# Examples of risk-seeking utility functions?

In the past, most literature assumed a risk-averse investor to model utility preferences. This includes the CRRA and CARA utility functions.

In recent papers, researchers state that investors may be actually risk-seeking, based on e.g. various studies on option pricing (options provide high leverage and therefore trade at a premium).

Can someone give examples of risk-seeking utility functions?

Thank you.

• The Prospect Theory utility or the Friedman-Savage utility are 2 ideas of how the utility curve would have to be shaped to allow some risk loving and some risk aversion lh6.ggpht.com/_1wtadqGaaPs/TG4xpo3FM-I/AAAAAAAATdA/cmGSTm9zeeQ/… – noob2 Oct 4 '16 at 13:59
• @noob2 Please state the functional form of the utility in your answer. – emcor Oct 4 '16 at 14:04
• Im not sure if thats what youre asking but what about the following. Say we have a market with mean vector $\mu \geq 0$ and a variance-covariance matrix $\Sigma$ that exists. Then, given a weight vector $x$, you could have $U(x) = x^T\mu$. That would be the "return is all that matters" utility funciton. You could also take $U(x) = x^T \Sigma x$, if you are only satisfied with lots of risk (notice the missing minus sign compared to some widely used variants!) – vanguard2k Oct 5 '16 at 8:16
• @vanguard2k $U(x)=x\mu$ is a linear function, it would represent risk-neutrality. $U(x)=x^2$ is therefore not risk-neutral, can you explain on the minus sign? Please also give a definition of risk-seeking in terms of the sign of the derivatives of the utility function. – emcor Oct 5 '16 at 10:42
• @emcor As for the second derivatives, we have $0$ for the first (risk-neutrality), and $\Sigma$ (or $2\Sigma$) for the Hessian matrix of the second one which is positive definite so $U$ is convex (on a convex set). A very common utility function is $U(x) = \mu^T x - \frac{C}{2} x^T \Sigma x$ from Markowitz optimization with the risk-aversion parameter $C$. Its Hessian is $H=-C\Sigma$. Now if $C>0$ have a concave utility function, since $H$ is negative definite and thus $U$ is risk-averse, if $C<0$ we have a convex utilityfunction since $H$ is positive definite and $U$ is risk-seeking. – vanguard2k Oct 6 '16 at 11:50