# Utility-based portfolio optimization

I think I can't get the idea of optimization based on utility. For some reasons, I should choose one of several common utility functions (exponential, isoelastic function and some others). Obviously, in it's formal representation as 1-exp(-a*Profit)- a (risk-aversion) doesn't make any sense in maximization. So, should any function be expanded by Taylor series, for example until 4th moment of distribution? Like this:

Would be glad if you could help me with this. I'm especially interested in isoelastic function. Thank you in advance!

• It is the expected utility that you maximize, not the utility itself. In general you will have an integral of the product of utility times probability. That can be complicated, so people sometimes do a Taylor expansion in terms of moments, like you are suggesting. Another way is a Monte Carlo simulation where you generate 1000 utilities of random returns and take the average. – noob2 Apr 26 at 20:51
• Thank you! I got the idea with the expectation, now it is pretty clear. Can you please expand more on Monte-Carlo method for this problem? What do you mean by random returns? In my study I have simulated 10,000 returns from copula-garch method for 50 weeks. Can I somehow use them here? – Gcube Apr 26 at 21:36