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yvr - OR&IE 320/520 Prof Bland Balancing Return and Risk A...

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OR&IE 320/520 10/11/07 Prof. Bland Balancing Return and Risk: A Simplified Example of Parametric Programmimg This problem is adapted from Bradley, Hax, and Magnanti. An investor has $5,000,000 and two potential investments. Let x j for j = 1 , 2 denote her allocation to investment j in millions of dollars. From historical data, the expected annual rates of return on investments 1 and 2 have been estimated at 20% and 16%, respectively. Also, we have estimated risk, measured by the variance in total return, of 3 x 1 + 2 x 2 . The investor in interested in both maximizing total return and minimizing risk, but these different objectives are in conflict. She is looking at two different approaches toward gathering information that will aid her in analyzing the return versus risk trade-off. Approaches 1. In this approach the investor will attempt to combine the two objectives into a single parametric objective function: (return) - θ (risk), where θ is a nonnegative parameter that is used to allow for different trade-offs between return and risk. For θ equal to zero, risk is ignored and return is max- imized. For sufficiently large θ , the weighted risk overwhelms return, and the investor would invest nothing. You need to ascertain what happens in between those extremes. The investor will then examine the information you provide and make a determination of x 1 and x 2 based on secondary considerations not described here. (a) Use the simplex method parametrically to determine z * ( θ ) and x * ( θ ) for all θ 0. (b) Plot the optimal objective function value, the expected return, and the risk as functions of θ .
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