**N-SECURITY CASE**

In the *N-*security case, we present the general theory of optimal portfolio selection developed by Markowitz (1952). You will learn how to characterize the set of undominated portfolios for the general *N-*security case.

To apply this theory, you need to solve for the optimal portfolio weights. With a large number of securities, an analytical solution is always easier than a numerical search. You will see that when short selling is permitted, an analytical solution is available. Without short selling, however, you do have to rely on numerical methods to obtain the solution. Advances in computer technology now make the application of numerical methods more feasible, and several researchers have developed portfolio selection models that are simpler than the general Markowitz problem. We study these in Chapter 8, Index Models.

The objective of the portfolio selection problem is to minimize portfolio variance 1) subject to attaining a target expected return, and 2) subject to the sum of portfolio weights equaling one. As a result, the general form of the problem has two constraints. We motivated the first constraint in the topic describing the investment opportunity set. It guarantees that for a given expected return a portfolio is not dominated by some other portfolio with lower portfolio standard deviation. The second constant requires that the portfolio weights sum to one. If we do not place any further restriction on the portfolio weights, we are allowing short selling. If we constrain all the weights to be non-negative, short selling is prohibited.

The general problem is formally specified in topic 4.7, The Markowitz Problem. We also show you how to identify the entire efficient frontier. The theory is then applied to the Three-Firm Case in topic 4.9, titled Three-Firm Case: Application of Mean Variance Theory. The solution to the three-firm problem reduces to a system of five linear equations with five unknowns. The first three unknowns are the optimal portfolio weights, while the last two unknown variables provide a measure of how sensitive the minimum-variance is to relaxing each of the two constraints. The solution to this three-firm problem is then generalized to the *N*-security case.

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