The Portfolio Selection Problem: Problem Formulations

If you understand the concept of efficient portfolios, we are now ready to talk about problem formulations. We will first talk about two commonly used formulations that may not produce efficient portfolios. The first is to minimize variance subject to achieving a specified level of return and the other is to maximize return subject to achieving a specified level of variance. Let the portfolio have an expected return of z = r^Tw and variance of .
In model 1, r_* is the minimal acceptable return.
In model 2, is the maximal acceptable variance.

Minimize Variance subject to Given Return

The model of the first can be given as follows:

Maximize Return subject to Given Variance

The model of the second can be given as follows:

These models will not necessarily give efficient portfolios. The first model will provide a portfolio having the smallest standard deviation for a specified minimum level of return. However, there may exist a portfolio having a greater return and an equivalent standard deviation. In such a case, the portfolio returned by the model would not be efficient. These can happen only if the matrix Q is not strictly positive definite.

Combining the Models:
Balancing Risk and Return

Each investor is willing to take a certain amount of risk to earn another dollar in returns. As the total return goes up, the investor is less and less willing to risk more to earn just one more dollar. Each investor has a certain utility for money, which determines exactly how much risk he is willing to take in order to obtain an expected amount of money. We assume that this utility can be measured by a utility function, u(x). One commonly used utility function is:
u(x) = 1 - exp(-kx), where k > 0 is a risk aversion constant. This function describes the relationship between risk and return for an investor.

We assume that the return vector is normally distributed with mean r and covariance matrix Q. Therefore, z is also normally distributed with mean z = r^Tw and variance . The expected value of utility can then be computed as

Since f(x)=1-exp(-x) is a strictly increasing function in x, maximizing utility is equivalent to maximizing

Now, given a covariance matrix, Q, a vector of expected returns, r, and a risk-aversion parameter, k, we can select a portfolio that maximizes expected utility by solving the following optimization problem:

The optimal portfolio is determined by solving for the weighting parameter, w.

Note: In our models,

• return is the average annual percentage increase in value, and
• risk is the standard deviation of the annual percentage returns.

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