[Journal of Economic Theory]
Optimum Consumption and Portfolio Rules
in a Continuous-Time Model
Robert C. Merton (1971)
Yuna Rhee
Seyong Park
MEIE811D Advanced Topics in Finance
Contents
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
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I. Optimal Portfolio and Consumption rules
Contents
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
MEIE811D Advanced Topics in Finance
3
J
I. Optimal Portfolio and
Consumption rules
01. Procedures
Construct the optimal equation
Derive the partial differential
equation
First Order Conditions of the
PDE
Optimal portfolio and
consumption rules expressed
by J and U
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I. Optimal Portfolio and
Consumption rules
02. Optimal Equation & PDE
 The problem of choosing the optimal portfolio and consumption rules,
T
max E0{ U (C (t ), t )dt  B[W (T ), T ]}
0
 The dynamic programming equation is
T
J (W , P, t )  max Et { U (C , s )ds  B[W (T ), T ]}
{c , w}
 Define
t
 (C , w;W , P, t )  U (C , t ) L [J ]
 Then the partial differential equation becomes,
n
n
J
J
J
 (C , w;W , P, t )  U (C , t )   ( wi iW  C )
   i Pi
t
W 1
Pi
1
2
1 n n
1 n n
2J
2  J
   ij wi w jW
  Pi Pj ij 2
2 1 1
W 2 2 1 1
P
1 n n
2 J
  PiWw j ij
2 1 1
WP i
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I. Optimal Portfolio and
Consumption rules
02. Optimal Equation & PDE
 By the Hamilton-Jacobi-Bellman (HJB) equation (I),
0   (C * , w* ;W , P, t )   (C, w;W , P, t )
In other words,
0  max  (C, w;W , P, t )
{c , w}
*
where C is the optimal consumption and
w*
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.
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I. Optimal Portfolio and
Consumption rules
03. First Order Conditions of the PDE
 Define the lagrangian
n
L     (1   wi )
1
 Then first order conditions
for maximum are
0  LC (C * , w* )  U C  JW ,...................(1)
n
n
0  Lwk (C , w )   kWJ W  JW W   kj w W   J jW  kj PjW ,........(2)
*
*
*
j
2
1
1
n
0  L  1   wi* ................................(3)
1
 The sufficient conditions for interior maximum
LCC  CC  U CC  0, LCwk  Cwk  0,
Lwk wk   k2W 2 J W W , Lwk w j  J W WW 2 kj , k  j.
 s.c.J W W
C *
 0. 
0
W
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04.
C *, w* expressed by
I. Optimal Portfolio and
Consumption rules
J and U
 Optimal Consumption rules expressed by J ,
C *  G ( J W , t ), G  [U C ]1
 Optimal Portfolio rules expressed by J ,
wk*  hk ( P, t )  m( P,W , t ) g k ( P, t )  f k ( P,W , t ), k  1,2,..., n
1 hk  1, 1 g k 0, 1 f k  0,
n
n
n
hk ( P, t )  
n
vkj
1

, m( P, W , t ) 
 JW
,
WJ W W
n
n
1 n
g k ( P, t )   vkl ( l   vij j )
 1
1
1
n
n
1
1
f k ( P, t )  (J kw Pk   J iw Pi  vkj ) / WJ W W
 If the nth asset is risk free,
J
w  W
JWWW
*
k
m
 vkj ( j  r ) 
1
MEIE811D Advanced Topics in Finance
J kW Pk
, k  1,2,..., m
JWWW
8
II. Explicit Solutions for A Particular Class of Utility Functions
Contents
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)
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01. Procedures
II. Explicit Solutions
Substitute C *, w* expressed by
J and U into the PDE
Solve the PDE for
J
Substitute J into the optimal
consumption and weight
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02. HARA (Hyperbolic Absolute Risk-Aversion)
 HARA family: Hyperbolic Absolute Risk Aversion.
U (C , t )  exp(  t )V (C ),
1   C
V (C ) 
(
  ) ,
 1 
C 
A(C )  V ' ' / V '  1 /(
 )  0,
1  
C
s.t.  1,   0,
   0,  1if .  ,
1 
 Optimal Portfolio and Consumption rules expressed by
1   exp( t ) JW 1/( 1) (1   )
C (t ) 
(
)




JW   r
*
w (t )  
JW WW  2
*
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J
II. Explicit Solutions
02. HARA (Hyperbolic Absolute Risk-Aversion)
 PDE for
II. Explicit Solutions
J
exp( t ) JW  /( 1)
JW2 (  r ) 2
(1   ) 2
(1   )
0
exp(  t )(
)
 Jt  [
 rW ]JW 



JW W 2 2
 Guess the solution to the PDE for
J
 W
1 

r ( t T ) 
J (W , t ) 
F (t )
 [1  e
] 

 1   


 Then we can get the solution to the above PDE

  (1  e
J (W , t )    e  t 
  


   

 (T  t )
  



)  W 
 r (T  t ) 

(
1

e
)
   

 

where   1   and   r  (a  r ) 2 / 2 2
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02. HARA (Hyperbolic Absolute Risk-Aversion)
 The optimal portfolio and consumption rules



[    ]W (t )  (1  e r (t T ) )


  
*
C (t ) 


 (    )

 1  exp 
(t  T ) 
 


w* (t )W (t ) 
(  r )
 (  r )
r ( t T )
W
(
t
)

(
1

e
)
2
2

r
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II. Explicit Solutions
03. CRRA (Constant Relative Risk-Aversion)
II. Explicit Solutions
 CRRA family: Constant Relative Risk Aversion.
C
U (C )  ,   1 and   0 or U (C )  log C ,   0

U '' (C )C
where  '
 1     is Pr att ' s measure of relative risk aversion
U (C )
 Optimal Portfolio and Consumption rules expressed by
1
 1
 t I t 
C (t )  e

 W 
I
 (a  r ) t
W
w* (t ) 
2

It
 2W
W 2
*
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I
03. CRRA (Constant Relative Risk-Aversion)
 I t 

 t

2 


1
(a  r )  W 
 I  1   I t I t
0  e1  t 


rW 
 2 It

t W
2 2
 W 
Guess the solution to the PDE for I
W 2
b(t )  t

I (W , t ) 
e W (t )

 PDE for

I
 Then we can get the solution to the above PDE


1  (  1)e
I (W , t ) 
 ( t T )

 / 
1
e  t W (t )

(a  r ) 2
1 
where      {r  2
},  
2 (1   )

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II. Explicit Solutions
2
03. CRRA (Constant Relative Risk-Aversion)
II. Explicit Solutions
 The optimal portfolio and consumption rules


C * (t )   /(1  (  1)e (t T ) ) W (t ), for   0
C * (t )  1 /(T  t   )W (t ), for   0
(  r )
w (t )  2
 (1   )
*
*
 In this, the consumption C (t ) is a constant proportion of wealth and the optimal portfolio
*
rules w (t ) is a constant independent of W or t.
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04. CARA (Constant Absolute Risk-Aversion)
II. Explicit Solutions
 CARA family: Constant Absolute Risk Aversion.
e C
U (C )  
,  0

U '' (C )
where  '
  is Pr att' s measure of absolute risk aversion
U (C )
 Optimal Portfolio and Consumption rules expressed by
1
C * (t )   ln J ' (W )

 (a  r ) J ' (W )
w* (t )  2
 WJ ' ' (W )
MEIE811D Advanced Topics in Finance
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I
04. CARA (Constant Absolute Risk-Aversion)
 PDE for
II. Explicit Solutions
J
J ' (W )
J ' (W )
(a  r ) 2 [ J ' (W )]2
0
 J (W )  rWJ ' (W ) 
ln J ' (W ) 


2 2 J ' ' (W )
 Guess the solution to the PDE for
J
p qW
J (W )   e
q
 Then we can get the solution to the above PDE
p
J (W )   e qW
q
r    (a  r ) 2 / 2 2
where p  exp(
), q  r
r
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04. CARA (Constant Absolute Risk-Aversion)
II. Explicit Solutions
 The optimal portfolio and consumption rules
  r  (a  r ) 2 / 2 2
C (t )  rW (t ) 
r
(  r )
w* (t ) 
r 2W (t )
*
 In this unlike the CRRA case, that consumption
of wealth although it is still linear in wealth.
C * (t )
is no longer a constant proportion
*
 Instead of the proportion of wealth invested in the risky asset being constant (i.e., w
(t )
a constant), the total dollar value of wealth invested in the risky asset is kept constant (i.e.,
w* (t )W (t )
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.
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III. The effects of alternative asset price dynamics
Contents
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)
MEIE811D Advanced Topics in Finance
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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
Y (1  e r (t T ) ) 
[    ][W 
 (1  e r (t T ) )]

r

*
C (t ) 

 (1  exp[(    )(t  T ) /  ])

(  r )
Y (1  er (t T ) ) (  r )
r ( t T )
w (t )W (t ) 
(
W

)

(
1

e
)
2
2

r
r
*
 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.
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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
0  U (C * , t )  J t  [ J ( w*W , t )  J (W , t )]  JW {[ w* (  r )  r ]W  C *}
1
 JW W 2 w*2W 2
2
 First order conditions with respect to
C*
and
w* are
0  U C (C * , t )  JW (W , t )
0  JW ( w*W , t )  JW (W , t )(  r )  JW W (W , t ) 2 w*W

C
Consider the particular case when U (C , t ) 
, for   1 , in other words the CRRA

case.
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02. Poisson Processes(Case1)
III. Alternative dynamics
 Then the optimal portfolio and consumption rules are
C * (t )  AW (t ) /(1   )(1  exp[ A(t  T ) / 1   )
 (  r ) 2
  (2   ) *  (  r ) * 1 
 r    1 
w  2
w 
where A   
2

2 (1   )

 2 (1   )  
(  r )

w*  2
 2
( w* ) 1
 (1   )  (1   )
 As might be expected, the demand for the common stock is an increasing function of  .
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02. Poisson Processes(Case2)
III. Alternative dynamics
 Consider an individual who receives a wage, Y (t ) , which is incremented by a constant
amount  at random points in time. In other words, dY   dq . Suppose further that the
 t
individual’s utility function is of the form U (C , t )  e V (C ) and that his time horizon is
infinite (i.e.,T   ).
 For the two-asset case, the optimality equation can be written as
0  V (C * )  I (W , Y )  [ I (W , Y   )  I (W , Y )]  IW (W , Y )
1
2 *2
 [( w (  r )  r )W  Y  C ]  IW W (W , Y ) w W 2
2
Consider the particular case when V (C )   exp( C ) / 
*

MEIE811D Advanced Topics in Finance
*
24
, in other words the CARA case.
02. Poisson Processes(Case2)
III. Alternative dynamics
 Then the optimal portfolio and consumption rules are
2
Y
(
t
)

1

exp(


)
1
(


r
)
C * (t )  r[W (t ) 
 2
]  [  r 
]
2
r
r

r
2
 r
*
w (t )W (t )  2
 r

 In this, [W (t )  Y (t ) / r   (1  e ) / r ] is the general wealth term, equal to the sum of
present wealth and capitalized future wage earnings.
2
 If   0 , then the above optimal consumption
constant wage case.
MEIE811D Advanced Topics in Finance
25
C * (t )
is same as the result of previous
02. Poisson Processes(Case2)

 1  exp(  )
When   0 , 2
r

III. Alternative dynamics
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.

Et  exp( r ( s  t ))(Y ( s)  Y (t )) d s 
t
 [1  exp(  )]

, if   0
2
2
r
r
 The individual, in computing the present value of future earnings, determines the Certaintyequivalent flow and the capitalizes this flow at the (certain) market rate of interest.


0
exp( rs ) X ( s) 
Y (0) [1  exp(  )]

r
r 2
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02. Poisson Processes(Case3)
III. Alternative dynamics
 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

max E0{ U (C, t )dt  B[W ( ), ]}
0
 The associated optimality equation is
0  U (C * , t )  [ B(W , t )  J (W , t )]  L( J )
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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, P (t ) , such that
lim ET ( P(t ) / P (t ))  1, 0  T  t  
t 
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
P (t )  P (0)et
and
dP
  {  vt  log( P(t ) / P(0))}dt  dz,
P
where
  k  v /    2 / 4 , k  log( P (0) / P(0))
 This implies an exponentially-regressive price adjustment toward a normal price, adjusted
for trend.
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03. Alternative price dynamics(Case1)
III. Alternative dynamics
Y  log( P(t ) / P(0)) . Then by Ito’s Lemma, we can write the dynamics for Y as
dY   (u  vt  Y )dt  dz
 Let
where u
    2 / 2 . 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 Y ,

P (t ) is log-normal and Markov.
The Y (t ) , conditional on knowing Y (T ) , as
2
t

Y (t )  Y (T )  [k  vT 
 Y (T )](1  exp(  )  v   exp( t ) exp( s)dz
T
4

where   t  T  0 .
 The instantaneous conditional variance of Y (t ) is
2
var(Y (t ) | Y (T ))  [1  exp( 2 )]
2
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03. Alternative price dynamics(Case1)
III. Alternative dynamics
 The conditional expected price can be derived
ET ( P(t ) / P(T ))  ET {exp( Y (t )  Y (T ))}
2
 exp{[ k  vT 
 Y (T )](1  exp(   ))
4
2
 v   [1  exp( 2 )]})]
4
 Consider the two-asset model is used with the individual having an infinite time horizon
and a constant absolute risk-aversion utility function, U (C , t )   exp( C ) /  .
 The fundamental optimality equation then is written as
0   exp( C * ) /   J t  JW {w*[  (  vt  Y )  r ]W  rW  C *}
1
1
*2 2 2
 JW W w W   J Y  (u  vt  Y )  J YY  2  J YW w*W 2
2
2
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03. Alternative price dynamics(Case1)
III. Alternative dynamics
 From the FOC, the associated equations for the optimal rules expressed by
log( JW )
C 

J [  (  vt  Y )  r ] J YW
w*W   W

2
JW W
JW W
After solving the previous PDE for J , we obtain the optimal rules in explicit form as
  2 
2 2
 
C*  rW  2 Y 
t    r      r  Y  a(t )
2 
2 r
r 
2


*

J

1   
 2  2
w *W 
1  ( ( P, t )  r )  2    r 
2 
r  r 
r  2

MEIE811D Advanced Topics in Finance
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03. Alternative price dynamics(Case1)
III. Alternative dynamics
 Recall the optimal rules when the geometric Brownian motion hypothesis is assumed
  r  (a  r ) 2 / 2 2
C (t )  rW (t ) 
r
(  r )
*
w (t ) 
r 2W (t )
*
 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   r , unlike in the geometric Brownian motion case, a positive amount of the
risky asset is held.
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03. Alternative price dynamics(Case2)
III. Alternative dynamics
 Second case is the same type of price-dynamics equation as was assumed for the
geometric Brownian motion, namely,
dP
 dt  dz
P
 However, instead of the instantaneous expected rate of return  being a constant, it is
assumed that  is itself generated by the stochastic differential equation,
dP  dt  dz,
d   (u   )dt  dz
 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.
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03. Alternative price dynamics(Case2)
 The
 (t ) conditional on knowing  (T )
 (t )   (T )  (   (T ))(1  e
where   t  T  0 .
 
III. Alternative dynamics
is
)   e
 Then, the conditional mean and variance of
 t

t
T
e s dz,
 (t )   (T )
are
ET ( (t )   (T ))  (   (T ))(1  e   )
and
 2 2
var[ (t )   (T ) |  (T )] 
(1  e2  )
2
 Thus, the
Y (t )
conditional on knowing
P(T )
and  (T ) ,
t s
t
1 2
(    (T ))
 
  ( s ' s )
Y (t )  Y (T )  (    ) 
(1  e )     e
dz ( s' )ds    dz
T
T
T
2

MEIE811D Advanced Topics in Finance
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03. Alternative price dynamics(Case2)
 Therefore, the conditional mean and variance of Y (t )  Y (T ) are
III. Alternative dynamics
1 2
(    (T ))
ET [Y (t )  Y (T )]  (    ) 
(1  e   )
2

and
 2 2
1
2
 
 2 
var[Y (t )  Y (T ) | Y (T )]    
[


2
(
1

e
)

(
1

e
)]
2
2
2
2 2
 2 [   (1  e   )]

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03. Alternative price dynamics(Case2)
III. Alternative dynamics
 Again, the two-asset model is used with the individual having an infinite time horizon and a
constant absolute risk-aversion utility function, U (C , t )   exp( C ) /  .
 The fundamental optimality equation is written as
0
e
C *

 J t  JW [ w* (  r )W  rW  C * ]
1
1
*2
2 2
 JW W w W   J   (    )  J   2 2  JW  2 w*W
2
2
 From the FOC, the associated equations for the optimal rules expressed by
log( JW )
C 

J (  r ) JW 
w*W   W

2
JW W
JW W
*
MEIE811D Advanced Topics in Finance
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J
03. Alternative price dynamics(Case2)
 After solving the previous PDE for
III. Alternative dynamics
J , we obtain the optimal rules in explicit form as

1
 (   r ) 
wW 
(r    2 )(  r ) 

2
r (r  2  2 ) 
r     
*
 Assuming
r
, compare this with the Brownian motion case
w* (t ) 
(  r )
, then
2
r W (t )
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 w*W 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.
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03. Alternative price dynamics(Case3)
III. Alternative dynamics
 Last case assumed that prices satisfy the geometric Brownian motion,

dP
 dt  dz
P
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  , ˆ (t ) ,
1
ˆ (t ) 
t 
dP
 P , ˆ ( )  0
t
 Then E(ˆ (t ))   , and if we define the error term  t
written as
dP
 ˆdt  dzˆ
P
dzˆ  dz   t dt / 
By differentiating ˆ (t ) , we have the dynamics for ̂ ,

dˆ 
dzˆ
t 
where

MEIE811D Advanced Topics in Finance
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   ˆ (t ) , then
dP
can be reP
03. Alternative price dynamics(Case3)
III. Alternative dynamics
 By differentiating ˆ (t ) , we have the dynamics for ̂ ,
dˆ 

dzˆ
t 
 We see that this “learning” model is equivalent to the special case of the De Leeuw
hypothesis of pure extrapolation (i.e.,  0 ), 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., r  0 ), then the optimal portfolio rule
t 
T 
w W  2 log(
)ˆ (t )

t 
*
and the optimal consumption rule is
C* 
W 1
2
T  t  (T   ) log( T   )  (t   ) log( t   )
 [log( T   ) 
T t 
T t
ˆ 2 (t   ) 2
T   (T   )
 2[
log(
)
]]
2 (T  t )
t 
t 
MEIE811D Advanced Topics in Finance
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03. Alternative price dynamics(Case3)
 By differentiating w*W with respect to
time for t  t , reached a maximum at
III. Alternative dynamics
t , we find that w*W is an increasing function of
t  t , and then is a decreasing function of time for
t  t  T , where t is defined by
t  [T  (1  e) ] / e
 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 w W 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
w*W 
w*W a    , we have that the optimal portfolio rule is
T t
 , as   
2

MEIE811D Advanced Topics in Finance
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03. Alternative price dynamics(Case3)
III. Alternative dynamics
*
 Consider the effect on w W 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
w*W a   , we have that the optimal portfolio rule is
T t
w W  2  , as   

*
which is the optimal rule for the geometric Brownian motion case when  is known with
certainty.
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Questions
&
Comments
MEIE811D Advanced Topics in Finance