# How to derive Expected utility approximation with power function

My question concern how to derive the expected utility of a power function in the following model:

I have two normally distributed risky assets X and Y and a risk-free asset B, for which :

rA = 0.5y + 0.5x

I have the following allocation problem in which I invest W0 = 100 \$ between A and B :

mrA + (100 − m)rB ∼ N

Then, for a negative exponential utility I have

U(w) = -exp(-θW) where W = mrA + (100 − m)rB and θ = A(W)

By doing the FOC, I can find m* which is the EU Maximizing weight to invest in the risky portfolio A if θ is known.

Now, I want to do the same thing with a power function so I have tried to understand how to derive the exponential function and I have found the proof in the paper of Garlappi and Skoulakis (2008).

The general model to approximate E(U(R)) around the center cp of the Mth order is:

Because we are working with normal distributions, odd moments are null and we are only interest by mean and variance (only the two first).

For exponential we have :

Now, I understand that the parameter is only the general model for the mth order derivative. On this formula, there are two things that I don't understand quite well:

(1) When I compute (1) for a 2nd order approximation, I obtain : but I don't see how to continue the proof in order to find the final answer and from where the summation term from k=0 to K comes?

(2) The term is developped for even moments (ie: second-order in this case) as :

i) Here, I cannot figure out what is the meaning of the parameter K and how to derive the proof. Even though I have found on Wikipédia that : (2k-1)!! = [(2k)! / (2k)!!] = [(2k)! / (2^n) * n!] but this is not the same thing because of the numerator that is different...

In brief, I hope that by understanding rigorously the exponential derivation, I will be able to apply as well for the power function in my little ptf problem.

Thanks in advance for this great help,

Gabriel