= 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 loss
$$
L(d,\sigma)=\sigma\varphi(d/\sigma)-d\Phi(-d/\sigma)\quad(\sigma>0),
$$
where $\varphi,\Phi$ are the standard normal density and distribution function. Differentiation at fixed $d$ yields $\partial L/\partial\sigma=\varphi(d/\sigma)>0$; the expression extends continuously to $\sigma=0$. Therefore \b[minimizing expected shortfall at a fixed mean chooses exactly the minimum-variance portfolio $\theta^*$] from part (a).
For $m\geq b$ 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 $m$.
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