Last Uncertainty Wednesday, I introduced Jensen’s Inequality. I mentioned briefly that it explains a lot of things and today we will look at the first one of these, which goes by the name of risk aversion. This is simply economists way of saying that most people prefer a smaller guaranteed payment over a large but uncertain one. We will now see that this follows directly from diminishing marginal utility of money via Jensen’s inequality.
So what is this “diminishing marginal utility” of money? Well it is generally assumed that the more money you make, the less an additional say 100 dollars will mean to you. This seems, for most people anyhow, a pretty safe assumption. If you are currently making $1,000 per month then getting an extra $100 per month let’s you have a lot more benefit than if you are already making $10,000 per month. But of course you are still somewhat better off making $10,100 per month.
Putting that together would suggest a function that’s increasing but at a decreasing rate and that’s exactly a concave function. Since we are talking about utility here, we will use U(w) to denote this with w standing for wage or wealth. Then U(w) being concave immediately get us the following from Jensen’s inequality:
U[EV(w)] ≥ EV[U(w)]
The left hand side is the utility of the expected value of the wage, whereas the right hand side is the so-called expected utility. So anyone with diminishing marginal utility will prefer say $1,000 per month guaranteed over the possibility of $1,100 per month with 50% probability and $900 per month with 50% probability (expected value also $1,000). That is known as risk aversion. The following image from Wikipedia nicely illustrates the situation:
In the image we can also graphically see two values: the so-called certainty equivalent and the risk premium. The certainty equivalent (CE) is the amount that would make the person indifferent between the risky payoff and the certain payoff. We see that the certainty equivalent is less than the expected value.
Meaning in the example above, someone with risk aversion would in fact accept less then $1,000 (the expected value) with certainty and still feel as good as having the uncertainty of $1,100 with 50% and $900 with 50%. The difference between the certainty equivalent and the expected value is known as the risk premium (RP). That is the amount someone would be willing to pay to not to face the uncertainty.
So if you are currently making $1,000 per month and your employer says that next month if the company does well you will make $1,100 but if it does poorly you will make $900, then a risk averse individual would be willing to pay some money, say $20 to make $1,000 with certainty (which after paying the risk premium will be $980). If you read my series on insurance fundamentals you will recall that this is the basis for the existence of insurance.
Next Wednesday we will talk about risk seeking and get into the ideas of convex tinkering and antifragility.
