Merton Problem with Taxes: Characterization, Computation, and Approximation

Abstract

Abstract. We formulate a computationally tractable extension of the classical Merton optimal consumptioninvestment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583-616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779-1801. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation.

Who should sell stocks?

“…We focus on two common methods: With the weighted average cost basis (adopted in Canada and Italy, and one of the options in the United States), the cost (and hence the tax basis) of each share sold is the average cost of all shares previously acquired (cf. Dammon et al., ; Tahar et al., ). With the specific share identification , also an option in the United States, an investor specifies which shares are being sold, freely choosing among lots with different costs.…”

“…For numerical studies compare, for example, Dybvig and Koo (), Dammon, Spatt, and Zhang (), DeMiguel and Uppal (), Gallmeyer, Kaniel, and Tompaidis (), Tahar, Soner, and Touzi (), and Dai, Liu, Yang, and Zhong (); also compare Cai, Chen, and Dai () for an asymptotic analysis of a particular model.…”

Optimal Asset Allocation for Passive Investing with Capital Loss Harvesting

“…Gallmeyer et al, 2006) and geometric Brownian motion models (see e.g. Ben Tahar et al, 2007Tahar et al, , 2010. Of course, for the case of American mutual funds, it is preferable not to use the average cost basis, because it is always better to sell mutual fund holdings with the highest purchase price first while keeping the low purchase price holdings as long as possible; in other words, using the full purchasing history provides for a superior, though computationally more challenging, strategy.…”

Modeling Capital Gains Taxes for Trading Strategies of Infinite Variation

“…A crucial observation is the following. If the investor buys, e.g., 100 General Motors stocks at time t 1 , another 100 at time t 2 , and sells 100 at time t 3 , it matters which of the stocks she sells, as in general α · 100(S t 3 − S t 2 ) = α · 100(S t 3 − S t 1 ). When the portfolio is liquidated at some date t 4 the difference of the accumulated tax payments disappears because * The authors thank Christoph Czichowsky and Teemu Pennanen for fruitful discussions and an anonymous referee for valuable comments.…”

“…Although in practice capital gains taxes may be the most relevant market friction, there is only little literature on capital gains taxes in advanced continuous time models. Ben Tahar, Soner, and Touzi [3,4] solve the Merton problem with proportional transaction costs and a tax based on the average of past purchasing prices. This approach has the advantage that the optimization problem is Markov with the one-dimensional tax basis as additional state variable.…”