Consider a market with no riskless asset and $d$ risky stocks where the price of stock $i \in\{1, \ldots, d\}$ at time $t \in\{0,1\}$ is denoted $S_{t}^{i}$. We assume the vector $S_{0} \in \mathbb{R}^{d}$ is not random, and we let $\mu=\mathbb{E} S_{1}$ and $V=\mathbb{E}\left[\left(S_{1}-\mu\right)\left(S_{1}-\mu\right)^{T}\right]$. Assume $V$ is not singular.

Suppose an investor has initial wealth $X_{0}=x$, which he invests in the $d$ stocks so that his wealth at time 1 is $X_{1}=\pi^{T} S_{1}$ for some $\pi \in \mathbb{R}^{d}$. He seeks to minimize the $\operatorname{var}\left(X_{1}\right)$ subject to his budget constraint and the condition that $\mathbb{E} X_{1}=m$ for a given constant $m \in \mathbb{R}$.

Illustrate this investor's problem by drawing a diagram of the mean-variance efficient frontier. Write down the Lagrangian for the problem. Show that there are two vectors $\pi_{A}$ and $\pi_{B}$ (which do not depend on the constants $x$ and $m$ ) such that the investor's optimal portfolio is a linear combination of $\pi_{A}$ and $\pi_{B}$.

Another investor with initial wealth $Y_{0}=y$ seeks to maximize $\mathbb{E} U\left(Y_{1}\right)$ his expected utility of time 1 wealth, subject to his budget constraint. Assuming that $S_{1}$ is Gaussian and $U(w)=-e^{-\gamma w}$ for a constant $\gamma>0$, show that the optimal portfolio in this case is also a linear combination of $\pi_{A}$ and $\pi_{B}$.

[You may use the moment generating function of the Gaussian distribution without derivation.]

Continue to assume $S_{1}$ is Gaussian, but now assume that $U$ is increasing, concave, and twice differentiable, and that $U, U^{\prime}$ and $U^{\prime \prime}$ are of exponential growth but not necessarily of the form $U(w)=-e^{-\gamma w}$. (Recall that a function $f$ is of exponential growth if $|f(w)| \leqslant a e^{b|w|}$ for some constants positive constants $a, b$.) Prove that the utility maximizing investor still holds a linear combination of $\pi_{A}$ and $\pi_{B}$.

[You may use the Gaussian integration by parts formula

$$\mathbb{E}[\nabla f(Z)]=\mathbb{E}[Z f(Z)]$$

where $Z=\left(Z_{1}, \ldots, Z_{d}\right)^{T}$ is a vector of independent standard normal random variables, and $f$ is differentiable and of exponential growth. You may also interchange integration and differentiation without justification.]