This project is concerned with the convergence of (discrete-time) recursive utility for vanishing grid size (in time dimension). From Kraft and Seifried (2013) it is well-known that in a general semi-martingale setting recursive utility converges to stochastic differential utility as introduced by Duffie and Epstein (1992b) if the certainty equivalent is of standard Kreps-Porteus type (linear certainty equivalents). Our project is concerned with studying nonstandard (non-linear) certainty equivalents such as divergence preferences (robust CEs), second-order expected utility, source-dependent risk aversion, or exotic CEs (loss aversion, weighted utility, Chew-Dekel). We expect that in some cases the non-linear effects do not disappear in the limit, which can lead to very interesting new insights on continuous-time preference functional. On the other hand, there might be cases (such as the cases suggested by Skiadas (2013a, b)) where the non-linear effects vanish in the limit. This is also important, since it would show that the corresponding preference specifications are not stable in the time dimension.
Studying preference functionals that are not nested by the expected utility paradigm is very relevant since the drawbacks of expected utility become apparent in an important area of research: the theory of equilibrium asset pricing. In particular, the implications of expected utility are known to be incompatible with various stylized facts in empirical findings; for instance, the excess return of stocks implied by expected utility is much too high for realistic risk aversion parameters (”equity premium puzzle”). In the last 25 years, recursive preferences have therefore become a key ingredient in the asset pricing literature. A prominent recent contribution, the so-called long-run risk approach of Bansal and Yaron (2004), combines a long-run risk factor affecting consumption growth with recursive preferences. This model is among the few that are capable of generating realistic risk premia and other stylized facts. Building on the long-run risk approach, disaster models such as Gabaix (2012) and Wachter (2013) are based on recursive utility and the possibility of persistent shocks to aggregate consumption. Despite their empirical relevance, the literature lacks rigorous results relating discrete-time recursive utility to stochastic differential utility especially if certainty equivalents are non-linear. Our project will close this crucial gap.
So far we have studied an asset pricing framework with recursive preferences and unspanned risk. We show that the value function of the representative agent can be characterized by a specific semi-linear partial differential equation. We have developed a novel approach that rigorously constructs the solution by a fixed point argument. We prove that under regularity conditions a solution exists and establish a fast and accurate numerical method to solve consumption-portfolio and asset pricing problems with recursive preferences and unspanned risk.
|Holger Kraft, Thomas Seiferling, Frank Thomas Seifried||Optimal Consumption and Investment with Epstein-Zin Recursive Utility|
Finance and Stochastics
|2017||Financial Markets||consumption-portfolio choice, asset pricing, stochastic differential utility, incomplete markets, fixed point approach, FBSDE|
|Holger Kraft, Frank Thomas Seifried||Stochastic Differential Utility as the Continuous-Time Limit of Recursive Utility|
Journal of Economic Theory
|2014||Financial Markets||stochastic differential utility, recursive utility, convergence, backward stochastic differential equation|