Chp.4 Lifetime Portfolio Selection Under Uncertainty

Hai Lin

Department of Finance, Xiamen University,361005

1.Introduction

• Examine the combined problem of optimal portfolio selection and consumption rules for individual in a continuous time model.

• The rates of return are generated by Wiener Brownian-motion process.

• Particular case: – Two asset model with constant relative risk av

ersion or isoelastic marginal utility.– Constant absolute risk aversion.

2.Dynamics of the Model: The Budget Equation

• W(t): the total wealth at time t;

• Xi(t): the price of ith asset at time t, i=1,2,…,m;

• C(t): the consumption per unit time at time t;

• wi(t):the proportion of total wealth invested in the ith asset at time t, i=1,2,…,m.

m

ii tw

1

1)(

The budget equation

• At time t0, the investment between t0 and t(t0+h) is :

• The value of this investment at time t is:htCtW )()( 00

m

i i

ii

m

i i

ii

htCtWtX

tXtw

tX

tXhtCtWtw

100

00

1 0000

))()(()(

)()(

)(

)())()()((

m

iii

m

i i

ii

htChtCtWhtgtw

htChtCtWtX

tXtwtWtW

10000

0001 0

00

)())()(}](1])(){exp[([

)())()()](1)(

)()(([)()(

The process of g(t)

• Suppose g(t) is the geometric Brownian motion. In discrete time,

• :the expected return of asset i;• : the volatility of asset i;

;)2

()(2

ii

ii Yhhtg

ii

),0( 2hNY ii

m

ii

iii htChtCtWYhtwtWtW

1000

2

00 )())()(}(1])2

){exp[(()()(

Momentum

)()()()((

)()()()()((

)())()()()((

)())()()(1))(exp(()}()(){(

001

1

20000

1

10000

1000000

hOhtCtWtw

htCtwhtCtWtw

htChtCtWhtw

htChtCtWhtwtWtWtE

i

m

ii

m

iiii

m

ii

m

iii

m

iii

)(

)(}){()()(})]()(){[( 02

1 1000

200

hO

tWYYtEtwtwtWtWtEm

i

m

jjiji

Continuous time

.)()()()]()()([

,)(

110

m

iiii

m

iii

iii

dttZtWtwdttCtWtwdW

dttZdY

m

iii

h

m

iii

tCtWtwh

tWtWtEtW

OtChtCtWtwh

tWtWtE

1000

00

00

10000

00

)()()())()(

)((lim)(

)1()(])()([)())()(

)((

3. The two asset model

• :the proportion invested in the risky asset;

• :the proportion invested in the sure asset.

• : the return on risky asset.

)(1)(2 twtw

)()(1 twtw

)()(1 tgtg

rtg )(2

Yhhtg )2/()( 2

Two asset model(2)

)()()))((()(

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

)()(

)())(()()(}))()(){((

);()]()()))(([(

)()]()())](1[)([())()()((

;)())()(])(1)[exp(

)](1[}1])2/){exp[((()()(

20

20

2

200

20

2200

000

000000

000

02

00

tCtWrrtwtW

dttWtZtwdttCtWrrtwdW

htWtw

hOYtEtWtwtWtWtE

hOhtCtWrrtw

hOhtCtWrtwtwtWtWtE

htChtCtWrh

twYhtwtWtW

The objective problem

0)('';0)('

;0)0(;0)(,0)(

..

)},),(())(()exp({max

0

0

CUCU

WWtWtC

ts

TTWBdttCUtET

The dynamic programming form

• Define

• Then the objective function can be written:

);),(()),((

};),(())(()exp(){(max)),(()(),(

TTWBTTWI

TTWIdssCUstEttWIT

tsWsC

})),(())(()exp({max)0,(00 t

ttWIdssCUsEWI

The dynamic programming(2)

• If ,then by the Mean Value Theorem and Taylor Rule,

htt 0

],[

)},()]()([)),((

2

1

)]()([)),((

)),(()),(())(()){exp(()),((

0

202

002

000

00000

},{00 max

ttt

hOtWtWW

ttWI

tWtWW

ttWI

ht

ttWIttWIhtCUttEttWI

wc

The dynamic programming(3)

• Take the conditional expectation on both sides and use the previous results, divide the equation by h and take the limit as

))()(2

1

)}()(]))(({[)]([)(exp(max0

2222

2

))(),((

tWtwW

I

tCtWrrtwW

I

t

ItCUt tt

twtc

0h

The solution

• Define

0),;,(max

),()(2

1

)}()(]))(({[)()exp(),;,(

},{

2222

2

tWCw

tWtwW

I

tCtWrrtwW

I

t

ICUttWCw

wc

t

tt

First order condition

22*2

2**

***

)(0);;,(

)(')exp(0);;,(

WwW

I

W

IWrtWCw

W

ICUttWCw

ttw

tC

Second order condition

• If is concave in W,•

)),(( ttWI

.0det,0,0

wwwC

CwCCCCww

.0)(

,0)('')exp(

,0

2

222

W

ItW

CUt

tww

CC

CwwC

Summary

• The maximum problem can be rewritten as:

)),(()),((

;0

;0

;0),;,( **

TTWBTTWI

tWCw

w

C

4.A special case: constant relative risk aversion

• The above mentioned nonlinear partial equation coupled with two algebraic equations is difficult to solve in general.

• But for the utility function with constant relative risk aversion, the equations can be solved explicitly.

1)('/)(''

.0,1,/)1()(

CUCCU

CCU

Optimality conditions

)4......(..................../

/)(

)3.......(..........)(

,)(0

)2........(..........,.........])[exp()(

)1.....(..............................)exp(

,)(')exp(0

222*

22*2

2

22*2

2

)1/(1*

1*

*

WI

WI

W

rtw

WwW

I

W

IWr

WwW

I

W

IWr

W

IttC

CtW

IW

ICUt

t

t

tt

ttw

t

t

tC

Optimality conditions(2)

22

2

2

2

)1/(

2222

2

/

)/(

2

)(

)1

exp()(1

0

),()(2

1

)}()(]))(({[)()exp(0

WI

WIrrW

W

I

t

It

W

I

tWtwW

I

tCtWrrtwW

I

t

ICUt

t

tt

tt

t

tt

Bequest value function

• The boundary condition can cause major changes in the solution.

• means no bequest.• A slightly more general form which can be u

sed as without altering the resulting solution substantively is

0

/)]()[exp()),(()),(( 1 TWTTTWBTTWI

/)]()[(]),([ TWTGTTWB

The trial solution

• Suppose

)]()[exp(1

)(

/

)/(

)]()[exp()()1(

)]()[exp()(

)]()[exp()(

)]()[exp()(

,)]()[exp()(

)),((

22

2

22

2

1

tWttb

WI

WI

tWttbW

I

tWttb

tWttb

t

I

tWttbW

I

tWttb

ttWI

t

t

t

t

t

The trial solution(2)

1

2*

)1/(1*

1

22

)1/(

}))(exp()1(1

{)(

)1()(

)()]([)(

)(

],)1(2/)[(

,)]()[1()()(

v

Ttvvtb

rtw

tWtbtC

Tb

rru

tbtubtb

Sufficient condition for the solution

• be real (feasibility);

• To ensure the above conditions,

0)(

,0

*

2

2

tC

W

I t

]),([ ttWI t

Ttv

Ttvv

0,0

)](exp[)1(1

The optimal consumption and portfolio selection rules

*2

*

*

)1()(

.0),(1

;0),()](exp[)1(1

)()(/1)(

wr

tw

vtWtT

vtWTtvv

v

tWtbtC

The Bequest valuation function

• The economic motive is that the true function for no bequest

• Then when

• This does not mean the infinite rate of consumption, but because the wealth is driven to 0.

00]),([ TTWB

WCTt /, *

Dynamic properties of consumption

• Then the instantaneous marginal propensity to wealth is an increasing function of time.

0)](exp[)]([

))(exp(()])(exp[1(

)(

),(/)()(,0

2

2

*

TtvtV

TtvvTtv

vtV

tWtCtV

Dynamic properties of consumption

• Define•

),0()(],0[ nVVT

Tn

n

vT

nT

vv

nnvT

nnvTv

vTnvn

vTTvnn

vT

vn

Tv

v

1

,0,11

0,}/1)/11)(log{exp(

/1)/11)(exp()exp(

1)exp()1()exp(

)exp(1))(exp(

,)exp(1))(exp(1

Dynamic behavior of wealth

• Remember that

• Then

),()(]))(([)( tCtWrrtwtW

.0)())(

)((

,)1(

)(

),()(

)(

,)1(

)(

),(]))(([)(

)(

2

2*

*

2*

tVtW

tW

dt

d

rr

tVtW

tW

rtw

tVrrtwtW

tW

Dynamic behavior of consumption(2)

• This implies that, for all finite-horizon optimal paths, the expected rate of growth of wealth is diminishing function of time.

• : the investor save more than expected return.• : the investor consume more than expected

return.• Then, if

:0)(

)(

tW

tW

:0)(

)(

tW

tW

0,1

,0),log(1

.sin,

,.,0

,)(.)0(

..sin)0(

*

*

*

*

**

*

vT

tvv

vTt

vestdiTtt

wealthincreasett

tVifV

morecomsumevestdiV

6. Infinite time horizon

• Consider the infinite time horizon case,

• Suppose

• It is independent of time, can be rewritten as J(W).• Remark: conditional expectation or unconditional

expectation?

),()(2

1

)}()(]))(({[)()exp(0

2222

2

tWtwW

I

tCtWrrtwW

I

t

ICUt

t

tt

0)()exp(max

)()](exp[)(max]),([)exp()),((

dvCUvE

dsCUtstEttWItttWJt

The ordinary differential equation

• Then the partial differential equation can be changed into a ordinary differential equation by J(W).

t

t

tt

dssCUstEt

IW

ItWJ

W

ItWJ

,)]([)exp()(

,)exp()('',)exp()('2

2

))(''2

1

}])(){[(')()((max0

222

),(

WwWJ

CWrrwWJWJCUwc

The ordinary equation(2)

• Then,

• First order conditions are:

)('

)(''

)]('[

2

)()()]('[

10

,)(''

)(')(

,)]([)(

2

2

2)1/(

2*

)1/(1*

WrWJ

WJ

WJrWJWJ

WJ

WJ

W

rw

WJtC

.0)]}([){exp(lim,0)}),(({lim,0]),([lim

)('')(')(0

)(')('22

tWJtEttWIETTWB

wWWJWWJr

WJCU

ttT

The additional conditions

• Similar to case of finite time horizon, to ensure the solution to be maximum,

• The boundary condition is satisfied.

• Using ito theorem, we can get

TtWtCV

rrvV

),(/)(

0]1)1(2

)([

1**

22

2*

/)]()[exp(]),([lim tWtvttWI tT

)exp()]0([})]()[exp({ vtWv

tWtv

E

remark

• Note that:

• The second item on the right side is very similar to a return or yield.

• Then it is a generalization of the usual consumption required in deterministic optimal consumption growth models when the production function is linear.

])1(2

)([0

2

2

rr

v

The consumption and portfolio selection under infinite time horizon

• Summary: in the case of infinite time horizon, the partial differential equation is reduced to an ordinary differential equation.

)1()(

)(]}1)1(2

)([

1{)(

2*

22

2*

r

rtw

tWrr

tC

7. Economic interpretation

• Samuelson(1969) proved by discrete time series, for isoelastic marginal utility, the portfolio-selection decision is independent of the consumption decision.

• For special case of Bernoulli logarithmic utility, the consumption is independent of financial parameters and is only dependent upon level of wealth.

• Two assumption:– Constant relative risk aversion which implies that one’s attitude t

oward financial risk is independent of one’s wealth level– The stochastic process which generate the price changes.

• Under the two assumptions, the only feedbacks of the system, the price change and resulting level of wealth have zero relevance for the optimal portfolio decision and is hence constant.

The relative risk aversion

• The optimal proportion in risky asset can be rewritten in terms of relative risk aversion,

• Then the mean and variance of optimal composite portfolio are

2* r

w

22

222*2

*

2

2**

*

)(

,)(

)1(

rw

rr

rww

Phelps-Ramsey problem

• Then after determining the optimal proportion, we can think of the original problem as a simple Phelps-Ramsey problem which we seek an optimal consumption rule given that the income is generated by the uncertain yield of an asset.

)()()}2

)(1({)(2*** tVWtWtC

Comparative analysis

VWVW

VV

W

Vb

WbW

b

dW

WbWb

dI

Wb

WI

I

I

I

1

)0(/)0()(

1

1]

)0([

,)0(

0])0(

)[0()0()0(

1

1

,0)])0(

[)]0()[0()]0([)0(1

(

)]0([)0(

))((

1

10

0

0

0

0

Comparative analysis(2)

• Consider the case

• Remark: the substitution effect is minus and the income effect is plus.

.0)0()(

),0(1

.0)0(

))0(()(

,)0(

1,

0

00

*

*

*

*

*

*

*

0

**

*

0

*

**

WCC

WC

WWVW

VC

V

WW

VV

I

II

Comparative analysis(3)

• One can see that,• The individuals with low risk aversion,

• The substitution effect dominates the income effect and the investor chooses to invest more.

• For high risk aversion,• The income effect dominates the substitution

effect.• For log utility, the income effect and substitution

effect offset each other.

10

1

The other case

• Consider

effectincWCC

effectsubWC

V

WW

V

I

I

I

..0)0(2

))(

()(

.,02

)0()

)((

,2

)0()(

,2

1

)(

,

0

0

0

2*

*

2*

*

2*

*

2*

2*

Elasticity analysis

• The elasticity of consumption to the mean is

• The elasticity of consumption to the variance is

VC

CE

1/

**

*

*1

VC

CE

2

1/ 2**

2*

*2*2

Elasticity analysis(2)

• When 21 EE

2/

,2,,1

,2

11

2*

2*

2*

*

k

or

Some cases

• For relatively high variance, high risk averter will be more sensitive to the variance change than to the mean.

• For relatively low variance, low risk averter will be sensitive to the mean.

• The sensitivity is depending on the size of k since the investors are all risk averters. For large k, risk is the dominant factor, the risk has more effect. If k is small, it is not the dominant factor, the yield has more effect.

8.Extension to many assets

• The two asset model can be extended to m asset model without any difficulty. Assume the mth asset to be certain asset, and the proportion in ith asset is wi(t).

1],[

,]',...,[ˆ,]',...,[,)]'(),...,(),([)('

)()()()(')]()()[(

)()()(])ˆ)(('[)]()()[(

1002010

02

002

00

00000

mn

rrrrtwtwtwtw

hOhtWtwtwtWtWtE

hOhtChtWrrtwtWtWtE

ij

nn

Solution

• Under the infinite time horizon, the ordinary differential equation becomes

• The optimal decision rules are:

}')(''2/1

])ˆ('){[(')()((max0

2

),(

wWwWJ

CWrrwWJWJCUwc

)ˆ(1

1)(

),(]}1)1(2

)()'ˆ([

1{)(

1*

2

1*

rtw

tWrrr

tC

9.Constant absolute risk aversion

• The other special case of utility function which can be solved explicitly is the constant absolute risk aversion.

)(/)(''

,0,/)exp()(

CUCU

CCU

The optimal problem

• After some mathematics, the optimal system can be written by

0)]}([){exp(lim..

),(''/))((')(

)],('log[1

)(

,)(''

)]('[

2

)(

)]('log[)('

)(')()('

0

2*

*

2

2

2

tWJtEts

WWJrWJtw

WJtC

WJ

WJr

WJWJ

rWWJWJWJ

t

solution

• Take a trial solution:

• Then, we can get:

)exp()( qWq

pWJ

)()(

,2/)(

)()(

],2/)(

exp[

,

2*

22*

22

tWr

rtw

r

rrtrWtC

r

rrp

rq

Implications

• The differences between constant relative risk aversion and constant absolute risk aversion are:

• The consumption is no longer a constant proportion of wealth although it is still linear in wealth.

• The proportion invested in the risky asset is no longer constant, although the total dollar value invested in risky asset is constant.

• As a person becomes wealthier, the proportion invested in risky falls. If the wealth becomes very large, the investor will invest all his wealth in certain asset.

10. Other extensions

• The model can be extended to the other cases.• Simple Wiener model can be generalized to multi Wiener

model.• A more general production function, Mirrless(1965).• Requirements:

– The stochastic process must be Markovian;– The first two moments of distribution must be proportional to delt

a t and higher moments on o(delt).

• Remark: although this model can be generalized in large amount, the computational solution is quite difficult since it involves a partial differential equation.