Portfolio Choice with House Value Misperception

Transcription

1 Portfolio Choice with House Value Misperception Stefano Corradin ECB José L. Fillat FRB Boston Carles Vergara-Alert IESE May 5, 2016 The views expressed in this paper are those of the authors and do not represent those of the Federal Reserve System, Federal Reserve Bank of Boston, or European Central Bank. 1 / 35

2 Misperception 2 / 35

3 Contribution: Misperception Matters! Misperception In this paper We present evidence on housing value misperception, sign, and size 3 / 35

4 Contribution: Misperception Matters! Misperception In this paper We present evidence on housing value misperception, sign, and size Develop a model of portfolio allocation with costly acquisition of information, 3 / 35

5 Contribution: Misperception Matters! Misperception In this paper We present evidence on housing value misperception, sign, and size Develop a model of portfolio allocation with costly acquisition of information, which results in households misvaluing their houses misperception matters for portfolio, housing, and consumption decisions (spoiler: increases risk aversion) 3 / 35

6 Contribution: Misperception Matters! Misperception In this paper We present evidence on housing value misperception, sign, and size Develop a model of portfolio allocation with costly acquisition of information, which results in households misvaluing their houses misperception matters for portfolio, housing, and consumption decisions (spoiler: increases risk aversion) Test model implications with household level data on financial wealth, housing, and portfolio allocation. 3 / 35

7 Contribution: Misperception Matters! Misperception In this paper We present evidence on housing value misperception, sign, and size Develop a model of portfolio allocation with costly acquisition of information, which results in households misvaluing their houses misperception matters for portfolio, housing, and consumption decisions (spoiler: increases risk aversion) Test model implications with household level data on financial wealth, housing, and portfolio allocation. Evidence on misperception (too long list), but evidence on sign is mixed (and very relevant for portfolio allocations) Benitez-Silva et al. (2008), Agarwal (2007) overvaluation Follain and Malpezzi (1981), Goodman and Ittner (1992) undervaluation 3 / 35

8 Misperception Definition Misperception Evidence on misperception: self-reported housing values vs market housing values market values built from purchase date (=zero misperception) using price index perceived housing wealth rarely equals market housing wealth 4 / 35

9 Misperception Definition Misperception Evidence on misperception: self-reported housing values vs market housing values market values built from purchase date (=zero misperception) using price index perceived housing wealth rarely equals market housing wealth Data PSID at zipcode level self reported house value CoreLogic at zipcode level market value 4 / 35

10 Misperception Definition Misperception Evidence on misperception: self-reported housing values vs market housing values market values built from purchase date (=zero misperception) using price index perceived housing wealth rarely equals market housing wealth Data PSID at zipcode level self reported house value CoreLogic at zipcode level market value Use the CL HPI index to inflate purchase price of house. Misperception = ( ) ( ) H PH,t PSID H P PSID H,0 HPI0 t CL 4 / 35

11 Histogram Time Series Tenure Risky Share Results 5 / 35

12 Distribution of Misperception Histogram Time Series Tenure Risky Share Results Density Distribution of the variable Final Misperception misper_final 6 / 35

13 Misperception cyclicality Histogram Time Series Tenure Risky Share Results Mean (%) p5 (%) p95 (%) HPI 7 / 35

14 Misperception is Persistent Average / 35

15 Risky stock holdings are persistent too Histogram Time Series Tenure Risky Share Results Median = / 35

16 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results 10 / 35

17 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results Households value of their houses differs from market value 10 / 35

18 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results Households value of their houses differs from market value less risky investments Misvaluation 10 / 35

19 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results Households value of their houses differs from market value less risky investments lower consumption and lower leverage Misvaluation 10 / 35

20 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results Households value of their houses differs from market value less risky investments lower consumption and lower leverage Misvaluation larger housing relative to total wealth 10 / 35

21 Preview of Results Misperception of housing wealth affects portfolio, consumption, and housing decisions Histogram Time Series Tenure Risky Share Results Households value of their houses differs from market value less risky investments lower consumption and lower leverage Misvaluation larger housing relative to total wealth more frequent acquisition of information 10 / 35

22 PSID CL 11 / 35

23 PSID Data household sample PSID CL 12 / 35

24 PSID Data household sample PSID CL Financial wealth = house value (first and second), business value, other assets, stock holdings, checking and savings valances, IRAs and annuities, less the mortgage principal on primary residence 12 / 35

25 PSID Data household sample PSID CL Financial wealth = house value (first and second), business value, other assets, stock holdings, checking and savings valances, IRAs and annuities, less the mortgage principal on primary residence All net of debt 12 / 35

26 PSID Data household sample PSID CL Financial wealth = house value (first and second), business value, other assets, stock holdings, checking and savings valances, IRAs and annuities, less the mortgage principal on primary residence All net of debt Only owners 12 / 35

27 PSID Data household sample PSID CL Financial wealth = house value (first and second), business value, other assets, stock holdings, checking and savings valances, IRAs and annuities, less the mortgage principal on primary residence All net of debt Only owners Identify movers, start measuring misperception at purchase time misperception is assumed to be zero at purchase 12 / 35

28 CoreLogic House Prices Repeat sales index (monthly, starting 1975), single family combined PSID CL 13 / 35

29 CoreLogic House Prices Repeat sales index (monthly, starting 1975), single family combined public record files from First American PSID CL 13 / 35

30 CoreLogic House Prices Repeat sales index (monthly, starting 1975), single family combined PSID CL public record files from First American Representative of all loans (not just GSEs) 13 / 35

31 CoreLogic House Prices Repeat sales index (monthly, starting 1975), single family combined PSID CL public record files from First American Representative of all loans (not just GSEs) Limited coverage at the zipcode level Use the index to inflate purchase price of house, starting at purchase time Misperception = ( ) ( ) H PH,t PSID H P PSID H,0 HPI0 t CL 13 / 35

32 Model Illustration Equilibrium Calibration Boundaries Sensitivity 14 / 35

33 Model Model Illustration Equilibrium Calibration Boundaries Sensitivity Notation: u(c,h) = 1 1 γ (Cβ H 1 β ) 1 γ dh = δhdt dp = Pµdt+PσdZ 2 db = rbdt ds = S α S dt+s σ S dz 1 W = B +Θ+HP P house price S stock price Θ financial wealth in risky stock B financial wealth in safe assets φ o cost of acquiring info φ a cost of moving m i market value surprise 15 / 35

34 Model cont d Value function for acquiring information Model Illustration Equilibrium Calibration Boundaries Sensitivity V(W,H,P) = max C,Θ,H,τ E [ τ 0 u(c,he δt )dt + I H >He ρτ (1 π)v ( W(τ),He δτ,p(τ) ) +πṽ (W(τ),H(τ),P(τ)) + I H <He ρτ πv ( W(τ),He δτ,p(τ) ) +(1 π)ṽ (W(τ),H(τ),P(τ)) ] W(τ) = W(τ ) φ o P(τ)H(τ )+m i P(τ )H(τ ) P(τ) = P(τ )(1+m i ) H(τ) = H and H(τ ) = He δτ 16 / 35

35 Model cont d Value function for acquiring information Model Illustration Equilibrium Calibration Boundaries Sensitivity V(W,H,P) = max C,Θ,H,τ E [ τ 0 u(c,he δt )dt + I H >He ρτ (1 π)v ( W(τ),He δτ,p(τ) ) +πṽ (W(τ),H(τ),P(τ)) + I H <He ρτ πv ( W(τ),He δτ,p(τ) ) +(1 π)ṽ (W(τ),H(τ),P(τ)) ] W(τ) = W(τ ) φ o P(τ)H(τ )+m i P(τ )H(τ ) P(τ) = P(τ )(1+m i ) H(τ) = H and H(τ ) = He δτ Value function of adjusting housing Ṽ(W,H,P) = max C,Θ,H,τ E [ τ 0 ] u(c,he δt )dt+e ρτ Ṽ (W(τ),H(τ),P(τ)), where W(τ) = W(τ ) φ a P(τ)H(τ ) φ o P(τ)H(τ )+m i P(τ )H(τ ). 16 / 35

36 Illustration of equilibrium 3,5 Overvaluation Undervaluation Overvaluation Undervaluation Model Illustration Equilibrium Calibration Boundaries Sensitivity Wt/(PtHt) 3.5 3, ,5 0 Move big * No No move move Subjective W t /(P th t ) ratio Market W t /(P t H t ) ratio 5 * Upper bound for moving Upper bound for information acquisition z Inaction región for information acquisition Optimal return point z * 2.5 2,0 4 4 No move 5 Lower bound for information acquisition z Lower bound for moving 2.0 1,5 5 Move small Time 17 / 35

37 Equilibrium Model Illustration Equilibrium Calibration Boundaries Sensitivity The value function of this problem, V(W(t),H(t),P(t)), satisfies the following Hamilton-Jacobi-Bellman (HJB) partial differential equation sup C,Θ,H,τ E(dV (W,H,P)+u(C,H)dt) = / 35

38 Equilibrium Model Illustration Equilibrium Calibration Boundaries Sensitivity The value function of this problem, V(W(t),H(t),P(t)), satisfies the following Hamilton-Jacobi-Bellman (HJB) partial differential equation sup C,Θ,H,τ E(dV (W,H,P)+u(C,H)dt) = 0. Thanks to homogeneity properties, we can rewrite the problem in terms of the wealth-to-housing ratio, z = W/(P H) ( ) W V(W,H,P) = H 1 γ P β(1 γ) V PH,1,1 = H 1 γ P β(1 γ) v(z). and solve for v(z). c denotes the scaled control c = C/(PH) and θ the scaled control θ = Θ/(PH). Solution 18 / 35

39 Solution: Portfolio Allocation and Consumption Given a wealth-to-housing ratio z, where v(z) > M (z+1 φ o) 1 γ 1 γ consumption c (z) and portfolio θ (z) and b (z), the agent chooses a optimal Model Illustration Equilibrium Calibration Boundaries Sensitivity c (z) = ( vz (z) β ) 1/(β(1 γ) 1) θ (z) = ω v z(z) v zz (z) + ρ PSσ P σ S (z 1) b (z) = z θ (z) for the constant ω defined as ω = [α S r+(1 β(1 γ))ρ PS σ P ]/σ 2 S. 19 / 35

40 Baseline Calibration Model Illustration Equilibrium Calibration Boundaries Sensitivity Variable Symbol Value Curvature of the utility function γ 2 House flow services 1 β 0.4 Time preference ρ Risk free rate r Housing stock depreciation δ 0.02 Transaction cost φ a 0.06 information cost φ o 0.06 Risky asset drift α S Standard deviation risky asset σ S Correlation house price - risky asset ρ PS 0.25 Standard deviation house price σ P 0.14 House price drift µ P 0.03 Overvaluation m H 20% Undervaluation m L 20% Probability π / 35

41 Graphical Solution Model Illustration Equilibrium Calibration Boundaries Sensitivity Information Adjustment Lower bound Adjustment Upper bound Adjustment Return point z + 1 = W / HP 21 / 35

42 Sensitivity to Misperception Model Illustration Equilibrium Calibration Boundaries Sensitivity Table 1: Acquisition of information, housing adjustments, and misperception. Model outcomes for the information acquisition boundaries, the housing adjustment boundaries, and the return points under different parameterizations. Adjust Info. Return Info. Adjust LB LB Point UB UB GL (no misperception) Benchmark (+5%/-5%) Increase misperception Overvaluation - π Undervaluation - π with respect to GL, inaction region is lower and smaller 22 / 35

43 Sensitivity to Misperception Model Illustration Equilibrium Calibration Boundaries Sensitivity Table 1: Acquisition of information, housing adjustments, and misperception. Model outcomes for the information acquisition boundaries, the housing adjustment boundaries, and the return points under different parameterizations. Adjust Info. Return Info. Adjust LB LB Point UB UB GL (no misperception) Benchmark (+5%/-5%) Increase misperception Overvaluation - π Undervaluation - π with respect to GL, inaction region is lower and smaller wider misperception, lowers inaction region even more 22 / 35

44 Sensitivity to Misperception Model Illustration Equilibrium Calibration Boundaries Sensitivity Table 1: Acquisition of information, housing adjustments, and misperception. Model outcomes for the information acquisition boundaries, the housing adjustment boundaries, and the return points under different parameterizations. Adjust Info. Return Info. Adjust LB LB Point UB UB GL (no misperception) Benchmark (+5%/-5%) Increase misperception Overvaluation - π Undervaluation - π with respect to GL, inaction region is lower and smaller wider misperception, lowers inaction region even more more undervaluation, widens inaction region for information 22 / 35

45 Sensitivity to Misperception Model Illustration Equilibrium Calibration Boundaries Sensitivity Table 1: Acquisition of information, housing adjustments, and misperception. Model outcomes for the information acquisition boundaries, the housing adjustment boundaries, and the return points under different parameterizations. Adjust Info. Return Info. Adjust LB LB Point UB UB GL (no misperception) Benchmark (+5%/-5%) Increase misperception Overvaluation - π Undervaluation - π with respect to GL, inaction region is lower and smaller wider misperception, lowers inaction region even more more undervaluation, widens inaction region for information more overvaluation, narrows inaction region for information 22 / 35

46 m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B 23 / 35

47 Risky assets and misperception 1.25 m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B GL Benchmark Misperception +5%/ 5% z + 1 = W/(H x P) 24 / 35

48 Risky assets and probabilities of over/undervaluation m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% z + 1 = W/(H x P) 25 / 35

49 Risky assets and probabilities of over/undervaluation m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% Misperception +5%/ 5% Undervaluation Overvaluation z + 1 = W/(H x P) z + 1 = W/(H x P) 25 / 35

50 Misperception and Risky Stock Holdings - Empirics Panel A: Misperception (dispersion) θ it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (1) [1] [2] [3] m it [ 2.11] [ 2.77] [ 1.88] z it [5.10] [1.79] [1.15] m it z [1.43] [1.93] [1.39] constant [0.1] [ 0.51] [ 0.33] R % 57.36% 57.14% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, / 35

51 Misperception and Risky Stock Holdings - Empirics Panel A: Misperception (dispersion) θ it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (1) [1] [2] [3] m it [ 2.11] [ 2.77] [ 1.88] z it [5.10] [1.79] [1.15] m it z [1.43] [1.93] [1.39] constant [0.1] [ 0.51] [ 0.33] R % 57.36% 57.14% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, / 35

52 Misperception and Risky Stock Holdings - Empirics θ it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (1) Panel A: Misperception (dispersion) [1] [2] [3] m it [ 2.11] [ 2.77] [ 1.88] z it [5.10] [1.79] [1.15] m it z [1.43] [1.93] [1.39] constant [0.1] [ 0.51] [ 0.33] R % 57.36% 57.14% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, 198 Panel B: Misperception (overvaluation) [1] [2] [3] m it [ 2.55] [ 3.29] [ 1.99] z it [6.34] [2.84] [1.82] m it z [1.28] [2.45] [1.62] constant [0.72] [ 0.23] [ 0.15] R % 57.4% 57.18% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, / 35

53 Misperception and Risky Stock Holdings - Empirics θ it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (1) Panel A: Misperception (dispersion) [1] [2] [3] m it [ 2.11] [ 2.77] [ 1.88] z it [5.10] [1.79] [1.15] m it z [1.43] [1.93] [1.39] constant [0.1] [ 0.51] [ 0.33] R % 57.36% 57.14% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, 198 Panel B: Misperception (overvaluation) [1] [2] [3] m it [ 2.55] [ 3.29] [ 1.99] z it [6.34] [2.84] [1.82] m it z [1.28] [2.45] [1.62] constant [0.72] [ 0.23] [ 0.15] R % 57.4% 57.18% FE/RE RE FE FE Cluster No No Zip Obs. 4, 225 4, 225 4, / 35

54 Misperception and Consumption m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B GL Benchmark Misperception +5%/ 5% z + 1 = W/(H x P) 27 / 35

55 Probabilities of Over/Undervaluation and Consumption m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% z + 1 = W/(H x P) 28 / 35

56 Probabilities of Over/Undervaluation and Consumption m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% Misperception +5%/ 5% Undervaluation Overvaluation z + 1 = W/(H x P) z + 1 = W/(H x P) 28 / 35

57 Misperception and Consumption - Empirics Panel A: Misperception (dispersion) C it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (2) [1] [2] [3] m it [ 2.99] [ 4.45] [ 3.64] z it [ 27.55] [ 25.67] [ 17.73] m it z it [3.12] [3.25] [3.20] constant [10.62] [8.08] [5.26] R % 81.85% 81.76% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, / 35

58 Misperception and Consumption - Empirics Panel A: Misperception (dispersion) C it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (2) [1] [2] [3] m it [ 2.99] [ 4.45] [ 3.64] z it [ 27.55] [ 25.67] [ 17.73] m it z it [3.12] [3.25] [3.20] constant [10.62] [8.08] [5.26] R % 81.85% 81.76% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, / 35

59 Misperception and Consumption - Empirics C it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (2) Panel A: Misperception (dispersion) [1] [2] [3] m it [ 2.99] [ 4.45] [ 3.64] z it [ 27.55] [ 25.67] [ 17.73] m it z it [3.12] [3.25] [3.20] constant [10.62] [8.08] [5.26] R % 81.85% 81.76% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, 028 Panel B: Misperception (overvaluation) [1] [2] [3] m it [ 15.18] [ 14.29] [ 8.73] z it [ 31.03] [ 28.49] [ 19.52] m it z it [5.03] [5.38] [4.22] constant [8.40] [6.88] [4.40] R % 82.67% 82.58% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, / 35

60 Misperception and Consumption - Empirics C it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (2) Panel A: Misperception (dispersion) [1] [2] [3] m it [ 2.99] [ 4.45] [ 3.64] z it [ 27.55] [ 25.67] [ 17.73] m it z it [3.12] [3.25] [3.20] constant [10.62] [8.08] [5.26] R % 81.85% 81.76% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, 028 Panel B: Misperception (overvaluation) [1] [2] [3] m it [ 15.18] [ 14.29] [ 8.73] z it [ 31.03] [ 28.49] [ 19.52] m it z it [5.03] [5.38] [4.22] constant [8.40] [6.88] [4.40] R % 82.67% 82.58% FE/RE RE FE FE Cluster No No Zip Obs. 8, 192 8, 192 8, / 35

61 Misperception and Leverage 0.3 m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B GL Benchmark Misperception +5%/ 5% z + 1 = W/(H x P) 30 / 35

62 Probabilities of Over/Undervaluation and leverage 0.3 m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% z + 1 = W/(H x P) 31 / 35

63 Probabilities of Over/Undervaluation and leverage m and θ Over/Under and θ m and C Over/Under and C m and B Over/Under and B Misperception +5%/ 5% Misperception +15%/ 15% Misperception +5%/ 5% Undervaluation Overvaluation z + 1 = W/(H x P) z + 1 = W/(H x P) 31 / 35

64 Misperception and Leverage - Empirics Panel A: Misperception (dispersion) B it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (3) [1] [2] [3] m it [ 7.11] [ 6.92] [ 3.90] z [ 14.24] [ 13.98] [ 9.14] m it z [7.91] [7.70] [4.03] constant [2.78] [1.57] [0.85] R % 71.16% 71.01% FE/RE RE FE FE Cluster No No Zip Obs. 3, 828 3, 828 3, / 35

65 Misperception and Leverage - Empirics Panel A: Misperception (dispersion) B it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (3) [1] [2] [3] m it [ 7.11] [ 6.92] [ 3.90] z [ 14.24] [ 13.98] [ 9.14] m it z [7.91] [7.70] [4.03] constant [2.78] [1.57] [0.85] R % 71.16% 71.01% FE/RE RE FE FE Cluster No No Zip Obs. 3, 828 3, 828 3, / 35

66 Misperception and Leverage - Empirics B it z it =γ 0 +γ 1 z it +γ 2 m it +γ 3 z it m it +Γ X it +u it, (3) Panel A: Misperception (dispersion) [1] [2] [3] m it [ 7.11] [ 6.92] [ 3.90] z [ 14.24] [ 13.98] [ 9.14] m it z [7.91] [7.70] [4.03] constant [2.78] [1.57] [0.85] R % 71.16% 71.01% Panel B: Misperception (overvaluation) [1] [2] [3] m it [0.46] [ 0.54] [ 0.31] z [ 12.81] [ 12.14] [ 6.05] m it z [ 4.47] [ 2.41] [ 1.44] constant [2.70] [1.71] [0.90] R % 70.68% 70.51% FE/RE RE FE FE Cluster No No Zip Obs. 3, 828 3, 828 3, 857 FE/RE RE FE FE Cluster No No Zip Obs. 3, 828 3, 828 3, / 35

67 Next 33 / 35

68 Conclusions Next House price misperception affects the optimal behavior of households (via risk aversion). The more misperception, less investment in risky assets larger housing wealth relative to total wealth acquire information more frequently Overvaluation less risky asset near downsizing Overvaluation narrower bands of inaction In this paper Showed evidence of misperception Build misperception into a portfolio choice model Tested implications with household level data (PSID) 34 / 35

69 Next Steps On the model: Next Extend to a richer model for misperception as a function of tenure (to match data) On the empirical implications: Extend the analysis to include tenure. Robustness: Census Data. Better understanding of the drivers behind misperception and implication on other markets. 35 / 35

70 Solution: Inaction Region 36 / 35

71 Solution: Inaction Region Solution: Inaction Region v(z) satisfies ρv(z) = sup c,θ {u(c)+dv(z)}, z (z o,z o ), 37 / 35

72 Solution: Inaction Region Solution: Inaction Region v(z) satisfies where ρv(z) = sup c,θ {u(c)+dv(z)}, z (z o,z o ), Dv(z) =(z(r+δ µ P +σ 2 P(1+β(γ 1))) +θ(α S r (1+β(γ 1))ρ PS σ S σ P ) c)v z (z) (z2 σ 2 P 2zˆθ ρ PS σ P σ S +θ 2 σ 2 S)v zz (z), 37 / 35

73 Solution: Inaction Region Solution: Inaction Region v(z) satisfies where ρv(z) = sup c,θ {u(c)+dv(z)}, z (z o,z o ), Dv(z) =(z(r+δ µ P +σ 2 P(1+β(γ 1))) +θ(α S r (1+β(γ 1))ρ PS σ S σ P ) c)v z (z) (z2 σ 2 P 2zˆθ ρ PS σ P σ S +θ 2 σ 2 S)v zz (z), v(z) = M (z +1 φ o) (1 γ) 1 γ, z / (z o,z o ) 37 / 35

74 Solution: Inaction Region Solution: Inaction Region v(z) satisfies where ρv(z) = sup c,θ {u(c)+dv(z)}, z (z o,z o ), Dv(z) =(z(r+δ µ P +σ 2 P(1+β(γ 1))) +θ(α S r (1+β(γ 1))ρ PS σ S σ P ) c)v z (z) (z2 σ 2 P 2zˆθ ρ PS σ P σ S +θ 2 σ 2 S)v zz (z), v(z) = M (z +1 φ o) (1 γ) 1 γ ṽ(z) = M (z +1 φ a φ 0 ) (1 γ) 1 γ, z / (z o,z o ), z / (z a,z a ) 37 / 35

75 Solution: Inaction Region Solution: Inaction Region v(z) satisfies where ρv(z) = sup c,θ {u(c)+dv(z)}, z (z o,z o ), Dv(z) =(z(r+δ µ P +σ 2 P(1+β(γ 1))) +θ(α S r (1+β(γ 1))ρ PS σ S σ P ) c)v z (z) (z2 σ 2 P 2zˆθ ρ PS σ P σ S +θ 2 σ 2 S)v zz (z), v(z) = M (z +1 φ o) (1 γ) 1 γ ṽ(z) = M (z +1 φ a φ 0 ) (1 γ) 1 γ, z / (z o,z o ), z / (z a,z a ) and M is defined as M = (1 γ)sup(z +1) γ 1 ṽ(z), z ǫ 37 / 35

76 Solution: Return Point The return point z a attains the maximum in ṽ(z ) = M (z a +1) (1 γ) 1 γ. Solution: Inaction Region 38 / 35

77 Solution: Information Acquisition and Transaction Boundaries Value matching and smooth pasting conditions hold at the two thresholds (z a,z a ) ṽ(z) = M (ẑ +1 φ a φ o ) (1 γ) 1 γ ṽ z (z) = M(ẑ φ a φ o ) γ Solution: Inaction Region 39 / 35

78 Solution: Information Acquisition and Transaction Boundaries Value matching and smooth pasting conditions hold at the two thresholds (z a,z a ) ṽ(z) = M (ẑ +1 φ a φ o ) (1 γ) 1 γ ṽ z (z) = M(ẑ φ a φ o ) γ Solution: Inaction Region for ẑ a = z a,z a and at the two thresholds (z o,z o ) ( zo v(z) = πv 1+m +1 φ h o )+(1 π)m (z a +1 φ a φ o ) (1 γ), 1 γ ( zo v(z) = (1 π)v 1+m +1 φ l o )+πm (z a +1 φ a φ o ) (1 γ) 1 γ ( zo v(z) = πv 1+m +1 φ h o )+(1 π)m (z a +1 φ a φ o ) (1 γ) 1 γ if z > 0, if z 0, 39 / 35