$4.$ 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.$)$

$$ Expected Run Worst Case Return Duration

Bond $1.12\mathrm{.}5\%8\mathrm{.}0\%8$

Bond$211\mathrm{.}5\%7\mathrm{.}5\%7$

Bond $310\mathrm{.}5\%6\mathrm{.}8\%6$

Bond $49\mathrm{.}5\%7\mathrm{.}0\%5$

Bond $58\mathrm{.}5\%7\mathrm{.}4\%3$

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 $40$ per$-$ cent 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.$)$

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.$)$