16-Port Selection Mean Var Page 473 Wednesday, February 4, 2004 1:09 PM
Portfolio Selection Using Mean-Variance Analysis 473Markowitz considered an investor who, at time t, decides what
portfolio of investments to choose; the time horizon of the investor is
∆t. The investor makes decisions on the gains and losses he or she will
make at time t + ∆t, without considering eventual gains and losses either
during or after the period ∆t. At time t + ∆t, the investor will reconsider
the situation and decide anew; this last condition is called myopic.
Nonmyopic investment strategies must be adopted when it is necessary
to make trade-offs at future dates between consumption and investment or
when significant trading costs related to specific subsets of investments are
incurred. We will handle these issues later in this chapter and when we dis-
cuss bond portfolio management in Chapter 21 where we apply the multi-
stage optimization technology discussed in Chapter 7.^2
Markowitz reasoned that investors should decide on the basis of a
trade-off between risk and return. He made the assumption that returns
are normally distributed and that risk is measured by the variance of the
return distribution. In the 1950s when asset pricing theories were not
yet developed, the assumption of joint normality of returns was a rea-
sonable statistical assumption. It was based on the fact that asset
returns are influenced by many different independent facts. Recall from
Chapter 6 on probability theory that the sum of many small random
disturbances tends to a normal distribution.
Markowitz argued that for any given level of expected returns
investors should choose the portfolios with minimum variance from
amongst the set of all possible portfolios that can be constructed. The
set of all possible portfolios that can be constructed is called the feasible
set. In this simple one-period model, variance of returns is a measure of
uncertainty and thus of risk. Minimum variance portfolios are called
mean-variance-efficient portfolios. The set of all mean-variance efficient
portfolios is called the efficient frontier.
Exhibit 16.1 presents the MPT investment process (mean-variance
optimization or the theory of portfolio selection). Notice in the exhibit
that the result of the analysis is the selection of the optimal portfolio.
We describe what is meant by an optimal portfolio later in this chapter.
Though its implementation can get quite complicated, the theory is
relatively straightforward. Here we want to give an intuitive and practi-
cal view of MPT. The theory dictates that given estimates of the returns,
volatilities, and correlations of a set of investments, and constraints on
investment choices (for example, maximum exposures and turnover(^2) There are applications of multistage optimization in equity portfolio management
though these are not as common in the bond portfolio management area. See, for ex-
ample, John M. Mulvey and Hercules Vladimirou, “Stochastic Network Optimization
Models for Investment Planning,” Management Science 38, no. 11, pp. 1642–1664.