The dynamic programming equation for the problem of optimal investment under capital gains taxes

This paper considers an extension of the Merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes. We derive the dynamic programming equation in the sense of constrained viscosity solutions. We next introduce a family of functions, which converges to our value function uniformly on compact subsets, and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation. In particular, this result justifies the numerical results reported in the accompanying paper by the authors

Modeling continuous-time financial markets with capital gains taxes

We formulate a model of continuous-time financial market consisting of a bank account with constant interest rate and one risky asset subject to capital gains taxes. We consider the problem of maximizing expected utility from future consumption in infinite horizon. This is the continuous-time version of the model introduced by Dammon, Spatt and Zhang [11]. The taxation rule is linear so that it allows for tax credits when capital gains losses are experienced. In this context, wash sales are optimal. Our main contribution is to derive lower and upper bounds on the value function in terms of the corresponding value in a tax-free and frictionless model. While the upper bound corresponds to the value function in a tax-free model, the lower bound is a consequence of wash sales. As an important implication of these bounds, we derive an explicit first order expansion of our value function for small interest rate and tax rate coefficients. In order to examine the accuracy of this approximation, we provide a characterization of the value function in terms of the associated dynamic programming equation, and we suggest a numerical approximation scheme based on finite differences and the Howard algorithm. The numerical results show that the first order Taylor expansion is reasonably accurate for reasonable market data

Optimization Makes Estimation Much More Worse

Mean-variance optimization as a modern portfolio theory is a major model for theoretical purposes, however, in practice portfolio managers don’t have enough interest despite some other ad hoc methods for many reasons such as estimation errors. Recently, the significance of modern portfolio theory has been analyzed that it doesn’t beat the simple naïve 1/N rule not only in many real empirical databases but also in a simulation. By this paper, due to inherent weakness of Sharpe ratio we first express more common use and adjusted measurements such as adjusted expected utility of portfolio under ambiguity aversion to analyze their effects on portfolio optimization after this consideration, because using only sample mean and variance (Sharpe ratio) to evaluate performance value for the portfolio models may be subject to considerable bias. Second, we propose a new model based on the new measurement (adjusting ambiguity Sharpe ratio) to improve portfolio optimization problem. Our result states that by using the new measurement mean- variance optimization beats the naïve rule by applying the adjusted measurement and also the novel model outperforms Markowitz in terms of Sharpe ratio while the interesting is that for adjusting Sharpe ratio inverse result exists. Therefore, our study expresses optimization makes estimation almost worse when we try to use a measurement as an optimization target. Keywords: portfolio selection, optimization, measurement, Sharpe ratio

Bitcoin mining and electricity consumption

We propose a dynamic industry equilibrium model for Bitcoin electricity consumption in a general framework, including Bitcoin miners’ optimal entry and exit with technology innovation. By adopting average operating costs as an approximation to the true operating costs, we overcome the difficulty of strong path-dependency due to the interaction among entry, exit, and technology innovation. The model can capture both the upside and downside co-movements of miners’ computing power, electricity consumption, and mining revenue. Our model shows that the Bitcoin electricity consumption will not grow indefinitely, with the ratio of Bitcoin electricity consumption to the miners’ revenue fluctuating within a range.First author draf

Portfolio Investment with the Exact Tax Basis via Nonlinear Programming

Computing the optimal portfolio policy of an investor facing capital gains tax is a challenging problem: because the tax to be paid depends on the price at which the security was purchased (the tax basis), the optimal policy is path dependent and the size of the problem grows exponentially with the number of time periods. Dammon et al. (2001, 2002, 2004), Garlappi et al. (2001), and Gallmeyer et al. (2001) address this problem by approximating the exact tax basis by the weighted average purchase price. Our contribution is threefold. First, we show that the structure of the problem has several attractive features that can be exploited to determine the optimal portfolio policy using the exact tax basis via nonlinear programming. Second, we characterize the optimal portfolio policy in the presence of capital gains tax when using the exact tax basis. Third, we show that the certainty equivalent loss from using the average tax basis instead of the exact basis is very small: it is typically less than 1% for problems with up to 10 periods, and this result is robust to the choice of parameter values and to the presence of transaction costs, dividends, intermediate consumption, labor income, tax reset provision at death, and wash-sale constraints.portfolio choice, capital gains tax, optimization, nonlinear programming

Tax effects on investments

This doctoral thesis investigates empirically and theoretically the effect of tax on the composition of the optimal allocation of wealth to risky assets from various points of view. The first empirical chapter considers the effect of tax on a U.K. personal investor targeting domestic financial products. This research helps investors estimate the impact of a future tax change and maximize their portfolio return using a newly proposed optimization model and solution method. Following Bonami and Lejeune (2009), personal portfolios are constrained to meet or exceed a prescribed return threshold with a high confidence level and satisfy buy-in threshold and diversification constraints. Their model is improved by incorporating complex tax trading rules with withdrawal features that enhance those considered by Osorio et al. (2004, 2008). A solution based on Greedy methods is developed to deal with the proposed large-scale portfolio optimization problem. The empirical results report substantial non-linear tax effects on riskier assets and enhanced effects of withdrawal tax only when tax rates are high. The developed framework better enables investors to react to tax changes, and tax policy makers to quantify the influence of tax changes on private investment preferences. The second empirical chapter investigates the effect of an international transaction tax, the so-called ‘Tobin tax’, from the point of view of U.K., U.S., and E.U. personal investors targeting international financial products. This empirical research helps the policy maker to estimate the impact of Tobin tax on international capital flows and, therefore, assess the optimal way to introduce the new tax. An optimization model is proposed to maximize the expected net Sharpe ratio and find the optimal risky portfolio internationally. Complex trading and tax rules are considered. To examine the precise effects of different investment and transaction tax rules, a comparison of four tax settings is presented: source only, residence only, mixed with credit and mixed with double taxation. The experimental results show that a source only tax union has more capital transits in international markets than a residence only tax union, and its optimal market portfolio is more sensitive to regional tax policy. In a mixed tax system, double taxation between residence- and source-taxed markets significantly reduces the attraction of the latter while its attraction is maintained with the credit method. Tobin tax can reduce the volatility of the market but the effect varies with tax rate, certain market specifications (e.g., expected returns and correlations with overseas markets) and investment tax rules. It does not depend on which side of the capital flow (inflow or outflow) is subject to Tobin tax. Finally, an agreement among countries to produce a consistent Tobin tax rate globally can significantly reduce the negative effect of Tobin tax on capital flows while retaining its positive effect on market stability in comparison to heterogeneous Tobin tax rates. Finally, the third analytical chapter investigates theoretically the effect of tax from the point of view of an arbitrageur. This theoretical research addresses the condition of the existence of arbitrage opportunities on an after-tax basis, helping the policy maker improve the fairness and efficiency of markets by addressing effective tax policy. To track tax arbitrage, continuous time optimization models are developed with heterogeneous taxation between investors programmed with continuous rather than static income and capital gains (or losses). It is proved analytically that arbitrage opportunities exist for both perfectly correlated and non-perfectly correlated assets. For perfectly correlated assets, the analysis shows that tax arbitrage may exist, with the investor’s top tax rate and some static asset parameters determining the existence of arbitrage opportunities. It is also proved that many of the equilibria obtained under income tax only are not optimal if investors are also subject to capital gains tax. For non-perfectly correlated assets, however, it is the market prices of cap and floor options on asset returns that decide the existence of tax arbitrage. In the government fixed income bond market, tax arbitrage between investors is difficult to eliminate unless investors are all subject to the same tax rates. But the return from this arbitrage can be limited if the government applies the same top tax rate to all investors