Optimal Risky PortfolioPortfolio of 2 Risky Assets (Debt & Equity)

Bacaan :Zvi Bodie, Alex Kane, Alan J.Marcus, Investments, 7th ed, McGraw-Hill, Boston, 2008Chapter 7 , pages 208-221Chapter 6, pages 175

Diketahui :Sumber : Bodie (2008 : 208)

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 ) = 4

Diminta : 1.Gambarkan indifference curve dan tabel utility values 2.Gambarkan efficient frontier & global minimum variance portfolio, Kemudian hitunglah : a. covariance (rD , rE) b. alokasi dana reksadana pendapatan tetap / debt / wD saat global minimum variance portfolio c. alokasi dana reksadana saham / equity / wE saat global minimum variance portfolio d. expected return saat global minimum variance portfolio E(rP) e. total risk saat global minimum variance portfolio P

Diminta : 1.Gambarkan indifference curve dan tabel utility values 2.Gambarkan efficient frontier & global minimum variance portfolio 3.Alokasi dana optimal reksadana pendapatan tetap / debt / wD 4.Alokasi dana optimal reksadana saham / equity / wE 5.Expected return portfolio optimal E(rP) 6.Total risk portfolio optimal P 7.Alokasi dana complete reksadana ( y ) 8.Alokasi dana complete treasury bill (1 y) 9.Expected return complete portfolio reksadana & treasury bill E(rC)10.Total risk complete portfolio reksadana & treasury bill C11.Alokasi dana reksadana pendapatan tetap (wD . y)12.Alokasi dana reksadana saham (wE . y)13.Slope of CAL14.Persamaan CAL15.Gambarkan CAL rf, optimal portfolio (P), complete portfolio (C)16.Gambarkan efficient frontier , indifference curve pada gambar CAL

Expected return (%)P Standard deviation (%) E D1. indifference curve12 0Indifference curve, A=4, U=0.0582013 Sumber : Bodie (2008 : 175), Figure 6.2Indifference curve : Portfolio points with the same utility value.

Table 6.3 Utility valuesE(r)A2U=E(r) -1/2A20.080.1240.01440.05120.130.2040.040.05

Expected return (%)P Standard deviation (%) E D12 0A=4, U=0.0582013 Sumber : Bodie (2008 : 175), Figure 6.2 the indifference curveA=3, U=0.07A=2, U=0.09

AU=E(r) -1/2A220.0930.0740.05

Diketahui : E(rD)=8%; E(rE)=13%; D=12%; E=20%; DE= 0.3Sumber : Bodie (2008 : 214)Cov(rD,rE)=DEDE 2E - Cov(rD,rE)wD = -------------------------------- 2D + 2E - 2Cov(rD,rE)wE = 1 - wDE(rP) = wD.E(rD)+wE.E(rE)P = [w2D2D + w2E2E + 2wDwECov(rD,rE)]1/2Cov(rD,rE)= 0.3(12)(20)=72 400 - 72wD= ----------------------- = 0.82 144 + 400 - 2(72)wE = 1 0.82 = 0.18E(rP) = 0.82(8)+0.18(13)=8.9%P = [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 portfolio

Diketahui : E(rD)=8%; E(rE)=13%; 2D=144%; 2E=400%; Cov(rD,rE) = 72%E(rP) = wD.E(rD)+wE.E(rE)P = [w2D2D + w2E2E + 2wDwECov(rD,rE)]1/2Global minimum variance portfolioTabel untuk membuat Efficient Frontier

wDwEE(rP)P 1.00.01.0(8)+0.0(13)=8.0[1.0(144)+0.0(400)+2(1.0)(0.0)(72)] 1/2 =120.90.10.9(8)+0.1(13)=8.5[0.81(144)+0.01(400)+2(0.9)(0.1)(72)] 1/2 =11.550.80.29.011.290.70.39.511.480.60.410.012.030.50.50.5(8)+0.5(13)=10.5[0.25(144)+0.25(400)+2(0.5)(0.5)(72)] 1/2 =13.110.40.611.013.990.30.711.515.300.20.812.016.760.10.912.518.340.01.013.020.000.820.188.911.44

Expected return (%) P Standard deviation (%) E DFigure 7.5 : E(rP) = f( P) 0Efficient frontier102013 Sumber : Bodie (2008 : 215) Efficient Portolio D = Minimum & E( r ) tertentuEfficient Portfolio E= Maximum E ( r ) & tertentu812=1=0.3=-1=0.25

Expected return (%) P Standard deviation (%) E D2. Efficient frontier, = 0.3 0Efficient frontier8.92013 Sumber : Bodie (2008 : 215) Efficient Portolio D = Minimum & E( r ) tertentuEfficient Portfolio E= Maximum E ( r ) & tertentu812=0.3 Global minimum variance portfolio11.44

2. Diketahui : E(rD)=8%; E(rE)=13%; D=12%; E=20%; DE= 0.3Sumber : Bodie (2008 : 214)Cov(rD,rE)=DEDE 2E - Cov(rD,rE)wD = -------------------------------- 2D + 2E - 2Cov(rD,rE)wE = 1 - wDE(rP) = wD.E(rD)+wE.E(rE)P = [w2D2D + w2E2E + 2wDwECov(rD,rE)]1/2Cov(rD,rE)= 0.3(12)(20)=72 400 - 72wD= ----------------------- = 0.82 144 + 400 - 2(72)wE = 1 0.82 = 0.18E(rP) = 0.82(8)+0.18(13)=8.9%P = [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 portfolio

3. Optimal risky portfolio wDDiketahui : E(rD) = 8%; E(rE) =13%; rf = 5% ; D=12%; E= 20%; Cov(rD,rE) = 72

[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 wD adalah alokasi dana optimal reksadana pendapatan tetap (debt) [3]400 [8]72 1200-576 624wD = -------------------------------------- = ---------------------- = --------- = 0.4 [3]400 + [8]144 - [3 + 8]72 1200+1152-792 1560

3. Optimal risky portfolio wDDiketahui : E(rD) = 0.08; E(rE) =0.13; rf = 0.05 ; D=0.12; E= 0.20; Cov(rD,rE) = 72

[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 wD adalah alokasi dana optimal reksadana pendapatan tetap (debt) [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)

4. Optimal risky portfolio 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 .. ?

5 : Expected Return Optimal Risky PortfolioDiketahui : wD = 0.4; wE = 0.6; E(rD)= 8%; E(rE)=13%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 %

6 : Total Risk Optimal Risky PortfolioDiketahui : wD= 0.4; wE= 0.6D=12%; E=20%; Cov(rD,rE) = 72

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

6 : Total Risk Optimal Risky PortfolioDiketahui : wD= 0.4; wE= 0.6; D=12%; E=20%; Cov(rD,rE) = 72

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=100

7.Alokasi dana Complete 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%

7.Alokasi dana Complete 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 100

8. Alokasi dana Complete Treasury bill ( 1 y ) Treasury bill (1 y) = 0.2561 atau 25.61 %

9.Expected return complete portfolio 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%

10.Total risk complete portfolio 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)

11. Alokasi dana Complete Portfolio reksadana pendapatan tetapAlokasi dana Complete Portfolio reksadana pendapatan tetap wD . y = 0.4(0.7439 )= 0.2976 atau 29.76 %

12. Alokasi dana Complete Portfolio reksadana saham Alokasi dana Complete Portfolio reksadana saham wE . y = 0.60(0.7439)=0.4463 atau 44.63 %

13. Slope of CAL (Capital Allocation Line) E(rP) rfSP = ------------ P

11 5SP = ------------ = 0.42 14.2

14.Persamaan CALE( r ) = Rf + SP P

E( r ) = 5 + 0.42 P

Expected return (%) PStandarddeviation (%)CAL11 14.2PP = Optimal risky portfolio15 CAL 0 F520E( r ) = 5 + 0.42 P

Expected return (%) PStandarddeviation (%)8CAL11 14.2PP = Optimal risky portfolioD & E = efficient portfolioEfficient frontier15 CAL & Efficient frontier 0 F11.44 D E G G global minimum variance portfolio 58.92013 12

Expected return (%) PStandarddeviation (%)8CAL11 14.2PC P = Optimal risky portfolioC = Complete portfolioD & E = efficient portfolioEfficient frontierIndifference curve16 CAL, efficient frontier & Indiference curve 010.569.46 F11.44 D E G G global minimum variance portfolio 58.92013 12