MP03-Optimal Portfolio 09

7/31/2019 MP03-Optimal Portfolio 09

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Optimum Portfolio 1

Selecting optimal portfolio

lIndifference curve and efficient

frontierlSingle index model


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Optimum Portfolio 3

Risk aversion and Indifference map

E(r)

A

B

Figure 1. Indifference curve with different

risk-aversion (A is more risk-

averse than B)

E(r)

IC1

IC2

Figure 2. Indifference map. IC1 gives higher

utility than IC2


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Optimum Portfolio 5

How to identify and graph indifference curve

of investor?

l The problem with indifference curve is it is a curvilinear, hence it

is not possible to draw it just by using two points. Therefore

Sharpe modify the horizontal axis into variance of return rather

than standard deviation of return. By doing so, the curvilinear ofIC could be changed into linear line.

l Then we could ask an investor to select two investment

opportunities (among so many opportunities) that equally

attractive. If he or she could do that, then we would be able to

identify and graph his or her indifference curvel Therefore in practice it is often done indirectly by estimating the

investors level of risk tolerance, denoted by .


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Optimum Portfolio 6

Suppose we offer the investor the following portfolios,

and ask him or her to select the desirable portfolio

Proportion of

risk free asset (%)

Proportion of

risky asset (%)

E(r)

(%)

(%)

100

90

8070

60

50

40

3020

10

0

0

10

2030

40

50

60

7080

90

100

13.00

14.20

15.4016.60

17.80

19.00

20.20

21.4022.60

23.80

25.00

0

3

69

12

15

18

2124

27

30


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Optimum Portfolio 7

Indifference curve . . . . (contd)

l Since the IC now has changed into a straight line, theline could be expressed as Y = a + bX where Y is theE(r) and X is the 2. At the point of tangency (i.e. the

portfolio has been selected by the investor), the slopeof efficient frontier and the IC is identical. The slope ofIC (i.e. the b) is equal to 1/, or b = 1/. Whereas thevalue of is (Sharpe and Alexander, 1990,Investment, p.718 or the newest edition),

2[(rC - rf)]S2

(rS - rf)2


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Optimum Portfolio 8

Indifference curve . . . . (contd 2)

l Where

rC is the expected return selected by the investor

rS is the expected return of the risky assets

rf is the risk free rate of return

S2 is the variance of the risky assets

(If the portfolio selected is 60% risk free and 40% risky assets,

what is the value?)

l Since the value of Y, X, and b are known, then the value ofacould be calculated. Assuming that the investor has a constant

risk aversion , the indifference map of the investor could be

drawn (For the selected portfolio, how is the equation of the IC?)


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Optimum Portfolio 9

Reason to introduce SIM

l Notice that for less (more) risk-averse investor, he orshe would select higher (lower) risk portfolio.

l The problem with the Mean-Variance Model is that the

portfolio risk depends on standard deviation ofindividual stock and the correlation among thosestocks. Therefore it is rather difficult to apply the Modelsince the correlation between pairs of stocks would bevery high if the number of stocks in the portfolio is

rather big (say 20-30 stocks).l To deal with this problem, the Single Index Model is

introduced.


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Optimum Portfolio 10

The equation of IC is;

E(r) = 15.4 + (1/60)2

IC and EF

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

SD

Return (IC)

(EF)


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Single Index Model (SIM)

l Single Index model (SIM) is a model used to simplify portfolioanalysis, particularly in estimating the portfolio risk. The mean-

variance model requires estimate of correlation matrix among

securities returns in order to estimate the portfolio risk.

However;q

the number of correlation between the pairs of securities increasessignificantly as the number of stocks increases. The number of

correlation follows the following formula,

ij = [(N(N-1)]/2, where N is the number of stock in portfolio.

q Moreover, using historical correlation is rather unreliable since

correlation usually is not stable over time.

l The idea of the SIM is that there must be a factor that affect all

securities returns (or excess returns). The factor selected

usually is market return (or excess market return).


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SIM: return or excess return?l Elton & Gruber, 2003 (Modern Portfolio Theory and Investment

Analysis), use return in the model, while Bodie, Kane, and Marcus,

2009 (Investments) use excess return.

l Elton and Gruber use the following formula.

ri = ai + irm + ei. Where ri is return of stock i, and rm is market return.The formula for individual security and portfolio could be compared as

follows

Individual security Portfolio .

E(r) E(ri ) = ai + iE(rm ) E(r p ) = ap + pE(rm )

Variance i2 = i2m2 + ei2 p2 = p2m2 + Xi2ei2

Covariance ij = ijm2


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SIM: return or excess return? (contd)

l Bodie, Kane and Marcus (2009) use the following formula.

Ri = i + iRm + ei. Where Ri is excess return of stock i(where Ri = ri

rf), and Rm is excess return of market (where RM = rM rf).

The formula for individual security and portfolio could be compared as

follows

Individual security Portfolio .

E(R) E(Ri ) = i + iE(Rm ) E(Rp ) = p + pE(Rm )

Variance i2

= i2

m2

+ ei2

p2

= p2

m2

+ Xi2

ei2

Covariance ij = ijm2


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Diversification reduces risk

l The equation for variance of portfolio p2 = p

2m2 + Xi

2ei2 is

also interpreted as follows,

p2 = Total risk

p2m

2 = Systematic risk

Xi2ei

2 = Unsystematic risk

Notice that if we invest with equal proportion (xi = 1/N), the

variance of portfolio would be

p2 = p2m2 + (1/N)[(1/N)ei2]

If N approaches infinity then

p2 = p

2m2 . In other words, unsystematic risk could be

diversified away by increasing the number of stocks in portfolio


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Figure 3. Diversification reduces risk

Av. SD

Unsystematic risk

Total risk

Systematic riskN in portfolio


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Portfolio risk depends on individual stock

beta

l The equation p2 = p

2m2 also show that Portfolio risk

depends on individual stock beta since portfolio beta is

simply the weighted average of individual stocks in the

portfolio.

l Thus if an investor would like to have low (high) p2 ,

he (or she) should form a portfolio consisting stocks

with low (high) betas


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What factors affecting Beta?

lBeta is a measure of equity risk. Therefore it

is affected by;q

Business risk (Brealey, Myers, and Allen, 2006) Measured by cyclicality of sales. How sensitive sales is

affected by macro economic conditions.

Measured by operating leverage. What is the proportion

of fixed cost in cost structure?

q Financial risk Measured by financial leverage. How much firm borrow?


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How to estimate Beta by using SIM?

lIf we use Elton and Gruber model, simply

regress returns of stock i(ri,t) to returns of

market portfolio (rm,t). The regressioncoefficient represents beta of the stock.

lIf we use Bodie, Kane and Marcus model,

simply regress excess returns of stock i(Ri,t

)

to excess returns of market portfolio (Rm,t).

The regression coefficient represents beta of

the stock.


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Beta estimation by using returns and excess returns:

A comparison.

lThe following is beta estimation;q Using returns, i.e. ri = ai + irm + ei, and

q Using excess returns, i.e. Ri = i + iRm + ei.


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Table 1.

Price and Returns of ASII and BBRI, together with Market Index (IHSG) and

Return of the Index, Jan 2006 Dec 2007

Date ASII r.ASII BBRI r.BBRI IHSG r.IHSG

2006.01

2006.02

2006.03

2006.04

2006.05

2006.06

2006.07

2006.08

2006.09

2006.10

2006.112006.12

10,400

9,800

11,450

11,950

9,800

9,750

9,600

11,100

12,450

13,400

15,95015,700

-0.0594

0.1556

0.0427

-0.1983

-0.0051

-0.0155

0.1451

0.1147

0.0735

0.1742

-0.0158-0.0557

3,400

3,250

3,975

4,625

3,950

4,100

4,275

4,350

4,900

4,900

5,3505,150

-0.0451

0.2013

0.1514

-0.1577

0.0372

0.0418

0.0174

0.1190

0.0000

0.0879

-0.03810.0287

1,232.32

1,230.66

1,322.97

1,464.41

1,330.00

1,310.26

1,351.65

1,431.26

1,534.61

1,582.63

1,718.961,805.52

-0.0013

0.0723

0.1015

-0.0962

-0.0149

0.0311

0.0572

0.0697

0.0308

0.0826

0.0491-0.0271


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Table 1.

Price and Returns of ASII and BBRI, . . . . . .

(contd)

Date ASII r.ASII BBRI r.BBRI IHSG r.IHSG

2007.01

2007.02

2007.03

2007.04

2007.05

2007.06

2007.07

2007.08

2007.09

2007.10

2007.112007.12

2008.01

14,850

14,050

13,200

14,400

16,400

16,900

18,750

17,850

19,250

25,600

25,00027,300

27,250

-0.0554

-0.0624

0.0870

0.1300

0.0300

0.1039

-0.0492

0.0755

0.2850

-0.0237

0.0880-0.0018

na

5,300

4,750

5,050

5,250

6,100

5,750

6,300

6,250

6,600

7,750

7,8007,400

7,000

-0.1095

0.0612

0.0388

0.1500

-0.0591

0.0913

-0.0079

0.0545

0.1606

0.0064

-0.0526-0.0555

na

1,757.26

1,740.97

1,830.92

1,999.17

2,084.32

2,139.28

2,348.67

2,194.34

2,359.21

2,643.49

2,688.332,745.83

2,627.00

-0.0093

0.0503

0.0879

0.0417

0.0260

0.0933

-0.0679

0.0724

0.1137

0.0168

0.0211-0.0442

na


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Using the data on Table 1, the estimates of Beta of ASII and BBRI

are presented in the following print-outs. Note that market index is

represented by IHSG.

LS // Dependent Variable is r.ASII

Date: 4-23-2009 / Time: 10:25

SMPL range: 2006.01 - 2007.12

Number of observations: 24

========================================================================VARIABLE COEFFICIENT STD. ERROR T-STAT. 2-TAIL SIG.

========================================================================

C -0.0067704 0.0154951 -0.4369395 0.6664

r.IHSG 1.4872146 0.2501852 5.9444550 0.0000

========================================================================

R-squared 0.616301 Mean of dependent var 0.040135

Adjusted R-squared 0.598860 S.D. of dependent var 0.103150

S.E. of regression 0.065330 Sum of squared resid 0.093898

Log likelihood 32.46872 F-statistic 35.33654

Durbin-Watson stat 2.188026 Prob(F-statistic) 0.000006

========================================================================


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The results show that ASII Beta is 1.49, while BBRI is 1.20. Notice

that statistically the betas are significant at 5% level.

LS // Dependent Variable is r.BBRI

Date: 4-23-2009 / Time: 10:25

SMPL range: 2006.01 - 2007.12


========================================================================VARIABLE COEFFICIENT STD. ERROR T-STAT. 2-TAIL SIG.

========================================================================

C -0.0078660 0.0146464 -0.5370592 0.5966

r.IHSG 1.2034154 0.2364811 5.0888430 0.0000

========================================================================






========================================================================


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Table 2.

Excess Returns (ri rf) of ASII, BBRI, and the Market (IHSG), Jan 2006 Dec

2007.

Risk free is assumed constant of 0.75% per month.

Date R.ASII R.BBRI R.IHSG

2006.01

2006.02

2006.03

2006.04

2006.05

2006.06

2006.07

2006.08

2006.09

2006.10

2006.11

2006.12

-0.066923

0.148107

0.035242

-0.205849

-0.012615

-0.023004

0.137682

0.107276

0.066034

0.166704

-0.023298

-0.063161

-0.052620

0.193870

0.143952

-0.165261

0.029771

0.034297

0.009892

0.111559

-0.007500

0.080361

-0.045600

0.021210

-0.008848

0.064829

0.094073

-0.103774

-0.022453

0.023600

0.049729

0.062221

0.023312

0.075131

0.041629

-0.034593


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Table 2.

Excess Returns (ri rf) of ASII, BBRI, and the Market (IHSG),

. . . . . . . . . (contd)

Date R.ASII R.BBRI R.IHSG

2007.01

2007.02

2007.03

2007.04

2007.05

2007.06

2007.07

2007.08

2007.09

2007.10

2007.11

2007.12

-0.062877

-0.069906

0.079511

0.122553

0.022532

0.096380

-0.056690

0.068008

0.277581

-0.031217

0.080511

-0.009333

-0.117062

0.053744

0.031340

0.142561

-0.066589

0.083850

-0.015468

0.046988

0.153123

-0.001069

-0.060144

-0.063070

-0.016813

0.042876

0.080414

0.034211

0.018527

0.085880

-0.075468

0.064945

0.106273

0.009320

0.013663

-0.051741


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Beta of ASII and BBRI are estimated by using excess return.

Notice that they are identical with the model that uses return. The

betas are identical if we have risk free rate.

LS // Dependent Variable is R.ASII

Date: 4-23-2009 / Time: 10:27

SMPL range: 2006.01 - 2007.12


========================================================================

VARIABLE COEFFICIENT STD. ERROR T-STAT. 2-TAIL SIG.

========================================================================

C -0.0031163 0.0146290 -0.2130235 0.8333

R.IHSG 1.4872146 0.2501852 5.9444549 0.0000

========================================================================






========================================================================


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Similar to the previous model, the betas are significant at 5%.

LS // Dependent Variable is R.BBRI

Date: 4-23-2009 / Time: 10:28

SMPL range: 2006.01 - 2007.12


========================================================================

VARIABLE COEFFICIENT STD. ERROR T-STAT. 2-TAIL SIG.========================================================================

C -0.0063404 0.0138277 -0.4585257 0.6511

X.IHSG 1.2034154 0.2364811 5.0888430 0.0000

========================================================================






========================================================================

MP03-Optimal Portfolio 09

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