Problem $6.($The price is right$)$ A mutual fund invests a proportion of $$ of its

funds at the risk$-$free rate of $r_f$ and a proportion of $1-$ in the market portfolio

with the return rate $r_M,$ where Suppose the current unit price of

this fund is $P$ and all the return rates are annual.

a$)$ What is the beta of this fund?

b$)$ What is the expected price of this fund after one year?

c$)$ Hence, show that both CAPM pricing formulae yield that the current price

of this fund is indeed $P.$