Introduction Model Calibration Results Portfolio Choice with Borrowing Constraints I NSTITUTE OF C OMPUTATIONAL E CONOMICS Heng Chen - University of Zurich Fabian Kinderman - University of Würzburg Robert Sarama - Ohio State University Daniel Shoag - Harvard University Xuan Tam - University of Virginia
Introduction Model Calibration Results Model Predictions S HARE OF RISKY ASSETS IN PORTFOLIO DECREASES OVER THE LIFE - CYCLE WHEN THERE IS LABOR INCOME UNCERTAINTY . C ORRELATION BETWEEN LABOR INCOME FLUCTUATIONS AND RISKY ASSET RETURN FLUCTUATIONS INDUCES AGENTS TO HOLD MORE SAFE ASSETS .
Introduction Model Calibration Results Basic Idea Life-cycle model Partial equilibrium Retirement with no bequest motive Life begins at age 20 and ends at age 80
Introduction Model Calibration Results Timing t———————————————— t+1 ↑ ↑ A GENTS OBSERVE : Wealth: W t Age: t Income: Y t
Introduction Model Calibration Results Timing A GENTS CHOOSE : Consumption: C t Risky Asset Holding: A t Risk Free Asset Holding: S t ↓ t———————————————— t+1 ↑ ↑ A GENTS OBSERVE : Wealth: W t Age: t Income: Y t
Introduction Model Calibration Results Timing A GENTS CHOOSE : Consumption: C t Risky Asset Holding: A t Risk Free Asset Holding: S t ↓ t———————————————— t+1 ↑ ↑ A GENTS OBSERVE : S TATES EVOLVE : W t + 1 = ( 1 + r f ) S t + ( 1 + r a Wealth: W t t ) A t r a t + 1 = f ( r a t , ǫ a ) Age: t Y t + 1 = f ( Y t , ǫ y ) Income: Y t
Introduction Model Calibration Results Households u ( C t , K t ) = ( C t ) 1 − γ Period Utility 1 − γ Budget Constraint C t + A t + S t = W t + Y t Nonnegativity Constraint C t ≥ 0 Borrowing Constraint S t ≥ S Short-selling Constraint A t ≥ 0
Introduction Model Calibration Results State Transitions ( 1 + r f ) S t + ( 1 + r a W t + 1 = t ) A t r a f ( r a t , ǫ a ) = t + 1 f ( Y t , ǫ y ) Y t + 1 =
Introduction Model Calibration Results Dynamic Decision Problem In period t the agent chooses a vector x = [ S t A t ] ′ to maximize expected life-time utility given a state vector s : � ˆ V t ( s ) = max u t ( x , s ) + β V t ( s’ ; a ) dF ( s’ | s , x ) x One continuous state: Wealth (W) Two discrete states: Risky asset return ( r a ) and Labor Income ( Y )
Introduction Model Calibration Results Value Function Approximation Approximate using n Chebyshev nodes z and n Chebyshev basis functions T : n ˆ � V t = a i T i ( z ) i = 0
Introduction Model Calibration Results Method We approximate the value function and solve the problem 1 via backward recursion using AMPL software. Within AMPL we call the KNITRO nonlinear optimization 2 solver to compute the optimal policy functions of the agents in each period. We run Monte Carlo simulations and generate graphics in 3 MATLAB.
Introduction Model Calibration Results Baseline Calibration P ARAMETER V ALUE D ESCRIPTION γ 3.000 Coefficient of relative risk aversion r f 0.025 Risk free rate β 0.990 Time discount factor n 35 Order of approximation S 0.000 Borrowing constraint Discrete states r a and Y take on two values each with i.i.d. shocks.
Introduction Model Calibration Results Income in First Period Consumption Share of risky asset in total invested amount 0.26 0.35 0.25 0.3 0.24 0.25 0.23 0.2 0.22 0.15 0.21 0.2 0.1 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Share of risky asset in total PDV of income Financial Wealth 0.26 1.5 0.25 0.24 1 0.23 0.22 0.5 0.21 0.2 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Introduction Model Calibration Results Deterministic Pre-retirement Income Stream Consumption Share of risky asset in total invested amount 1.5 1.3 1.2 1 1.1 1 0.9 0.5 0.8 0.7 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Share of risky asset in total PDV of income Financial Wealth 0.14 4 0.12 3 0.1 2 0.08 1 0.06 0.04 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Introduction Model Calibration Results Income Risk and Retirement Payments Share of risky asset in total invested amount Consumption 1.5 1.3 1.2 1 1.1 1 0.9 0.5 0.8 0.7 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Share of risky asset in total PDV of income Financial Wealth 0.14 3 0.12 2.5 0.1 2 0.08 1.5 0.06 1 0.04 0.5 0.02 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Introduction Model Calibration Results Borrowing Against Risky Asset Allowed Consumption Share of risky asset in total invested amount 4 1.3 3 1.2 1.1 2 1 0.9 1 0.8 0.7 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Share of risky asset in total PDV of income Financial Wealth 0.14 3 0.12 2.5 0.1 2 0.08 1.5 0.06 1 0.04 0.5 0.02 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Introduction Model Calibration Results Correlation between Labor Income and Asset Risk Consumption Share of risky asset in total invested amount 1.5 1.3 1.2 1 1.1 1 0.9 0.5 0.8 0.7 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Share of risky asset in total PDV of income Financial Wealth 4 0.12 3 0.1 2 0.08 1 0.06 0.04 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12
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