Tables and Figures for Dynamic Asset Allocation with Ambiguous Return Predictability

Table 1: Ambiguity Premium as a Percentage of the Expected Value of the Bet γ/η 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 A. Prize-wealth ratio = 1.0% 0.5 2.0 5.0 10.0 15.0 9.8 9.4 8.6 7.4 6.2 12.2 11.8 11.1 9.8 8.6 14.6 14.2 13.5 12.2 11.0 17.0 16.6 15.8 14.6 13.4 19.3 18.9 18.2 16.9 15.7 21.6 21.2 20.4 19.2 18.0 23.8 23.4 22.7 21.4 20.2 26.0 25.6 24.9 23.6 22.4 B. Prize-wealth ratio = 0.5% 0.5 2.0 5.0 10.0 15.0 4.9 4.7 4.4 3.7 3.1 6.2 6.0 5.6 5.0 4.3 7.4 7.2 6.8 6.2 5.6 8.6 8.4 8.1 7.4 6.8 9.8 9.7 9.3 8.7 8.0 11.1 10.9 10.5 9.9 9.3 12.3 12.1 11.7 11.1 10.5 13.5 13.3 12.9 12.3 11.7 Notes: This table reports ambiguity premium as a percentage of the expected value of the bet for various different values of γ and η. The expression for ambiguity premium is given by equation (8). Table 2: Estimations of the IID and VAR Models A. IID Model m σ1 rf 0.0577 (0.0219) 0.1971 (0.0156) 0.0078 (0.0045) B. VAR Model m b ρ σ2 σ23 σ3 R2 ldy 0.0577 (0.0219) 0.1055 (0.0527) 0.9452 (0.0416) 0.1935 (0.0154) -0.0199 (4.01E-3) 0.1527 (0.0121) 0.0488 ltp 0.0577 (0.0219) 0.2037 (0.0742) 0.8501 (0.0589) 0.1894 (0.0151) -0.0191 (3.86E-3) 0.1504 (0.0120) 0.0881 lnp 0.0577 (0.0219) 0.7526 (0.1418) 0.6539 (0.0857) 0.1700 (0.0135) -0.0057 (2.07E-3) 0.1028 (0.0082) 0.2653 Variable Notes: This table reports the results from estimating the riskfree rates and the IID and VAR models of stock returns. The numbers in brackets are standard errors. The variable ldy denotes the log dividend yield. The variables ltp and lnp denote the total payout yield and the net payout yield series constructed by Boudoukh et al. (2007). The sample period is 1926-2005. All variables are annualized when applicable. Table 3: Optimal Portfolio Weights in percentage (γ, η) µ0 = 0.2 x20 x30 x10 µ0 = 0.5 x20 x30 x10 x10 µ0 = 0.8 x20 x30 2, 2, 2, 2, 2 60 80 100 57.1 52.0 51.0 50.3 100.0 100.0 100.0 100.0 100.0 98.0 94.4 93.3 69.5 55.3 52.4 50.3 100.0 100.0 100.0 100.0 100.0 98.5 97.6 97.4 84.5 64.5 59.5 55.6 100.0 100.0 100.0 100.0 100.0 99.1 98.9 98.9 5, 5, 5, 5, 5 60 80 100 28.6 28.1 27.9 27.7 49.7 46.4 45.7 45.2 57.2 42.6 39.8 37.8 30.0 27.7 27.0 26.3 43.2 42.2 42.0 41.9 48.8 41.3 40.1 39.3 33.8 31.1 30.1 29.2 40.4 40.3 40.3 40.2 43.0 40.3 39.8 39.6 15.5 15.4 15.4 15.3 25.2 24.1 23.8 23.5 27.2 23.3 22.1 21.1 15.4 14.9 14.7 14.5 21.6 21.3 21.2 21.2 23.8 21.7 21.1 20.7 16.9 16.3 16.1 15.9 20.1 20.1 20.1 20.0 21.2 20.4 20.2 20.1 10, 10, 10, 10, 10 60 80 100 γ 2 5 10 15 IID Model 99.3 39.6 19.7 13.1 x10 VAR Model x20 x30 48.8 27.1 15.2 10.5 100.0 63.0 34.4 23.7 100.0 98.3 54.9 37.8 Notes: This table presents the optimal portfolio weights in percentage allocated to the stock. Column 1 denotes various values of γ and η. Columns 2-10 report the poptimal portfolio weights in 2 1 2 percentage p for different combinations of µ0 and x0 where x0 = −σ3 / 1 − ρ = −0.4677, x0 = 0.0, 3 2 x0 = σ3 / 1 − ρ = 0.4677. The predictive variable x is the price-dividend ratio. Table 4: Percentage Hedging Demand over Total Stock Demand (γ, η) x10 µ0 = 0.2 x20 x30 x10 µ0 = 0.5 x20 x30 x10 µ0 = 0.8 x20 x30 2, 2, 2, 2, 2 60 80 100 18.1 26.3 27.6 28.5 0.6 0.6 0.6 0.6 0.0 -2.1 -5.9 -7.2 4.3 15.6 17.4 18.8 0.6 0.6 0.6 0.6 0.0 -1.5 -2.5 -2.4 -1.9 6.9 9.0 10.6 0.7 0.7 0.7 0.7 0.0 -0.9 -0.7 -0.5 5, 5, 5, 5, 5 60 80 100 35.2 40.1 41.3 42.3 20.1 14.6 13.3 12.2 -6.7 -22.3 -23.9 -24.5 12.3 19.4 21.2 22.7 8.2 6.1 5.7 5.3 -7.6 -9.3 -8.7 -8.0 -1.2 4.0 6.1 8.0 1.9 1.6 1.5 1.5 -3.9 -2.9 -2.6 -2.2 40.4 42.9 43.7 44.5 21.8 18.2 17.1 16.2 -11.5 -21.7 -24.3 -26.2 15.2 18.9 20.2 21.3 8.9 7.6 7.3 6.9 -9.4 -11.1 -11.1 -11.0 -0.8 1.6 2.7 3.8 2.1 2.0 1.9 1.8 -4.4 -4.0 -3.8 -3.5 10, 10, 10, 10, 10 60 80 100 Notes: This table presents the ratio of the hedging demand to the total stock demand in percentage. Column 1 denotes various values of γ and η. Columns 2-10 report the percentage hedging p demand 1 = −σ / 1 − ρ2 = over the total stock demand for different combinations of µ and x where x 0 0 3 0 p −0.4677, x20 = 0.0, x30 = σ3 / 1 − ρ2 = 0.4677. The predictive variable x is the price-dividend ratio. Table 5: Welfare Costs of Following Suboptimal Investment Strategies γ/η γ 60 100 γ 60 100 γ 60 µ0 = 0.2 2 5 10 2 5 10 11.8 7.1 3.5 IID 13.6 5.7 3.0 0.4 1.6 1.1 BM 0.5 2.5 1.3 VAR 0.2 22.7 44.6 0.3 16.6 43.7 0.6 2.8 1.4 Bayesian 0.0 0.2 0.3 0.0 0.9 1.5 0.0 0.2 0.4 0.1 23.6 44.5 9.1 4.4 2.2 IID 13.0 4.1 2.0 0.2 0.6 0.4 BM 0.7 1.2 0.5 µ0 = 0.6 6.6 3.1 1.5 IID 12.5 3.5 1.6 13.4 3.7 1.6 2.4 20.7 35.8 2 5 10 0.1 0.3 0.2 BM 1.4 0.6 0.3 1.9 0.8 0.3 0.0 0.0 0.0 γ 60 100 13.6 4.2 2.0 1.1 19.6 39.4 VAR 0.3 22.9 40.5 0.2 23.3 40.7 0.9 1.3 0.6 Bayesian 0.0 0.5 0.7 0.0 0.5 0.7 0.0 0.1 0.2 µ0 = 0.4 13.8 5.5 2.9 2 5 10 100 µ0 = 0.8 VAR 0.8 22.6 37.2 0.5 22.7 37.8 4.1 2.1 1.1 IID 11.5 2.9 1.2 13.0 3.3 1.4 4.5 21.4 32.9 Bayesian 1.1 1.6 0.3 0.5 0.0 0.1 0.0 0.1 0.1 BM 2.4 0.4 0.1 3.4 0.7 0.2 0.0 0.0 0.0 VAR 1.5 22.2 34.2 0.8 22.2 35.0 Bayesian 1.9 2.9 0.2 0.4 0.0 0.1 Notes: This table reports the welfare costs of following four suboptimal investment strategies: the IID strategy, the VAR strategy, the Bayesian myopic (BM) strategy, and the Bayesian strategy, when the investor is averse to model uncertainty. Column 1 denotes various values of γ. Welfare costs are measured by percentage of initial wealth. Row 1 denotes various values of η. When η = γ, the model reduce to the standard Bayesian framework. The predictive variable is the dividend yield. Table 6: Welfare Costs of Following Suboptimal Investment Strategies γ/η γ 60 100 γ 60 100 γ 60 100 µ0 = 0.2 2 5 10 12.4 10.4 5.3 IID 14.2 7.6 4.4 14.5 7.3 4.1 0.4 18.6 136.7 2 5 10 0.4 2.1 1.6 BM 0.7 4.2 2.1 0.9 4.9 2.5 0.0 0.0 0.0 6.0 3.5 1.7 IID 13.1 3.6 1.7 14.5 4.0 1.7 3.0 26.1 120.4 2 5 10 0.1 0.4 0.3 BM 1.7 1.1 0.5 2.6 1.5 0.6 0.0 0.0 0.0 60 100 VAR 0.7 30.7 132.8 0.6 31.6 133.6 µ0 = 0.4 VAR 0.3 30.4 141.9 0.4 32.5 142.1 Bayesian 0.4 0.7 2.1 3.2 0.4 0.9 8.9 5.7 2.9 IID 13.5 4.7 2.4 14.4 4.8 2.4 1.5 23.7 128.0 0.2 0.9 0.7 BM 0.9 2.1 1.0 1.4 2.5 1.1 0.0 0.0 0.0 µ0 = 0.6 2 5 10 γ Bayesian 0.9 1.5 1.2 1.8 0.2 0.5 µ0 = 0.8 VAR 1.1 30.2 126.2 0.7 30.7 128.1 Bayesian 1.7 2.7 0.7 1.2 0.1 0.3 3.4 1.9 1.0 IID 12.3 2.9 1.1 14.4 3.5 1.3 5.2 27.6 113.4 0.0 0.1 0.1 BM 3.2 0.7 0.2 4.6 1.1 0.3 0.0 0.0 0.0 VAR 1.9 29.7 119.8 1.1 29.9 122.8 Bayesian 3.0 4.5 0.5 1.0 0.1 0.2 Notes: This table reports the welfare costs of following four suboptimal investment strategies: the IID strategy, the VAR strategy, the Bayesian myopic (BM) strategy, and the Bayesian strategy, when the investor is averse to model uncertainty. Welfare costs are measured by percentage of initial wealth. Column 1 denotes various values of γ. Row 1 denotes various values of η. When η = γ, the model reduce to the standard Bayesian framework. The predictive variable is the total payout yield. Table 7: Welfare Costs of Following Suboptimal Investment Strategies γ/η γ 60 100 γ 60 100 γ 60 100 µ0 = 0.2 2 5 10 48.6 29.2 14.3 IID 48.9 21.5 11.5 49.3 21.0 10.9 1.7 18.9 165.7 2 5 10 0.1 1.1 1.0 BM 5.3 8.6 3.7 8.1 10.5 4.6 0.0 0.0 0.0 22.1 12.4 6.3 IID 45.3 14.4 6.5 48.8 15.8 6.9 7.3 30.0 173.1 2 5 10 0.1 0.6 0.4 BM 7.3 3.5 1.2 10.5 5.0 1.7 0.0 0.0 0.0 60 100 VAR 3.7 35.6 185.4 3.2 35.4 185.3 µ0 = 0.4 VAR 3.3 36.5 189.7 3.6 37.4 190.4 Bayesian 5.3 8.2 5.2 7.2 1.3 2.2 34.2 19.1 9.5 IID 46.4 16.9 8.4 48.4 17.4 8.3 4.2 25.9 170.9 0.1 0.9 0.7 BM 6.2 5.4 2.1 9.1 6.9 2.7 0.0 0.0 0.0 µ0 = 0.6 2 5 10 γ Bayesian 6.4 9.4 3.4 5.0 0.8 1.4 µ0 = 0.8 VAR 4.1 35.1 183.1 3.1 34.5 182.7 12.0 7.1 3.8 IID 44.1 12.2 4.8 49.3 14.5 5.7 11.1 33.2 174.6 Bayesian 7.6 10.9 2.6 4.3 0.5 1.1 0.0 0.3 0.2 BM 9.4 2.5 0.7 13.2 4.2 1.1 0.0 0.0 0.0 VAR 5.0 35.1 181.7 3.3 34.2 181.2 Bayesian 9.9 13.9 2.3 4.2 0.4 0.9 Notes: This table reports the welfare costs of following four suboptimal investment strategies: the IID strategy, the VAR strategy, the Bayesian myopic (BM) strategy, and the Bayesian strategy. Welfare costs are measured by percentage of initial wealth, when the investor is averse to model uncertainty. Column 1 denotes various values of γ. Row 1 denotes various values of η. When η = γ, the model reduce to the standard Bayesian framework. The predictive variable is the net payout yield. A. excess return (r−r ) B. log dividend yield (ldy) f 0.5 1 0.5 0 0 −0.5 −0.5 −1 −1 1926 1941 1956 1971 1986 2001 −1.5 1926 C. log total payout yield (ltp) 1941 1956 1971 1986 2001 D. log net payout yield (lnp) 1 0.4 0.2 0.5 0 0 −0.2 −0.5 −1 1926 −0.4 1941 1956 1971 1986 2001 −0.6 1926 1941 1956 1971 1986 2001 Figure 1: Returns and predictors. This figure plots the historical data of the real return from the CRSP value-weighted market portfolio, the demeaned log dividend yield, and the demeaned log total and net payout yields from 1926-2005. The latter two payout yields are constructed by Boudoukh et al. (2007) and downloaded from Michael Roberts’ homepage. 0.7 ldy ltp lnp 0.6 0.5 0.4 0.3 0.2 0.1 0 1930 1940 1950 1960 1970 1980 1990 2000 Figure 2: Posterior probabilities of the IID model. This figure plots the posterior probabilities of the IID model using the historical annual data of stock returns and the three payout yields as predictors from 1926-2005. The prior is set at µ0 = 0.5. The log dividend yield (ldy): solid line; the log total payout yield (ltp): dashdot line; the log net payout yield (lnp): dashed line. A. µ0=0.2 B. µ0=0.4 80 60 100 IID VAR Robust Baysian MP Stock Allocation (%) Stock Allocation (%) 100 40 20 0 −2 −1 0 1 Dividend−Price Ratio 80 60 40 20 0 −2 2 C. µ =0.6 100 80 80 Stock Allocation (%) Stock Allocation (%) 0 100 60 40 20 −1 0 1 Dividend−Price Ratio 2 D. µ =0.8 0 0 −2 −1 0 1 Dividend−Price Ratio 2 60 40 20 0 −2 −1 0 1 Dividend−Price Ratio 2 Figure 3: Alternative investment strategies: market-timing effect. This figure plots the initial portfolio weights on the stock for five alternative investment strategies as functions of the initial observation of the dividend yield given four different values of initial beliefs about the IID model. We set the risk aversion parameter γ = 5 and the investment horizon T = 40. Other parameter values are estimated using annual data as reported in Table 2. Bayesian strategy: dashdot line; Robust strategy for η = 60: dashed line; VAR strategy: solid line; IID strategy: horizontal dot line; Multiple-prior (MP) strategy: dotted line with x-marks. A. x =−1.12 B. x =−0.37 0 0 40 Stock Allocation (%) Stock Allocation (%) 40 30 20 10 0 0 0.2 0.4 0.6 0.8 Beliefs about IID Model 38 36 34 32 30 1 0 C. x0=0.37 100 Stock Allocation (%) 90 Stock Allocation (%) 1 D. x0=1.12 100 80 70 60 50 40 30 0.2 0.4 0.6 0.8 Beliefs about IID Model 0 0.2 0.4 0.6 0.8 Beliefs about IID Model 1 IID Baysian Robust 80 60 40 20 0 0 0.2 0.4 0.6 0.8 Beliefs about IID Model 1 Figure 4: Alternative investment strategies: uncertainty effect. This figure plots the portfolio weights on the stock for three alternative investment strategies as functions of the initial beliefs about the p IID model given four different initial values of the dividend yield. One standard deviation of xt is σ3 / 1 − ρ2 = 0.4677. We set the risk aversion parameter γ = 5 and the investment horizon T = 40. Other parameter values are estimated using annual data as reported in Table 2. Bayesian strategy: dashdot line; Robust strategy for η = 60: dashed line; IID strategy: horizontal dotted line. A. x =−0.4677 Stock Allocation (%) 0 40 30 20 10 0 5 10 15 20 25 30 35 40 25 30 35 40 Stock Allocation (%) B. x0=0 70 60 50 40 30 0 5 10 15 20 Stock Allocation (%) C. x =0.4677 0 100 VAR Bayesian Robust 80 60 40 0 5 10 15 20 Horizon 25 30 35 40 Figure 5: Alternative investment strategies: horizon effect. This figure plots the portfolio weights on the stock for three alternative investment strategies as functions of the investment horizon. One p standard deviation of xt is σ3 / 1 − ρ2 = 0.4677. We set the risk aversion parameter γ = 5 and the initial belief µ0 = 0.5. Other parameter values are estimated using annual data as reported in Table 2. Bayesian strategy: dashdot line; Robust strategy for η = 60: dashed line; VAR strategy: solid line. 90 IID VAR Bayesian Robust 80 70 Stock Allocation (%) 60 50 40 30 20 10 0 1970 1975 1980 1985 1990 1995 2000 2005 Figure 6: Simulated stock allocations. This figure plots the simulated portfolio weights on the stock for four alternative investment strategies using the historical data of returns and dividend price ratios from 1926-2007. The investor starts in 1968 with initial prior µ0 = 0.5. We set the risk aversion parameter γ = 5 and the investment horizon T = 40. Other parameter values are estimated using annual data as reported in Table 2. IID strategy: horizontal solid line; VAR strategy: solid line with star; Bayesian strategy: solid line with circle; Robust strategy for η = 60: dashed line. 1 Bayesian Distorted 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1970 1975 1980 1985 1990 1995 2000 2005 Figure 7: Distorted beliefs about the IID model. This figure plots the beliefs of a Bayesian investor (µt ) and the distorted beliefs of an ambiguity-averse investor (µ̂t ) about the IID model using historical annual data of stock returns and dividend yields from 1968-2007. We set the initial belief µ0 = 0.5, the risk aversion parameter γ = 5, and the investment horizon T = 40. Other parameter values are estimated using annual data as reported in Table 2. Bayesian beliefs: solid line; distorted beliefs for η = 60: dashed line.