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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.
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