Equity Valuation, Risk, and Investment: A Practitioner's Roadmap

APPENDIX J

Additional Background on Mathematical Optimization Subject to Constraint Conditions

The methods for optimizing equations, subject to constraint conditions, are set forth in Chapter 9. This appendix supplements the treatment found there.

In the Chapter 9 solution systems, we introduced λ1 and λ2, the so-called LaGrange multipliers, as expedients necessary to reaching optimal solutions for portfolio weights: w1, w2, and w3. This section provides insight into the economic meaning of these multipliers.

The definition of portfolio variance is:

(J.1)

This equation permits us to write a total differential formula for instantaneous changes in the underlying variables, w1, w2, and w3. This is represented as:

(J.2)

From equation (9.30) in Chapter 9, we recollect that the solution system contains these expressions for j = 1, 2, and 3:

(J.3)

This permits the obvious rearrangement:

(J.4)

This rearranged expression, for each value of j, can be substituted into equation (J.2), resulting in:

(J.5)

The next step is a little combining of like terms.

(J.6)

The second term in parentheses ...