Chapter 4 Eﬃcient Portfolios and CAPM

Chapter 4

E cient Portfolios and CAPMffi

4.1 E cient Portfoliosffi

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

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,

σt (α0, α),

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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,

max µt(α0, α) −α0,α

A 22

where A measures the investor’s risk aversion.

Remark: The above optimization problem is equivalent to

max µt(α0, α)α0,α

A Tα Vart(Y t+1)α

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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

maxα αT EtY t+1 − 2 + R0,t.

tOptimal allocation: α ∗ = 1 Vart(YA t+1 )−1EtY t+1 and α0 ,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%,

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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

Σt = Vart(Rt+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

0.05

0.8722

0.50 0.2416

Suppose that an investor is willing to invest 20% in the riskless asset.

0.8 0.7346 = 0.3180 .

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Then, we have

α1 ∗ + α2 ∗ + α3 ∗ = (0.8722 + 0.7346 + 0.2416)/A = 1.848/A = 0.8,

or

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%.

108

t t

This portfolio has the variance

α T ∗ Σtα ∗ = 5.83%,

better than individual stock in terms the mean-variance e ciency.ffiCharacteristics of e cient portifilioffi : 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

and1

µ ∗ = R0,t + Pt2σ∗.

σt= Pt

A T A 2∗

2 t

α Vart(Y t+1)α T ∗ +A 2∗

2 t

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Sharpe ratio of the e cient portfolio:ffi

Sharpe ratio = µ t ∗ − Rt,0∗

1/2 .

This gives excess return per unit risk.

— expected excess gain divided by its standard deviation;

— used to compare the e ciency of two portfolios;ffi

— related to risk-adjusted return of the Bank Trust.

E cient Frontierffi : For any other portfolio with the same risk

αT Vart(Y t+1)α = (σt∗)2, its expected excess return is bounded by

αT EtY t+1 = αT EtY t+1 −2

α Vart(Y t+1)α + σ

≤ α T ∗ EtY t+1 −A T∗

2σ

1

= µ t ∗ = R0,t + Pt2 σt∗.

4 4

0

1

mean2

3

0

1

mean2

3

(a) (b)

110

variance

0 5 10 15 20 25

Mean-variance efficient frontier

SD

0 1 2 3 4 5

Mean-SD efficient frontier

Figure 4.1: Mean-variance e cient frontiers.ffi

bounded by the e cient frontierffi . As the percentage of risky

asset increases, the expected return increases (Figure 4.1).

αT Vart(Y t+1)αT =σt 2∗ σt

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Sharpe Ratio: The sharp ratio for any portfolio is defined by

S(α) = αT EtY t+1

(αT Vart(Y t+1)αT )1/2.

Note that S(α) is independent of a scaling of α. Thus,

max S(α) = maxα αT EtY t+1/σt ∗ ≤

µ t ∗ − Rt,0∗ .

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

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The Sharpe ratio is maximized at the e cient portfolio. For the op-ffi

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:

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• 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 erent investors have very di erent attitude:ff ff

— (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.

1.0

y

0.0

0.2

0.4

0.6

0.8

114

0 2 4 6 8 10

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.

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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 (µ∼ t, Σt), then αT Y t+1 N (α∼ T µt, αT Σtα). Thus,

A21 T

116

the expected utility is given by

Et exp(−A1αT Y t+1) = exp{−A1αT µt + 2 α Σtα}.

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.

117

4.3 The Capital Asset Pricing Model

Assumption: Each investor trades at the mean-variance optimal

portfolio, with the absolute risk aversion coe cient Affi i

t t A ti i

i

Equilibrium condition: Suppose that the total supply of shares

at the period t is b on all assets. Then

t tαD = 1 Σ−1µt = b µ⇐⇒ t = AΣtb.A(1)

mMarket portfolio: Yt+1 = bT Y t+1 (excessive return of the port-

folio of all invested wealth). From (1), it is a mean-variance e cientffiportfolio.

Vart(Yt+1= T

118

m

m

m m m

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

β =m

Covt(Y t+1, Yt+1)m )

Σtb

b Σtb.

= AΣtb − T

119

m

Hence, by (1),

α = EtY t+1 − βEtYt+1Σtb

b Σtb· bT AΣtb

= 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 e -⇐⇒ fficient portfolio.

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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.

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.

m

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;

122

♣ portfolio performance evaluation (mean-variance e ciency);ffi

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.

123

Statistical Model: Y t is a vector of excess returns of N assets. It

follows the linear model

Y t = α + βYtm + εt,

Eεt = 0, Var(εt) = Σ, Cov(Ytm, εt) = 0,

for t = 1, · · · , T periods.

Testing against CAPM: H0 : α =0

Additional assumption: εt i.i.d.N (0, Σ).∼

− (Y t − α − βYtm)T Σ−1(Y t − α − βYtm) .

124

The conditional likelihood function, given Y1m, . . . , YTm, is

f (Y 1, · · · , Y T |Y1m, · · · , YTm) =

T

t=1

(2π)−N/2|Σ|−21

× exp12

The log-likelihood function is given by

(α, β, Σ) = −NT

2log(2π) −

T2

log |Σ|

−12

T

t=1(Y t − α − βYtm)T Σ−1(Y t − α − βYtm).

125

β =

MLE for α and β:

α = Y¯ − βY¯m,T

t=1(Y t − Y¯ )(Ytm − Y¯m)/

T

i=1(Ytm − Y¯m)2,

−1T

t=1Y t and Y¯m = T−1

T

t=1Ytm.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 isT

t=1(Y t − α − βYtm)(Y t − α − βYtm)T .

126

— It is the covariance matrix of residuals from fitting OLS sep-

arately.

2

Wald test:

T0 = αT [Var(α)]−1αa

2

Exact distribution: Under the normal model, it can be shown that

T1 =T − N − 1

NT T0 F∼ N,T −N −1.

= T (log |Σ0| − log |Σ|) χ∼ 2N ,

T2 χ∼ 2N

127

Maximum likelihood ratio test:

T2 = 2{max − max }H0

a

where Σ0 is obtained under H0 with α = 0. It can be shown that

T1 =T − N − 1

N (exp(T2/T ) − 1).

— T1 and T2 are equivalent;

— exact distribution of T2 can be found via T1;

— adjusted version

T3 =T − N/2 − 2

Ta

has better performance at finite sample (see Table 3.1);

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 anda

P (T0 ≥ 18.31) ≈ 5%.

On the other hand, the exact size of the test is

P (T0 ≥ 18.31) = P (T1 ≥T − N − 1

NT 18.31) = P (T1 ≥ 1.495).

Since T1 F∼ 10,49, the actual probability is 17.0%. The approximation

is very poor. The following Table shows the size of the tests.

129

Table 4.1: Size of tests of the Sharpe-Lintner CAPM using asymptotic critical values.

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.

130

1. Basu(1977) reported PE e ectff : 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:

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

132

stocks traded at NYSE and ASE. Results:

Table 4.2: Empirical results for tests of the Shape-Lintner version of the CAPM .

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

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

134

> ls.print(lsfit(x,y[,2])) # GE




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




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%.

135

For modified maximum likelihood ratio test, we have

T3 = (T − N/2 − 2) log 1 +N T1

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: 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%

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 di ers from the actual averageff

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 erent period of data (e.g. 15 years data), the di erence is hard to quantify.ff ff

4.6 Cross-sectional regression∗

Method: Let µj = 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

µj = EYj,t = λβj, j = 1, . . . , N,

where λ = EYtm > 0 (risk premium).T

t=1

Yj,t and MLE

Cov{(Yjt, Ytm), t = 1, · · · , T }βj =

Var(Ytm, t = 1, · · · , T )

be the empirical estimate of µj and βj. Then, fit

µj = a0 + a1βj + εj, j = 1, . . . , N

138

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.

4.7 E cient-set Theoryffi

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 erenceff : Don’t assume the existence of the risk-free bonds.

The protfolio optimization in §3.1 is equivalent to

min αT Σα,α

α Σα + λ1(µp − αT µ) + λ2(1 − αT 1).

139

subject to αT µ = µp and αT 1 = 1, where 1 = (1, . . . , 1)T .

Lagrange multiplier method: Minimize

1 T2

or solve

Σα − λ1µ − λ21 = 0,

where λ1 and λ2 are determined by

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.

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 e cient frontier ffi 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

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-

142

variance portfolio p0 with

µp0 =AC

−D

C2(µp − A/C)

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=

σpDCµp − A

.

143

It can easily be verified that by (2)

µp −dµp

dσpσp = µp −

= µp −

σp2DCµp − AD{C−1 − C/D · (µp − A/C)2}

Cµp − A= µp0.

♣ Consider a multiple regression of the return on any portfolio Ra on

Rp and Rp0:

Ra = β1 + β2Rp0 + β3Rp + εp,

we have (homework)

β1 = 0, β2 = 1 − βap, β3 = Cov(Ra, Rp)/σp2 ≡ βap,

where βap is the beta of the portfolio with respect to portfolio p.

144

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.

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;

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

Eεt = 0, Var(εt) = Σ, and Cov(Rtm, εt) = 0

Black version of CAPM: H0 : α = γ(1 − β)

146

Log-likelihood:

(α, β, Σ) = −NT

2log(2π) −

T2

log |Σ|

−12

T

t=1(Y t − α − βRmt)T Σ−1(Y t − α − βRmt).

Σ0 = 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 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 byT

t=1[Rt − γ(1 − β0) − β0Rmt][Rt − γ(1 − β0) − β0Rmt]T .

γ = (1 − β0)T Σ−1(R − β0R¯m)/(1 − β0)T Σ−1(1 − β0).

(Rtm − γ)(R − γ1)/

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β0 =

For given Σ0 and β0, the estimated γ is

¯0 0

For given Σ0 and γ,T

¯ (Rtm − γ)2.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:

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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.

Chapter 4 Eﬃcient Portfolios and CAPM

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Transcript of Chapter 4 Eﬃcient Portfolios and CAPM