Dynamic programming and Bellman Equation

We consider the following maximization problem with CRRA utility function

subject to and

for

where is random return and Rft is risk-free return both of which are positive.

The value of can be either positive or negative or even zero.

We can construct Bellman equation

at time t

or

at time t

We guess

to find constant A for.

Plugging guess form into Bellman equation

There are two kinds of FOC’s.

st:

That is almost the same as what we got in 0512.

: 　　　

Since st and A are not zero, we can factor them out and erase and we get

That is

Using the rule

Thus the whole system is to get so as to satisfy

from the data of and together with parameter.

And then

is calculated and we get

from the same Bellman equation and solution as what we got in 0512.

Here, does not depend on Wt and it is independently determined.