Last week in Uncertainty Wednesday, I introduced functions of random variables as the third level in measuring uncertainty. Today I will introduce a beautiful result known as Jensen’s inequality. Let me start by stating the inequality:
f[EV(X)] ≤ EV[f(X)] where f is a convex function
In words, if we apply a convex function to the expected value of a random variable, then we get a lower value than if we take the expected value of the same function of the random variable. This turns out to be an extremely powerful result.
Jensen’s inequality explains, among other things, the existence of risk seeking and risk aversion (via the curvature of the utility function), why options have value and how we should structure (corporate) research. I will go into detail on these in future Uncertainty Wednesdays. Today, I want to show this wonderful picture from Wikipedia, which gives a visual intuition for the result:
And before we get into applications and implications of the inequality, I should mention for completeness that the inverse holds for concave functions, meaning
g[EV(X)] ≥ EV[g(X)] where g is a concave function
Next Wednesday we will look at utility functions and risk seeking / risk aversion as explained by these inequalities.
