Modern Portfolio Theory and Investment Analysis, 9th Edition

6

Techniques for Calculating the Efficient Frontier

In Chapters 4 and 5 we discussed the properties of the efficient frontier under alternative assumptions about lending and borrowing and alternative assumptions about short sales. In this chapter we describe and illustrate methods that can be used to calculate efficient portfolios. By necessity, this chapter is more mathematically complex than those that preceded it and most of those that follow. The reader who is concerned only with a conceptual approach to portfolio management can skip this chapter and still understand later ones. However, we believe that knowledge of the solution techniques to portfolio problems outlined here yields a better understanding and appreciation of portfolio management.

We have not followed the same order in presenting solution techniques for portfolio problems as was followed in describing the properties of the efficient set (Chapter 5). Rather, we have rearranged the order so that solution techniques are presented from the simplest to the most complex. The first four sections of this chapter discuss the solution to the portfolio problem when it is assumed in turn that

1. short sales are allowed and riskless lending and borrowing is possible
2. short sales are allowed but riskless lending or borrowing is not permitted
3. short sales are disallowed but riskless lending and borrowing exists
4. neither short sales nor riskless lending and borrowing is allowed

A fifth section shows how additional constraints, such ...