Ok guys, I'm reading a book and I'm not getting quite well a concept. If I have to expand $U'(Y_0(1+r_i))$ around $Y_0(1+r_f)$, why I get this:
$\mathbb{E}[U'(Y_0(1+r_i))(r_i-r_f)]=U'[Y_0(1+r_f)]\mathbb{E}(r_i-r_f)+U''[Y_0(1+r_f)]\mathbb{E}(r_i-r_f)^2Y_0$
Any step-by-step explanation would be greatly appreciated, since I can't get the same result by applying Taylor. Thanks in advance!
As far as I understand what is written here:
If $f(x) = U'(x)$ then
$E[\ f(x)(a-b) \ ] =$/expanding function under expectation/ = $E[(f(x_0)+f'(x_0)(x-x_0))(a-b)] = E[f(x_0)(a-b) + f'(x_0)(x-x_0)(a-b)]= f(x_0)E[(a-b)]+f'(x_0)E[(x-x_0)(a-b)]$;
$$x = Y_0(1+r_i), \ x_0 = Y_0(1+r_f), \ f(x_0) = U'(Y_0(1+r_f)),$$ $$f'(x_0)=U''(Y_0(1+r_f))$$ $$(x-x_0) = Y_0(1+r_i - 1-r_f)=Y_0(r_i-r_f)$$ $$E[(x-x_0)(a-b)]=E[(r_i-r_f)^2\cdot Y_0]$$