Topic and Objectives
In economic models, decision making of agents is described by utility functions. Representative agent models, that have dominated macroeconomics for the last three decades, usually use a von Neumann-Morgenstern utility function. The decision-maker who is faced with probabilistic outcomes of different choices will behave as if he/she is maximizing the expected value of the potential outcomes. In addition, most models assume that agents have access to a financial market that is complete. Both these specifications are potential reasons for the fact that the classical framework has shown to being unable to explain several empirical findings about asset prices. Economists have responded to these challenges by assuming that agents have more general preferences (e.g. recursive preferences) and/or by postulating more involved asset or endowment processes (e.g. jumps, disasters, unspanned diffusion) that lead to incomplete financial markets. In a representative agent economy, the stochastic discount factor is the key ingredient. This concept induces a pricing rule that determines all asset prices in an economy. It is thus of crucial importance that the agent's utility can be described in a tractable way. In every continuous-time model, this utility satisfies a certain partial differential equation that can be derived by applying a stochastic representation theorem for expectations. For recursive preferences, this can be reduced to an equation belonging to a particular class of semi-linear partial differential equations. Such equations are inherently difficult to solve and, in general, it is not even clear whether they admit (unique smooth) solutions. So far, researchers have usually resorted to approximation techniques of unclear precision and considered affine frameworks. In Working Paper No. 17 (published in Journal of Economic Theory), we establish a convergence theorem that shows that discrete-time recursive utility, converges to stochastic differential utility in the continuous-time limit of vanishing grid size. In Working Paper No. 52 we provide a significant contribution to the extensive literature on dynamic incomplete-market portfolio theory. This area is concerned with an agent's consumption-portfolio choice problem where returns are not necessarily independent and identically distributed. We study an incomplete-market consumption-portfolio problem that nests several classical frameworks. In contrast to the existing literature, we do not restrict our analysis to affine models but allow for recursive preferences. We reduce the Bellman equation to a partial differential equation that belongs to the same class as the above-mentioned equation in asset pricing. Researchers have so far relied on approximative methods in this context as well, although in general not even the issueof existence has been resolved.
- For possibly non-affine models, we prove the existence of a solution and develop a fast and accurate numerical method to compute this solution. Our scheme solves the nonlinear partial differential equation by iteratively solving certain linear partial differential equations. We also derive worst-case bounds for the accuracy of our methodology. Therefore, our results provide a solid basis for future research in asset pricing with recursive preferences.
- We provide a verification theorem demonstrating that a suitable smooth solution of the reduced Bellman equation is also the solution to the consumption-portfolio problem. Following the same agenda as outlined above, we then establish existence of a solution and construct this solution by fixed point arguments. Again, our numerical method provides a fast and accurate way of calculating theinvestor's indirect utility and optimal strategies. Our results thus also establish a tractable approach to incomplete-market consumption-portfolio choice problems with recursive preferences.
|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|
|17||Holger Kraft, Frank Thomas Seifried||Stochastic Differential Utility as the Continuous-Time Limit of Recursive Utility||2013||Financial Markets||stochastic differential utility, recursive utility, convergence, backward stochastic differential equation|
|52||Holger Kraft, Thomas Seiferling, Frank Thomas Seifried||Optimal Consumption and Investment with Epstein-Zin Recursive Utility||2014||Financial Markets||consumption-portfolio choice, asset pricing, stochastic differential utility, incomplete markets, fixed point approach, FBSDE|