Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 29J (Stochastic Financial Models)

Put $b=(1+r)w_0$, $a=\mu -(1+r)p$ and $d=m-b$. The terminal mean and variance are $b+{\theta }^{\mathsf{T}}a$ and ${\theta }^{\mathsf{T}}V\theta$. A nonsingular covariance matrix is positive definite. If $a\ne 0$, define $D=a^{\mathsf{T}}{V}^{-1}a>0$. The Lagrange multiplier equations for the fixed-mean constraint give $2V\theta =\lambda a$, and thusFor a direct global proof, every feasible portfolio is ${\theta }^{\ast }+h$ with $a^{\mathsf{T}}h=0$, so its variance is $d^2/D+h^{\mathsf{T}}Vh$, minimized uniquely at $h=0$. If $a=0$, only $m=b$ is feasible and its minimum variance is zero, attained by the all-bank portfolio; no other required mean is achievable.

When $a\ne 0$ and $n\ge 2$, the set of achievable mean-variance pairs isIndeed the decomposition in part (a) proves necessity. There is a nonzero $h$ orthogonal to $a$; scaling it makes $h^{\mathsf{T}}Vh$ any nonnegative number, proving sufficiency. For $n=1$ that orthogonal space is zero and only the boundary parabola is achievable, not its interior. If $a=0$, the set is the vertical half-line $(b,s^2)$, $s^2\ge 0$.

The mean-variance efficient frontier comprises undominated portfolios: no other portfolio has at least as high a mean and at most as large a variance, with one strict inequality. For $a\ne 0$ it is the upper branch $m\ge b$, $s^2=(m-b{)}^2/D$. Lower-branch portfolios have lower mean and larger variance than the all-bank portfolio, and points above the minimum-variance boundary are dominated at their own mean. On the upper branch increasing mean necessarily increases minimum variance, so no dominance is possible. If $a=0$, only the zero-variance all-bank point is efficient.

At the prescribed mean, $w_1-b$ has a normal distribution with mean $d=m-b$ and standard deviation $\sigma$. Integrating its negative part gives the expected losswhere $\phi ,\Phi$ are the standard normal density and distribution function. Differentiation at fixed $d$ yields $\partial L/\partial \sigma =\phi (d/\sigma )>0$; the expression extends continuously to $\sigma =0$. Therefore minimizing expected shortfall at a fixed mean chooses exactly the minimum-variance portfolio ${\theta }^{\ast }$ from part (a).

For $m\ge b$ it lies on the mean-variance efficient frontier. If “efficient frontier” has its usual undominated meaning, the printed claim needs this qualification: for $m<b$, the same optimizer lies on the lower minimum-variance branch and is dominated by the all-bank point $(b,0)$. It still solves the fixed-mean problem, but is not efficient in that unconstrained comparison. If the term instead means the full fixed-mean minimum-variance boundary, the claim holds for every feasible $m$.