Managing a Portfolio. A local bank wants to build a bond portfolio from a set of five bonds with $$1$ million available for investment. The expected annual return, the worst$-$case annual

return on each bond, and the "duration" of each bond are given in the following table. $($The duration of a bond is a measure of the bond's sensitivity to changes in interest rates.$)$

The bank wants to maximize the expected return from its bond investments, subject to three conditions:

The average worst$-$case return for the portfolio must be at least $7\mathrm{.}2$ percent.

The average duration of the portfolio must be at most $6.$

Because of diversification requirements, at most $40$ percent of the total amount invested can be invested in a single bond.

a$.$ What is the maximum return on the $$1$ million investment? How should the investment be distributed among the bonds to achieve this return? $($Assume that bonds can be purchased

in fractional amounts.$)$

b$.$ What is the qualitative pattern in the optimal solution?

c$.$ What is the marginal rate of return on the investment amount? That is$,$ what would be the percentage return on an additional dollar invested? $($Give the percentage to four significant

figures.$)$