error variance. Test also γ j = 0, j = 2, 3 using t-tests or Wald statistic. FM do not reject Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 49 / 64

50 Standard errors FM estimate the cross-sectional regressions (for t = 1,..., T ) R it R ft = γ 0t + β i γ 1t + u it Then average the estimates over time (gives the same as the single regression of the average returns on the β i ) [ ] γ0 γ = = 1 [ ] γ0t γ 1 T γ 1t T t=1 They estimate the asymptotic variance matrix of γ by V = 1 ([ ] [ ]) ([ γ0t γ0 γ0t T γ 1t γ 1 T t=1 γ 1t ] [ γ0 γ 1 Standard errors are widely used, but wrong unless combined with the portfolio grouping of a large original set of assets. Errors in variables/generated regressor issue. Shanken (1992) suggests an analytical correction necessary for individual stocks. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 50 / 64 ])

51 Actual methodology even more complicated. There are four stages: 1 Time series estimation of pre-ranking individual stock beta in period A 2 Portfolio formation based on estimated double sorted individual stock beta and size 3 Estimate portfolio betas from the time series in period B 4 Cross-sectional regressions for each time period in B from which time series of estimated risk premia are obtained γ t. Test hypothesis on average of time series of risk premia using standard errors from above Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 51 / 64

52 Testing on portfolios (composed of a large number of individual stocks) rather than individual stocks can mitigate the errors-in-variable problem as estimation errors cancel out each other. Sorting by beta reduces the shrinkage in beta dispersion and statistical power; sorting by size takes into account correlation between size and beta; performing pre-ranking and estimation in different periods avoids selection bias. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 52 / 64

53 Figure: Risk Return Relation Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 53 / 64

54 Empirical Evidence in the Literature Many tests and many rejections of the CAPM!! Size Effect. Market capitalization Firms with a low market capitalization seem to earn positive abnormal returns (α > 0), while large firms earn negative abnormal returns (α < 0) Value effect. Dividend to price ratio (D/P) and book to market ratio (B/M). Value firms (low value metrics relative to market value) have α > 0 while growth stocks (high value metrics relative to market value) have α < 0 Momentum effect. Winner portfolios outperform loser portfolios over medium term. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 54 / 64

55 Active vs. Passive Portfolio Management Active portfolio management: attempts to achieve superior returns α through security selection and market timing Security selection = picking misspriced individual securities, trying to buy low and sell high or short-sell high and buy back low Market timing = trying to enter the market at troughs and leave at peaks Conflicts of interest between owners and managers Passive portfolio management: tracking a predefined index of securities with no security analysis whatsoever, just choose β (smart beta). Index funds, ETF s. No attempt to beat the market, in line with EMH, which says this is not possible. Much cheaper than active management since no costs of information acquisition and analysis, lower transaction costs (less frequent trading), also generally greater risk diversification (the only free lunch around). Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 55 / 64

56 If an active manager overseeing a 5 billion portfolio could increase the annual return by 0.1%, her services would be worth up to 5 million. Should you invest with her? Role of luck Imagine 10,000 managers whose strategy is to park all assets in an index fund but at the end of every year use a quarter of it to make (independently) a single bet on red or black in a casino. After 10 years, many of them no longer keep their jobs but several survivors have been very successful ((1/2) 10 1/1000). The infinite monkey theorem says that if one had an infinite number of monkeys randomly tapping on a keyboard, with probability one, one of them will produce the complete works of Shakespeare. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 56 / 64

57 Time Varying Parameters The starting point of the market model and CAPM testing was that we have a sample of observations independent and identically distributed from a fixed population. This setting was convivial for the development of statistical inference. However, much of the practical implementations acknowledge time variation by working with short, say 5 year or 10 year windows. A number of authors have pointed out the variation of estimated betas over time. We show estimated betas for IBM (against the SP500) computed from daily stock returns using a five year window over the period We present the rolling window estimates. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 57 / 64

58 Figure: Time Varying Betas Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 58 / 64

59 Figure: Time Varying Alphas Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 59 / 64

60 We next consider a more general framework where time variation is explicitly considered. Suppose that Z it = α it + β it Z mt + ε it, where α it, β it vary over time, but otherwise the regression conditions are satisfied, i.e., E (ε it Z mt ) = 0. We may allow σ 2 it = var(ε it Z mt ) to vary over time and asset. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 60 / 64

61 One framework that may work here is that α it = α i (t/t ), β it = β i (t/t ), where α i and β i are smooth, i.e., differentiable functions. We can reconcile this with the CAPM by testing whether α i (u) = 0 for u [0, 1]. This can be done using the rolling window framework. Note that it is not necessary to provide a complete specification for σ 2 it as one can construct heteroskedasticity robust inference. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 61 / 64

62 An alternative framework is the unobserved components model α it = α i,t 1 + η it, β it = β i,t 1 + ξ it, where η it, ξ it are i.i.d. shocks with mean zero and variances σ 2 η, σ 2 ξ, Harvey (1990). One may take α i0 = 0 and initialize β i0 in some other way. In this framework, the issue is to test whether σ 2 η = 0 (which corresponds to constant and zero α) versus the general alternative. In both of these models, the regression parameters evolve in some deterministic or autoregressive way. CCAPM and other intertemporal models drive the evolution of risk premia through macro and other state variables Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 62 / 64

63 Criticisms of Mean-Variance analysis and the CAPM Theoretically, it is claimed that expected utility is invalid, some of the other assumptions which underline these models are invalid, and that the Mean-Variance criterion may lead to paradoxical choices. Allais (1953) shows that using EUT in making choices between pairs of alternatives, particularly when small probabilities are involved, may lead to some paradoxes. Roy (1952). Even if EUT is intact some fundamental papers question the validity of the risk aversion assumption: Friedman and Savage (1948), Markowitz (1952) and Kahneman and Tversky (1979) claim that the typical preference must include risk-averse as well as risk-seeking segments. Thus, the variance is not a good measure of risk, which casts doubt on the validity of the CAPM. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 63 / 64

64 Roll critique. Cant observe the market portfolio. So rejections of CAPM are not valid Normality (or an Elliptic distribution) is crucial to the derivation of the CAPM. The Normal distribution is statistically strongly rejected in the data. Furthermore, the CAPM has only negligible explanatory power. Ex ante versus ex post betas. Conditional capm. Time varying risk premia. For example recession indicators. Will discuss later. Financial Econometrics Short Course Lecture 2 Portfolio Choice Renminand University CAPM 64 / 64