# Certainty Equivalents and Risk Premiums in Expected Utility Theory for Asymmetric Distributions

I want to calculte risk-premiums in order to assess how much risk-averse customers would be willing to pay for an insurance against an uncertain loss modeled by a random variable $X$. How would a risk-premium be calculated, if $X$ does not follow a normal distribution? My reading of the literature is as follows:

1) Expected utility theory: Assumes CARA (CRRA) like utility functions with well behaved properties that are more or less agreed to model the behavior of rational risk-averse decision makers by satisfying certain axioms. The risk-aversion of decision makers can be modelled by Arrow-Pratt with $\alpha=R(X)=-\frac{u''(x)}{u'(x)}$. Using Arrow Pratt, one can calculate a risk-premium $\pi$ if $X$ is normally distributed such that $\pi(X)=\frac{\alpha}{2}\sigma^2(X)$ which derives from the formula of the certainty equivalent.

2) Downside risk measures from finance: Since losses (or returns) may not be normally distributed, finance has developed other measures to capture the concept of risk. One of these measures is the semi-variance as a special case of lower partial moments, which is very similar to the idea of mean-variance principle as outlined above: $SV=E((max[0,E(X)-X])^2)$.

Now the problem is as follows:

Aiming to calculate a risk-premium I can either follow utility theory. If $X$ is normally distributed we get a well contained formula to calculate the risk-premium $\pi(X)$. If not, then what? Would the certainty equivalent be calculated based on numerical integrat? If so, how would this look like for an arbitrary distribution of $X$?

On the other hand, I could follow a down-side risk measure approach: Given that I use semi-variance as a concept to measure risk, how could you account for the risk-aversion of customer in order to derive a risk-premium? I would just weight the semi variance with a factor $\tau^{~}SV$ (with $\tau$ being some kind of a proxy for risk-aversion) this wouldn't do the trick, would it? Essentially it would probabily not satisfy the expected utility axioms such as CARA or CRRA do correct?

How to approach this problem then? Are there any alternative strategies that I have overlooked?

Now I have also found this post on the matter which would give a solution to my problem - as far as I understand - since it allows $X$ to have a non-zero expected value? This is due to the fact that the taylor approximation is - as it says - only an approximation of the risk premium with regard to various moments of the given distribution $X$. This also explains why the risk-premium can be correctly calculated for the normal distribution which is completely defined trough first and second moment.
This brings me to my last question: Considering the risk premium for non-zero expected values this would according to the post listed above also be approximated by the formula $\pi \approx \frac{1}{2}R(w)\sigma^{2}_X$? And the approxmation error for non-normal distributions is then depending on the question how well first and second moment capture the true nature of a distribution?