Problem description
The investor may also look at his portfolio selection problem from a different angle: instead of maximizing the estimated return and limiting the portion of high-risk investments he now wishes to minimize the risk whilst obtaining a certain target yield. He adopts the Markowitz idea of getting estimates of the variance/covariance matrix of estimated returns on the securities. (For example, hardware and software company worths tend to move together, but are oppositely correlated with the success of theatrical production, as people go to the theater more when they have become bored with playing with their new computers and computer games.) The return on theatrical productions are highly variable, whereas the treasury bill yield is certain. The estimated returns and the variance/covariance matrix are given in the following table:
| treasury | hardw. | theater | telecom | brewery | highways | cars | bank | softw. | electr. | |
|---|---|---|---|---|---|---|---|---|---|---|
| treasury | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| hardware | 0 | 19 | -2 | 4 | 1 | 1 | 1 | 0.5 | 10 | 5 |
| theater | 0 | -2 | 28 | 1 | 2 | 1 | 1 | 0 | -2 | -1 |
| telecom | 0 | 4 | 1 | 22 | 0 | 1 | 2 | 0 | 3 | 4 |
| brewery | 0 | 1 | 2 | 0 | 4 | -1.5 | -2 | -1 | 1 | 1 |
| highways | 0 | 1 | 1 | 1 | -1.5 | 3.5 | 2 | 0.5 | 1 | 1.5 |
| cars | 0 | 1 | 1 | 2 | -2 | 2 | 5 | 0.5 | 1 | 2.5 |
| bank | 0 | 0.5 | 0 | 0 | -1 | 0.5 | 0.5 | 1 | 0.5 | 0.5 |
| software | 0 | 10 | -2 | 3 | 1 | 1 | 1 | 0.5 | 25 | 8 |
| electronics | 0 | 5 | -1 | 4 | 1 | 1.5 | 2.5 | 0.5 | 8 | 16 |
Question 1: Which investment strategy should the investor adopt to minimize the variance subject to getting some specified minimum target yield?
Question 2: Which is the least variance investment strategy if the investor wants to choose at most four different securities (again subject to getting some specified minimum target yield)?
The first question leads us to a Quadratic Programming problem, that is, a Mathematical Programming problem with a quadratic objective function and linear constraints. The second question necessitates the introduction of discrete variables to count the number of securities, and so we obtain a Mixed Integer Quadratic Programming problem. The two cases will be discussed separately in the following two sections.
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