Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 29J / Solution

The terminal wealth of portfolio $\psi$ is the scalar normal distribution $\psi \cdot X$, with mean $\psi \cdot \mu$ and variance $\psi \cdot V\psi$. Its Gaussian exponential moment givesSince $-e^{-\gamma q}$ is strictly increasing in $q$, strict preference is exactly $q(\psi )>q(\theta )$. The nonsingular covariance matrix is positive definite. Completing the square givesso the unique optimum is $\boxed{{\psi }^{\ast }={\gamma }^{-1}{V}^{-1}\mu }$.

Paying the deterministic transaction cost reduces the certainty equivalent of the new portfolio by $ϵ\Vert z{\Vert }_1$. Put $g=\mu -\gamma V\theta$. Its gain over the current portfolio is exactlyIf every $|g_i|\le ϵ$, then $g\cdot z\le ϵ\Vert z{\Vert }_1$, so every nonzero trade has strictly negative gain, by positive definiteness. Conversely, if $|g_i|>ϵ$, choose $z=t\mathrm{sign}(g_i)e_i$. Its gain isfor $0<t<2(|g_i|-ϵ)/(\gamma V_{ii})$. Therefore