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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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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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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Optimum Portfolio 13
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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Optimum Portfolio 14
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
Number of observations: 24
========================================================================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
========================================================================
R-squared 0.540675 Mean of dependent var 0.030089
Adjusted R-squared 0.519796 S.D. of dependent var 0.089112
S.E. of regression 0.061752 Sum of squared resid 0.083893
Log likelihood 33.82072 F-statistic 25.89632
Durbin-Watson stat 2.881297 Prob(F-statistic) 0.000042
========================================================================
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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
Number of observations: 24
========================================================================
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
========================================================================
R-squared 0.616301 Mean of dependent var 0.032635
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 2.188027 Prob(F-statistic) 0.000006
========================================================================
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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
Number of observations: 24
========================================================================
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
========================================================================
R-squared 0.540675 Mean of dependent var 0.022589
Adjusted R-squared 0.519796 S.D. of dependent var 0.089112
S.E. of regression 0.061752 Sum of squared resid 0.083893
Log likelihood 33.82072 F-statistic 25.89632
Durbin-Watson stat 2.881297 Prob(F-statistic) 0.000042
========================================================================
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