I am very close to showing part b of this question and think that the reason my solution doesn't match up is because I don't understand a piece of notation.The notation I don't understand is $\textbf{$e^T$}$ on the bottom line inside the function $h$.
I am at the point:
$$E[f(W)] = \frac{1}{a}\huge{[}\normalsize 1 - \exp\{-aw\xi_0 r_0 + aw\big(\frac{1}{2}aw\xi V\xi^T - \xi r^T\big)\}\huge{]} $$
