Prove the approximation for risk premium

Question

Exercise comes from Bowers' "Actuarial Mathematics". I'm self-studying and have virtually no clue how to approach it. We make no assumptions about either utility function or distribution of $X$.

Let $w$ - wealth level, $G$ - premium, $X$ - random loss, $u$ - utility function of individual. We know that: $$u(w-G)=E(u(w-X))$$ $$E(X)=\mu$$ Use the approximations:

$$u(w-G)\approx u(w-\mu)+(u-G)\mu'(w-\mu)$$

$$u(w-x)\approx u(w-\mu)+(u-x)\mu'(w-\mu)+\frac{1}{2}(\mu-x)^2\mu''(w-\mu)$$

and derive the following approximation for G:

$$G \approx \mu - \frac{1\mu''(w-\mu)}{2\mu'(w-\mu)}\sigma^2 $$

Answer 1

HINT: In the topmost equation $u(w-G)=E(u(w-X))$, replace the LHS with the first approximation. For the RHS of the topmost equation, calculate $E(u(w-X))$ using the second approximation (details below):

In the second approximation, put the random variable $X$ in place of $x$, and take expectations. The term involving $\mu-X$ has expectation zero, so it gets knocked out.

I think we can assume that $E(X-\mu)^2=\sigma^2$.

Now solve for $G$.

Note that your notation is messed up. In some of your equations you have written $\mu$ instead of $u$, and vice versa.