Consider the general case

\[\alpha_{M}-r=\frac{M}{A} \sigma_{M}^{2}-\frac{H}{A} \sigma_{M r}\]

where interest rate is stochastic.

(a) Explain how, in the case of a constant interest rate, it leads to CAPM and the two-fund separation theorem.

(b) Explain how, in the case of a stochastic interest rate, it leads to the three-fund separation theorem. Discuss the connection with

\[w_{k}^{*}=h_{k}(P, t)+m(P, W, t) g_{k}(P, t)+f_{k}(P, W, t)\]

in Eq. (5.8). [Hint: see Merton (1990) Chapter 15, at the end of Sec. 15.7, Theorem 15.2.]