The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 493 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 493

wealth is invested at a rate Rt. An infinite stream of consumption is possible

if the return rate is positive. We will write utility in the following form:

∞

Ut()C = ∑diUC( ti+ )

i = 0

where C is a shorthand for a realization of the consumption process and

d < 1 is the time discount factor of utility. In the following formulation

we will consider an infinite horizon, i.e., consumption extends over an

infinite stream at all future dates. It is also possible to consider only a

finite number of steps ahead; in this case, one needs to write a utility

function for final wealth in order to establish a trade-off between con-

sumption and final wealth. As in the previous single-period case, utility is

a random variable as consumption is a stochastic process. We will there-

fore define utility as the expected value of stochastic utility as follows:

∞

Ut = Et ∑d

i

UC( ti+ )

i = 0

The process dynamics are given by the following equation:

Wt + 1 = ( 1 + Rt)[Wt – Ct]

where Rt is the portfolio stochastic return. The investor’s portfolio

selection consists of maximizing his expected utility given a return rate

process for the portfolio and an initial endowment. The solution of this

problem can be obtained through the methods of stochastic multistage

optimization. The solution of the infinite horizon problem implies that

first-order conditions, called Euler conditions, are satisfied for each

asset. Euler conditions are the following:

U′()Ct = dEt[( 1 + Rit , + 1 )U′(Ct + 1 )]

where Ri,t is the period t return of the i-th asset. The left hand side of

the equation is the utility the investor derives from consuming one unit

less at time t while the right hand side is the additional expected utility

that derives from consuming at time t + 1 the unit saved at time t and

invested at rate Rt. Optimality implies that the two coincide.

If we take the unconditional expectation and divide by U′()Ct we

can write the above equations in the following form: