Portfolio Choice with Borrowing Constraints I NSTITUTE OF C - - PowerPoint PPT Presentation

Introduction Model Calibration Results

Portfolio Choice with Borrowing Constraints

INSTITUTE OF COMPUTATIONAL ECONOMICS

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

SHARE OF RISKY ASSETS IN PORTFOLIO DECREASES OVER

THE LIFE-CYCLE WHEN THERE IS LABOR INCOME UNCERTAINTY.

CORRELATION 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 ↑ ↑

AGENTS OBSERVE:

Wealth: Wt Age: t Income: Yt

Introduction Model Calibration Results

Timing

AGENTS CHOOSE:

Consumption: Ct Risky Asset Holding: At Risk Free Asset Holding: St

↓ t———————————————— t+1 ↑ ↑

AGENTS OBSERVE:

Wealth: Wt Age: t Income: Yt

Introduction Model Calibration Results

Timing

AGENTS CHOOSE:

Consumption: Ct Risky Asset Holding: At Risk Free Asset Holding: St

↓ t———————————————— t+1 ↑ ↑

AGENTS OBSERVE: STATES EVOLVE:

Wealth: Wt Wt+1 = (1 + rf)St + (1 + r a

t )At

Age: t ra

t+1 = f(r a t , ǫa)

Income: Yt Yt+1 = f(Yt, ǫy)

Introduction Model Calibration Results

Households

Period Utility u(Ct, Kt) = (Ct )1−γ

1−γ

Budget Constraint Ct + At + St = Wt + Yt Nonnegativity Constraint Ct ≥ 0 Borrowing Constraint St ≥ S Short-selling Constraint At ≥ 0

Introduction Model Calibration Results

State Transitions

Wt+1 = (1 + rf)St + (1 + r a

t )At

r a

t+1

= f(r a

t , ǫa)

Yt+1 = f(Yt, ǫy)

Introduction Model Calibration Results

Dynamic Decision Problem

In period t the agent chooses a vector x = [St At]′ to maximize expected life-time utility given a state vector s: Vt(s) = max

x

ut(x, s) + β

• ˆ

Vt(s’; a)dF(s’|s, 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: ˆ Vt =

n

• i=0

aiTi(z)

Introduction Model Calibration Results

Method

1

We approximate the value function and solve the problem via backward recursion using AMPL software.

2

Within AMPL we call the KNITRO nonlinear optimization solver to compute the optimal policy functions of the agents in each period.

3

We run Monte Carlo simulations and generate graphics in MATLAB.

Introduction Model Calibration Results

Baseline Calibration

PARAMETER VALUE DESCRIPTION γ 3.000 Coefficient of relative risk aversion rf 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

2 4 6 8 10 12 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Share of risky asset in total invested amount 2 4 6 8 10 12 0.1 0.15 0.2 0.25 0.3 0.35 Consumption 2 4 6 8 10 12 0.2 0.21 0.22 0.23 0.24 0.25 0.26 Share of risky asset in total PDV of income 2 4 6 8 10 12 0.5 1 1.5 Financial Wealth

Introduction Model Calibration Results

Deterministic Pre-retirement Income Stream

2 4 6 8 10 12 0.5 1 1.5 Share of risky asset in total invested amount 2 4 6 8 10 12 0.7 0.8 0.9 1 1.1 1.2 1.3 Consumption 2 4 6 8 10 12 0.04 0.06 0.08 0.1 0.12 0.14 Share of risky asset in total PDV of income 2 4 6 8 10 12 1 2 3 4 Financial Wealth

Introduction Model Calibration Results

Income Risk and Retirement Payments

2 4 6 8 10 12 0.5 1 1.5 Share of risky asset in total invested amount 2 4 6 8 10 12 0.7 0.8 0.9 1 1.1 1.2 1.3 Consumption 2 4 6 8 10 12 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Share of risky asset in total PDV of income 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 Financial Wealth

Introduction Model Calibration Results

Borrowing Against Risky Asset Allowed

2 4 6 8 10 12 1 2 3 4 Share of risky asset in total invested amount 2 4 6 8 10 12 0.7 0.8 0.9 1 1.1 1.2 1.3 Consumption 2 4 6 8 10 12 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Share of risky asset in total PDV of income 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 Financial Wealth

Introduction Model Calibration Results

Correlation between Labor Income and Asset Risk

2 4 6 8 10 12 0.5 1 1.5 Share of risky asset in total invested amount 2 4 6 8 10 12 0.7 0.8 0.9 1 1.1 1.2 1.3 Consumption 2 4 6 8 10 12 0.04 0.06 0.08 0.1 0.12 Share of risky asset in total PDV of income 2 4 6 8 10 12 1 2 3 4 Financial Wealth