Last Uncertainty Wednesday, we saw how diminishing marginal utility of wealth provides an explanation of risk aversion via Jensen’s inequality. Why would it be then that lots of people seem to like small gambles, like a game of poker among friends. One possible explanation is that the utility function is locally convex around your current endowment. So this would look something like the following:

In the immediate area around the endowment (marked with dotted lines for two different levels) the utility function is convex, but for larger movements it is concave. 

In the convex area someone would be risk seeking. Why? Well because Jensen’s inequality now gives us

U[EV(w)] ≤  EV[U(w)]

Again, the left hand side is the utility of the expected value of the wealth, whereas the right hand side is the expected utility, meaning the expected value of the utility. Now the inequality says that someone would prefer an uncertain amount over a certain one. Here is a nice illustration from Wikipedia:

We see clearly that the Certainty Equivalent (CE) is now larger than the expected value of wealth, meaning the Risk Premium (RP) works the other way: in order to make a risk seeker as well off as accepting the bet, you have to pay them more than the expected value.

Next Uncertainty Wednesday we will look more at how incredibly powerful convexity is in the face of uncertainty. 

Last Uncertainty Wednesday, we saw how diminishing marginal utility of wealth provides an explanation of risk aversion via Jensen’s inequality. Why would it be then that lots of people seem to like small gambles, like a game of poker among friends. One possible explanation is that the utility function is locally convex around your current endowment. So this would look something like the following:

In the immediate area around the endowment (marked with dotted lines for two different levels) the utility function is convex, but for larger movements it is concave. 

In the convex area someone would be risk seeking. Why? Well because Jensen’s inequality now gives us

U[EV(w)] ≤  EV[U(w)]

Again, the left hand side is the utility of the expected value of the wealth, whereas the right hand side is the expected utility, meaning the expected value of the utility. Now the inequality says that someone would prefer an uncertain amount over a certain one. Here is a nice illustration from Wikipedia:

We see clearly that the Certainty Equivalent (CE) is now larger than the expected value of wealth, meaning the Risk Premium (RP) works the other way: in order to make a risk seeker as well off as accepting the bet, you have to pay them more than the expected value.

Next Uncertainty Wednesday we will look more at how incredibly powerful convexity is in the face of uncertainty.