Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 30K (Stochastic Financial Models)

The time-zero budget constraint isso ${\theta }^0=w_0-{\theta }^T{S}_0$. The terminal wealth is thereforePutThenand the target mean is the linear constraint ${\theta }^{T}b=c$.

Since the invertible covariance matrix $V$ is positive definite, the Lagrange multiplier equation for minimizing ${\theta }^{T}V\theta$ isThusThe constraint determinesThis is the one-period Gaussian minimum-variance portfolio; the riskless holding is then ${\theta }^0=w_0-{\theta }^T{S}_0$.

Because $S_1$ is Gaussian, the terminal wealth $W_1$ is normal. For constant absolute risk aversion utilitythe moment-generating function of a normal variable givesMaximizing this expected utility is equivalent to maximizingafter dropping a constant. The unique first-order condition isand strict concavity givesIt agrees with part (a) exactly whenFor the usual risk-averse convention $\gamma >0$, the asserted correspondence assumes $w_1>(1+r)w_0$; under that natural target-return condition the choice is unique.