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Optimal Risky PortfolioPortfolio of 2 Risky Assets (Debt & Equity)Sumber ; Zvi Bodie, Alex Kane, Alan J.Marcus, Investments, 7th ed, McGraw-Hill, Boston, 2008, Chapter 6, pages 175, Chapter 7 pages 208-221. Diketahui :Table 7.1 : Descriptive statistics for two Mutual FundsDebt (D)reksadana pendapatan tetapEquity (E)

reksadana saham Expected Return, E ( r ) [ % ]813Standard Deviation, [ % ]1220Covariance,Cov(rD,rE)72Correlation coefficient,DE0.30T-bill rate ( rf ) = 5 %Coefficient of risk aversion ( A ) = 4Sumber : Bodie (2008 : 208)Pertanyaan : 1. Gambarkan indifference curve dan tabel utility values, titik D & titik E 2. Tentukan global minimum variance efficient portfolio dengan menghitung : a. covariance (rD , rE) b. alokasi dana reksadana pendapatan tetap / debt / vD c. alokasi dana reksadana saham / equity / vE d. expected return E(r) e. total risk 3. Buatah tabel koordinat efficient frontier versi sederhana (3 titik) yaitu : titik D dengan vD = 1, titik E dengan vD = 0, titik G dengan vD = sesuai jawaban soal 2b. 4. gambarkan efficient frontier berdasarkan tabel koordinat (jawaban soal 3).

Pertanyaan : 5. Tentukan optimal portfolio dengan menghitung : a. Alokasi dana reksadana pendapatan tetap / debt / wD b. Alokasi dana reksadana saham / equity / wE c. Expected return E(rP) d. Total risk P 6. Tentukan complete portfolio dengan menghitung : a.alokasi dana reksadana ( y ) b.Alokasi dana treasury bill (1 y) c.Expected return reksadana & treasury bill E(rC) d.Total risk reksadana & treasury bill C e.Alokasi dana reksadana pendapatan tetap (wD . y) f..Alokasi dana reksadana saham (wE . y) Pertanyaan : 7. Gambarkan Capital Allocation line ( CAL ) dengan menghitung : a. Slope of CAL b. Persamaan CAL c. Gambar CAL 8. Gabungkan CAL dengan efficient frontier. 9. Gambarkan Indifference Curve saat Complete portfolio10. Cantumkan Indifference Curve saat complete portfolio pada CAL11. Buatlah tabel utility values pada titik : a. D b. E c. G d. P e. C f. rF dan buatlah kesimpulan.

Expected return (%) Standard deviation (%) E D1. indifference curve12 0Indifference curve, A=4, U=0.0582013 Sumber : Bodie (2008 : 175), Figure 6.2Table 6.3 Utility valuestitikE(r)A2U=E(r) -1/2A2D0.080.1240.01440.0512E0.130.2040.040.05Indifference curve : Portfolio points withthe same utility value.2. Global minimum variance efficient portfolioSumber : Bodie (2008 : 214)Cov(rD,rE)=DEDE 2E - Cov(rD,rE)vD = -------------------------------- 2D + 2E - 2Cov(rD,rE)vE = 1 - wDE(r) = vD.E(rD)+vE.E(rE) = [v2D2D + v2E2E + 2vDvECov(rD,rE)]1/2Cov(rD,rE)= 0.3(12)(20)=72 400 - 72vD= ----------------------- = 0.82 144 + 400 - 2(72)vE = 1 0.82 = 0.18E(r) = 0.82(8)+0.18(13)=8.9% = [0.6724(144)+0.0324(400)+2(0.82)(0.18)(72)]1/2 = = (131.04)1/2 =11.447 %a.b.c.d.e.diketahuiglobal minimum variance portfolioTitikvDvEE(r) = vD.E(rD)+vE.E(rE) = [v2D2D + v2E2E + 2vDvECov(rD,rE)]1/2 D1.00.01.0(8)+0.0(13)=8.0[1.0(144)+0.0(400)+2(1.0)(0.0)(72)] 1/2 =12.00 G0.820.188.911.44E0.01.013.020.00Global minimum variance portfolio3.Tabel koordinat Efficient Frontier, versi sederhana (3 titik), = 0.3E( r )Expected return (%) Standarddeviation (%)8G & E = efficient portfolioD = Inefficient portfolioEfficient frontier, =0.3

4. Efficient frontier, = 0.3. versi sederhana 3 titik 011.44 D E G G global minimum variance portfolio 8.92013 12Efficient Portolio G = Minimum & E( r ) tertentuEfficient Portfolio E= Maximum E ( r ) & tertentuSumber : Bodie (2008 : 215) Pertanyaan : 5. Tentukan optimal portfolio dengan menghitung : a. Alokasi dana reksadana pendapatan tetap / debt / wD b. Alokasi dana reksadana saham / equity / wE c. Expected return E(rP) d. Total risk P 6. Tentukan complete portfolio dengan menghitung : a.alokasi dana reksadana ( y ) b.Alokasi dana treasury bill (1 y) c.Expected return reksadana & treasury bill E(rC) d.Total risk reksadana & treasury bill C e.Alokasi dana reksadana pendapatan tetap (wD . y) f..Alokasi dana reksadana saham (wE . y) 5a. Optimal portfolio : alokasi dana reksadana pendapatan tetap / debt [ wD ]

[E(rD) rf] 2E [E(rE) rf]Cov(rD,rE)wD = ------------------------------------------------------------------------ [E(rD) rf]2E+[E(rE) rf]2D-[E(rD)rf+E(rE)rf]Cov(rD,rE) [8 5]202 [13 5]72wD = ---------------------------------------------------------- = 0.40 [8 5]202 + [13 5] 122 - [8 5 + 13 5]72 [3]400 [8]72 1200-576 624wD = -------------------------------------- = ---------------------- = --------- = 0.4 [3]400 + [8]144 - [3 + 8]72 1200+1152-792 15605a. Optimal portfolio : alokasi dana reksadana pendapatan tetap / debt [ wD ]

[E(rD) rf] 2E [E(rE) rf]Cov(rD,rE)wD = ------------------------------------------------------------------------ [E(rD) rf]2E+[E(rE) rf]2D-[E(rD)rf+E(rE)rf]Cov(rD,rE) [0.08 0.05]0.202 [0.13 0.05]72wD = -------------------------------------------------------------------------------------------- = 0.40 [0.08 0.05]0.202 + [0.13 0.05] 0.122 - [0.08 0.05 + 0.13 0.05]72 [0.03]0.04 [0.08]72 0.0012-5.76 6.24wD = ---------------------------------------------------------- = -------------------------------- = -------- = 0.4 [0.03]0.04 + [0.08]0.0144 - [0.03 + 0.08]72 0.0012 + 0.001152-7.92 15.6 12-5.76 6.24wD = ----------------------- = ------------ = 0.4 12 + 11.52-7.92 15.6(angka merah kali 10.000)5b. Optimal portfolio : alokasi dana reksadana saham / equity. [ wE ] wE = 1 - wDwE = 1 0.40 = 0.60wE adalah alokasi dana optimal reksadana saham (equity)wD adalah alokasi dana optimal reksadana pendapatan tetap (debt)1 adalah .. ?

5c. Optimal portfolio : Expected return E(rP)E(rP) = wD.E(rD) + wE.E(rE)

E(rP) = 0.4 x 0.08 + 0.6 x 0.13 = 0.11E(rP) = 11 % atau E(rP) = 0.4 x 8% + 0.6 x 13% = 11 %

5d. Optimal portfolio : Total risk [ P ]P = [w2D2D + w2E2E +2wDwECov(rD,rE)]1/2 P = [0.42122 + 0.62202 +2(0.4)(0.6)72]1/2 P = [0.16(144) + 0.36(400) +2(0.4)(0.6)72]1/2P = [23.04 + 144 + 34.56]1/2 (angka merah dalam %)P = [201.6]1/2 P = 14.198 = 14.2 %Portfolio Variance 2P = w2D2D + w2E2E +2wDwECov(rD,rE)Portfolio Standard Deviation : P = 2P = [ 2P ]1/2 5d. Optimal portfolio : Total risk [ P ]P = [w2D2D + w2E2E +2wDwECov(rD,rE)]1/2 P = [0.42(0.12)2 + 0.62(0.20)2 +2(0.4)(0.6)72]1/2 P = [0.16(0.0144) + 0.36(0.04) +2(0.4)(0.6)72]1/2P = [0.002304 + 0.0144 + 34.56]1/2 (angka merah kali 10.000)P = [23.04 + 144 + 34.56]1/2 P = [201.6]1/2 P = 14.198 = 14.2 %Portfolio Variance 2P = w2D2D + w2E2E +2wDwECov(rD,rE)Portfolio Standard Deviation : P = 2P = [ 2P ]1/2 (10.000)1/2=1006a. Complete portfolio : Alokasi dana reksadana ( y ) E(rP) rfy = ------------- A2P y adalah alokasi dana Complete Portfolio reksadanaE(rP) adalah expected return portfoliorf adalah riskfree rate of returnA adalah coefficient of risk aversion. A = 42P adalah variance portfolio 0.11 - 0.05y = ---------------- = 0.7439 4 (0.142)2

y = 74.39%6a.Complete portfolio : Alokasi dana reksadana ( y ) E(rP) rfy = -------------- x 100 A2P y adalah alokasi dana Complete Portfolio reksadanaE(rP) adalah expected return portfoliorf adalah riskfree rate of returnA adalah coefficient of risk aversion. A = 42P adalah variance portfolio 11 5 6 y = ----------- x 100 = ----------- x 100 = 0.7439 4(14.2)2 806.56

y = 74.39% bila bukan decimal, rumus kali 1006b. Complete portfolio : Alokasi dana Treasury bill ( 1 y ) (1 y) = 1 0.7439 = 0.2561 atau 25.61 %

6c. Complete portfolio : expected return reksadana & treasury bill E(rC)E(rP) = 11%; P = 14.2% ; rf = 5%; y = 74.39%; (1 y) = 0.2561 With a proportion, y, in the risky portfolio and 1-y in the risk-free asset, the rate of return on the complete portfolio rC = yrP + (1-y)rf The expectation of complete portfolio rate of return E(rC) = yE(rP) + (1-y)rf E(rC) = yE(rP) + rf yrf E(rC) = y{E(rP) rf} + rf

Bodie (2008:178)E(rC) = rf + y{E(rP) rf} E(rC) = 0.05+0.7439(0.06) = 0.0946 = 9.46% 6d. Complete portfolio ; total risk reksadana & treasury bill CE(rP) = 11%; P = 14.2% ; rf = 5%; y = 74.39%; (1 y) = 0.2561

When we combine a risky asset and a risk-free asset in a portfolio, the standard deviation of complete portfolio is C = y P C= 0.7439(0.142) = 0.1056 = 10.56%C = y PBodie (2008:178)6e. Complete portfolio : alokasi dana reksadana pendapatan tetap (wD . y)wD . y = 0.4(0.7439 ) = 0.2976 atau 29.76 %

6f.Complete portfolio : alokasi dana reksadana saham (wE . y)wE . y = 0.60(0.7439)=0.4463 atau 44.63 %

Pertanyaan : 7. Gambarkan Capital Allocation line ( CAL ) dengan menghitung : a. Slope of CAL b. Persamaan CAL c. Gambar CAL 8. Gabungkan CAL dengan efficient frontier. 9. Gambarkan Indifference Curve saat Complete portfolio10. Cantumkan Indifference Curve saat complete portfolio pada CAL11. Buatlah tabel utility values pada titik : a. D b. E c. G d. P e. C f. rF dan buatlah kesimpulan.

7a.Capital Allocation Line : Slope of CAL E(rP) rfSP = ------------ P

11 5SP = ------------ = 0.42 14.2 7b.Capital Allocation Line : Persamaan CALE( r ) = Rf + SP P

E( r ) = 5 + 0.42 P

E( r )Expected return (%) Standarddeviation (%)Capital Allocation LineE( r ) = 5 + 0.42 P11 14.2 PP = Optimal risky portfolio7c. Capital Allocation Line : Gambar CAL 0 rF5E( r )Expected return (%) Standarddeviation (%)8G & E = efficient portfolioD = Inefficient portfolioEfficient frontier, =0.3

4. Efficient frontier, = 0.3. versi sederhana 3 titik 011.44 D E G G global minimum variance portfolio 8.92013 12Efficient Portolio G = Minimum & E( r ) tertentuEfficient Portfolio E= Maximum E ( r ) & tertentuSumber : Bodie (2008 : 215) E( r )Expected return (%) Standarddeviation (%)8Capital Allocation LineE( r ) = 5 + 0.42 P11 14.2 PP = Optimal risky portfolioG & E = efficient portfolioD = Inefficient portfolioEfficient frontier8. CAL & efficient frontier 0 rF11.44 D E G G global minimum variance portfolio 58.92013 12E( r )Expected return (%) Standarddeviation (%)C = Complete portfolio

Indifference curveU = E(rC) -1/2A2C9. Indiference curve saat Complete portfolio 010.569.46CE( r )Expected return (%) Standarddeviation (%)8Capital Allocation LineE( r ) = 5 + 0.42 P11 14.2 PP = Optimal risky portfolioC = Complete portfolioG & E = efficient portfolioD = Inefficient portfolioEfficient frontierIndifference curveU = E(rC) -1/2A2C10 CAL, efficient frontier & Indiference curve 010.569.46 rF11.44 D E G G global minimum variance portfolio 58.92013 12C11. Tabel utility values

Tabel Utility valuesTitikportfolioE(r)A2U=E(r) -1/2A2DEfficient0.080.1240.01440.0512EEfficient0.130.2040.040.05GGlobal Minimum Variance Efficient0.0890.114440.013087360.088POptimal0.110.14240.0201640.109CComplete0.09460.105640.011151360.072rFRisk-free0.050400.05Kesimpulan : Utility tertinggi adalah pada saat optimal portfolioSumber : Bodie (2008 : 175), Figure 6.2Tabel 7.1 : Global minimum variance efficient , optimal, complete portfolioDIKETAHUIrD 8%rE 13%D 12%E 20%DE 0.3Covariance,Cov(rD,rE)=DEDE 72T-bill rate ( rf ) 5%Coefficient of risk aversion ( A ) 41Indifference curve & table untiulity values2aCovariance,Cov(rD,rE)=DEDE Cov(rD,rE)= 0.3(12)(20)=72b 2E - Cov(rD,rE)vD = -------------------------------- 2D + 2E - 2Cov(rD,rE) 400 - 72vD= ----------------------- = 0.82 144 + 400 - 2(72)cvE = 1 - wDvE = 1 0.82 = 0.18dE(r) = vD.E(rD)+vE.E(rE)E(r) = 0.82(8)+0.18(13)=8.9%e = [v2D2D + v2E2E + 2vDvECov(rD,rE)]1/2= [0.6724(144)+0.0324(400) + 2(0.82)(0.18)(72)]1/2 = = (131.04)1/2 =11.447 %3Tabel koordinat efficient frontier4Efficient frontier5a [E(rD) rf] 2E [E(rE) rf]Cov(rD,rE)wD = -------------------------------------------------------------------------- [E(rD) rf]2E+[E(rE) rf]2D-[E(rD)rf+E(rE)rf]Cov(rD,rE) [8 5]202 [13 5]72wD = ---------------------------------------------------------- = 0.40 [8 5]202 + [13 5] 122 - [8 5 + 13 5]72 bwE = 1 - wDwE = 1 0.40 = 0.60cE(rP) = wD.E(rD) + wE.E(rE)E(rP) = 0.4 x 0.08 + 0.6 x 0.13 = 0.11 = 11 %dP = [w2D2D + w2E2E +2wDwECov(rD,rE)]1/2

P = [0.42122 + 0.62202 +2(0.4)(0.6)72]1/2 P = [0.16(144) + 0.36(400) +2(0.4)(0.6)72]1/2P = [23.04 + 144 + 34.56]1/2 (angka merah dalam %)P = [201.6]1/2 = 14.198 = 14.2 %6a E(rP) rfy = ------------- A2P 0.11 - 0.05y = ---------------- = 0.7439 = 74.39% 4 (0.142)2 bTreasury bill = (1 y) (1 y) = 0.2561 atau 25.61 %cE(rC) = rf + y{E(rP) rf} E(rC) = 0.05+0.7439(0.06) = 0.0946 = 9.46% dC = y PC= 0.7439(0.142) = 0.1056 = 10.56%ewD . y = 0.4(0.7439 )= 0.2976 atau 29.76 %fwE . y = 0.60(0.7439)=0.4463 atau 44.63 %7a E(rP) rfSlope CAL, SP = -------------- P 11 5SP = ------------ = 0.42 14.2 bPersamaan CAL, E( r ) = Rf + SP PE( r ) = 5 + 0.42 PcGambar CAL8CAL & Efficient Frontier9Indifference Curve saat complete portfolioU = E(rC) -1/2A2CU = 9.46% - (4)(10.56%)2 = 0.0710CAL, Efficient Frontier, Indifference Curve11Tabel utility valuesExpected return (%) Standard deviation (%) E D12 0A=4, U=0.0582013 Sumber : Bodie (2008 : 175), Figure 6.2 the indifference curveAU=E(r) -1/2A220.0930.0740.05A=3, U=0.07A=2, U=0.09Indifference mapDiketahui : E(rD)=8%; E(rE)=13%; 2D=144%; 2E=400%; Cov(rD,rE) = 72%TitikvDvEE(r) = vD.E(rD)+vE.E(rE) = [v2D2D + v2E2E + 2vDvECov(rD,rE)]1/2 D1.00.01.0(8)+0.0(13)=8.0[1.0(144)+0.0(400)+2(1.0)(0.0)(72)] 1/2 =12.00 0.90.10.9(8)+0.1(13)=8.5[0.81(144)+0.01(400)+2(0.9)(0.1)(72)] 1/2 =11.55G0.820.188.911.440.80.29.011.290.70.39.511.48J0.60.410.012.030.50.50.5(8)+0.5(13)=10.5[0.25(144)+0.25(400)+2(0.5)(0.5)(72)] 1/2 =13.110.40.611.013.99P11.014.2H0.30.711.515.300.20.812.016.760.10.912.518.34E0.01.013.020.00Global minimum variance portfolioTabel koordinat Efficient Frontier, DE= 0.3 versi lengkap Optimal portfolioExpected return (%) Standard deviation (%) E D4. Efficient frontier, = 0.3 0Efficient frontier8.92013 Sumber : Bodie (2008 : 215) Efficient Portolio G = Minimum & E( r ) tertentuEfficient Portfolio E= Maximum E ( r ) & tertentu812=0.3G = Global minimum variance efficient portfolio11.44 GD = Inefficient portfolioH H = Efficient portfolio J 10 Expected return (%) Standard deviation (%) E DFigure 7.5 : Efficient Frontier E(r) = f( ) 0Efficient frontier102013 Sumber : Bodie (2008 : 215) Efficient Portolio = portfolio dengan minimum & E( r ) tertentu atau portfolio dengan maximum E ( r ) & tertentu812=1=0.3=-1=0.25Efficient frontier :Portfolio efficient points