chp.4 lifetime portfolio selection under uncertainty hai lin department of finance, xiamen...
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Chp.4 Lifetime Portfolio Selection Under Uncertainty
Hai Lin
Department of Finance, Xiamen University,361005
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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.
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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)(
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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)(
)()(([)()(
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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[(()()(
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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
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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()(])()([)())()(
)((
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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
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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
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The objective problem
0)('';0)('
;0)0(;0)(,0)(
..
)},),(())(()exp({max
0
0
CUCU
WWtWtC
ts
TTWBdttCUtET
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The dynamic programming form
• Define
• Then the objective function can be written:
);),(()),((
};),(())(()exp(){(max)),(()(),(
TTWBTTWI
TTWIdssCUstEttWIT
tsWsC
})),(())(()exp({max)0,(00 t
ttWIdssCUsEWI
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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
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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
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The solution
• Define
0),;,(max
),()(2
1
)}()(]))(({[)()exp(),;,(
},{
2222
2
tWCw
tWtwW
I
tCtWrrtwW
I
t
ICUttWCw
wc
t
tt
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First order condition
22*2
2**
***
)(0);;,(
)(')exp(0);;,(
WwW
I
W
IWrtWCw
W
ICUttWCw
ttw
tC
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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
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Summary
• The maximum problem can be rewritten as:
)),(()),((
;0
;0
;0),;,( **
TTWBTTWI
tWCw
w
C
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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
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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
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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
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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
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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
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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
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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
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The optimal consumption and portfolio selection rules
*2
*
*
)1()(
.0),(1
;0),()](exp[)1(1
)()(/1)(
wr
tw
vtWtT
vtWTtvv
v
tWtbtC
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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 /, *
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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
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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
![Page 29: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/29.jpg)
Dynamic behavior of wealth
• Remember that
• Then
),()(]))(([)( tCtWrrtwtW
.0)())(
)((
,)1(
)(
),()(
)(
,)1(
)(
),(]))(([)(
)(
2
2*
*
2*
tVtW
tW
dt
d
rr
tVtW
tW
rtw
tVrrtwtW
tW
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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
![Page 31: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/31.jpg)
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
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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
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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
![Page 34: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/34.jpg)
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
![Page 35: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/35.jpg)
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
![Page 36: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/36.jpg)
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
![Page 37: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/37.jpg)
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.
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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
![Page 39: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/39.jpg)
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
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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
![Page 41: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/41.jpg)
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
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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
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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*
![Page 44: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/44.jpg)
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
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Elasticity analysis(2)
• When 21 EE
2/
,2,,1
,2
11
2*
2*
2*
*
k
or
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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.
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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
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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
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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
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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
![Page 51: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/51.jpg)
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
![Page 52: Chp.4 Lifetime Portfolio Selection Under Uncertainty Hai Lin Department of Finance, Xiamen University,361005](https://reader036.vdocuments.us/reader036/viewer/2022062417/551b196f55034607418b5718/html5/thumbnails/52.jpg)
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.
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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.