### Abstract

This article examines how to quantify and optimally utilize the beneficial effect that capital loss harvesting generates in a taxable portfolio. We explicitly determine the optimal initial asset allocation for an investor who follows the continuous time dynamic trading strategy of Constantinides (1983). This strategy sells and re-buys all stocks with losses, but is otherwise passive. Our model allows the use of the stock position's full purchase history for computing the cost basis. The method can also be used to rebalance at later times. For portfolios with one stock position and cash, the probability density function for the portfolio's state corresponds to the solution of a 3-space+1-time dimensional partial differential equation (PDE) with an oblique reflecting boundary condition. Extensions of this PDE, including to the case of multiple correlated stocks, are also presented. We detail a numerical algorithm for the PDE in the single stock case. The algorithm shows the significant effect capital loss harvesting can have on the optimal stock allocation, and it also allows us to compute the expected additional return that capital loss harvesting generates. Our PDE-based algorithm, compared with Monte Carlo methods, is shown to generate much more precise results in a fraction of the time. Finally, we employ Monte Carlo methods to approximate the impact of many of our model's assumptions.

Original language | English (US) |
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Pages (from-to) | 291-329 |

Number of pages | 39 |

Journal | Applied Mathematical Finance |

Volume | 18 |

Issue number | 4 |

DOIs | |

State | Published - Sep 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Finance
- Applied Mathematics

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## Cite this

*Applied Mathematical Finance*,

*18*(4), 291-329. https://doi.org/10.1080/1350486X.2010.513499