# Extremal Behavior of Long-Term Investors with Power Utility

## TL;DR

This paper analyzes the long-term behavior of investors with power utility in a Bayesian market, showing they act optimistically or pessimistically depending on the utility coefficient, aligning with the best or worst drift scenarios.

## Contribution

It proves the conjectured long-term behavior of optimal strategies for power utility investors in Bayesian markets, clarifying their optimism or pessimism.

## Key findings

- Positive utility coefficient leads to optimistic long-term behavior.
- Negative utility coefficient results in pessimistic long-term behavior.
- Investor behavior aligns with best or worst drift scenarios depending on utility sign.

## Abstract

We consider a Bayesian financial market with one bond and one stock where the aim is to maximize the expected power utility from terminal wealth. The solution of this problem is known, however there are some conjectures in the literature about the long-term behavior of the optimal strategy. In this paper we prove now that for positive coefficient in the power utility the long-term investor is very optimistic and behaves as if the best drift has been realized. In case the coefficient in the power utility is negative the long-term investor is very pessimistic and behaves as if the worst drift has been realized.