Utility Functions and the Theory of Choice

When payouts are deterministic, investor preferences are easy to determine, and if the payout of asset A is twice the payout of asset B, then the price of A is twice the price of B. Now, when payouts are random, determining the criteria of choice between two investments is more complex, and if the expected payout of asset A is twice the expected payout of asset B, the price of A is not necessarily twice the price of B. Furthermore, when choosing between two assets, beyond the mathematical expectations of the payouts, the variances and the whole distribution of the payouts are usually considered. For a random payout X, this analysis leads us to look not only at its expectation E(X) but at something of the form E(u(X)), where u is called a utility function Utility function . In this context, E(u(X)) is the objective to maximise when choosing between different random payouts X. The function u is chosen to reflect the preferences of the investors and it will be shown in this chapter that the appetite or aversion to risk is linked to the convexity or concavity of the function u. An extensive literature exists in economics about utility functions and their applications. John von Neumann and Oskar Morgenstern proved that, from a mathematical point of view, individuals whose preferences satisfy four particular axioms have a utility function. Gerard Debreu (Nobel price in Economics 1983) and Kenneth Arrow (Nobel price in Economics 1972) produced additional landmark results, defining the equilibrium in an economy where agents act upon some utility functions. This chapter is only a very brief introduction to the topic, and is included to make the link with the mean-variance criteria which is used in the rest of the book.