Bothersome Mean/Variance Analysis
The equality in question follows from the expression for expectation of a log-normal distribution.
For example, if $X \stackrel{d}{\sim} N(\mu, \sigma^2)$, then
$$
E[e^{a X}] = e^{a \mu + \frac12 a^2 \sigma^2}.
$$
This is the reason that CARA/normal or CRRA/log-normal (agent utility/asset return distribution pair) set-ups reduce to the mean-variance case, up to certainty equivalence.