Optimal investment under capital gains taxes
Alexander Dimitrov, Christoph Kühn
TL;DR
The paper studies portfolio optimization under capital gains taxes with limited use of losses (LUL) in discrete time, showing that NA alone may not ensure closedness of attainable terminal wealth due to tax-induced nonlinearity. It introduces the stronger no unbounded non-substitutable investment with bounded risk (NUIBR) condition, proving that this ensures closedness in probability of the attainable wealth set. Using this closedness, it establishes existence of optimal strategies for utility maximization under fairly weak utility assumptions, and extends foundational results from frictionless markets to taxed settings with short-selling constraints. A by-product includes a generalized bound for integrable utilities and a constructive dominance argument, along with a demonstration that closedness holds automatically in finite probability spaces. The work links tax-based wealth dynamics to transaction-cost methodologies, providing a rigorous framework for optimal investment under realistic tax rules.
Abstract
We generalize classical results on the existence of optimal portfolios in discrete time frictionless market models to models with capital gains taxes. We consider the realistic but mathematically challenging rule that losses do not trigger negative taxes but can only be offset against potential gains in the future. Central to the analysis is a well-known phenomenon from arbitrage-free markets with proportional transaction costs that does not exist in arbitrage-free frictionless markets: an investment in specific quantities of stocks that is completely riskless but may provide an advantage over holding money in the bank account. As a result of this phenomenon, on an infinite probability space, no-arbitrage does not imply that the set of attainable terminal wealth is closed in probability. We show closedness under the slightly stronger {\em no unbounded non-substitutable investment with bounded risk} condition. As a by-product, we provide a proof that in discrete time frictionless models with short-selling constraints, no-arbitrage implies that the set of attainable terminal wealth is closed in probability -- even if there are redundant stocks.