Percival Hygiene has $$10$ million invested in long$-$term corporate bonds. This bond portfolio$$s expected annual rate of return is $11\%,$ and the annual standard deviation is $12\%.$ Amanda Reckonwith, Percival$$s financial adviser, recommends that Percival consider investing in an index fund that closely tracks the Standard & Poor$$s $500$ index. The index has an expected return of $16\%,$ and its standard deviation is $17\%.$

a$.$ Suppose Percival puts all his money in a combination of the index fund and Treasury bills. Can he thereby improve his expected rate of return without changing the risk of his portfolio? The Treasury bill yield is $6\%$

b$.$ Could Percival do even better by investing equal amounts in the corporate bond portfolio and the index fund? The correlation between the bond portfolio and the index fund is $+0\mathrm{.}3.$