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Chapter 4 Efficient Portfolios and CAPM

Chapter 4 Efficient Portfolios and CAPM. 4.1. Efficient 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 }.

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Chapter 4 Efficient Portfolios and CAPM

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  1. Chapter 4 Efficient Portfolios and CAPM 4.1 Efficient Portfolios Problem: Suppose that we have n risky securities at time t with return {Ri,t+1} at the next period, which includes dividend payment. Suppose that there exists a riskless bond earning interest {R0,t}. What is the optimal portfolio allocation? 101

  2. 102 Portfolio: Characterized by an allocation vector (α0, α1, . . . , αn)T with proportion αi amount invested on security i. Denote by α = (α1, . . . , αn)T and Rt = (R1,t, . . . , Rn,t)T . The proportion satisfies α0 + α1 + · · · + αn = α0 + 1T α = 1. Note some αi can be negative (a short position). Portfolio return: At time t + 1, the return of the portfolio is rt+1 = α0R0,t + αT Rt+1. Expected value and volatility: µt(α0, α) = Etrt+1 = α0R0,t + αT EtRt+1 = R0,t + αT EtY t+1,

  3. 103 where Y t+1 = Rt+1 − R0,t1 is the excess return, and σt2(α0, α) = αT Σtα = αT Vart(Y t+1)α, where Σt = Vart(Rt+1). Mean-variance approach: Maximize the expected return while minimizing the risk (Markowitz 1952, Sharpe 1963). Criterion: Subject to the constraint α0 + αT 1 = 1, A 2 2 max µt(α0, α) − α0,α σt (α0, α), where A measures the investor’s risk aversion. Remark: The above optimization problem is equivalent to max µt(α0, α) α0,α

  4. 104 subject to σt2(α0, α) ≤ B and α0 + αT 1 = 1. It is also equivalent to α0,α subject to µt(α0, α) ≥ C and α0 + αT 1 = 1. Let us now solve the first optimization problem. Using α0 = 1 − αT 1, we have A T αT EtY t+1 − max α α Vart(Y t+1)α + R0,t. 2 Optimal allocation: α∗ = 1 Vart(Y A )−1EtY t+1 and α0∗,t = t+1 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. Thus, the covariance matrix is given by 106     0.12 0.12          Σt = Vart(Rt+1) =   R  0.4 0.4     1.1 1.1   0.0144 0.0336 0.0528       0.0528 0.2200 1.2100 The optimal portfolio allocation is given     0.05 0.8722             0.50 0.2416 Suppose that an investor is willing to invest 20% in the riskless asset.

  7. Then, we have α1∗ + α2∗ + α3∗ = (0.8722 + 0.7346 + 0.2416)/A = 1.848/A = 0.8, 107 or     0.8722 0.3775 0.8    1.848         α∗ = =  0.3180  .   0.7346  0.2416 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 This portfolio has the variance α∗T Σtα∗ = 5.83%, better than individual stock in terms the mean-variance efficiency. Characteristics of efficient portifilio: With the optimal port- folio allocation, the expected return is µ∗t = R0,t + α∗t T EtY t+1 = R0,t + Pt/A, and the variance is given by σt∗2 = α∗t T vart(Y t+1)α∗t T = Pt/A2, 1/2 and 1 µ∗ = R0,t + Pt2σ∗. t t

  9. Sharpe ratio of the efficient portfolio: 109 µ∗t − Rt,0 ∗ 1/2 Sharpe ratio = = Pt . σt This gives excess return per unit risk. — expected excess gain divided by its standard deviation; — used to compare the efficiency of two portfolios; — related to risk-adjusted return of the Bank Trust. Efficient Frontier: For any other portfolio with the same risk αT Vart(Y t+1)α = (σt∗)2, its expected excess return is bounded by A T A ∗2 αT EtY t+1 = αT EtY t+1 − α Vart(Y t+1)α + σ 2 t 2 A ∗T 2 A ∗2 ≤ α∗T EtY t+1 − α Vart(Y t+1)α∗T + σ 2 t 1 = µ∗t = R0,t + Pt2 σt∗.

  10. Mean-variance efficient frontier 110 Mean-SD efficient frontier 4 4 3 3 mean mean 2 2 1 1 0 0 0 5 10 15 20 25 0 1 2 3 4 5 variance SD (a) (b) Figure 4.1: Mean-variance efficient frontiers. bounded by the efficient frontier. As the percentage of risky asset increases, the expected return increases (Figure 4.1).

  11. Sharpe Ratio: The sharp ratio for any portfolio is defined by 111 αT EtY t+1 (αT Vart(Y t+1)αT )1/2 S(α) = . Note that S(α) is independent of a scaling of α. Thus, µ∗t − Rt,0 ∗ αT EtY t+1/σt∗ ≤ max S(α) = max α . σt αT Vart(Y t+1)αT =σt∗2 Example 1 (Continued). The risk-adjusted returns for 4 assets are summarized as follows. Stock 0 1 2 3 Optimal return 5% 10% 25% 55% 18.48% E-return 0% 5% 20% 50% 13.48% risk 0% 12% 40% 110% 24.15% Sharpe Ratio - 0.417 0.500 0.455 0.558

  12. 112 The Sharpe ratio is maximized at the efficient portfolio. For the op- timal portfolio, Pt = 0.3113 = 0.5582. 4.2 Optimizing 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). Different investors have very different 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. 114 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 1.0 0.8 0.6 y 0.4 0.2 0.0 0 2 4 6 8 10 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 wt+1 = w + (R0,t + αT Y t+1)w = w(1 + R0,t) + wαT Y t+1. Thus, he wishes to maximize α0,α α0,α This is the same as minimizing the function Et exp(−A1αT Y t+1), where A1 = Aw. If the conditional distribution of the excess return is Y t+1 ∼ N (µt, Σt), then αT Y t+1 ∼ N (αT µt, αT Σtα). Thus,

  16. the expected utility is given by Et exp(−A1αT Y t+1) = exp{−A1αT µt + 116 A21 T α Σtα}. 2 This is the same as maximizing αT µt − A1αT Σtα, 2 and explains the portfolio optimization from the optimizing the ex- pected utility point of view.

  17. 117 4.3 The Capital Asset Pricing Model Assumption: Each investor trades at the mean-variance optimal portfolio, with the absolute risk aversion coefficient Ai t t A t i i i Equilibrium condition: Suppose that the total supply of shares at the period t is b on all assets. Then αD = 1 Σ−1µt = b ⇐⇒ µt = AΣtb. A (1) t t Market portfolio: Yt+1 = bT Y t+1 (excessive return of the port- folio of all invested wealth). From (1), it is a mean-variance efficient portfolio. m

  18. Theorem 3 In the linear regression Y t+1 = α + βYt+1 + εt+1 with Etεt+1 = 0 and Covt(εt+1, Yt+1) = 0, the intercept α = 0. Proof: Note that Cov(Y t+1, Yt+1) = βCov(Yt+1, Yt+1). It follows that 118 m m m m m m Covt(Y t+1, Yt+1) m ) Σtb b Σtb β = = T . Vart(Yt+1

  19. Hence, by (1), α = EtY t+1 − βEtYt+1 119 m Σtb b Σtb · bT AΣtb = AΣtb − T = 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 = aT Y t+1, the intercepts in the regression Y t+1 = α(a) + β(a)Yta+1 + εat+1 are 0 ⇐⇒ a is proportional to b, i.e. it is a mean-variance effi- cient portfolio.

  20. 120 CAPM(Sharpe-Lintner version): Y t = βYtm + εt — derived by Sharpe (1964) and Lintner (1965) with the existence of the risk-free asset; ♠ the excess return of i-th security Et−1Yit = βiEt−1Ytm; ♣ quantify exactly the relationship between risk and return; ♠ βi = Covt−1(Yit, Ytm)/Vart−1(Ytm) measures the cross-sectional risk of the asset; ♣ market risk premium Et−1Ytm > 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 Yit = αi + βiYtm + εit, t = 1, . . . , T. Application: ♠ Estimating covariance matrix: var(Y ) = ββT var(Yt+1)+var(εt+1), in which var(εt+1) can be assumed to diagonal. ♣ capital budgeting decisions in corporate finance; m

  22. 122 ♣ portfolio performance evaluation (mean-variance efficiency); The expected return of a firm: rf +β(rm−rf ), where rf is the average risk-free rate, rm is the average return. With this estimate, decision makers can decide whether or not to carry out an investment. 4.4 Validating 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 + βYtm + εt, Cov(Ytm, εt) = 0, Eεt = 0, Var(εt) = Σ, for t = 1, · · · , T periods. Testing against CAPM: H0 : α =0 Additional assumption: εt ∼ i.i.d.N (0, Σ).

  24. The conditional likelihood function, given Y1m, . . . , YTm, is 124 T −N/2|Σ|−21 f (Y 1, · · · , Y T |Y1m, · · · , YTm) = (2π) t=1 1 2 − (Y t − α − βYtm)T Σ−1(Y t − α − βYtm) . × exp The log-likelihood function is given by NT 2 T 2 (α, β, Σ) = − log(2π) − log |Σ| T t=1 1 2 (Y t − α − βYtm)T Σ−1(Y t − α − βYtm). −

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

  26. 126 — It is the covariance matrix of residuals from fitting OLS sep- arately. 2 Wald test: T0 = αT [Var(α)]−1α a 2 Exact distribution: Under the normal model, it can be shown that T − N − 1 NT T0 ∼ FN,T −N −1. T1 =

  27. Maximum likelihood ratio test: T2 = 2{max − max } H0 a where Σ0 is obtained under H0 with α = 0. It can be shown that 127 = T (log |Σ0| − log |Σ|) ∼ χ2N , T − N − 1 N (exp(T2/T ) − 1). T1 = — T1 and T2 are equivalent; — exact distribution of T2 can be found via T1; — adjusted version T − N/2 − 2 T a T2 ∼ χ2N T3 = has better performance at finite sample (see Table 3.1);

  28. 128 ♠ for practical purpose, T1 or T3 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 T1 has an exact F- distribution. Using this as the golden standard, with N = 10 and a P (T0 ≥ 18.31) ≈ 5%. On the other hand, the exact size of the test is T − N − 1 NT P (T0 ≥ 18.31) = P (T1 ≥ 18.31) = P (T1 ≥ 1.495). Since T1 ∼ F10,49, the actual probability is 17.0%. The approximation is very poor. The following Table shows the size of the tests.

  29. Table 4.1: Size of tests of the Sharpe-Lintner CAPM using asymptotic critical values. 129 N = 10 N = 20 N = 40 Time T0 T2 T3 T0 T2 T3 T0 T2 T3 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 4.5 Empirical 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 effect: 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. stocks traded at NYSE and ASE. Results: Table 4.2: Empirical results for tests of the Shape-Lintner version of the CAPM . 132 Time T1 p-value T2 p-value T3 p-value 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 1/65-12/94 2.159 0.020 21.612 0.017 21.192 0.020

  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. > y <- returns[60:179, 2:4] > x <- returns[60:179,1] #last 120 months e-return > 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.err t.stat p.value Intercept 0.9435 0.4817 1.9588 0.0525 X 1.0678 0.1053 10.1446 0.0000 > 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 T0 = 8.26, d.f. = 3, p-value = 4.1%. T1 = 2.66, d.f. = (3, 116), p-value = 5.1%.

  35. For modified maximum likelihood ratio test, we have 135 N T1 T − N − 1 T3 = (T − N/2 − 2) log 1 + = 7.527, with degree of freedom 3, giving a p-value of 5.69%. Results: Weak evidence against CAPM Market β for Ford: β1 = 1.1967. To predict the monthly return of a firm, according to CAPM: rf + β(rm − rf ). We need to use a longer time-horizon to compute rm and rf . This produce more stable prediction. Average log-monthly-return of SP500 over last 15yrs: rm = 0.7612%

  36. 136 Average risk-free rate over last 15 yrs: rf = 0.3750% Expected monthly return for Ford: 0.8372% Similar quantities for GE and John and John and GE can be computed Ford GE JNJ β 1.1967 1.0678 0.1225 Expected return (monthly) 0.8372% 0.7834% 0.4223% Remark∗: For the linear model, yi = a + bxi + εi, 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(¯rm − r¯f ), or R¯ = a + r¯f + b(¯rm − 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(¯rm − r¯f ) and differs 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 different period of data (e.g. 15 years data), the difference is hard to quantify. Cross-sectional regression∗ 4.6 Blume and Friend (1973) and Fama and McBeth (1973) introduced the following cross-sectional regressions for the Sharpe-Lintner version of CAPM. Note that µj = EYj,t = λβj, j = 1, . . . , N, where λ = EYtm > 0 (risk premium). T −1 Method: Let µj = T Yj,t and MLE Cov{(Yjt, Ytm), t = 1, · · · , T } βj = Var(Ytm, t = 1, · · · , T ) t=1 be the empirical estimate of µj and βj. Then, fit µj = a0 + a1βj + εj, j = 1, . . . , N

  38. CAPM: If the CAPM holds, the following three properties should be true. (i) a0 is statistically insignificant (ii) a1 is statistically positive (iii) Multiple R2 should be large Drawback: Errors-in-variables create biases. 138 4.7 Efficient-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. Difference: Don’t assume the existence of the risk-free bonds. The protfolio optimization in §3.1 is equivalent to min αT Σα, α

  39. subject to αT µ = µp and αT 1 = 1, where 1 = (1, . . . , 1)T . Lagrange multiplier method: Minimize 1 T 2 or solve Σα − λ1µ − λ21 = 0, where λ1 and λ2 are determined by 139 α Σα + λ1(µp − αT µ) + λ2(1 − αT 1). αT µ = µp and αT 1 = 1. Solution: α = g + µph, where g = D−1[BΣ−11 − AΣ−1µ], h = D−1[CΣ−1µ − AΣ−11], A = 1T Σ−1µ, B = µT Σ−1µ, C = 1T Σ−11, and D = BC − A2.

  40. Hence, the optimal variance and mean satisfy σp2 = (g + µph)T Σ(g + µph) or Cσp2 − C2/D · (µp − A/C)2 = 1, 140 (2) which defines an efficient frontier in the space of (σp, µp). 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+µph)T Σ(g+µqh) = C/D·(µp−A/C)(µq−A/C)+C−1. (3) ♣ For each minimum-variance portfolio, there exists a unique minimum-

  42. variance portfolio p0 with 142 A C D C2(µp − A/C) − µp0 = that has zero covariance with p. This can easily be obtained by setting (3) to zero and solving for µq. p0 is called the zero-beta portfolio with respect to p. ♠ From (2), we have σpdσp − C/D · (µp − A/C)dµp = 0. The slope at point p is given by dµp dσp σpD Cµp − A = .

  43. It can easily be verified that by (2) 143 σp2D Cµp − A D{C−1 − C/D · (µp − A/C)2} Cµp − A dµp dσp µp − σp = µp − = µp − = µp0. ♣ Consider a multiple regression of the return on any portfolio Ra on Rp and Rp0: Ra = β1 + β2Rp0 + β3Rp + εp, we have (homework) β3 = Cov(Ra, Rp)/σp2 ≡ βap, β2 = 1 − βap, β1 = 0, where βap is the beta of the portfolio with respect to portfolio p.

  44. In other words, ERa − ERp0 = βapE(Rp − Rp0). ♠ The slope in Fig 3.3 is the Sharpe type of ratio ♣ This tangent portfolio is called market portfolio. 144 4.8 Black version of CAPM In absence of the risk-free asset, Black(1972) derived the following CAPM. Notation: Rt — vector of returns of individual stock or portfolio;

  45. 145 Rtm — 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 Et−1Rt = γ1 + β(Et−1Rtm − γ). — Same as the Sharpe-Lintner version when γ =risk free rate Statistical model: Rt = α + βRtm + εt Var(εt) = Σ, and Cov(Rtm, εt) = 0 Eεt = 0, α = γ(1 − β) Black version of CAPM: H0 :

  46. Log-likelihood: 146 NT 2 T 2 (α, β, Σ) = − log(2π) − log |Σ| T t=1 1 2 (Y t − α − βRmt)T Σ−1(Y t − α − βRmt). − 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 H0: α = γ(1 − β). The MLE can not be explictly found. It solves the following equations: For each given γ and β0, the estimated covariance Σ0 is given by T Σ0 = T −1 [Rt − γ(1 − β0) − β0Rmt][Rt − γ(1 − β0) − β0Rmt]T . t=1

  47. For given Σ0 and β0, the estimated γ is ¯ 0 0 For given Σ0 and γ, T 147 γ = (1 − β0)T Σ−1(R − β0R¯m)/(1 − β0)T Σ−1(1 − β0). (Rtm − γ)(R − γ1)/ (Rtm − γ)2. ¯ β0 = t=1 MLR test: Under H0, 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 Rt = α + βRtm + εt to obtain α, β and Σ. 2. Using these estimates as the initial value, obtain ¯ 3. Compute β0 and then Σ0. 4. Iterate between steps 2 and 3 if needed (from statistical point of view, this step is optional). 5. Compute the test statistics T5. 6. Compute the P-value.

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