Security Markets
VIII
Miloslav S. Vosvrda
Teorie financnich trhu
Portfolio Control Problem
We apply Ito‘s Lemma to the following portfolio
control problem. We assume that a risky security has
a price process S satisfying the stochastic differential
equation
dS t  mS t dt  vS t dBt ,
t  0,
and pays dividends at the rate of  St at any time t,
where m, v and  are strictly positive scalars. We
may scalars. We may think heuristically of m  
2

as the „instantaneous expected rate of return„ and
as the „instantaneous variance of the rate of return„.
A riskless security has a price that is always one,
and pays dividends at the constant interest rate r,
where 0  r  m   . Let X  X t : t  0
denote the stochastic process for the wealth of an
agent who may invest in the two given securities and
withdraw funds for consumption at the rate ct at any
time t  0 .
If at is the fraction of total wealth invested at time t
in the risky security, it follows that X satisfies the
stochastic differential equation:
dXt  at X (m  )dt  at Xt vdBt  (1 at ) Xt rdt  ct dt ,
which should be easily enough interpreted.
Simplifying,
dX t  at X t (m    r )  rX t  ct dt  at X t v dBt .
We assume that wealth constraint is X t  0 for
t
We suppose that our investor derives utility from a
consumption process c  ct : t  0
according to
   t

U c   E

0
e uct dt ,

where  0 is a discount factor, and u is a strictly
increasing, differentiable, and strictly concave
function. The problem of optimal choice of
portfolio at and consumption rate ct is solved as
follows. Of course, ct and at can only depend on the
information available at time t .
Because the wealth X t constitutes all relevant
information at any time t, we may limit ourselves
without loss of generality to the case at  AXt  and
ct  CXt  for some (measurable) functions A and C.
We suppose that A and C are optimal, and note that
dX t   X t dt    X t dBt ;X 0  w,
where
x  Axxm    r   rx  Cx, x  Axxv,
and w>0 is the given initial wealth.
Indirect utility
The indirect utility for wealth w is
t


V w  E  e uC  X t dt .
 0


For any time T 0 we can break this expression
into two parts:
T

t



V w  E  e uC X t dt  E  esuC X s ds
 0
  T

Taking   s  T
T

t
T 


V w  E  e uCXt dt  e E  e uCX d 
 0

 0

T
t
T


 E  e uCXt dt  e EV XT .
 0

eTVw
Adding and subtracting

e EVXT Vw e
T
T

T
T

1Vw E  e uCXt dt  0.
 0

We divide each term by T and take limits as T
converges to 0, using Ito‘s Lemma and l‘Hopital‘s
Rule to arrive at
DV  w    V  w   u C  w   0 ,
where
1
2
DV w  Awwm    r   rw  C wV w  V wAwwv .
2
A and C are optimal
If A and C are indeed optimal, that is, if they
maximize Vw , then they must maximize
 t
 T

E  e uC  X t dt  e V  X t 
 0

T
for any time T. This is equivalent to the problem:
DV w   V w   u C w .
max
   
A w ,C w
Necessary conditions
u C w  V w  0
m    r wV w V wAww v
2 2
 0.
Solving,
and
V ( w)(m    r )
A( w) 
2
V ( w)v w
C ( w )  g V ( w ) ,
where g is the function inverting u  .
An example
 

u ct  ct
If
for some scalar risk aversion coefficient   0,1,
1 /  1
then g  y    y /  
.Substituing C and A
from these expressions into
DV  w    V  w   u C  w   0 ,
leaves a second order differential equation for V that

has a general solution V w   kw . It follows that
2
Aw   m    r  / v 1    and C w    w,
where   (   r  )   ( m    r ) .
1
(1   )
2
In other words, it is optimal to consume at a rate
given by a fixed fraction of wealth and to hold a
fixed fraction of wealth in the risky asset. It is a key
fact that the objective function
max
  
A w ,C
w

DV
 w    V  w   u C  w  .
is quadratic in A(w). This property carries over to a
general constinuous-time setting. As the
Consumption-Based Capital Asset Pricing Model
(CCAPM) holds for quadratic utility functions, we
should not then be overly surprised to learn that a
version of the CCAPM applies in continuous-time,
even for agents whose preferences are not strictly
varience averse.