Num\'{e}raire-invariant preferences in financial modeling

TL;DR

This paper establishes an axiomatic basis for numéraire-invariant preferences in financial markets, providing a unified framework for static and dynamic environments and solving related investment problems.

Contribution

It introduces a novel axiomatic foundation for preferences invariant to numéraire changes, including a dynamic representation and explicit solutions for investment and consumption issues.

Findings

Preferences characterized by relative rate of return comparison

Expected logarithmic utility representation with transitivity

Explicit solution to investment-consumption problem with random horizon

Abstract

We provide an axiomatic foundation for the representation of num\'{e}raire-invariant preferences of economic agents acting in a financial market. In a static environment, the simple axioms turn out to be equivalent to the following choice rule: the agent prefers one outcome over another if and only if the expected (under the agent's subjective probability) relative rate of return of the latter outcome with respect to the former is nonpositive. With the addition of a transitivity requirement, this last preference relation has an extension that can be numerically represented by expected logarithmic utility. We also treat the case of a dynamic environment where consumption streams are the objects of choice. There, a novel result concerning a canonical representation of unit-mass optional measures enables us to explicitly solve the investment--consumption problem by separating the two aspects of investment and consumption. Finally, we give an application to the problem of optimal num\'{e}raire investment with a random time-horizon.