Optimal Investment with Deferred Capital Gains Taxes

Transcription

1 Optimal Investment with Deferred Capital Gains Taxes A Simple Martingale Method Approach Frank Thomas Seifried University of Kaiserslautern March 20, 2009 F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

2 Outline 1 Introduction: Taxation of Capital Gains 2 Optimal Investment Problem 3 Optimal Terminal Wealth 4 Optimal Strategy 5 Conclusion F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

3 Capital Gains Taxes I Optimal investment in the presence of taxes is complicated by the presence of implicit options: Typically, capital gains are charged upon realization, subject to rates depending on the source (stocks, dividends, bonds), the holding period, etc. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

4 Capital Gains Taxes I Optimal investment in the presence of taxes is complicated by the presence of implicit options: Typically, capital gains are charged upon realization, subject to rates depending on the source (stocks, dividends, bonds), the holding period, etc. Hence portfolio decisions are determined by short-term tactical aspects as well as long-term strategic considerations. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

5 Capital Gains Taxes I Optimal investment in the presence of taxes is complicated by the presence of implicit options: Typically, capital gains are charged upon realization, subject to rates depending on the source (stocks, dividends, bonds), the holding period, etc. Hence portfolio decisions are determined by short-term tactical aspects as well as long-term strategic considerations. As yet there has been no attempt to separate these two effects. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

6 Capital Gains Taxes II Novel legislation in Germany ( Abgeltungssteuer ) since 01/01/2009: Capital gains are treated equally and taxed with 25%. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

7 Capital Gains Taxes II Novel legislation in Germany ( Abgeltungssteuer ) since 01/01/2009: Capital gains are treated equally and taxed with 25%. In pension accounts of U.S. or German retirement plans taxes are charged on capital gains only; they are not due until retirement. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

8 Capital Gains Taxes II Novel legislation in Germany ( Abgeltungssteuer ) since 01/01/2009: Capital gains are treated equally and taxed with 25%. In pension accounts of U.S. or German retirement plans taxes are charged on capital gains only; they are not due until retirement. This motivates abstracting from short-term tactical aspects of investment: Idealization of Taxation Capital gains are taxed upon liquidation for consumption purposes, irrespective of their origin. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

9 Capital Gains Taxes III Our stylized model of capital gains taxation has a threefold purpose: F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

10 Capital Gains Taxes III Our stylized model of capital gains taxation has a threefold purpose: It is an exact model of certain real-world investments. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

11 Capital Gains Taxes III Our stylized model of capital gains taxation has a threefold purpose: It is an exact model of certain real-world investments. It helps distinguish tactical and strategic effects of taxation. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

12 Capital Gains Taxes III Our stylized model of capital gains taxation has a threefold purpose: It is an exact model of certain real-world investments. It helps distinguish tactical and strategic effects of taxation. It can serve as a benchmark for dynamic taxation. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

13 1 Introduction: Taxation of Capital Gains 2 Optimal Investment Problem 3 Optimal Terminal Wealth 4 Optimal Strategy 5 Conclusion F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

14 Financial Market I The financial market with time horizon [0, T ] is modeled on a standard filtered probability space (Ω, F, {F t }, P). It consists of a locally riskless bond P 0 = {P 0 t }, and risky assets P i = {P i t}, i = 1,..., d, modeled as semimartingales. Let Q be an equivalent martingale measure, and denote the state-price deflator Z = {Z t } by Z t 1 dq Pt 0 dp for t [0, T ]. Ft F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

15 Financial Market II We suppose that the market is complete: For any positive F T -measurable random variable X with x E[Z T X ] < there exists ϕ A(x) such that T X = X ϕ T x + ϕ t, dp t a.s. with A(x) the class of trading strategies admissible for initial wealth x. 0 F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

16 Utility and Taxes I Let u C 2 be the investor s (real-world, empirical) utility function for net wealth, and let x 0 > 0 denote her initial wealth. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

17 Utility and Taxes I Let u C 2 be the investor s (real-world, empirical) utility function for net wealth, and let x 0 > 0 denote her initial wealth. We assume that taxes are deferred until t = T, so we have the following fundamental observation: If x > 0 is the investor s untaxed wealth at t = T, then her after-tax wealth is x if x x 0 and x 0 + q(x x 0 ) if x > x 0 where k [0, 1) is the investor s personal tax rate and q 1 k. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

18 Utility and Taxes I Let u C 2 be the investor s (real-world, empirical) utility function for net wealth, and let x 0 > 0 denote her initial wealth. We assume that taxes are deferred until t = T, so we have the following fundamental observation: If x > 0 is the investor s untaxed wealth at t = T, then her after-tax wealth is x if x x 0 and x 0 + q(x x 0 ) if x > x 0 where k [0, 1) is the investor s personal tax rate and q 1 k. But the investor obtains utility u from taxed wealth; thus the effective utility from untaxed terminal wealth is û(x) u(x) if x x 0 and û(x) u(x 0 + q(x x 0 )) if x > x 0. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

19 Utility and Taxes II Thus taxation naturally induces a kink and a change of scale in utility: Figure: Effective utility function û F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

20 Optimal Portfolio Problem Optimal Portfolio Problem The optimal portfolio problem with deferred capital gains taxes is to maximize E[û(X ϕ T )] over ϕ A(x 0). (P) This is a standard portfolio problem, except for the fact that û / C 1. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

21 Optimal Portfolio Problem Optimal Portfolio Problem The optimal portfolio problem with deferred capital gains taxes is to maximize E[û(X ϕ T )] over ϕ A(x 0). (P) This is a standard portfolio problem, except for the fact that û / C 1. Alternatively, the tax code may be interpreted as follows: The investor is forced to write a call on the k th fraction of her own capital gains. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

22 1 Introduction: Taxation of Capital Gains 2 Optimal Investment Problem 3 Optimal Terminal Wealth 4 Optimal Strategy 5 Conclusion F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

23 Optimal Terminal Wealth I To solve the portfolio problem, we adapt the classical martingale method. Thus we define ˆι to be the inverse of dû dx, i.e.: Figure: Inverse marginal utility ˆι F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

24 Optimal Terminal Wealth II Theorem (Solution of the Portfolio Problem) Suppose there exists γ 0 > 0 with E[Z T ˆι(γ 0 Z T )] = x 0. Then the optimal terminal wealth in problem (P) is given by X ˆι(γ 0 Z T ). F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

25 Optimal Terminal Wealth II Theorem (Solution of the Portfolio Problem) Suppose there exists γ 0 > 0 with E[Z T ˆι(γ 0 Z T )] = x 0. Then the optimal terminal wealth in problem (P) is given by X ˆι(γ 0 Z T ). If x 0 and γ 0 are as above, then for any x 1 (0, x 0 ) there does exist some γ 1 (γ 0, ) such that E[Zˆι(γ 1 Z T )] = x 1. In typical applications, γ 0 is uniquely determined by x 0. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

26 Tax Burden I To assess the loss induced by taxes, we determine the fraction of initial wealth that the investor would be willing to give up in order to be exempt from taxation: F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

27 Tax Burden I To assess the loss induced by taxes, we determine the fraction of initial wealth that the investor would be willing to give up in order to be exempt from taxation: If x 1 > 0 is such that v(x 1 ) = sup E[û(X ϕ T )], ϕ A(x 0 ) where v denotes the value function of the untaxed portfolio problem, then the tax burden is x 0 x 1 x 0. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

28 Tax Burden II We illustrate the tax burden in a single-asset Black-Scholes model for an investor with CRRA preferences and risk aversion ρ > 0. Figure: Tax burden as a function of the tax rate k F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

29 Tax Burden III Figure: Tax burden as a function of relative risk aversion ρ F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

30 Tax Burden IV Figure: Tax burden as a function of the time horizon T F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

31 1 Introduction: Taxation of Capital Gains 2 Optimal Investment Problem 3 Optimal Terminal Wealth 4 Optimal Strategy 5 Conclusion F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

32 Optimal Portfolio Strategy We explicitly compute the optimal strategy in a d-dimensional standard Black-Scholes market, i.e. dp 0 t = P 0 t rdt, dp t = diag(p t )(r1 + η)dt + diag(p t )σdw t, where r R +, η R d +, and σ R d d are constant. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

33 Optimal Portfolio Strategy We explicitly compute the optimal strategy in a d-dimensional standard Black-Scholes market, i.e. dp 0 t = P 0 t rdt, dp t = diag(p t )(r1 + η)dt + diag(p t )σdw t, where r R +, η R d +, and σ R d d are constant. Theorem (Optimal Strategy) Under growth conditions on the utility function u, the optimal portfolio strategy ϕ = {ϕ t } in problem (P) is given by ϕ t = σ t θe r(t t) E Q [ γ0 Z T dˆι dλ (γ 0Z T ) Ft ]. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

34 Optimal Portfolio Strategy for CRRA Investors Consider a CRRA investor, i.e. u(x) = 1 1 ρ x 1 ρ, x > 0, with ρ > 0. Corollary (Optimal Portfolio for CRRA Investors) For the CRRA investor, the optimal portfolio strategy is given by ϕ t = π M (X t + x 0 C t ) for t [0, T ] where π M 1 ρ (σσt ) 1 η is the vector of Merton proportions and C t is the fair time-t price of the so-called tax derivative. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

35 Optimal Portfolio Strategy for CRRA Investors Consider a CRRA investor, i.e. u(x) = 1 1 ρ x 1 ρ, x > 0, with ρ > 0. Corollary (Optimal Portfolio for CRRA Investors) For the CRRA investor, the optimal portfolio strategy is given by ϕ t = π M (X t + x 0 C t ) for t [0, T ] where π M 1 ρ (σσt ) 1 η is the vector of Merton proportions and C t is the fair time-t price of the so-called tax derivative. Here, the tax derivative is the European derivative on optimal terminal wealth XT with payoff function g(x) 0 if x < x 0, g(x 0 ) 1, g(x) k 1 k if x > x 0. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

36 Tax Derivative It is possible to derive an explicit Black-Scholes formula for the price of the tax derivative as a function of the state-price deflator. Figure: Fair price C t of the tax derivative as a function of Z t F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

37 Ignorance Cost I To quantify the impact of taxation on portfolio choice, we compute the expected utility that the investor attains if she applies the classical Merton strategy in the taxed portfolio problem, and determine the initial wealth necessary to obtain the same level of utility: F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

38 Ignorance Cost I To quantify the impact of taxation on portfolio choice, we compute the expected utility that the investor attains if she applies the classical Merton strategy in the taxed portfolio problem, and determine the initial wealth necessary to obtain the same level of utility: If x 1 > 0 is such that E[û(X M T )] = sup E[û(X ϕ T )], ϕ A(x 1 ) where XT M is the terminal wealth achieved with the Merton strategy, then the ignorance cost is x 0 x 1 x 0. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

39 Ignorance Cost II The ignorance cost is well below 0.5% for realistic tax rates: Figure: Ignorance cost as a function of the tax rate k F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

40 1 Introduction: Taxation of Capital Gains 2 Optimal Investment Problem 3 Optimal Terminal Wealth 4 Optimal Strategy 5 Conclusion F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

41 Conclusion We have suggested a novel martingale approach to optimal investment with capital gains taxes that is mathematically simple and gives economic insight into the problem. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

42 Conclusion We have suggested a novel martingale approach to optimal investment with capital gains taxes that is mathematically simple and gives economic insight into the problem. Our results show that while the impact of deferred taxes on expected utility is considerable, F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

43 Conclusion We have suggested a novel martingale approach to optimal investment with capital gains taxes that is mathematically simple and gives economic insight into the problem. Our results show that while the impact of deferred taxes on expected utility is considerable, the Merton strategy is virtually optimal even in the presence of taxes, so the impact of deferred taxes on asset allocation is negligible. F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

44 Conclusion We have suggested a novel martingale approach to optimal investment with capital gains taxes that is mathematically simple and gives economic insight into the problem. Our results show that while the impact of deferred taxes on expected utility is considerable, the Merton strategy is virtually optimal even in the presence of taxes, so the impact of deferred taxes on asset allocation is negligible. If you are interested, the working paper is available at SSRN: F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27

45 The End Thank you very much for your attention! F. Seifried (Kaiserslautern) Deferred Capital Gains Taxes March 20, / 27