Forward utility and market adjustments in relative investment-consumption games of many players

TL;DR

This paper develops a forward performance process framework for many-player and mean field investment-consumption games, providing explicit solutions and insights into how competition influences consumption preferences and risk-tolerance.

Contribution

It introduces a novel FPP-based model for multi-agent investment and consumption, deriving closed-form solutions and analyzing the impact of market competition on preferences.

Findings

Closed-form solutions for many-player and mean field games.

Market competition alters consumption perception non-trivially.

FPP framework disentangles risk-tolerance and EIS like Epstein-Zin preferences.

Abstract

We study a portfolio management problem featuring many-player and mean field competition, investment and consumption, and relative performance concerns under the forward performance processes (FPP) framework. We focus on agents using power (CRRA) type FPPs for their investment-consumption optimization problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin "market-risk relative consumption preference". The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle her risk-tolerance and elasticity of intertemporal substitution (EIS) just like Epstein-Zin preferences under recursive utility framework and unlike the classical utility theory one. This, in turn, allows a finer analysis on the agent's consumption "income" and "substitution" regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent's perception of consumption in a non-trivial manner. We provide numerical illustrations of our results.