MF question 1: just math...
 

this one is just a clarification of something i dont get b/c i dont remember exactly how to reduce taylor expansions precisely. here's one example from my textbooks (Stochastic calculus for finance II and Introduction to Economics and Mathematics of Financial Markets):

let U be a utility function

CE is a certainty equivalent. this is thhe amount of money an investor is willing to take now to forgo a random positive investment Y.

so the utility of the certainty equivalent must be equal to the expected value of the investment Y. U(CE)=E[Y]. and we see that E[Y] - CE[Y] is the expected return over and above the cash amount, or the risk premium.

now denote uy=E[Y] and U(Y)=utility of Y. now we use taylor's expansion of U(Y) to get:

U(Y) ~~ U(uy) + (Y-uy)*U'(uy) + 1/2(Y-uy)^2*U''(uy)

then the book says "taking expectations:"

SIDENOTE: haha, i noted this as a problem to come back to...looked at it again and then t yping this out i see the easy answer...but i'll keep going so that anybody else interested can take a shot.

E[U(Y)]~~ U(uy) + 1/2 Var[Y]U''(uy)

so what happened to the middle term in the first expansion?

for those who dont immediately see it, the variance of f(x) is E[(x-f(x))^2] so the 2nd term above is clear.

Barron