How to compute this utility function's expected value.

Question

image here

This is a question of financial economics. There are N stocks,

1. The utility function U is the function of $W_1$ and X, $U(\tilde W_{1i},X_i)$,
2. $W_0$ is the investor's wealth at moment $0$, and $W_1$ is the wealth at moment $1$,
3. $r_f$ is the risk-free interest rate, $\tilde r$ be the N × 1 vector which is the excess rate of return,We assume $\tilde r$ is normally distributed,
4. Let $X_i$ denote an N × 1 vector whose nth element is the fraction of agent i’s wealth invested in stock n.
5. $a_i = A_i*W_{0i}$ , Both $a_i$ and $b_i$ are known constants.

How is the last line of formula derived? (the last Part with VAR (r))?

Answer 1

It uses the moment-generating function of a multivariate normal distribution.