INDUSTRIAL ENGINEERING APPLICATIONS IN FINANCIAL ASSET MANAGEMENT:THE OPTIMIZATION PROBLEM

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THE OPTIMIZATION PROBLEM

The specific mathematical formulation for the standard MV problem is straightforward:

Industrial Engineering Applications in Financial Asset Management-0009

Equation (1) incorporates the investor’s attitude to risk via the objective function U. Equation (2) represents the portfolio return. Equation (3) is portfolio risk. Constraint (4) requires that the portfolio weights sum to 100%, while constraint (5) requires that weights be positive. This latter restriction can be dropped if short selling is allowed, but this is not usually the case for most investors.

This formulation is a standard quadratic programming problem for which an analytical solution exists from the corresponding Kuhn–Tucker conditions. Different versions of the objective function are sometimes used, but the quadratic version is appealing theoretically because it allows investor preferences to be convex.

The set of efficient portfolios can be produced more simply by solving

Industrial Engineering Applications in Financial Asset Management-0010

A vast array of alternative specifications is also possible. Practitioners often employ additional ine- quality constraints on portfolio weights, limiting them to maximums, minimums, or linear combi- nations. For example, total equity exposure may be restricted to a percentage of the portfolio or cash to a minimum required level. Another common variation is to add inequality constraints to force solutions close to benchmarks. This minimizes the risk of underperforming.

With respect to computation, for limited numbers of assets (small J ), solutions are easily obtained (although not necessarily efficiently) using standard spreadsheet optimizers. This works for the vast majority of allocation problems because most applications typically include no more than a dozen assets. More specialized optimizers are sometimes necessary when there are many assets. For ex- ample, if MV is applied to select a stock portfolio, there may be hundreds of securities used as admissible ‘‘assets.’’

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