An Equilibrium Model with Restricted Stock Market...

An Equilibrium Model with Restricted Stock Market Participation

S. Basak, D. CuocoPresented by Yoel Krasny

Outline

Introduction

The ModelEconomyUnrestricted caseRestricted caseExample

Discussion

Limited Participation –Empirical findings

72.4% of the households held no stock at all (as of 1984) but accounted for 68% of aggregate food expenditures

52.3% of the HH, holdings other liquid assets in excess of 100,000$, held no stock at all

The fraction of HH owning stocks increases with average labor income and education

Aggregate consumption of Stock holders is more volatile and correlated with the equity risk premia

Main Insight

Two types of investors:Non-participants – trade only bondsParticipants – trade both stocks and bonds

The non-participants have a smooth consumption process stock holders are left alone to bear the aggregate risk of the equity market stock holders demand an higher equity premium

Outline

Introduction


Discussion

The Economy

Information StructureProbability Space – (Ω, f, F, P)One dimensional Brownian Motion –Information Set determined by W(t)

Consumption SpaceA single perishable goodThe consumption process c(t) is nonnegative and progressively measurable

],0[),( TttW ∈

The Economy - Securities

Exogenous positive dividend process

Endogenous bond price process

Endogenous Stock price – claim to dividend stream

dttBtrtdB )()()( =

The Economy – Trading Strategies

Admissible trading strategy is a vector process (α(t),θ(t)) – amounts invested in (bond, stock)

Trading strategy is said to finance the consumption plan c(t):

Two agents, each with

Agent 1:Has access to both the bond and stock marketU(c) satisfies the Inada conditionEndowed with (-β shares of bond, 1 share of stock)

Agent 2: Prevented from investing in the stock marketU(c) = log(c)Endowed with (β shares of bond, 0 shares of stock)

The Economy – Agents

The Economy – Equilibrium

Outline

Introduction


Discussion

The Unrestricted case –Representative agent

The second agent can trade the stockEquilibrium is constructed by replacing the two agents with a single representative agent as in Huang (1987)The representative agent is

Endowed with the aggregate supply of securities Has the following utility function:

Solving the Unrestricted case

)),0(()),(()(λδλδε ρ

c

ct

UtUet −=

The Marginal Rate of Substitution є(t) is the pricing kernel

tXX dWtdtttdX )()()( σμ +=Let X(t) be some security:

t

t

t

WttXtdMTGistXt

sXsEttX

sXtsEtXst

⋅=⇒⇒

=⇒⎭⎬⎫

⎩⎨⎧

=<∀

)()()()()(

)()()()(

)()()()(:

ψεε

εεεε


tXX dWtdtttdX )()()( σμ +=

[ ] tXX dWdtttXtttXtd ...)()()()()()( +++= εε σσμμεε

0)()()()( =++ εε σσμμε XX ttXtt

)()()()()()( trttdttBtrtdB ⋅−=⇒= εμε

tdWtStdttSttdS )()()()()( σμ +=

)()()()()(

ttrttt

σμεσε

−⋅−=⇒

ITO

K(t)

MTG

Bond

Stock

tdWtdtttd )()()( εε σμε +=


).),(()),0(()),(()( ttf

UtUet

c

ct δλδλδε ρ == −

))0((

))(()())(()())(()(2

21

δδρσδμδμ

ρ

εc

cdcccdcct

UtUttUttUet −+

=−

))0(()())(()(

δσδσ

ρ

εc

dcct

UttUet

−

=

tdWtdtttd )()()( εε σμε +=

tdWtdtttd )()()( δδ σμδ +=

ITO

)()()()()()( 22

1 ttPtAttAtr dd σμρ −+=

;)),(()),(()(;

)),(()),(()(

λδλδ

λδλδ

tUtUtP

tUtUtA

cc

ccc

c

cc −=−=

)()()( ttAtk dσ=


Interest Rate

Market Price of Risk

)()()()()()( 22

1 ttPtAttAtr dd σμρ −+=

)()()( ttAtk dσ=


Interest Rate


[ ] [ ]122

21

11 +++ Δ−Δ⋅+−= ttttf

t ccElanr σγγδ

[ ] [ ][ ] [ ]111

1

12

21

11 ),( ++++

+++ ΔΔ=+−

ttp

tttptt

ptt

ft

ptt crc

rrrrE σγρ

σσ

Discrete

Discrete

Solving the Unrestricted case –getting individual consumption

ITO

Solving the Unrestricted case –Choosing λ

λ is chosen such that the budget constraint for agent 2 is satisfied:

Outline

Introduction


Discussion

The Restricted case – Rep. agent

Second Agent does not face a complete marketEquilibrium consumption allocation is not efficientFollowing Cuoco and He (1994) – Aggregation in case of incomplete market

))(()())((max)),(( 2211 tcuttcutcu ⋅+= λλ

MRS is the SDF

MRS follows

Agent 1 is facing a complete market

Agent 2 invests in bond and has log utility (He and Pearson 1991)

Solving the Restricted case

Use the above to get the evolution of lamda

Lamda is negatively correlated with aggregate consumption

If the given dividend process is Markovian:Unrestricted case – δ(t) is the state of the economyRestricted case – (δ(t), λ(t)) is the state


;))(),(())(),(()(;

))(),(())(),(()(

ttUttUtP

ttUttUtA

cc

ccc

c

cc

λδλδ

λδλδ

−=−=


Interest Rate


;)1()(;)(c

tPc

tA +==

γγCRRA

Restricted case - Calibration

Following Mehra and Prescott (1985), calibrate:

Stockholders have CRRA utility Now, MPR and interest rate depend only on consumption share of non-stockholders Ф:

0357.0)(

,0183.0)(

==ttdd

δσ

δμ

)1(0357.0

)1)(()(0357.0

)1)(()( 1 φ

γφδδγ

φδσγσ

−=

−⋅

=−

⋅=⋅=

tt

tAtk d

d


)1(0357.0)(

φγ

−=tk

RA of 3.3, real interest rate of 1.3%

Ф

Mehra and Prescott’s estimated MPR

0.68

Interest Rate

Ф

Outline

Introduction


Discussion

Example

Both agents have log utility

Dividend growth follows

Now, Lamda represents consumption ratio

)()(

))(())(()(

1

2

2'2

1'1

tctc

tcutcut ==λ

Example - results

Unrestricted Restricted

r(t)

K(t)

dλ(t) 0

C1(t)

C2(t)

2dd σμρ −+ 2))(1(

dtd σλμρ +−+

( ) tdWttd

22)()( σλλ ⋅+−

λδ+1

)(t

λλδ+1

)(t

( ) ( )dd

tttttt cdc σλσσλλμμ )(1)(,)()()()( 22 +=++=

( ) 0)(,)()()()( 22 =+−= ttttt cdc dσσλλμμ

dσ ( )

dt σλ )(1+

Simulation

1.0001.0

000,1001

0357.00183.0

0

======

dt

T

d

d

ρ

λσμ

Simulated 1000 years with:

Unrestricted - consumption

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

250 500 750 1000

U_C1 U_C2 DIV

Securities

0

1

2

3

4

5

6

250 500 750 10000.0E+00

1.0E+07

2.0E+07

3.0E+07

4.0E+07

5.0E+07

6.0E+07

9000 9250 9500 9750 10000

Stock Bond

Restricted - consumption

StockHolder NonStock Agg

0.0

0.4

0.8

1.2

1.6

2.0

2.4

250 500 750 10000.00E+00

5.00E+06

1.00E+07

1.50E+07

2.00E+07

2.50E+07

3.00E+07

9000 9250 9500 9750 10000

Unrestricted

Consumption Growth -Shocks

-.04

-.03

-.02

-.01

.00

.01

.02

.03

10 11 12 13 14 15 16 17 18 19 20

restricted - agent 1restricted - agent 2unrestricted - agent 1/2DDIV

Consumption Growth –Correlations and Stdevs

Stdev

DDIV DC1 DC2 DC_U

DDIV 1.000000 0.993561 0.071014 1.000000

DC1 0.993561 1.000000 0.033538 0.993561

DC2 0.071014 0.033538 1.000000 0.071014

DC_U 1.000000 0.993561 0.071014 1.000000

0.023346 0.017383 0.000553 0.011673

Lamda

0.2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2500 5000 7500 10000

Unrestr ic ted Restr ic ted

MPR

.0 3

.0 4

.0 5

.0 6

.0 7

.0 8

.0 9

.1 0

2 5 0 0 5 0 0 0 7 5 0 0 1 0 0 0 0

R e s tr i c te d U n re s tr i c te d

Interest Rate

.0156

.0160

.0164

.0168

.0172

.0176

.0180

.0184

2500 5000 7500 10000

Unre stric ted Restric te d

Outline

Introduction


Discussion

Discussion - Theoretical

Participation in the model is determined only by the wealth held by each type of agent, however:

As financial markets become more accessible, participation increases (1989 2002: 74% increase)Participation might decrease in bad times

Model indirectly considers non-participants to be less risk-averse, however, the opposite is more reasonable

Dependence of the results on utility specification:Agent 1 utility – ARA must decrease with c Agent 2 log utility no hedging demands consumption of agent 2 even smoother

Discussion – Empirical

The following should vary with participation:Equity Premium (negative relationship)Interest Rate (positive)

Test implications:Time Series – historical dataCross Section – across countries

Endogeneity issues?

An Equilibrium Model with Restricted Stock Market...

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