# An investor has \$5000 and two potential investments. Let x j for j = 1 and j = 2 denote her…

An investor has \$5000 and two potential investments. Let x j for j = 1 and j = 2 denote her allocation to investment j in thousands of dollars. From historical data, investments 1 and 2 are known to have an expected annual return of 20 and 16 percent, respectively. Also, the total risk involved with investments 1 and 2, as measured by the variance of total return, is known to be given by 2×1 + x2 + (x1 + x2), so that risk increases with total investment (x1 + x2) and with the amount of each individual investment. The investor would like to maximize her return and at the same time minimize her risk. Figure E3.6 illustrates the conflict between these two objectives. Point A in this figure corresponds to investing nothing, point B to investing all \$5000 in the second alternative, and point C to investing completely in the first alternative. Every other point in the shaded region of the figure corresponds to some other investment strategy; that is, to a feasible solution to the constraints:

x1 + x2                               ≤ 5,

x1 ≥ 0,                               x2 ≥ 0.

To deal with the conflicting objectives, the investor decides to combine return and risk into a single    objective function.

She uses the parameter θ in this objective function to weigh the trade-off between the two objectives.

Because the ‘‘most appropriate’’ trade-off parameter is difficult to assess, the investor would like to maximize the objective function (28) subject to the constraints (27), for all values of θ, and then use secondary considerations, not captured in this simple model, to choose from the alternative optimal solutions.

a) Use parametric linear programming to solve the problem for all nonnegative values of θ.

b) Plot the optimal objective value, the expected return, and the risk as a function of θ.

c) Interpret the solutions on Fig.

d) Suppose that the investor can save at 6% with no risk. For what values of θ will she save any money using the objective function (28)?

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