Numerical Solution of Dynamic Quantile Models

Abstract

This paper studies dynamic programming for quantile preference models, in which the agent maximizes the stream of the future τ-quantile utilities, for τ∈(0,1). We suggest numerical methods, based on value function iterations, for solving the quantile recursive dynamic programming, and computing value and policy functions. In addition, we extend theoretical results to allow the dynamic quantile model to have a finite-horizon, instead of infinite-horizon. To illustrate the methods, we use an intertemporal consumption quantile model that has an explicit closed form solution for both the value and policy functions. Based on this example, we assess the accuracy of the numerical methods by computing and comparing theoretical and numerical value and policy functions, for several combinations of the parameters – discount factor, elasticity of intertemporal substitution, and risk attitude, which is measured by the quantile. Results document evidence that the suggested algorithm provides numerical solutions that are very close to theoretical counterparts, and also illustrate the usefulness and practicality of the proposed methods.