Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 2

Assume the covariance matrix $\Sigma$ is positive definite and the excess-mean vector $d=z-R1$ is nonzero. Put $A=d^t{\Sigma }^{-1}d>0$. Substituting $w=\lambda {\Sigma }^{-1}d$ in the mean constraint gives $\lambda A=\mu -R$. ThereforeThese weights are indeed the minimum-variance solution, not just a stationary point. Any other feasible vector is $w+v$ with $d^{t}v=0$. Its variance is $w^t\Sigma w+2\lambda d^{t}v+v^t\Sigma v=w^t\Sigma w+v^t\Sigma v$, which is strictly larger unless $v=0$. If $d=0$, only mean $R$ is feasible, and the minimum-risk portfolio invests entirely in the risk-free asset; the multiplier formula with zero denominator is not applicable.

Using $A$ from part (a), the portfolio variance isThus the minimum-variance correspondence for fixed means has the two branches $\mu =R\pm \sqrt{Av}$. The lower branch is dominated by the upper one with the same variance, and portfolios with mean below $R$ are also dominated by the risk-free position. Consequently the actual mean-variance efficient frontier is the upper capital market line:Its slope in standard-deviation/mean coordinates is the maximal Sharpe ratio $\sqrt{A}$. In variance/mean coordinates it is the upper branch of $v=(\mu -R{)}^2/A$.

The risk-free weight is $1-1^{t}w$, so a zero risk-free weight requires $1^{t}w=1$. Define $H=1^t{\Sigma }^{-1}d$. If $H\ne 0$, this condition gives $\lambda =1/H$ andThese are the requested tangent-portfolio mean and variance under the usual condition $H>0$. Then $\lambda >0$, the normalized portfolio lies on the upper efficient ray, and its Sharpe ratio is $\sqrt{A}$.

There is a real qualification if the data are allowed to be arbitrary. If $H=0$, the efficient risky direction has zero total risky weight, and no finite multiple makes its risky weights sum to one. If $H<0$, the formal normalized portfolio above has negative excess mean and lies on the lower, dominated branch; it is not an efficient tangent portfolio as described in the question. The sign of the normalized tangency portfolio must therefore be checked, rather than inferred from positive definiteness of $\Sigma$ alone.