Evolution of the Arrow–Pratt measure of risk-tolerance for predictable forward utility processes

We study the evolution of the Arrow–Pratt measure of risk-tolerance in the framework of discrete-time predictable forward utility processes in a complete semimartingale financial market. An agent starts with an initial utility function, which is then sequentially updated forward at discrete times under the guidance of a martingale optimality principle. We mostly consider a one-period framework and first show that solving the associated inverse investment problem is equivalent to solving some generalised integral equations for the inverse marginal function or for the conjugate function, both associated with the forward utility. We then completely characterise the class of forward utility pairs that can have a time-invariant measure of risk-tolerance and thus a preservation of preferences in time. Next, we show that in general, preferences vary over time and that whether the agent becomes more or less tolerant to risk is related to the curvature of the measure of risk-tolerance of the forward utility pair. Finally, to illustrate the obtained general results, we present an example in a binomial market model where the initial utility function belongs to the SAHARA class, and we find that this class is analytically tractable and stable in the sense that all the subsequent utility functions belong to the same class as the initial one.