Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 29J / i / Solution

Set $R=1+r$, $a=\mu -R{S}_0$, and $M=(\gamma V{)}^{-1}$. The terminal wealth is $W_1=R{w}_0+{\theta }^T(S_1-R{S}_0)$, a normal random variable with mean $R{w}_0+{\theta }^{T}a$ and variance ${\theta }^{T}V\theta$. Its moment-generating function givesMaximizing this expected exponential utility is equivalent to maximizing the strictly concave quadratic form ${\theta }^{T}a-(\gamma /2){\theta }^{T}V\theta$. Since the nonsingular covariance $V$ is positive definite, the unique stationary point is the unique maximum: