1. Chapter 4 E ffi cient Portfolios and CAPM 4.1E ffi cient Portfolios Problem: Suppose that we have n risky securities at time t with return {R i,t+1 } at the next period, which includes dividend payment. Suppose that there exists a riskless bond earning interest {R 0,t }. What is the optimal portfolio allocation? 101

2. 102 Portfolio: Characterized by an allocation vector (α 0, α 1,..., α n ) T with proportion α i amount invested on security i. Denote by α = (α 1,..., α n ) T and R t = (R 1,t,..., R n,t ) T. The proportion satisfies α 0 + α 1 + · · · + α n = α 0 + 1 T α = 1. Note some α i can be negative (a short position). Portfolio return: At time t + 1, the return of the portfolio is r t+1 = α 0 R 0,t + α T R t+1. Expected value and volatility: µ t (α 0, α) = E t r t+1 = α 0 R 0,t + α T E t R t+1 = R 0,t + α T E t Y t+1,

3. σ t (α 0, α), 103 where Y t+1 = R t+1 − R 0,t 1 is the excess return, and σ t2 (α 0, α) = α T Σ t α = α T Var t (Y t+1 )α, where Σ t = Var t (R t+1 ). Mean-variance approach: Maximize the expected return while minimizing the risk (Markowitz 1952, Sharpe 1963). Criterion: Subject to the constraint α 0 + α T 1 = 1, max µ t (α 0, α) − α 0,α A 2 2 where A measures the investor’s risk aversion. Remark: The above optimization problem is equivalent to max µ t (α 0, α) α 0,α

4. A T α Var t (Y t+1 )α 104 subject to σ t2 (α 0, α) ≤ B and α 0 + α T 1 = 1. It is also equivalent to α 0,α subject to µ t (α 0, α) ≥ C and α 0 + α T 1 = 1. Let us now solve the first optimization problem. Using α 0 = 1 − α T 1, we have max α α T E t Y t+1 − 2 + R 0,t. t Optimal allocation: α ∗ = 1 Var t (Y A t+1 ) −1 E t Y t+1 and α 0 ∗,t = 1 − α ∗ t 1. Example 3.1: Suppose that the riskless asset earns 5% interest and that the excess returns of 3 risky assets earn respectively 10%,25%,

5. 105 and 55% per year with volatility (standard deviation) 12%, 40 % and 110% respectively. In addition, suppose that the correlation matrix of the three risky assets is given by 1 0.7 0.4 0.4 0.5 1

6. Σ t = Var t (R t+1 ) = R 106 Thus, the covariance matrix is given by 0.12 0.4 1.1 0.12 0.4 1.1 0.0144 0.0336 0.0528 0.0528 0.2200 1.2100 The optimal portfolio allocation is given 0.050.8722 0.50 0.2416 Suppose that an investor is willing to invest 20% in the riskless asset.

7. 0.8 0.7346 = 0.3180. 107 Then, we have α 1 ∗ + α 2 ∗ + α 3 ∗ = (0.8722 + 0.7346 + 0.2416)/A = 1.848/A = 0.8, or α∗ =α∗ = 0.8722 0.2416 1.848 0.3775 0.1046 In other words, he should invest 37.5%, 31.8%, 10.46% in stocks 1, 2, and 3, respectively. With this allocation, the expected return of the portfolio is 5% + 37.75% · 5% + 31.80% × 20% + 10.46% × 50% = 18.48%.

8. 108 t This portfolio has the variance α ∗ T Σ t α ∗ = 5.83%, better than individual stock in terms the mean-variance e ffi ciency. Characteristics of e ffi cient portifilio: With the optimal port- folio allocation, the expected return is µ ∗ t = R 0,t + α ∗ t T E t Y t+1 = R 0,t + P t /A, and the variance is given by σ t ∗ 2 = α ∗ t T var t (Y t+1 )α ∗ t T = P t /A 2, 1/2 and 1 µ ∗ = R 0,t + P t 2 σ ∗.

9. σtσt = P t A T A ∗ 2 2 t α Var t (Y t+1 )α ∗ T + A ∗ 2 2 t 109 Sharpe ratio of the e ffi cient portfolio: Sharpe ratio = µ ∗ t − R t,0 ∗ 1/2. This gives excess return per unit risk. — expected excess gain divided by its standard deviation; — used to compare the e ffi ciency of two portfolios; — related to risk-adjusted return of the Bank Trust. E ffi cient Frontier: For any other portfolio with the same risk α T Var t (Y t+1 )α = (σ t ∗ ) 2, its expected excess return is bounded by α T E t Y t+1 = α T E t Y t+1 − 2 α Var t (Y t+1 )α + σ ≤ α ∗ T E t Y t+1 − A ∗ T 2 σ 1 = µ ∗ t = R 0,t + P t 2 σ t ∗.

10. 44 0 1 mean 2 3 0 1 2 3 (a)(b) 110 variance 0510152025 Mean-variance efficient frontier SD 012345 Mean-SD efficient frontier Figure 4.1: Mean-variance e ffi cient frontiers. bounded by the e ffi cient frontier. As the percentage of risky asset increases, the expected return increases (Figure 4.1).

11. α T Var t ( Y t+1 ) α T =σ t ∗ 2 σtσt 111 Sharpe Ratio: The sharp ratio for any portfolio is defined by S(α) = α T E t Y t+1 (α T Var t (Y t+1 )α T ) 1/2. Note that S(α) is independent of a scaling of α. Thus, max S(α) = max α α T E t Y t+1 /σ t ∗ ≤ µ ∗ t − R t,0 ∗. Example 1 (Continued). The risk-adjusted returns for 4 assets are summarized as follows. Stock0123Optimal return5% 10% 25% 55% 18.48% E-return0% 5%20% 50% 13.48% risk0% 12% 40% 110% 24.15% Sharpe Ratio - 0.417 0.500 0.4550.558

12. 112 The Sharpe ratio is maximized at the e ffi cient portfolio. For the op- timal portfolio, P t = 0.3113 = 0.558 2. 4.2Optimizing expected utility function Let U (w) = 1−exp(−Aw), a utility function of wealth. The absolute risk aversion −U (w)/U (w) = A is independent of w. It is a commonly-used utility function in invest- ment decision. To understand better the utility function, consider the following example:

13. 113 Action 1: Win $ 100 with certainty. Action 2: Win $ 10,000 with probability a and loss $ 1,000 with probability (1 − a). Set U (−1000) = 0 and U (10000) = 1 (The scale is arbitarily). Di ff erent investors have very di ff erent attitude: — (a) If a = 10%, which action do you take? — (b) If a = 20%, which action do you take? If you think that for a = 0.1, action 1 and action 2 are about same, then your utility function at 100 is U (100) = 0.1U (10000) + 0.9U (−1000) = 0.1.

14. 1.0 y 0.0 0.2 0.4 0.6 0.8 114 0246810 If another investor feels that a = 0.2, action 1 and action 2 are equivalent to his decision, then his utility function is U (100) = 0.2U (10000) + 0.8U (−1000) = 0.2. Apparently, the second investor is more conservative. Exponential utility functions x Figure 4.2: Exponential utility functions with A = 0.5 and A = 0.10.

15. 115 The exponential function is a risk aversion utility function. An in- vestor may maximize his expected utility under budget constraints. Assume that his initial wealth is w. Then, his wealth in the next period is w t+1 = w + (R 0,t + α T Y t+1 )w = w(1 + R 0,t ) + wα T Y t+1. Thus, he wishes to maximize α 0,α This is the same as minimizing the function E t exp(−A 1 α T Y t+1 ), where A 1 = Aw. If the conditional distribution of the excess return is Y t+1 ∼ N (µ t, Σ t ), then α T Y t+1 ∼ N (α T µ t, α T Σ t α). Thus,

16. A 21 T 116 the expected utility is given by E t exp(−A 1 α T Y t+1 ) = exp{−A 1 α T µ t + 2 α Σ t α}. This is the same as maximizing α T µ t − A 1 α T Σ t α, 2 and explains the portfolio optimization from the optimizing the ex- pected utility point of view.

17. 117 4.3The Capital Asset Pricing Model Assumption: Each investor trades at the mean-variance optimal portfolio, with the absolute risk aversion coe ffi cient A i t t A t i Equilibrium condition: Suppose that the total supply of shares at the period t is b on all assets. Then t α D = 1 Σ −1 µ t = b ⇐⇒ µ t = AΣ t b. A (1) m Market portfolio: Y t+1 = b T Y t+1 (excessive return of the port- folio of all invested wealth). From (1), it is a mean-variance e ffi cient portfolio.

18. Var t (Y t+1 = T 118 m m m m m Theorem 3 In the linear regression Y t+1 = α + βY t+1 + ε t+1 with E t ε t+1 = 0 and Cov t (ε t+1, Y t+1 ) = 0, the intercept α = 0. Proof: Note that Cov(Y t+1, Y t+1 ) = βCov(Y t+1, Y t+1 ). It follows that β = m Cov t (Y t+1, Y t+1 ) m ) Σ t b b Σ t b.

19. = AΣ t b − T 119 m Hence, by (1), α = E t Y t+1 − βE t Y t+1 Σ t b b Σ t b · b T AΣ t b = 0. Following the same steps of the proof, we have the following inverse of Theorem 1 (Homework). Theorem 4 Given a portfolio a with excess gain Y at+1 = a T Y t+1, the intercepts in the regression Y t+1 = α(a) + β(a)Y ta+1 + ε at+1 are 0 ⇐⇒ a is proportional to b, i.e. it is a mean-variance e ffi - cient portfolio.

20. 120 CAPM(Sharpe-Lintner version): Y t = βY tm + ε t — derived by Sharpe (1964) and Lintner (1965) with the existence of the risk-free asset; ♠ the excess return of i-th security E t−1 Y it = β i E t−1 Y tm ; ♣ quantify exactly the relationship between risk and return; ♠ β i = Cov t−1 (Y it, Y tm )/Var t−1 (Y tm ) measures the cross-sectional risk of the asset; ♣ market risk premium E t−1 Y tm > 0.

21. 121 Determination of market β: — S&P500 index or CRSP as a proxy of the market portfolio; — the US T-bill rates as proxies of the riskless return; — monthly returns over 5 years (T = 60) are used to determine the beta via the regression Y it = α i + β i Y tm + ε it, t = 1,..., T. m Application: ♠ Estimating covariance matrix: var(Y ) = ββ T var(Y t+1 )+var(ε t+1 ), in which var(ε t+1 ) can be assumed to diagonal. ♣ capital budgeting decisions in corporate finance;

22. 122 ♣ portfolio performance evaluation (mean-variance e ffi ciency); The expected return of a firm: r f +β(r m −r f ), where r f is the average risk-free rate, r m is the average return. With this estimate, decision makers can decide whether or not to carry out an investment. 4.4Validating CAPM Ingredients of CAPM: — The intercepts are 0. — Market β’s completely capture the cross-sectional variation of expected excess returns. — The market risk premium EY m is positive.

23. 123 Statistical Model: Y t is a vector of excess returns of N assets. It follows the linear model Y t = α + βY tm + ε t, Eε t = 0,Var(ε t ) = Σ, Cov(Y tm, ε t ) = 0, for t = 1, · · ·, T periods. Testing against CAPM: H 0 : α =0 Additional assumption: ε t ∼ i.i.d.N (0, Σ).

24. − (Y t − α − βY tm ) T Σ −1 (Y t − α − βY tm ). 124 The conditional likelihood function, given Y 1m,..., Y Tm, is f (Y 1, · · ·, Y T |Y 1m, · · ·, Y Tm ) = T t=1 (2π) −N/2 |Σ| − 21 × exp 1212 The log-likelihood function is given by (α, β, Σ) = − NT 2 log(2π) − T2T2 log |Σ| − 1212 T t=1 (Y t − α − βY tm ) T Σ −1 (Y t − α − βY tm ).

25. 125 β = MLE for α and β: α = Y¯ − βY¯ m, T t=1 (Y t − Y¯ )(Y tm − Y¯ m )/ T i=1 (Y tm − Y¯ m ) 2, −1 T t=1 Y t and Y¯ m = T −1 T t=1 Y tm. where Y¯ = T Remark: Σ = T −1 — The estimators of α and β are the same as those fitting the OLS separately. — The cross-sectional covariance estimator is T t=1 (Y t − α − βY tm )(Y t − α − βY tm ) T.

26. 126 — It is the covariance matrix of residuals from fitting OLS sep- arately. 2 Wald test: T 0 = α T [Var(α)] −1 α a 2 Exact distribution: Under the normal model, it can be shown that T 1 = T − N − 1 NT T 0 ∼ F N,T −N −1.

27. = T (log |Σ 0 | − log |Σ|) ∼ χ 2N, T 2 ∼ χ 2N 127 Maximum likelihood ratio test: T 2 = 2{max − max } H 0 a where Σ 0 is obtained under H 0 with α = 0. It can be shown that T 1 = T − N − 1 N (exp(T 2 /T ) − 1). — T 1 and T 2 are equivalent; — exact distribution of T 2 can be found via T 1 ; — adjusted version T 3 = T − N/2 − 2 T a has better performance at finite sample (see Table 3.1);

28. 128 ♠ for practical purpose, T 1 or T 3 is good enough, as long as the data follows normal distribution (recall aggregational Gaussianity). Size of test. To see how good the asymptotic approximations are, let us assume that the data are normal so that T 1 has an exact F- distribution. Using this as the golden standard, with N = 10 and a P (T 0 ≥ 18.31) ≈ 5%. On the other hand, the exact size of the test is P (T 0 ≥ 18.31) = P (T 1 ≥ T − N − 1 NT 18.31) = P (T 1 ≥ 1.495). Since T 1 ∼ F 10,49, the actual probability is 17.0%. The approximation is very poor. The following Table shows the size of the tests.

29. 129 Table 4.1: Size of tests of the Sharpe-Lintner CAPM using asymptotic critical values. N = 10N = 20N = 40 Time T0T0 T2T2 T3T3 T0T0 T2T2 T3T3 T0T0 T2T2 T3T3 60 120 180 240.170.099.080.072.096.070.062.059.051.462.050.200.050.136.050.109.211.105.082.073.057.985.051.610.051.368.050.257.805.141.275.059.164.053.124.052 4.5Empirical Studies Summary: — Early evidence was largely positive on CAPM (Black, Jensen, Scholes, 1972, Fama and MacBeth 1973) ♠ Anomalies can be thought of as firm characteristics which can be grouped to create a portfolio that has higher Sharpe ratio than that of the proxy of the market portfolio.

30. 130 1. Basu(1977) reported PE e ff ect: Firms with low PE ratios have higher sample returns than those predicted by CAPM. 2. Low market capitalization firms have higher sample mean re- turns (Banz, 1981). 3. Firms with high book-to-market ratio have higher average re- turns than those predicted by the CAPM. (Fama and French, 1992, 1993). 4. Buying losers and selling winners have higher average return than the CAPM predicts (DeBondt and Thaler, 1985, Jegadeesh and Titman 1985). ♣ Counter arguments:

31. 131 1. The proxy of market portfolio is not good enough (should include bonds, real-estate, foreign assets). 2. Issues of data-snooping, bias sampling. 3. Multi-period data are used instead of one-period of data. Example 3.2: To test Sharpe-Lintner version of CAPM — The CRSP value-weighted index is used as a proxy for market portfolio. — The one-month T-Bill return is used for risk-free return. — Periods: January 1965-December 1994. — Ten value-weighted portfolios (N = 10) were created based on

32. 132 stocks traded at NYSE and ASE. Results: Table 4.2: Empirical results for tests of the Shape-Lintner version of the CAPM. Time T1T1 p-value T2T2 T3T3 Five-year subperiods 1/65-12/69 1/70-12/74 1/75-12/79 1/80-12/84 1/85-12/89 1/90-12/94 overall 2.038 2.136 1.914 1.224 1.732 1.153 77.224 0.049 0.039 0.066 0.300 0.100 0.344 0.004 20.867 21.712 19.784 13.378 18.164 12.680 106.586 0.022 0.017 0.031 0.203 0.052 0.242 ** 18.432 19.179 17.476 11.818 16.045 11.200 94.151 0.048 0.038 0.064 0.297 0.098 0.342 0.003 Ten-year subperiods 1/65-12/74 1/75-12/84 1/85-12/94 overall 2.400 2.248 1.900 57.690 0.013 0.020 0.053 0.001 23.883 22.503 19.281 65.667 0.008 0.013 0.037 ** 22.490 21.190 18.157 61.837 0.013 0.020 0.052 0.001 Thirty-year period 1/65-12/942.1590.02021.6120.01721.1920.020

33. 133 Example 3.3: To test the Sharpe-Lintner version of CAPM, we took — the SP500 index as a proxy for market portfolio; — the 3-month T-bill return as a proxy for risk free return; — Periods: Feb. 1994 — Feb. 2004; — Stocks: Ford, Johnson and Johnson, General Electric. The least-squares fit of individual stocks are as follows. #last 120 months e-return> y <- returns[60:179, 2:4] > x <- returns[60:179,1] >ls.print(lsfit(x,y[,1]))#Ford Residual Standard Error = 9.605,Multiple R-Square = 0.247 N = 120,F-statistic = 38.6999 on 1 and 118 df, p-value = 0 coef std.err t.stat p.value Intercept 0.2318 X 1.1967 0.8804 0.2633 0.1924 6.2209 0.7928 0.0000

34. 134 >ls.print(lsfit(x,y[,2]))#GE Residual Standard Error = 5.2554,Multiple R-Square = 0.4659 N = 120,F-statistic = 102.9137 on 1 and 118 df, p-value = 0 coef std.errt.stat p.value Intercept 0.94350.48171.95880.0525 X 1.06780.1053 10.14460.0000 >ls.print(lsfit(x,y[,3]))# Johnson and Johnson Residual Standard Error = 6.1167,Multiple R-Square = 0.1402 N = 120,F-statistic = 19.2361 on 1 and 118 df, p-value = 0 coef std.err t.stat p.value Intercept 1.1368 X 0.5373 0.5606 2.0277 0.1225 4.3859 0.0448 0.0000 Clearly, the intercepts of three stocks are not very significant. We now combine them to test CAPM. We compute T 0 = 8.26, d.f. = 3,p-value = 4.1%. T 1 = 2.66, d.f. = (3, 116),p-value = 5.1%.

35. 135 For modified maximum likelihood ratio test, we have T 3 = (T − N/2 − 2) log 1 + N T 1 T − N − 1 = 7.527, with degree of freedom 3, giving a p-value of 5.69%. Results: Weak evidence against CAPM Market β for Ford: β 1 = 1.1967. To predict the monthly return of a firm, according to CAPM: r f + β(r m − r f ). We need to use a longer time-horizon to compute r m and r f. This produce more stable prediction. Average log-monthly-return of SP500 over last 15yrs: r m = 0.7612%

36. 136 Average risk-free rate over last 15 yrs: r f = 0.3750% Expected monthly return for Ford: 0.8372% Similar quantities for GE and John and John and GE can be computed FordGEJNJ β1.19671.06780.1225 Expected return (monthly) 0.8372% 0.7834% 0.4223% Remark ∗ : For the linear model, y i = a + bx i + ε i, the least-squares estimator satisfies y¯ = a + bx¯. Now, letting x and y be respectively the excess returns of the market portfolio and an asset, we have R¯ − r¯ f = a + b(¯r m − r¯ f ), or R¯ = a + r¯ f + b(¯r m − r¯ f ), where R¯ is the average return of the asset. Thus, if we use CAPM with r¯ f and r¯ m computed in the same period (1994-2004 in the Example 3.3), the predicted monthly return r¯ f + b(¯r m − r¯ f ) and di ff ers from the actual average

37. 137 only by a. CAPM prediction merely replaces a by its theoretical value 0. If we compute r¯ m and r¯ f using a di ff erent period of data (e.g. 15 years data), the di ff erence is hard to quantify. 4.6 Cross-sectional regression ∗ Method: Let µ j = T −1 Blume and Friend (1973) and Fama and McBeth (1973) introduced the following cross-sectional regressions for the Sharpe-Lintner version of CAPM. Note that µ j = EY j,t = λβ j, j = 1,..., N, where λ = EY tm > 0 (risk premium). T t=1 Y j,t and MLE Cov{(Y jt, Y tm ), t = 1, · · ·, T } β j = Var(Y tm, t = 1, · · ·, T ) be the empirical estimate of µ j and β j. Then, fit µ j = a 0 + a 1 β j + ε j, j = 1,..., N

38. 138 CAPM: If the CAPM holds, the following three properties should be true. (i) a 0 is statistically insignificant (ii) a 1 is statistically positive (iii) Multiple R 2 should be large Drawback: Errors-in-variables create biases. 4.7E ffi cient-set Theory Notation: n risky assets with mean return µ and covariance matrix Σ. See Huang, C.F and Litzenberger, R.H.(1988). Foundations for financial economics, North-Holland, N.Y. Di ff erence: Don’t assume the existence of the risk-free bonds. The protfolio optimization in §3.1 is equivalent to min α T Σα, α

39. α Σα + λ 1 (µ p − α T µ) + λ 2 (1 − α T 1). 139 subject to α T µ = µ p and α T 1 = 1, where 1 = (1,..., 1) T. Lagrange multiplier method: Minimize 1 T 2 or solve Σα − λ 1 µ − λ 2 1 = 0, where λ 1 and λ 2 are determined by α T µ = µ p and α T 1 = 1. Solution: α = g + µ p h, where g = D −1 [BΣ −1 1 − AΣ −1 µ], h = D −1 [CΣ −1 µ − AΣ −1 1], A = 1 T Σ −1 µ, B = µ T Σ −1 µ, C = 1 T Σ −1 1, and D = BC − A 2.

40. Hence, the optimal variance and mean satisfy σ p2 = (g + µ p h) T Σ(g + µ p h) or Cσ p2 − C 2 /D · (µ p − A/C) 2 = 1, 140 (2) which defines an e ffi cient frontier in the space of (σ p, µ p ). This curve is a parabola. ♠ There exists a portfolio g, which has the global minimum vari- ance C −1. ♣ Any portfolio has expected return less than that of g is not ad- missible solution

41. 141 Figure 4.3: Minimum-Variance Portfolios without Risk free Asset ♠ The covariance between two frontier portfolios p and q is (g+µ p h) T Σ(g+µ q h) = C/D·(µ p −A/C)(µ q −A/C)+C −1. (3) ♣ For each minimum-variance portfolio, there exists a unique minimum-

42. 142 variance portfolio p 0 with µ p 0 = ACAC − D C 2 (µ p − A/C) that has zero covariance with p. This can easily be obtained by setting (3) to zero and solving for µ q. p 0 is called the zero-beta portfolio with respect to p. ♠ From (2), we have σ p dσ p − C/D · (µ p − A/C)dµ p = 0. The slope at point p is given by dµ p dσ p = σ p D Cµ p − A.

43. 143 It can easily be verified that by (2) µ p − dµ p dσ p σ p = µ p − = µ p − σ p2 D Cµ p − A D{C −1 − C/D · (µ p − A/C) 2 } Cµ p − A = µ p 0. ♣ Consider a multiple regression of the return on any portfolio R a on R p and R p 0 : R a = β 1 + β 2 R p 0 + β 3 R p + ε p, we have (homework) β 1 = 0, β 2 = 1 − β ap, β 3 = Cov(R a, R p )/σ p2 ≡ β ap, where β ap is the beta of the portfolio with respect to portfolio p.

44. 144 In other words, ER a − ER p 0 = β ap E(R p − R p 0 ). ♠ The slope in Fig 3.3 is the Sharpe type of ratio ♣ This tangent portfolio is called market portfolio. 4.8Black version of CAPM In absence of the risk-free asset, Black(1972) derived the following CAPM. Notation: R t — vector of returns of individual stock or portfolio;

45. 145 R tm — return of market portfolio; γ — return of zero-beta portfolio, uncorrelated with the market portfolio. Black version of CAPM: The log-likelihood of the full model is E t−1 R t = γ1 + β(E t−1 R tm − γ). — Same as the Sharpe-Lintner version when γ =risk free rate Statistical model: R t = α + βR tm + ε t Eε t = 0, Var(ε t ) = Σ, and Cov(R tm, ε t ) = 0 Black version of CAPM: H 0 : α = γ(1 − β)

46. 146 Log-likelihood: (α, β, Σ) = − NT 2 log(2π) − T2T2 log |Σ| − 1212 T t=1 (Y t − α − βR mt ) T Σ −1 (Y t − α − βR mt ). Σ 0 = T −1 The maximum likelihood ratio test can be derived. In particular, the MLE under the full model is the same as the Sharpe-Lintoner version, resulting in an estimated covariance Σ. MLE under H 0 : α = γ(1 − β). The MLE can not be explictly found. It solves the following equations: For each given γ and β 0, the estimated covariance Σ 0 is given by T t=1 [R t − γ(1 − β 0 ) − β 0 R mt ][R t − γ(1 − β 0 ) − β 0 R mt ] T.

47. γ = (1 − β 0 ) T Σ −1 (R − β 0 R¯ m )/(1 − β 0 ) T Σ −1 (1 − β 0 ). (R tm − γ)(R − γ1)/ 147 β 0 = For given Σ 0 and β 0, the estimated γ is ¯ 0 For given Σ 0 and γ, T ¯ (R tm − γ) 2. t=1 MLR test: Under H 0, a The factor T can be replaced by (T − N/2 − 2) in an hope to improve the finite sample approximation: This gives a Implementation: We can following the following steps:

48. 148 1. Fit the linear model R t = α + βR tm + ε t to obtain α, β and Σ. 2. Using these estimates as the initial value, obtain ¯ 3. Compute β 0 and then Σ 0. 4. Iterate between steps 2 and 3 if needed (from statistical point of view, this step is optional). 5. Compute the test statistics T 5. 6. Compute the P-value.