Policonomics » LPsection » Risk and uncertainty I: Arrow-Pratt measures

Risk and uncertainty I: Arrow-Pratt measures

Summary

Everyone has to make decisions, but it is not always clear to us what outcomes can derive from these decisions. When this happens, we say we are making decisions in situations under risk or uncertainty. In this LP we learn about risk and uncertainty. We see how risk can be analysed by using expected utility instead of expected value, and how different kind of people will behave differently when facing risk.

Risk and uncertainty:

Expected utility:

Risk aversion analysis:

Risk aversion
Measuring aversion

It is sometimes important to know how averse to risk a certain individual is. To this effect there are a set of tools to measure risk in a quantitative way. The most common and frequently used measure of risk aversion are the Arrow-Pratt measures of absolute and relative risk-aversion. Named after John W. Pratt’s paper “Risk Aversion in the Small and in the Large”, 1964, and Kenneth Arrow’s “The Theory of Risk Aversion”, 1965, these are the measures:

Arrow-Pratt measure of absolute risk aversion:

Arrow-Pratt measure of relative risk aversion:

Where x is the payoff of a given lottery and U(x) the utility derived from that payoff.

In this LP we’ve started by seeing what the difference between risk and uncertainty is. Then, we learned about expected utility, how Daniel Bernoulli coined the term when solving the St. Petersburg paradox and how John von Neumann and Oskar Morgenstern developed a whole theory around it. Finally, using the expected utility theory, we learned how to analyse and measure risk aversion. In the next LP on risk and uncertainty we see other ways to analyse risk, especially Prospect theory, developed by Daniel Kahneman, who completely changed the analysis of decision making. We also see ways to analyse uncertainty.

Risk and uncertainty I: Risk aversion