Recursive Overbetting of a Satellite Investment Account

Abstract

This paper builds a core-satellite model of semi-static Kelly betting and log-optimal investment. We study the problem of a saver whose core portfolio consists in unlevered (1x) retirement plans with no access to margin debt. However, the agent has a satellite investment account with recourse to significant, but not unlimited, leverage; accordingly, we study optimal controllers for the satellite gearing ratio. On a very short time horizon, the best policy is to overbet the satellite, whereby the overriding objective is to raise the aggregate beta toward a growth-optimal level. On an infinite horizon, by contrast, the correct behavior is to blithely ignore the core and optimize the exponential growth rate of the satellite, which will anyways come to dominate the entire bankroll in the limit. For time horizons strictly between zero and infinity, the optimal strategy is not so simple: there is a key trade-off between the instantaneous growth rate of the composite bankroll, and that of the satellite itself, which suffers ongoing volatility drag from the overbetting. Thus, a very perspicacious policy is called for, since any losses in the satellite will constrain the agent's access to leverage in the continuation problem. We characterize the optimal feedback controller, and compute it in earnest by solving the corresponding HJB equation recursively and backward in time. This solution is then compared to the best open-loop controller, which, in spite of its relative simplicity, is expected to perform similarly in practical situations.