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)

enter image description here

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:

General model for Taylor Approximation

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 :

enter image description here

Now, I understand that the parameter enter image description here 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 : enter image description here 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 enter image description here is developped for even moments (ie: second-order in this case) as :

enter image description here

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,



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