1. Theory of Capital Markets
Security Markets IX
Miloslav S. Vosvrda
Theory of Capital Markets

2. 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
and pays dividends at the rate of at any time t,
where m, v and are strictly positive scalars. We
may scalars. We may think heuristically of
as the „instantaneous expected rate of return„ and
as the „instantaneous variance of the rate of return“.

3. A riskless security has a price that is always one,
and pays dividends at the constant interest rate r,
where Let
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 at any
time

4. If is the fraction of total wealth invested at time t
in the risky security, it follows that X satisfies the
stochastic differential equation:
which should be easily enough interpreted.
Simplifying,
We assume that wealth constraint is

5. We suppose that our investor derives utility from a
consumption process
according to
where is a discount factor, and u is a strictly
increasing, differentiable, and strictly concave
function. The problem of optimal choice of
portfolio and consumption rate is solved as
follows. Of course, and can only depend on the
information available at time t .

6. Because the wealth constitutes all relevant
information at any time t, we may limit ourselves
without loss of generality to the case and
for some (measurable) functions A and C.
We suppose that A and C are optimal, and note that
where
and w>0 is the given initial wealth.

7. Indirect utility
The indirect utility for wealth w is

8. For any time we can break this expression
into two parts:
Taking

9. Adding and subtracting
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
where

10. A and C are optimal
If A and C are indeed optimal, that is, if they maximize , then they must maximize
for any time T. This is equivalent to the problem:

11. Necessary conditions

12. Solving,
and
where g is the function inverting

13. An example If for some scalar risk aversion coefficient
then Substituing C and A
from these expressions into
leaves a second order differential equation for V that
has a general solution It follows that
and
where

14. 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
is quadratic in A(w).

15. Consumption-Based Capital Asset Pricing Model
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
variance averse.