Case Study 6: Portfolio Theory

Case Study : Portfolio Theory Contents 1 Simulation: Two-Asset Portfolios 2 2 US 2.1 2.2 2.3 Sector ETFs: 2009-2013 Mean, Variance, Correlation Statistics . . . . . . . . . . . . . . . Optimal Portfolios (Max Allocation=0.30) . . . . . . . . . . . . . Optimal Portfolios (Max Allocation=0.15) . . . . . . . . . . . . . 4 4 5 7 3 US 3.1 3.2 3.3 Sector ETFs: 2003-2006 Mean, Variance, Correlation Statistics . . . . . . . . . . . . . . . Optimal Portfolios (Max Allocation=0.30) . . . . . . . . . . . . . Optimal Portfolios (Max Allocation=0.15) . . . . . . . . . . . . . 9 9 10 12 1 1 Simulation: Two-Asset Portfolios Consider m = 2 assets: p R1 : E(R1 ) = 0.15 = α1 pV ar(R1 ) = 0.25 = σ1 R2 : E(R1 ) = 0.20 = α2 V ar(R1 ) = 0.30 = σ2 Corr(R1 , R2 ) = ρ Portfolio: Rw = (1 − w)R1 + wR2 , 0≤w≤1 αw = E[Rw ] = (1 − w)α1 + wα2 σw2 = V ar(R2 ) = (1 − w)2 σ12 + w2 σ22 + 2(1 − w)(w)ρσ1 σ2 Mean-Variance Analysis Feasible Portfolio Set: Π∗ = {(σw , αw ) : 0 ≤ w ≤ 1} Issues: • What is Π∗ ? • What portfolios are optimal / sub-optimal? • How to choose/specify an optimal portfolio? • Do optimal portfolios have special structure? Simulation: • Simulate 500 weekly returns with ρ = −.8, −.4, 0., +.4, +.8 • Examine – Cumulative returns of each asset 2 – Asset returns: means, volatilities, correlations – Plot of Π∗ – Cumulative returns of each asset and the minimumvariance portfolio. See the plots in the pdf file Simulation T woAsset P ortf olios.pdf. 3 2 2.1 US Sector ETFs: 2009-2013 Mean, Variance, Correlation Statistics Sector ETFs: Period: 2009-2013 Annualized Return and Volatility: Ret MATERIALS(XLB) 0.16 HEALTH CARE(XLV) 0.17 CONSSTAPLES(XLP) 0.15 CONSDISC(XLY) 0.24 ENERGY(XLE) 0.15 FINANCIAL(XLF) 0.13 INDUSTRIALS(XLI) 0.18 TECHNOLOGY(XLK) 0.18 UTILITIES(XLU) 0.10 Vol 0.24 0.15 0.12 0.21 0.25 0.33 0.23 0.19 0.16 Correlations: XLB MATERIALS(XLB) 1.00 HEALTH CARE(XLV) 0.70 CONSSTAPLES(XLP) 0.68 CONSDISC(XLY) 0.87 ENERGY(XLE) 0.88 FINANCIAL(XLF) 0.74 INDUSTRIALS(XLI) 0.90 TECHNOLOGY(XLK) 0.85 UTILITIES(XLU) 0.59 XLV 0.70 1.00 0.80 0.72 0.69 0.64 0.76 0.69 0.64 4 XLP 0.68 0.80 1.00 0.77 0.69 0.63 0.76 0.70 0.71 XLY 0.87 0.72 0.77 1.00 0.82 0.82 0.93 0.88 0.67 XLE 0.88 0.69 0.69 0.82 1.00 0.74 0.85 0.80 0.65 XLF 0.74 0.64 0.63 0.82 0.74 1.00 0.85 0.69 0.57 XLI 0.90 0.76 0.76 0.93 0.85 0.85 1.00 0.85 0.66 XLK 0.85 0.69 0.70 0.88 0.80 0.69 0.85 1.00 0.59 XLU 0.59 0.64 0.71 0.67 0.65 0.57 0.66 0.59 1.00 2.2 Optimal Portfolios (Max Allocation=0.30) Optimal Allocations for Selected Target Vols Max. Allocation = 0.30 target.vol0.009 target.vol0.099 target.vol0.153 MATERIALS(XLB) 0.000 0.000 0.000 HEALTH CARE(XLV) 0.009 0.157 0.300 CONSSTAPLES(XLP) 0.040 0.300 0.227 CONSDISC(XLY) 0.017 0.226 0.300 ENERGY(XLE) 0.000 0.000 0.000 FINANCIAL(XLF) 0.000 0.000 0.000 INDUSTRIALS(XLI) 0.000 0.000 0.000 TECHNOLOGY(XLK) 0.000 0.000 0.173 UTILITIES(XLU) 0.000 0.000 0.000 riskFree 0.935 0.317 0.000 Portfolio Statistics for Optimal Allocations target.vol0.009 target.vol0.099 target.vol0.153 Ann Return 0.011 0.124 0.187 Ann Volatility 0.009 0.099 0.153 Graphical displays of the optimal allocations are presented in plots 2 and 3 of ET F S 1 periodA 30.pdf • As the target return increases from zero, only XLY, XLP, XLV are in the model. They enter in the same proportion, i.e., the scaled (de-levered) optimal portfolio w/o constraints. • When the allocation constraint is hit, first for consumer staples, higher allocations given to XLY and XLV 5 • When the .30 allocations are reached for these three, then XLK (tech) is added. It has higher return than the other ETFs, so eventually allocations to XLY and XLP are reduced to allow for higher-return from XLK. • From the efficient frontier, all the ETFs (except XLY) are dominated by an optimal allocation with a 0.30 max constraint. • No allocation is ever given to Financials (XLF). 6 2.3 Optimal Portfolios (Max Allocation=0.15) Optimal Allocations for Selected Target Vols Max. Allocation = 0.15 target.vol0.009 target.vol0.099 target.vol0.163 MATERIALS(XLB) 0.000 0.000 0.098 HEALTH CARE(XLV) 0.009 0.150 0.150 CONSSTAPLES(XLP) 0.042 0.150 0.150 CONSDISC(XLY) 0.018 0.150 0.150 ENERGY(XLE) 0.000 0.000 0.000 FINANCIAL(XLF) 0.000 0.000 0.000 INDUSTRIALS(XLI) 0.000 0.000 0.150 TECHNOLOGY(XLK) 0.000 0.150 0.150 UTILITIES(XLU) 0.000 0.069 0.150 riskFree 0.931 0.331 0.002 Portfolio Statistics target.vol0.009 target.vol0.099 target.vol0.163 Ann Return 0.012 0.117 0.168 Ann Volatility 0.009 0.099 0.163 7 Graphical displays of the optimal allocations are presented in plots 2 and 3 of ET F S 1 periodA 15.pdf • The 0.15 maximum allocation constraint has no impact on low-return portfolios. The optimal portfolios allocate to XLP, XLY and XLV, initially until they hit their limits. • The allocations to XLK increases until its limit is reached. • The Allocation to XLU (utilities) is mixed with XLI (industrials), until their limits • Efficient frontier with Max Allocation=0.30 is above the EF for Max Allocation =0.15 Compare Plot 4 in the two files ET F S 1 periodA 30.pdf and ET F S 1 periodA 30.pdf 8 3 3.1 US Sector ETFs: 2003-2006 Mean, Variance, Correlation Statistics Sector ETFs: Period: 2003-2006 Annualized Return and Ret MATERIALS(XLB) 0.16 HEALTH CARE(XLV) 0.07 CONSSTAPLES(XLP) 0.08 CONSDISC(XLY) 0.14 ENERGY(XLE) 0.26 FINANCIAL(XLF) 0.15 INDUSTRIALS(XLI) 0.15 TECHNOLOGY(XLK) 0.12 UTILITIES(XLU) 0.19 Volatility: Vol 0.18 0.12 0.09 0.14 0.20 0.13 0.14 0.18 0.13 Correlations: XLB MATERIALS(XLB) 1.00 HEALTH CARE(XLV) 0.45 CONSSTAPLES(XLP) 0.57 CONSDISC(XLY) 0.74 ENERGY(XLE) 0.54 FINANCIAL(XLF) 0.68 INDUSTRIALS(XLI) 0.82 TECHNOLOGY(XLK) 0.66 UTILITIES(XLU) 0.50 XLV 0.45 1.00 0.55 0.54 0.24 0.61 0.59 0.47 0.42 9 XLP 0.57 0.55 1.00 0.69 0.16 0.71 0.65 0.50 0.45 XLY 0.74 0.54 0.69 1.00 0.31 0.82 0.85 0.79 0.48 XLE 0.54 0.24 0.16 0.31 1.00 0.24 0.37 0.21 0.51 XLF 0.68 0.61 0.71 0.82 0.24 1.00 0.78 0.70 0.54 XLI 0.82 0.59 0.65 0.85 0.37 0.78 1.00 0.78 0.49 XLK 0.66 0.47 0.50 0.79 0.21 0.70 0.78 1.00 0.40 XLU 0.50 0.42 0.45 0.48 0.51 0.54 0.49 0.40 1.00 3.2 Optimal Portfolios (Max Allocation=0.30) Optimal Allocations for Selected Target Vols Max. Allocation = 0.30 target.vol0.009 target.vol0.1 target.vol0.114 MATERIALS(XLB) 0.000 0.000 0.000 HEALTH CARE(XLV) 0.000 0.000 0.000 CONSSTAPLES(XLP) 0.004 0.088 0.093 CONSDISC(XLY) 0.000 0.000 0.000 ENERGY(XLE) 0.017 0.251 0.300 FINANCIAL(XLF) 0.013 0.213 0.270 INDUSTRIALS(XLI) 0.003 0.027 0.037 TECHNOLOGY(XLK) 0.000 0.000 0.000 UTILITIES(XLU) 0.042 0.300 0.300 riskFree 0.921 0.121 0.000 Portfolio Statistics for Optimal Allocations target.vol0.009 target.vol0.1 target.vol0.114 Ann Return 0.015 0.165 0.188 Ann Volatility 0.009 0.100 0.114 Graphical displays of the optimal allocations are presented in plots 2 and 3 of ET F S 1 periodB 30.pdf • As the target return increases from zero, only XLU, XLE, XLF, XLP, and XLI are in the model. They enter in the same proportion, i.e., the scaled (delevered) optimal portfolio w/o constraints. • The allocation constraint is hit first for utilities (XLU). • As the target return increases, the energy (XLE) and financials (XLF) increase to their limits. 10 • At high target return levels, allocation to industrials (XLI) substitutes for consumer stables (XLP). 11 3.3 Optimal Portfolios (Max Allocation=0.15) Optimal Allocations for Selected Target Vols Max. Allocation = 0.15 target.vol0.01 target.vol0.1 target.vol0.11 MATERIALS(XLB) 0.000 0.022 0.092 HEALTH CARE(XLV) 0.000 0.000 0.000 CONSSTAPLES(XLP) 0.004 0.150 0.150 CONSDISC(XLY) 0.000 0.150 0.150 ENERGY(XLE) 0.019 0.150 0.150 FINANCIAL(XLF) 0.014 0.150 0.150 INDUSTRIALS(XLI) 0.003 0.150 0.150 TECHNOLOGY(XLK) 0.000 0.000 0.000 UTILITIES(XLU) 0.044 0.150 0.150 riskFree 0.916 0.078 0.008 Portfolio Statistics target.vol0.01 target.vol0.1 target.vol0.11 Ann Return 0.016 0.149 0.16 Ann Volatility 0.010 0.100 0.11 Graphical displays of the optimal allocations are presented in plots 2 and 3 of ET F S 1 periodB 15.pdf • The 0.15 maximum allocation constraint has no impact on the relative allocations for low-return portfolios. • Technology (XLK) never enters the optimal allocation. • The allocations to XLK increases until its limit is reached. 12 • Efficient frontier with Max Allocation=0.30 is above the EF for Max Allocation =0.15. Compare Plot 4 in the two files ET F S 1 periodB 30.pdf and ET F S 1p eriodB 15.pdf The curve of the latter starts to bend lower as the maximum allocation constraints are hit. As a result greater return is achieved, only by those assets that result in marginal increases in the relative volatility. 13