meie811d advanced topics in finance optimum consumption and portfolio rules in a continuous-time...

MEIE811D Advanced Topics in FinanceMEIE811D Advanced Topics in Finance

Optimum Consumption and Portfolio Rules in a Continuous-Time Model

Yuna Rhee

Seyong Park

Robert C. Merton (1971)

[Journal of Economic Theory]

MEIE811D Advanced Topics in FinanceMEIE811D Advanced Topics in Finance 2

I. Optimal Portfolio and Consumption Rules

II. Explicit Solutions For A Particular Class of Utility Functions

III. The effects on the consumption and portfolio rules of

alternative asset price dynamics

Contents


ContentsContents

I. Optimal Portfolio and Consumption rules

1. Procedures of solving the optimal equation

2. Optimal Equation & PDE

3. First order conditions of the PDE

4. Optimal Portfolio and Consumption rules expressed by J


01. ProceduresI. Optimal Portfolio and

Consumption rules

Construct the optimal equation

Derive the partial differential equation

First Order Conditions of the PDE

Optimal portfolio and consumption rules expressed

by andJ U


02. Optimal Equation & PDEI. Optimal Portfolio and

Consumption rules

The problem of choosing the optimal portfolio and consumption rules,

The dynamic programming equation is

Define L

Then the partial differential equation becomes,

]}),([)),(({max00 TTWBdtttCUET

n

iij

n

ji

n n

ijjij

n n

iij

n n

iiiii

PW

JWwP

P

JPP

W

JWww

P

JP

W

JCWw

t

JtCUtPWwC

1

2

1

1 12

2

2

22

1 1

1 1

2

1

2

1

2

1

)(),(),,;,(

),(),,;,( tCUtPWwC ][J

}]),([),({max),,(},{

T

ttwc

TTWBdssCUEtPWJ


By the Hamilton-Jacobi-Bellman (HJB) equation (I),

In other words,

where is the optimal consumption and is the optimal weight.

In this paper, Theorem I is the HJB equation (I).

The Hamilton-Jacobi-Bellman (HJB) equation (I) is written on the Stochastic Differential

Equations by Bernt Øksendal.

02. Optimal Equation & PDEI. Optimal Portfolio and

Consumption rules

),,;,(),,;,(0 ** tPWwCtPWwC

*C*w

),,;,(max0},{

tPWwCwc


03. First Order Conditions of the PDEI. Optimal Portfolio and

Consumption rules

Define the lagrangian

Then first order conditions for maximum are

The sufficient conditions for interior maximum

)3..(..............................10

)2........(,),(0

)1(..........,.........),(0

1

*

1 1

2***

**

n

i

n n

jkjjWjkjWWWkwk

WCC

wL

WPJWwJWJwCL

JUwCL

0.0..

.,,

,0,0

*

222

W

CJcs

jkWJLJWL

LUL

WW

kjWWwwWWkww

CwCwCCCCCC

jkkk

kk

)1(1n

iwL


Optimal Consumption rules expressed by ,

Optimal Portfolio rules expressed by ,

If the nth asset is risk free,

04. , expressed by and I. Optimal Portfolio and

Consumption rules *C *w

n n

WWkjiiwkkwk

n n n

jijlklk

n

WW

Wkjk

n

k

n n

kk

kkkk

WJvPJPJtPf

vvtPg

WJ

JtWPm

vtPh

fgh

nktWPftPgtWPmtPhw

1 1

1 1 1

1

11 1

*

/)(),(

)(1

),(

,),,(,),(

,0,0,1

,...,2,1),,,(),(),,(),(

J

J

J

U

1* ][),,( CW UGtJGC

m

WW

kkWjkj

WW

Wk mk

WJ

PJrv

WJ

Jw

1

* ,...,2,1,)(


ContentsContents

II. Explicit Solutions for A Particular Class of Utility Functions

1. Procedures of the solving the optimal equation

2. HARA (Hyperbolic Absolute Risk-Aversion)

3. CRRA (Constant Relative Risk-Aversion)

4. CARA (Constant Absolute Risk-Aversion)


01. Procedures II. Explicit Solutions

Substitute , expressed by and into the PDE

Solve the PDE for J

Substitute into the optimal consumption and weight

J

*C *wJ U


02. HARA (Hyperbolic Absolute Risk-Aversion) II. Explicit Solutions

HARA family: Hyperbolic Absolute Risk Aversion.

Optimal Portfolio and Consumption rules expressed by

2*

)1/(1*

)(

)1()

)exp((

1)(

r

WJ

Jtw

JttC

WW

W

W

J

,.1,01

,0,1..

,0)1

/(1'/'')(

,)1

(1

)(

),()exp(),(

ifC

ts

CVVCA

CCV

CVttCU



PDE for

Guess the solution to the PDE for

Then we can get the solution to the above PDE

2

22)1/(

2

2

)(]

)1([)

)exp()(exp(

)1(0

r

J

JJrWJ

Jtt

WW

WWt

W

J

J

]1[1

)(1

),( )( TtreW

tFtWJ

)1()1(

),( )(

)(

tTr

tT

t eWe

etWJ

22 2/)(1 rarandwhere



The optimal portfolio and consumption rules

)()(

exp1

)1()(][

)(

)(

*

Tt

etW

tC

Ttr

)1()(

)()(

)()( )(22

* Ttrer

rtW

rtWtw


03. CRRA (Constant Relative Risk-Aversion) II. Explicit Solutions

CRRA family: Constant Relative Risk Aversion.


2

22

*

1

1

*

)()(

)(

W

IW

W

Ira

tw

W

IetC

t

t

tt

I

aversionriskrelativeofmeasuresattisCU

CCUwhere

CCUorandC

CU

'Pr1)(

)(

0,log)(01,)(

'

''



PDE for



2

2

2

2

211

2

)(10

WI

WI

rarW

W

I

t

I

W

Ie

t

t

ttt

t

I

)(

)(),( tWe

tbtWI t

)(/)1(1

),(1)(

tWee

tWI tTt

1},

)1(2

)({

2

2rarwhere

I




In this, the consumption is a constant proportion of wealth and the optimal portfolio

rules is a constant independent of W or t.

0),())1(1/()( )(* fortWetC Tt

)1(

)()(

2*

r

tw

0),()/(1)(* fortWtTtC

)(* tw)(* tC


04. CARA (Constant Absolute Risk-Aversion) II. Explicit Solutions

CARA family: Constant Absolute Risk Aversion.


)(''

)(')()(

)('ln1

)(

2*

*

WWJ

WJratw

WJtC

I

aversionriskabsoluteofmeasuresattisCU

CUwhere

eCU

C

'Pr)(

)(

0,)(

'

''



PDE for



)(''

)]('[

2

)()('ln

)(')(')(

)('0

2

2

2

WJ

WJraWJ

WJWrWJWJ

WJ

J

J

qWeq

pWJ )(

qWeq

pWJ )(

rqr

rarpwhere

),2/)(

exp(22




In this unlike the CRRA case, that consumption is no longer a constant proportion

of wealth although it is still linear in wealth.

Instead of the proportion of wealth invested in the risky asset being constant (i.e.,

a constant), the total dollar value of wealth invested in the risky asset is kept constant (i.e.,

a constant).

As one becomes wealthier, the proportion of his wealth invested in the risky asset falls,

and

asymptotically, as W goes to infinity, one invests all his wealth in the certain asset and

consumes all his (certain) income.

r

rartrWtC

22

* 2/)()()(

)(

)()(

2*

tWr

rtw

)(* tC

)(* tw

)()(* tWtw


ContentsContents

III. The effects of alternative asset price dynamics

1. Noncapital Gains Income: Wages (Constant Case)

2. Poisson processes (Case1, Case2, Case3)

3. Alternative Price Expectations to the Geometric

Brownian Motion (Case1, Case2, Case3)


01. Noncapital Gains Income: Wages III. Alternative dynamics

In the previous sections, it was assumed that all income was generated by capital gains. If

a (certain) wage income flow (constant case) in introduced, then the optimal consumption

and portfolio rules are as the follwoing

Comparing these results with the HARA case’s one, we finds that the individual capitalizes

the lifetime flow of wage income at the market (risk-free) rate of interest and then treats the

capitalized value as an addition to the current stock of wealth.

])/))(exp[(1(

)]1()1(

][[)(

)()(

*

Tt

ereY

WtC

TtrTtr

)1()(

))1(

()(

)()( )(2

)(

2* Ttr

Ttr

er

r

r

eYW

rtWtw


02. Poisson Processes(Case1) III. Alternative dynamics

Consider first the two-asset case. Assume that one asset is a common stock whose price is

log-normally distributed and that the other asset is a “risky” bond which pays an

instantaneous rate of interest r when not in default but, in the event of default, the price of

the bond becomes zero.

The optimality equation can be written as

First order conditions with respect to and are

Consider the particular case when , in other words the CRRA

case.

22*2

****

2

1

}])({[)],(),([),(0

WwJ

CWrrwJtWJtWwJJtCU

WW

Wt

),(),(0 * tWJtCU WC

*C*w

WwtWJrtWJtWwJ WWWW*2* ),())(,(),(0

1,),(

forC

tCU



Then the optimal portfolio and consumption rules are

where

As might be expected, the demand for the common stock is an increasing function of .

)1/)(exp[1)(1/()()(* TtAtAWtC

1*

2*

2

2

)1(2

)()2(1

)1(2

)(

w

rwr

rA

1*22

* )()1()1(

)(

wr

w



Consider an individual who receives a wage, , which is incremented by a constant

amount at random points in time. In other words, . Suppose further that the

individual’s utility function is of the form and that his time horizon is

infinite (i.e., ).

For the two-asset case, the optimality equation can be written as

Consider the particular case when , in other words the CARA

case.

2*2**

*

2

),(2

1]))([(

),()],(),([),()(0

WwYWICYWrrw

YWIYWIYWIYWICV

WW

W

dqdY )(tY

)(),( CVetCU t

T

/)exp()( CCV



Then the optimal portfolio and consumption rules are

In this, is the general wealth term, equal to the sum of

present wealth and capitalized future wage earnings.

If , then the above optimal consumption is same as the result of previous

constant wage case.

r

rtWtw

rr

rrr

tYtWrtC

2*

2

2

2*

)()(

]2

)([

1]

)exp(1)()([)(

]/)1(/)()([ 2rertYtW

0 )(* tC



When , is the capitalized value of (expected) future increments to

the wage rate, capitalized at a some what higher rate than the risk-free market rate

reflecting the risk-aversion of the individual.

The individual, in computing the present value of future earnings, determines the Certainty-

equivalent flow and the capitalizes this flow at the (certain) market rate of interest.

0

)exp(12

r

0,)]exp(1[

))()())((exp(22

if

rrsdtYsYtsrE

tt

20

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

rr

YsXrs



Consider an individual whose age of death is a random variable. Further assume that the

event of death at each instant of time is an independent Poisson process with parameter .

Then, the age of death, , is the first time that the event (of death) occurs and is an

exponentially distributed random variable with parameter .

The optimality criterion is to

The associated optimality equation is

}]),([),({max00

WBdttCUE

)()],(),([),(0 * JLtWJtWBtCU


03. Alternative price dynamics(Case1) III. Alternative dynamics

First case is the “asymptotic ‘normal’ price-level” hypothesis which assumes that there

exists a “normal” price function, , such that

i.e., independent of the current level of the asset price, the investor expects the “long-run”

price to approach the normal price.

A particular example which satisfies the hypothesis is that

and

where

This implies an exponentially-regressive price adjustment toward a normal price, adjusted

for trend.

)(tP

tTtPtPETt

0,1))(/)((lim

tePtP )0()(

,))}0(/)(log({ dzdtPtPvtP

dP

))0(/)0(log(,4// 2 PPkvk



Let . Then by Ito’s Lemma, we can write the dynamics for Y as

where . This process is called Ornstein-Uhlenbeck process.

The Y is a normally-distributed random variable generated by a Markov process which is

not stationary and does not have independent increments. And from the definition of ,

is log-normal and Markov.

The , conditional on knowing , as

where .

The instantaneous conditional variance of is

)(tP

))0(/)(log( PtPY dzdtYvtudY )(

2/2u

Y

)(tY )(TY

t

TdzstvTYvTkTYtY )exp()exp()exp(1)]((

4[)()(

2

0 Tt)(tY

)]2exp(1[2

))(|)(var(2

TYtY



)]})]2exp(1[4

))exp(1)]((4

exp{[

))}()({exp())(/)((

2

2

v

TYvTk

TYtYETPtPE TT

The conditional expected price can be derived

Consider the two-asset model is used with the individual having an infinite time horizon

and a constant absolute risk-aversion utility function, .

The fundamental optimality equation then is written as

/)exp(),( CtCU

2*2222*

***

2

1)(

2

1

}])([{/)exp(0

WwJJYvtuJWwJ

CrWWrYvtwJJC

YWYYYWW

Wt



From the FOC, the associated equations for the optimal rules expressed by

After solving the previous PDE for , we obtain the optimal rules in explicit form as

J

WW

YW

WW

W

W

J

J

J

rYvtJWw

JC

2*

*

])([

)log(

rr

rtPrr

Ww

taYrrtr

Yr

rWC

2)),((1

1*

)(22

*

2

2

2

2

2

22

2

2

J



Recall the optimal rules when the geometric Brownian motion hypothesis is assumed

We find that the proportion of wealth invested in the risky asset is always larger under the

“normal price” hypothesis than under the geometric Brownian motion hypothesis.

Even if , unlike in the geometric Brownian motion case, a positive amount of the

risky asset is held.

r

rartrWtC

22

* 2/)()()(

)(

)()(

2*

tWr

rtw

r



Second case is the same type of price-dynamics equation as was assumed for the

geometric Brownian motion, namely,

However, instead of the instantaneous expected rate of return being a constant, it is

assumed that is itself generated by the stochastic differential equation,

The first term implies a long-run, regressive adjustment of the expected rate of return

toward a “normal” rate of return, , where is the speed of adjustment.

The second term implies a short-run, extrapolative adjustment of the expected rate of

return of the “error-learning” type, where is the speed of adjustment.

This assumption is called the “De Leeuw” hypothesis.

dzdtP

dP

dzdtud

dzdtdP

)(

,



The conditional on knowing is

where .

Then, the conditional mean and variance of are

and

Thus, the conditional on knowing and ,

t

T

st dzeeeTTt ,)1))((()()( 0 Tt

t

T

s

T

t

T

ss dzdssdzeeT

TYtY )'()1(

))(()

2

1()()( )'(2

)(tY

)(t

)(T

)(T

)()( Tt )1))((())()(( eTTtET

)1(2

)](|)()(var[ 222

eTTt

)(TP



Therefore, the conditional mean and variance of are

and

)()( TYtY )1(

))(()

2

1()]()([ 2

e

TTYtYET

)]1([2

)]1(2

1)1(2[

2)](|)()(var[

2

2

22

222

e

eeTYTYtY



Again, the two-asset model is used with the individual having an infinite time horizon and a

constant absolute risk-aversion utility function, .

The fundamental optimality equation is written as

From the FOC, the associated equations for the optimal rules expressed by

WwJJJWwJ

CrWWrwJJe

WWW

Wt

C

*22222*

**

2

1)(

2

1

])([0

2

*

/)exp(),( CtCU

WW

W

WW

W

J

J

J

rJWw

2

* )(

)log(* WJC

J



After solving the previous PDE for , we obtain the optimal rules in explicit form as

Assuming , compare this with the Brownian motion case , then

we find that under the De Leeuw hypothesis, the individual will hold a smaller amount of

the risky asset than under the geometric Brownian motion hypothesis.

Note that is a decreasing function of the long-run normal rate of return . This is

because as increases for a given , the probability increases that future “ s” will be

more favorable relative to the current , and so there is a tendency to hold more of one’s

current wealth in the risk-free asset as a “reserve” for investment under more favorable

conditions.

r

rrr

rrWw

)())(2(

)22(

12

*

J

)(

)()(

2*

tWr

rtw

r

Ww* '



Last case assumed that prices satisfy the geometric Brownian motion,

However, it is also assumed that the investor does not know the true value of the

parameter,

, but must estimate it from past data.

Suppose the investor has price data back to time . Then, the best estimator

for , ,

Then , and if we define the error term , then can be re-

written as

where

By differentiating , we have the dynamics for ,

dzdtP

dP

0)(ˆ,1

)(ˆ

t

P

dP

tt

)(ˆ t

))(ˆ( tE )(ˆ tt P

dP

zddtP

dPˆˆ

/ˆ dtdzzd t)(ˆ t ̂

zdt

d ˆˆ



By differentiating , we have the dynamics for ,

We see that this “learning” model is equivalent to the special case of the De Leeuw

hypothesis of pure extrapolation (i.e., ), where the degree of extrapolation is

decreasing over time.

If the two-asset model is assumed with an investor who lives to time T with a constant

absolute risk-aversion utility function, and if (for computational simplicity) the risk-free

asset is money (i.e., ), then the optimal portfolio rule

and the optimal consumption rule is

0

)(ˆ)log(2

* tt

TtWw

0r

]])(

)log()(

)([

2

ˆ

)log()()log()(2

)[log(1

2

2

2

*

t

T

t

T

tT

t

ttTTtTtT

TtT

WC

)(ˆ t ̂zd

td ˆˆ



By differentiating with respect to , we find that is an increasing function of

time for , reached a maximum at , and then is a decreasing function of time for

, where is defined by

In early life, the investor learns more about the price equation with each observation, hence

investment in the risky asset becomes more attractive.

But as he approaches the end of life, he is generally liquidating his portfolio to consume a

larger fraction of wealth.

Consider the effect on of increasing the number of available previous observations

(i.e. increase ). As expected, the dollar amount invested in the risky asset increases

monotonically.

Taking the limit a , we have that the optimal portfolio rule is

eeTt /])1([

Ww* t Ww*

tt tt Ttt t

Ww*

Ww*

astT

Ww ,2

*



Consider the effect on of increasing the number of available previous observations

(i.e. increase ). As expected, the dollar amount invested in the risky asset increases

monotonically.

Taking the limit a , we have that the optimal portfolio rule is

which is the optimal rule for the geometric Brownian motion case when is known with

certainty.

Ww*

Ww*

astT

Ww ,2

*

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