Instructions for Optimization Worksheet

Inputs

The Inputs box should contain all the needed input information other than the investor's risk tolerance. Each row provides the inputs for a single decision variable (for example, an asset class). The columns must be in the order indicated below. Each must be given a short heading, although these are not actually checked for content.

The first three columns indicate the Minimum proportions that may be invested in the items, the Initial portions invested, and the Maximum proportions that may be invested. The initial proportions will usually sum to 1.0 although this is not necessary. Each initial proportion must lie within or at an end of the range given by the minimum and maximum allowed values. The optimizer will consider only portfolios that are feasible (with each proportion greater than or equal to its minimum value and less than or equal to its maximum value) and for which the proportions sum to the same value as the sum of the proportions in the initial portfolio.

The next two columns provide the Expected Returns (ExpRet) and Standard Deviations (StdDev) for the decision variables. These are usually stated in terms of return per year (for example, 10.5 for 10.5% per year). The remaining columns provide the Correlation Coefficients for the variables. The order must be the same as that used for the rows in the table.

The next input box provides the investor's Risk Tolerance. This indicates the marginal rate of substitution of variance for expected return in the investor's utility function. It is normally positive, but a zero value is allowed for the special case in which the objective is to find the feasible portfolio with the smallest possible standard deviation.

The final input box provides the Marginal Utility Cutoff. This is used to terminate the algorithm. When the marginal utility of the best feasible swap falls below this amount, the program stops. The smaller this amount, the more precise the solution but the longer the time required to obtain it.

Algorithm

The worksheet uses the gradient quadratic programming method of Sharpe [ "An Algorithm for Portfolio Improvement", in "Advances in Mathematical Programming and Financial Planning", K.D. Lawrence, J.B. Guerard, Jr., and Gary D. Reeves, Editors, JAI Press, Inc., 1987, pp. 155-170 ] to find the feasible portfolio with the maximum possible Utility:

Up = ep - ( ( sdp ^ 2 ) / t )

where:

Up = the utility of the portfolio

ep = the portfolio's expected return

sdp = the portfolio's standard deviation of return

t = the investor's risk tolerance

Output

All output is provided in a single box. Two tables are included. The first table shows the Initial Portfolio, the Optimal Portfolio and the Change (Optimal - Initial). The second table shows the characteristics of the Initial and Optimal Portfolio and the associated changes in the level of each one (Optimal - Initial). Unless the risk tolerance is zero, the third line shows the utility of the portfolios and the change therein.

Notes

You may enter any desired text in this box to describe the source of the input data, etc...