Transcript Chp.4 Lifetime Portfolio Selection Under Uncertainty

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
aversion 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
 w (t )  1
i 1
i
The budget equation
• At time t0, the investment between t0 and t(t0+h)
is :
W (t0 )  C(t0 )h
• The value of this investment at time t is:
m
 wi (t0 )(W (t0 )  C (t0 )h)
i 1
m
  wi (t0 )
i 1
X i (t )
X i (t0 )
X i (t )
(W (t0 )  C (t0 )h)
X i (t0 )
m
X i (t )
W (t )  W (t0 )  [ wi (t0 )(
 1)](W (t0 )  C (t0 )h)  C (t0 )h
X i (t0 )
i 1
m
 [ wi (t0 ){exp[gi (t )h]  1}](W (t0 )  C (t0 )h)  C (t0 )h
i 1
The process of g(t)
• Suppose g(t) is the geometric Brownian
motion. In discrete time,
g i (t )h  ( i 
 i2
2
)h  Yi ;
• i :the expected return of asset i;
•  i : the volatility of asset i;
Yi  N (0, i2h)
m
 i2
i 1
2
W (t )  W (t0 )   wi (t0 ){exp[( i 
)h  Yi ]  1}(W (t0 )  C (t0 )h)  C (t0 )h
Momentum
m
E (t0 ){W (t )  W (t0 )}   wi (t0 )(exp( i h)  1)(W (t0 )  C (t0 )h)  C (t0 )h
i 1
m
  ( wi (t0 ) i h)(W (t0 )  C (t0 )h)  C (t0 )h
i 1
m
m
i 1
i 1
  wi (t )( iW (t0 )  C (t0 )h   wi (t0 ) i C (t0 )h 2
m
  wi (t )( iW (t0 )  C (t0 )h  O(h)
i 1
m
m
E (t0 ){[W (t )  W (t0 )] }   wi (t0 ) w j (t0 ) E (t0 ){Yi Y j }W 2 (t0 )
2
i 1 j 1
 O(h)
Continuous time
dYi   i Z i (t ) dt ,
m
m
i 1
i 1
dW  [ wi (t0 ) iW (t )  C (t )]dt   wi (t )W (t ) i Z i (t ) dt.
m
W (t )  W (t0 )
E (t0 )(
)   wi (t0 ) i [W (t0 )  C (t0 )h]  C (t0 )  O(1)
h
i 1
m
W
(
t
)

W
(
t
)
0
W (t0 )  lim E (t0 )(
)   wi (t0 ) iW (t0 )  C (t0 )
h 0
h
i 1
3. The two asset model
• w1 (t )  w(t ) :the proportion invested in the risky
asset;
• w (t )  1  w(t ) :the proportion invested in the sure
asset.
• g1 (t )  g (t ) : the return on risky asset.
2
g 2 (t )  r
g (t )h  (   2 / 2)h  Y
Two asset model(2)
W (t )  W (t0 )  ( w(t0 ){exp[(   2 / 2)h  Y ]  1}  [1  w(t0 )]
 [exp(rh)  1])(W (t0 )  C (t0 )h)  C (t0 )h;
E (t0 )(W (t )  W (t0 ))  [(w(t0 )  [1  w(t0 )]r )W (t0 )  C (t0 )]h  O(h)
 [(w(t0 )(  r )  r )W (t0 )  C (t0 )]h  O(h);
E (t0 ){(W (t )  W (t0 ))2 }  w2 (t0 )W 2 (t0 ) E (t0 )(Y ) 2  O(h)
 w2 (t0 )W 2 (t0 ) 2 h
dW  ((w(t )(  r )  r )W (t )  C (t ))dt  w(t )Z (t )W (t ) dt ,
W (t )  ( w(t )(  r )  r )W (t )  C (t )
The objective problem
T
max E{ exp( t )U (C (t ))dt  B(W (T ),T )},
0
s.t.
C (t )  0,W (t )  0;W (0)  W0  0;
U ' (C )  0;U ' ' (C )  0
The dynamic programming form
• Define
T
I (W (t ), t )  max E (t ){ exp( s)U (C (s))ds  I (W (T ),T };
t
C ( s ),W ( s )
I (W (T ),T )  B(W (T ),T );
• Then the objective function can be written:
t
I (W0 ,0)  max E{ exp( s)U (C (s))ds  I (W (t ), t )}
0
The dynamic programming(2)
• If t  t0  h ,then by the Mean Value Theorem
and Taylor Rule,

I (W (t0 ), t0 )
I (W (t0 ), t0 )  max E (t0 ){exp( t )U (C (t ))h  I (W (t0 ), t0 ) 
h
t
{c , w}
I (W (t0 ), t0 )

[W (t )  W (t0 )]
W
1  2 I (W (t0 ), t0 )
2

[
W
(
t
)

W
(
t
)]
 O(h)},
0
2
2
W

t  [t0 , t ]
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 h  0
I t I t
0  max (exp( t )U [C (t )]   {[ w(t )(  r )  r ]W (t )  C (t )}
( c ( t ), w( t ))
t W
2
1 I 2 2
2


w
(
t
)
W
(t ))
2
2 W
The solution
• Define
 ( w, C ;W , t )  exp( t )U (C ) 
1  2 It 2 2
2


w
(
t
)
W
(t ),
2
2 W
max ( w, C ;W , t )  0
{c , w}
I t I t

{[ w(t )(  r )  r ]W (t )  C (t )}
t W
First order condition
I t
C ( w , C ;W ; t )  0  exp( t )U ' (C ) 
W
2

I

It * 2 2
*
*
t
w ( w , C ;W ; t )  0  (  r )W

wW 
2
W W
*
*
*
Second order condition
ww  0, CC
• If
•
I (W (t ), t )
CC
 0, det 
wC
Cw 
 0.

ww 
is concave in W,
wC  Cw  0,
CC  exp( t )U ' ' (C )  0,
ww
2

It
2
2
 W (t )
 0.
2
W
Summary
• The maximum problem can be rewritten as:
 ( w* , C * ;W , t )  0;
C  0;
w  0;
I (W (T ),T )  B(W (T ),T )
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.
U (C )  (C   1) /  ,   1,   0.
 U ' ' (C )C / U ' (C )  1    
Optimality conditions
C  0  exp( t )U ' (C * ) 
I t
,
W
 1
I t
 exp( t )C * ...................................(1)
W
I t 1/( 1)
C * (t )  [exp(t )
]
,...........................(2)
W
I t
 2 It
*
2
2
w  0  (  r )W

w
W

,
2
W
W
I t
 2 It
*
2
2
(  r )W

w
W

.................(3)
2
W
W
  r I t / W
*
w (t )  
..........................(4)
2
2
2
W  I t / W
Optimality conditions(2)
  0  exp( t )U (C ) 
I t I t

{[ w(t )(  r )  r ]W (t )  C (t )}
t W
1  2 It 2 2
2


w
(
t
)
W
(t ),
2 W 2
1   I t  /( 1)
 t I t
0
(
)
exp(
)
 W
1 
t
I t
(  r ) 2 (I t / W ) 2

rW 
W
2 2  2 I t / W 2
Bequest value function
I (W (T ),T )  B(W (T ),T )   1 exp(T )[W (T )] / 
• The boundary condition can cause major
changes in the solution.
  0 means no bequest.
•
• A slightly more general form which can be
used as without altering the resulting
solution substantively is
B[W (T ),T ]  G(T )[W (T )] / 
The trial solution
• Suppose
I (W (t ), t ) 
b(t )

exp( t )[W (t )] ,
I t
 b(t ) exp( t )[W (t )] 1
W
I t
b(t )
b(t )


exp( t )[W (t )] 
exp( t )[W (t )]
t


 2 It
 2

(


1
)
b
(
t
)
exp(


t
)[
W
(
t
)]
W 2
(I t / W ) 2
b(t )


exp(


t
)[
W
(
t
)]
 2 I t / W 2
 1
The trial solution(2)
b(t )  ub(t )  (1   )[b(t )] /(1 ) ,
u     [(  r ) 2 / 2 2 (1   )  r ],
b(T )   1
C * (t )  [b(t )]1/( 1) W (t )
 r
*
w (t )  2
 (1   )
1  (v  1) exp(v (t  T )) 1
b(t )  {
}
v
Sufficient condition for the solution
• I t [W (t ), t ] be real (feasibility);
 2 It
 0,
2
W
C * (t )  0
• To ensure the above conditions,
1  (v  1) exp[ v(t  T )]
 0,0  t  T
v
The optimal consumption and
portfolio selection rules
C * (t )  1 / b(t )W (t )
v

W (t ), v  0;
1  (v  1) exp[v(t  T )]
1

W (t ), v  0.
T t 
 r
*
*
w (t )  2
w
 (1   )
The Bequest valuation function
• The economic motive is that the true function for
no bequest
B[W (T ),T ]  0    0
• Then when
t  T,C /W  
*
• This does not mean the infinite rate of
consumption, but because the wealth is driven to
0.
Dynamic properties of consumption
  0, V (t )  C * (t ) / W (t ),
V (t )  
v
(v exp(v(t  T ))
2
(1  exp[v(t  T )])
 [V (t )]2 exp[v(t  T )]  0
• Then the instantaneous marginal
propensity to wealth is an increasing
function of time.
Dynamic properties of consumption
• Define
•
 [0, T ]  V ( )  nV (0),
v
v
n
,
1  exp(v (  T ))
1  exp(vT )
n  n exp(v (  T ))  1  exp(vT )
n exp(v )  ( n  1) exp(vT )  1
exp(v )  exp(vT )(1  1 / n)  1 / n
log{exp(vT )(1  1 / n)  1 / n}
 
,v  0
v
1
1
 n , v  0,
T 
T
n 1
 
T
n
Dynamic behavior of wealth
• Remember that
W (t )  [w(t )(  r )  r ]W (t )  C(t ),
• Then
 (t )
W
 [ w(t )(  r )  r ]  V (t ),
W (t )
 r
w* (t )  2
,
 (1   )
 (t )
W
  *  V (t ),
W (t )
(  r ) 2
  2
 r,
 (1   )
 (t )
d W
(
)  V (t )  0.
dt W (t )
*
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.

W (t )
• W (t )  0 : : the investor save more than expected return.
 (t )
 0 : : the investor consume more than expected
• W
W (t )
return. *
  V (0)  di sin vest.  com sum .em ore
• Then, if *
*
  V (0)  if .V (t )   ,

0  t  t ,  increase.wealth,
t  t  T ,  di sin vest.
1
* v
 *T  1
t  T  log( * ), v  0, t 
,v  0
*
v


6. Infinite time horizon
• Consider the infinite time horizon case,
0  exp( t )U (C ) 
I t I t

{[ w(t )(  r )  r ]W (t )  C (t )}
t W
1  2 It 2 2

 w (t )W 2 (t ),
2
2 W
• Suppose

J (W (t ), t )  exp(t ) I [W (t ), t ]  max E (t ) exp[  ( s  t )]U (C )ds
t

 max E  exp( v)U (C )dv
0
• It is independent of time, can be rewritten as J(W).
• Remark: conditional expectation or unconditional
expectation?
The ordinary differential equation
I t
 2 It
J ' (W )  exp(t )
, J ' ' (W )  exp(t )
,
W
W 2

I t
  E (t )  exp( s )U [C ( s )]ds,
t
t
• Then the partial differential equation can be
changed into a ordinary differential equation by
J(W).
0  max(U (C )  J (W )  J ' (W ){[w(  r )  r ]W  C}
( c , w)

1
J ' ' (W ) 2 w2W 2 )
2
The ordinary equation(2)
• Then,
C * (t )  [ J (W )]1/( 1) ,
w* 
0
 (  r ) J ' (W )
,
 2W J ' ' (W )
1 

 /( 1)
[ J ' (W )]
(  r ) 2 [ J ' (W )]2
  J (W ) 
2 2
J ' ' (W )
 rWJ ' (W )
• First order conditions are:
U ' (C )  J ' (W )
0  (  r )WJ ' (W )  J ' ' (W ) 2 wW 2
lim B[W (T ),T ]  0, lim E{I (W (t ), t )}  0, lim E{exp( t ) J [W (t )]}  0.
T 
t 
t 
The additional conditions
• Similar to case of finite time horizon, to ensure
the solution to be maximum,

(  r ) 2
r
V v
[ 2

] 0
2
1 
2 (1   ) 1  
*
V *  C * (t ) / W (t ),T  
• The boundary condition is satisfied.
lim I t [W (t ), t ]  v exp(  t )[W (t )] / 
T 
• Using ito theorem, we can get
v
v
E{ exp( t )[W (t )] }  [W (0)] exp(vt)


remark
• Note that:
(  r ) 2
v  0    [ 2
 r]
2 (1   )
• 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.
  
The consumption and portfolio
selection under infinite time horizon

2
(


r
)
r
*
C (t )  {
[ 2

]}W (t )
2
1 
2 (1   ) 1  
 r
*
w (t )  2
 (1  r )
• Summary: in the case of infinite time
horizon, the partial differential equation is
reduced to an ordinary differential
equation.
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
toward 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,
w* 
 r
 2
• Then the mean and variance of optimal
composite portfolio are
 *  w  (1  w )r 
*
 *2  w*2 2 
*
(  r ) 2
 2 2
(  r ) 2
 
2
 r,
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



C * (t )  {  (  1)( *  * )}W (t )  VW (t )


2
Comparative analysis
I 0 (W ( )) 
b(0)

[W (0)]
1 b(0)

 1 W (0)
dI0  (
[W (0)]  b(0)[W (0)] [
]I 0 )d  0,
 

1 b(0)
W (0)

W (0)  b(0)[
]I 0  0
  1 

b(0)  V  ,
W (0)
1
 W (0) V
 1 V

[
]I 0 
( )V
W (0) / V 

 1

  1 
Comparative analysis(2)
• Consider the case
V
  * , 


V
 1

 *

W
W (0)
 0 ,
 *
V
W0
C *
V
W (0)
(
) I0  (
W (0)  V
) I0  
 0.
 *
 *
 *

C *
 1

W (0),
 *

C *
C *
(
) I  W (0)  0.
 *
 * 0
• Remark: the substitution effect is minus and the
income effect is plus.
Comparative analysis(3)
• One can see that,
• The individuals with low risk aversion,
0   1
• The substitution effect dominates the income
effect and the investor chooses to invest more.
• For high risk aversion,   1
• The income effect dominates the substitution
effect.
• For log utility, the income effect and substitution
effect offset each other.
The other case
• Consider
   *2 ,
V
 1

,
 ( *2 )
2
W
 W (0)
) I0 
,

2V
*
C
W (0)
(
)


 0,  sub.effect
I
 ( *2 ) 0
2
(
C *
C *


(
)

W (0)  0.  inc.effect
I
 ( *2 )  ( *2 ) 0 2
Elasticity analysis
• The elasticity of consumption to the mean is
C * /  *
 1
E1   *
 *
C
V
• The elasticity of consumption to the variance
is
*
2

C
/


2  1
*
E2   *2



*
C*
2V
Elasticity analysis(2)
• When
E1  E2
 1
2  1
 *
,
*
2
  1, or,  *2  2 ,
k   *2 / 2
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).
E (t0 )[W (t )  W (t0 )]  [ w' (t0 )(  rˆ)  r ]W (t0 )h  C (t0 )h  O(h)
E (t0 )[W (t )  W (t0 )]2  w' (t0 )w(t0 )W 2 (t0 )h  O(h)
w' (t0 )  [ w1 (t0 ), w2 (t0 ),...,wn (t0 )]',   [1 ,..., n ]', rˆ  [r , r ,...r ]',
  [ ij ], n  m  1
Solution
• Under the infinite time horizon, the ordinary
differential equation becomes
0  max(U (C )  J (W )  J ' (W ){[w' (  rˆ)  r ]W  C
( c , w)
 1 / 2 J ' ' (W ) w' wW 2 }
• The optimal decision rules are:
(  rˆ)'  1 (  r )
r
C (t )  {
[

]}W (t ),
2
1 
2(1   )
1 
1
*
w (t ) 
 1 (  rˆ)
1 
*


9.Constant absolute risk aversion
• The other special case of utility function which
can be solved explicitly is the constant absolute
risk aversion.
U (C )   exp(C ) /  ,  0,
 U ' ' (C ) / U (C )  
The optimal problem
• After some mathematics, the optimal system
can be written by
0
 J ' (W )

 J (W )  J ' (W )rW 
(  r ) 2 [ J ' (W )]2

,
2
2
J ' ' (W )
1
C * (t )   log[J ' (W )],

w* (t )   J ' (W )(  r ) /  2WJ ' ' (W ),
s.t. lim E{exp( t ) J [W (t )]}  0
t 
J ' (W )

log[J ' (W )]
solution
• Take a trial solution:

p
J (W ) 
exp(qW )
q
• Then, we can get:
q  r ,
r    (  r ) 2 / 2 2
p  exp[
],
r
r    (  r ) 2 / 2 2
*
C (t )  rW (t ) 
,
r
 r
w* (t ) 
r 2W (t )
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
delta 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.