The numeraire property and long-term growth optimality for drawdown-constrained investments

TL;DR

This paper studies long-term investment strategies under drawdown constraints, introducing the numeraire property, and shows how constrained portfolios relate to unconstrained ones over long horizons.

Contribution

It introduces the numeraire property for drawdown-constrained portfolios and characterizes their long-term growth optimality and explicit construction.

Findings

Existence and uniqueness of drawdown-constrained numeraire portfolios.

Explicit model-independent transformation for long-term constrained portfolios.

Asymptotic equivalence of constrained and unconstrained growth-optimal strategies.

Abstract

We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained numeraire portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of numeraire strategies on finite horizons.