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 :
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,