The annual return per dollar for two different investment instruments is the outcome of a bivariate random variable \(\left(X_{1}, X_{2}ight)\) with joint moment-generating function \(M_{\mathbf{x}}(\mathbf{t})=\exp \left(\mathbf{u}^{\prime} \mathbf{t}+.5 \mathbf{t}^{\prime} \mathbf{\Sigma} \mathbf{t}ight)\), where

\(\mathbf{t}=\left[\begin{array}{l}t_{1} \\ t_{2}\end{array}ight], \mathbf{u}=\left[\begin{array}{l}.07 \\ .11\end{array}ight]\) and \(\boldsymbol{\Sigma}=\left[\begin{array}{cc}.225 \times 10^{-3} & -.3 \times 10^{-3} \\ -.3 \times 10^{-3} & .625 \times 10^{-3}\end{array}ight]\)

(a) Find the mean annual return per dollar for each of the projects.

(b) Find the covariance matrix of \(\left(X_{1}, X_{2}ight)\).

(c) Find the correlation matrix of \(\left(X_{1}, X_{2}ight)\). Do the outcomes of \(X_{1}\) and \(X_{2}\) satisfy a linear relationship \(x_{1}=\alpha_{1}+\alpha_{2} x_{2}\) ?

(d) If an investor wishes to invest \(\$ 1,000\) in a way that maximizes her expected dollar return on the investment, how should she distribute her investment dollars between the two projects? What is the variance of dollar return on this investment portfolio?

(e) Suppose the investor wants to minimize the variance of her dollar return. How should she distribute the \(\$ 1,000\) ? What is the expected dollar return on this investment portfolio?

(f) Suppose the investor's utility function with respect to her investment portfolio is \(U(M)=5 M^{b}\), where \(M\) is the dollar return on her investment of \(\$ 1,000\). The investor's objective is to maximize the expected value of her utility. If \(b=1\), define the optimal investment portfolio.
(g) Repeat (f), but let \(b=2\).
(h) Interpret the investment behavior differences in

(f) and \((\mathrm{g})\) in terms of investor attitude toward risk.