 Solution

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 $m$.