Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 1 / b / Solution

Write $d=\mu -r1$ and let $x$ denote the risky holdings as fractions of total wealth. The remaining fraction $1-1^{T}x$ is in the risk-free asset. Consequently their excess expected return is $d^{T}x$ and their variance is $x^T\Sigma x$. The Cauchy-Schwarz inequality in the inner product defined by $\Sigma$ givesEquality with positive excess return holds precisely for $x=cz$, $c>0$, where $z={\Sigma }^{-1}d$. Thus solving $\Sigma z=d$ determines both the efficient risky direction and the upper capital market line slope. Provided $s=1^{T}z>0$, the normalized market portfolio and its Sharpe ratio areHere the excess means are $(7,14,21{)}^T$, so the required equations areThe leading principal minors of $\Sigma$ are $4$, $35$ and $539$, so Sylvester's criterion verifies positive definiteness and the solution is unique.