Given the individual demand function [d_{i}^{k}=w_{i}^{k} W^{k}=-frac{J_{W}^{k}}{J_{W W}^{k}} sum_{j=1}^{m} u_{i j}left(alpha_{j}-r ight)-frac{J_{W r}^{k}}{J_{W W}^{k}} sum_{j=1}^{m} u_{i j}

Given the individual demand function

\[d_{i}^{k}=w_{i}^{k} W^{k}=-\frac{J_{W}^{k}}{J_{W W}^{k}} \sum_{j=1}^{m} u_{i j}\left(\alpha_{j}-r\right)-\frac{J_{W r}^{k}}{J_{W W}^{k}} \sum_{j=1}^{m} u_{i j} \sigma_{j r}\]

(a) Drive the aggregate demand \(\underline{D}\) below

\[\begin{aligned} \underline{D} & =A \Omega^{-1}(\underline{\alpha}-r)+H \Omega^{-1} \underline{\sigma}_{r} \\ A & \equiv \sum_{1}^{K} A^{k} \quad \text { and } \quad H \equiv \sum_{1}^{K} H^{k} \\ A^{k} & =-\frac{J_{W}^{k}}{J_{W W}^{k}}, \quad H^{k}=-\frac{J_{W r}^{k}}{J_{W W}^{k}} \end{aligned}\]

(b) The individual asset excess returns \[\alpha_{i}-r=\frac{M}{A} \sigma_{i M}-\frac{H}{A} \sigma_{i r}\]
and interpret this solution using your own words. [Hint: see Merton (1990) Chapter 15, at the end of Sec. 15.5.]