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Modeling continuous-time ﬁnancial markets with capital gains taxes ∗ Imen Ben Tahar Nizar Touzi e CEREMADE, Universit´ Paris Dauphine Centre de Recherche e e and Universit´ L´onard De Vinci en Economie et Statistique imen.ben tahar@devinci.fr e and CEREMADE, Universit´ Paris Dauphine touzi@ensae.fr January 2003 Abstract We formulate a model of continuous-time ﬁnancial market with risky asset subject to capital gains taxes. We study the problem of maximizing expected utility of future consumption within this model both in the ﬁnite and inﬁnite horizon. Our main result is that the maximal utility does not depend on the taxation rule. This is shown by exhibiting maximizing strategies which tracks the classical Merton optimal strategy in tax-free ﬁnancial markets. Hence, optimal investors can avoid the payment of taxes by suitable strategies, and there is no way to beneﬁt from tax credits. Key Words and phrases: Optimal consumption and investment in continuous- time, capital gains taxes. ∗ We are grateful to Stathis Tompaidis for numerous discussions on the modelization issue of this paper. We have also beneﬁted from interesting comments and discussions with Jose Scheinkman and Mete Soner. In particular, Jose suggested the introduction of the ﬁxed delay in the tax basis, and Mete gave a clear motivation to push forward the approximating strategies introduced in this paper. 1 1 Introduction Since the seminal papers of Merton [6, 7], there has been a continuous interest in the theory of optimal consumption and investment decision in ﬁnancial markets. A large literature has focused particularly on the eﬀect of market imperfections on the optimal consumption and investment decision, see e.g. Cox and Huang [1] and Karatzas, Lehoczky and Shreve [5] c for incomplete markets, Cvitani´ and Karatzas [2] for markets with portfolio constraints, Davis and Norman [3] for markets with proportional transaction costs. However, there is a very limited literature on the capital gains taxes which apply to ﬁnancial securities and represent a much higher percentage than transaction costs. Com- pared to ordinary income, capital gains are taxed only when the investor sells the security, allowing for a deferral option. One may think that the taxes on capital gains have an appreciable impact on individuals consumption and investment decisions. Indeed, under taxation of capital gains, an investor supports supplementary charges when he rebalances his portfolio, which alters the available wealth for future consumption, possibly depreciat- ing consumption opportunities compared to a tax-free market. On the other hand, since taxes are paid only when embedded capital gains are actually realized, the investor may choose to defer the realization of capital gains and liquidate his position in case of capital losses, particularly when the tax code allows for tax credits. Previous works attempted to characterize intertemporal consumption and investment decisions of investors who have permanently to choose between two conﬂicting issues : realize the transfers needs for an optimally diversiﬁed portfolio, or use the ability to defer capital gains taxes. The taxation code speciﬁes the basis to which the price of a security has to be compared in order to evaluate the capital gains (or losses). The tax basis is either deﬁned as (i) the speciﬁc share purchase price, or (ii) the weighted average of past purchase prices. In some countries, investors can chose either one of the above deﬁnitions of the tax basis. A deterministic model with the above deﬁnition (i) of the tax basis, together with the ﬁrst in ﬁrst out rule of priority for the stock to be sold, has been introduced and studied by Jouini, Koehl and Touzi [9, 10]. The case where the tax basis is deﬁned as the weighted average of past purchase prices is easier to analyze, as the tax basis can be described by a controlled Markov dynamics. Therefore, it can be treated as an additional state variable in a classical stochastic control problem. A discrete-time formulation of this model with short sales constraints and lin- ear taxation rule has been studied by Dammon, Spatt and Zhang [4]. They considered the problem of maximizing the expected discounted utility of future consumption, and provided a numerical analysis of this model based on the dynamic programming principle. In partic- ular, they showed that investors may optimally sell assets with embedded capital gains, and that the Merton tax-free optimal strategy is approximately optimal for ”young investors”. We refer to Gallmeyer, Kaniel and Tompaidis [11] for an extension of this analysis to the multi-asset framework. The ﬁrst contribution of this paper is to provide a continuous-time formulation of the utility maximization problem under capital gains taxes, see Section 2. The ﬁnancial market consists of a tax exempt riskless asset and a risky one. Transfers are not subject to any 2 transaction costs. The holdings in risky assets are subject to the no-short sales constraints, and the total wealth is restricted by the no-bankruptcy condition. The risky asset is subject to taxes on capital gains. The tax basis is deﬁned as the weighted average of past purchase prices. We also introduce a possible ﬁxed delay in the tax basis. In contrast with [4], we consider a general nonlinear taxation rule. Our results hold both for ﬁnite and inﬁnite horizon models. Section 3 shows that the reduction of our model to the tax-free case produces the same indirect utility than the classical Merton model. The main result of our paper states that the value function of the continuous-time utility maximization problem with capital gains taxes coincides with the Merton tax-free value function. In other words, investors can optimally avoid taxes and realize the same indirect utility as in the tax-free market. We also provide a maximizing strategy which shows how taxes can be avoided. This result is ﬁrst proved in Section 4 in the case where no tax credits are allowed by the taxation rule. It is then extended to general taxation rules in Section 5 by reducing the problem to the linear taxation rule. The particular tractability of the linear taxation rule case allows to prove that it is optimal to take advantage of the tax credits by realizing immediately capital losses. From an economic viewpoint, our result shows that capital gains taxes do not induce any tax payment by optimal investors. This suggests that the incorporation of capital gains taxes in ﬁnancial market models should be accompanied by another market imperfection, as transaction costs, which prevents optimal investors from implementing the maximizing strategies exhibited in this paper. This aspect is left for future research. 2 The Model 2.1 The ﬁnancial Market We consider a ﬁnancial market consisting of one bank account with constant interest rate r > 0, and one risky asset with price process evolving according to the Black and Scholes model: dSt = µSt dt + σSt dWt , (2.1) where µ is a constant instantaneous mean rate of return, σ > 0 is a constant volatility parameter, and the process W = {Wt , 0 ≤ t} is a standard Brownian motion deﬁned on the underlying probability space (Ω, F, P). Let F be the P-completion of the natural ﬁltration of the Brownian motion. In order for positive investment in the risky asset to be interesting, we shall assume throughout that µ > r. (2.2) We also assume that the ﬁnancial market is not subject to any transaction costs, and the shares of the stock are inﬁnitely divisible. 2.2 Relative tax basis The sales of the stock are subject to taxes on capital gains. The amount of tax to be paid for each sale of risky asset is computed by comparison of the current price to the weighted 3 average of price of the assets in the investor portfolio. We therefore introduce the relative tax basis process Bt which records the ratio of the weighted average price of the assets in the investor portfolio to the current price. When Bt is less than 1, the current price of the risky asset is greater than the weighted-average purchase price of the investor so if she sells the risky asset, she would realize a capital gain. Similarly, when Bt is larger then 1, the sale of the risky asset corresponds to the realization of a capital loss. Example 2.1 Let 0 ≤ t0 < t1 < t2 < t3 be some given trading dates, and consider the following discrete portfolio strategy. : - buy 5 units of risky asset at time t0 , - sell 1 unit of risky asset at time t1 , - buy 2 units of risky assets at time t2 , - sell 4 units of risky asset at time t3 , - buy 2 units of risky assets at time t3 . The relative tax basis is not deﬁned strictly before the ﬁrst purchase date t0 , and is equal to one exactly at t0 . We set by convention Bt = 1 for t ≤ t0 . Sales do not alter the basis. Therefore, we only care about purchases in order to determine the basis at each time. At times t2 and t4 , the relative tax basis is given by 5St0 + 2St2 5St0 + 2St2 + 2St3 Bt2 = and Bt3 = . 7St2 9St3 Although no purchases occur in the time intervals (t0 , t2 ), (t2 , t3 ), the relative tax basis moves because of the change of the current price : (BS)t0 for t0 ≤ t < t2 , (BS)t = (BS)t2 for t2 ≤ t < t3 (BS) for t ≥ t . t3 3 2.3 Taxation rule Each monetary unit of stock sold at some time t is subject to the payment of an amount of tax computed according to the relative tax basis observed at the prior time tδ := (t − δ)+ = max{0, t − δ} . (2.3) Here δ ≥ 0 is a ﬁxed characteristic of the taxation rule. Another characteristic of the taxation rule is the amount of tax to be paid per unit of sale. This is deﬁned by f (Btδ − ) , (2.4) where f is a map from R+ into R satisfying f (b) f non-increasing, f (1) = 0, and lim inf > −∞ . (2.5) b↑1 b−1 4 Example 2.2 (Proportional tax on non-negative gains) Let f (b) := α(1 − b)+ for some constant 0 < α < 1 . When the relative tax basis is less than unity, the investor realizes a capital gain, and pays the amount of tax α(1 − Bt ) per unit amount of sales. Example 2.3 (Proportional tax with tax credits) Let f (b) := α(1 − b) for some constant 0 < α < 1 . When the relative tax basis is less than unity, the investor realizes a capital gain, and pays the amount of tax α(1 − Bt ) per unit amount of sales. When the relative tax basis is larger then unity, the investor receives the tax credit α(Bt − 1) per unit amount of sales. Remark 2.1 When there are no tax credits, i.e. f ≥ 0, it is clear that the total tax paid by the investor is non-negative, and the investor can not do better than in a tax-free market. However, when f is not non-negative, it is not obvious that the investor can not take advantage of the tax credits, and do better than in a tax-free model. Of course, this would not be acceptable from the economic viewpoint. Our analysis of this situation in Section 5 shows that the presence of tax credits does not produce such a non-desirable eﬀect. 2.4 Investment-consumption strategies An investor start trading at time t = 0 with an initial capital x in cash and y monetary units in the risky asset. At each time t ≥ 0, trading occurs by means of transfers between the two investment opportunities. ˜ ˜ We denote by L := (Lt , t ≥ 0) the process of cumulative transfers form the bank account ˜ ˜ to the risky assets one, and M := (Mt , t ≥ 0) the process of cumulative transfers from the ˜ ˜ risky assets account to the bank. Here, L and M are two F−adapted, right-continuous, ˜ ˜ non-decreasing processes with L0− = M0− = 0. In addition to the trading activity, the investor consumes in continuous-time at the rate C = {Ct , t ≥ 0}. The process C is F−adapted and nonnegative. ˜ ˜ Given a consumption-investment strategy (C, L, M ), we denote by Xt the position on the bank, Yt the position on the risky assets account, and Bt the relative tax basis at time t. We also introduce the total wealth Zt := Xt + Yt , t ≥ 0 . 2.5 Portfolio constraints We ﬁrst restrict the strategies to satisfy the no-bankruptcy condition Zt ≥ 0 P − a.s. for all t ≥ 0 , (2.6) 5 i.e. the total wealth of the portfolio at each time has to be non-negative. We also impose the no-short sales constraint Yt ≥ 0 P − a.s. for all t ≥ 0 , (2.7) together with the absorption condition Yt0 (ω) = 0 for some t0 =⇒ Yt (ω) = 0 for a.e. ω ∈ Ω. (2.8) The latter is a technical condition which is needed for a rigorous continuous-time formula- tion of our problem. ˜ ˜ The consumption-investment strategy (C, L, M ) is said to be admissible if the resulting state variables (X, Y, B) satisfy the above conditions (2.6)-(2.7)-(2.8). In particular, the ¯ process (Z, Y ) is valued in the closure S of the subset of R2 : S := (0, ∞) × (0, ∞). (2.9) ˜ ˜ Finally, given an admissible strategy (C, L, M ), we introduce the stopping time : τ := inf{t ≥ 0 : Yt ∈ (0, ∞)} = inf{t ≥ 0 : Yt = 0} , where the last equality follows from (2.7). In view of (2.8), it is clear that the trading strat- egy can be described by means of the non-decreasing right-continuous processes (Lt , Mt )t≥0 ˜ ˜ which are related to (Lt , Mt )t≥0 by t t Lt := Yt−1 dLt and Mt := ˜ Yt−1 dMt , t < τ . ˜ 0 0 Here, dLt and dMt represent the proportion of transfers of risky assets. 2.6 Controlled dynamics ˜ ˜ Let (C, L, M ) be an admissible strategy, and deﬁne (L, M ) as in the previous paragraph. We shall denote ν := (C, L, M ), and (X ν , Y ν , B ν ) := (X, Y, B) the corresponding state variables Given an initial capital x on the bank account, the evolution of the wealth on this account is described by the dynamics : ν ν ν ν ν dXt = (rXt − Ct )dt − Yt− dLt + Yt− 1 − f (Btδ − ) dMt and X0− = x , (2.10) recall from (2.3) that tδ := (t − δ)+ . Given an initial endowment y on the risky assets account, the evolution of the wealth on this account is also clearly given by dSt ν dYtν = Yt− + dLt − dMt and Y0− = y . (2.11) St This implies that the total wealth evolves according to ν ν ν ν ν dZt = r(Zt − Ct )dt + Yt− [(µ − r)dt + σdWt ] − Yt− f (Btδ − )dMt (2.12) and Z0− = y + x . 6 In order to specify the dynamics of the relative tax-basis, we introduce the auxiliary process K ν := B ν Y ν . By deﬁnition of B ν , we have : ν ν ν ν dKt = Yt− dLt − Kt− dMt and K0− = y , (2.13) ν since B0− = 1. Observe that the contribution of the sales in the dynamics of Kt is evaluated at the basis price. We then deﬁne the relative basis process B ν by Ktν ν Bt = 1{Ytν =0} + 1 ν . (2.14) Ytν {Yt =0} Hence, the position of the investor resulting from the strategy ν is described by the triple (Z ν , Y ν , B ν ). We call (Z ν , Y ν , B ν ) the state process associated with the control ν. Proposition 2.1 Let ν = (C, L, M ) be a triple of adapted process such that A1 L, M are right-continuous, non-decreasing and L0− = M0− = 0 A2 the jumps of M satisfy ∆M ≤ 1 t A3 C ≥ 0 and 0 Cs ds < ∞ a.s. for all t ≥ 0 Then, there exists a unique solution (Z ν , Y ν , B ν ) to (2.12)-(2.11)-(2.13)-(2.14). Moreover, (Z ν , Y ν , B ν ) satisﬁes conditions (2.7)-(2.8). Proof. Equation (2.11) clearly deﬁnes a unique solution Y ν . Given Y ν , it is also clear that (2.13) has a unique solution Y ν B ν , and (2.14) deﬁnes B ν uniquely. Finally, given (Y ν , B ν ), it is an obvious fact that equation (2.10) has a unique solution X ν . 2 Remark 2.2 The statement of the above proposition is still valid when A2 is replaced by the following weaker condition A2’ the jumps of the pair process (L, M ) satisfy ∆L − ∆M ≥ −1. However in the case where tax credits are allowed by the taxation rule, see Section 5, it is easy to construct consumption-investment strategies, satisfying A1-A2’-A3, which increase without bound the value function of the problem (2.16) deﬁned below, starting from some ﬁxed positive initial holding in stock. Hence such a model allows for a weak notion of arbitrage opportunities. Indeed, for each ε > 0 and λ > 0, let ∆Lt = ∆Mt := 0 for t = τ , ∆Lτ = ∆Mτ := Λ where τ := inf{t : Bt > 1 + ε}. By sending Λ to inﬁnity, the value function of the problem (2.16) converges to +∞. ¯ Deﬁnition 2.1 Let ν = (C, L, M ) be a triple of F−adapted processes, and (z, y) ∈ S. We say that ν is a (z, y)−admissible consumption-investment strategy if it satisﬁes Conditions A1-A2-A3 together with the no-bankruptcy condition (2.6). We shall denote by Af (z, y) the collection of all (z, y)−admissible consumption-investment strategies. Remark 2.3 We used the absorption at zero condition in order to express an investment d˜ ˜ strategy by means of the proportions dLt = YLt and dMt = dMt , instead of the volume of t− Yt− ˜ ˜ transfers, dLt and dMt . This modiﬁcation was needed for the speciﬁcation of the tax basis 7 ˜ ˜ by means of the process K deﬁned above. Indeed, in terms of (L, M ), the dynamics of the state variables X and Y are given by : dSt ˜ ˜ ˜ ˜ dYt = Yt− + dLt − dMt and dXt = r(Xt − ct )dt − dLt + (1 − f (Btδ − )) dMt , St but the relative basis is deﬁned by means of the process K whose dynamics are given by ˜ Kt− ˜ dKt = dLt − d Mt . Yt− Since the event {Y = 0} has positive probability, this may cause trouble for the deﬁnition of the model. 2.7 The consumption-investment problem Throughout this paper, we consider a power utility function : cp U (c) := for all c ≥ 0, p where 0 < p < 1 is a given parameter. We next consider the investment-consumption criterion t Jtf (z, y; ν) := E e−βt U (Ct )dt + U (Xt )1{t<∞} ν (2.15) 0 ¯ for t ∈ R+ ∪ {+∞}, (z, y) ∈ S and ν ∈ Af (z, y). Let T ∈ R+ ∪ {+∞} be a given time horizon, so that our analysis holds for both ﬁnite and inﬁnite horizon. The consumption investment problem is deﬁned by f f ¯ VT (z, y) := sup JT (z, y; ν) , (z, y) ∈ S . (2.16) ν∈Af (z,y) In the context of ﬁnancial markets without taxes, i.e. f ≡ 0, a slight modiﬁcation of this problem has been solved by Merton [7, 6] by means of a veriﬁcation argument. In the ﬁnite horizon case (T < ∞), the tax-free problem can be solved directly by passing to a dual formulation, [8, 1, 5]. In the inﬁnite horizon tax-free problem, Merton [6] singled out the condition 2 1 1 p µ−r γ := β − rp − > 0 whenever T = +∞ , (2.17) 1−p 21−p σ in order to ensure that the value function is ﬁnite. The (explicit) solution in this context is simply obtained by sending the time horizon to inﬁnity in the solution of the ﬁnite horizon problem. We conclude this section by the following easy result which states that, the value function V f is non-increasing in f . 8 Proposition 2.2 Let (z, y) be some initial holdings pair in S, and let f ≥ f be two maps f g from R+ into R. Then VT (z, y)T ≤ VT (z). Proof. Consider some admissible consumption-investment strategy (C, L, M ) ∈ Af (z, y). We denote by (Z f , Y f , B f ) and (Z g , Y g , B g ) the corresponding state process respectively under the taxation rule implied by f and g. Observing that (Y f , B f ) = (Y g , B g ), we directly compute that : t t g f g f f f g g Zt − Zt = r(Zs − Zs )ds + [f (Bsδ − )Ys− − g(Bsδ − )Ys− ]dMs 0 0 t t 0 f f f ≥ r(Zs − Zs )ds + (f − g)(Bsδ − )Ys− dMs 0 0 t 0 f ≥ r(Zs − Zs )ds , 0 since f ≥ g. This shows that Z g ≥ Z f so that (C, L, M ) is also an admissible strategy in Ag (z, y). Hence Af (z, y) ⊂ Ag (z, y) by arbitrariness of (C, L, M ) ∈ Af (z, y), and the required result follows. 2 3 Financial market without taxes In this section, we review the solution of the consumption-investment problem in a ﬁnancial market without capital gain taxes, i.e. when f ≡ 0. Since the reduction of our problem to this case is slightly diﬀerent from the classical Merton model, we shall study both problems. We will show that they have essentially the same value functions, and discuss the issue of optimal strategies. 3.1 The classical Merton model In the classical formulation of the tax-free consumption-investment problem, the investment control variable is described by means of unique process π which represents the proportion of wealth invested in risky assets at each time. Given a consumption plan C, the total wealth process is then deﬁned by the dynamics : ¯(C,π) ¯(C,π) ¯ (C,π) ¯ (C,π) = z.(3.1) dZt = rZt − Ct dt + πt Zt [(µ − r)dt + σdWt ] , and Z0 In this context, a consumption-investment strategy is a pair of F−adapted processes (C, π), where C is non-negative and T T Cs ds + |πs |2 ds < ∞ P − a.s. 0 0 ¯ We shall denote by A(z) the collection of all such consumption-investment strategies which satisfy the additional no-bankruptcy condition ¯ (C,π) ≥ 0 P − a.s. for all 0 ≤ t ≤ T . Zt 9 The relaxed tax-free consumption-investment problem is then deﬁned by : T ¯ VT (z) := sup E ¯ (π,C) 1{T <∞} . e−βt U (Ct )dt + U ZT (3.2) ¯ (C,π)∈A(z) 0 ¯ Set ∂ zS := {(z, y) ∈ S : z = 0}, and let (z, y) be an arbitrary initial data in S ∪ ∂ zS ¯ = {(z, y) ∈ S : y > 0}. Clearly, for any admissible consumption-investment strategy ¯ ¯ ν = (C, L, M ) ∈ A0 (z, y), one can deﬁne a pair (C, π) ∈ A(z) such that Z ν = Z (C,π) . This shows that 0 ¯ VT (z, y) ≤ VT (z) for all (z, y) ∈ S ∪ ∂ zS , (3.3) ¯ 0 and justiﬁes the name of the problem VT . The value function VT on the boundary ∂ yS := ∂S \ ∂ zS will be studied separately. We shall prove later on (Proposition 3.2) that equality holds in (3.3) by exhibiting a 0 maximizing strategy for the problem VT . In preparation to this, let us ﬁrst recall the explicit solution of the relaxed tax-free consumption-investment problem. Theorem 3.1 Let Condition (2.17) hold. Then, for all z > 0 : 1−p zp 1 1 ¯ VT (z) = + 1− e−γT . p γ γ ¯ Moreover, existence holds for the problem VT (z) with optimal consumption-investment strat- egy given by : µ−r ¯ πt = π := ¯ ¯ , ¯ ¯ Ct := c(t)Zt , (1 − p)σ 2 ¯ where c(.) is the deterministic function −1 1 1 c(t) := ¯ + 1− e−γ(T −t) , γ γ ¯ ¯ ¯π ¯ ¯ and Z := Z (C,¯ ) is the wealth process deﬁned by the strategy (C, π ) : ¯ ¯ ¯ µ−r µ−r Z0 = z, dZt = Zt (r − c(t)) dt − ¯ dt + dWt . (1 − p)σ σ Observe that ¯ - the optimal investment strategy π is constant both in the ﬁnite and inﬁnite horizon cases, - the optimal consumption process is a linear deterministic function of the wealth process, ¯ ¯ with slope deﬁned by the function c(t); in the inﬁnite horizon case, the function c reduces to the constant γ, ¯ Remark 3.1 Consider the inﬁnite horizon case T = +∞. Then c(t) = γ. By direct computation, we see that e−βt E U (Zt ) = z p e−γt for all t ≥ 0 . ¯ ¯ Hence Condition (2.17) guarantees that e−βt E U (Zt ) −→ 0 as t → ∞, and therefore : t ¯ V∞ (z) = lim E e−βs U (γ Zs )ds + e−βt U (Zt ) . ¯ ¯ t→∞ 0 10 3.2 Connection with our tax-free model We now focus on the reduction of the model of Section 2 to the tax-free case, i.e. f ≡ 0. In this context, the state variable B is not relevant any more. Given an initial data (z, y) ∈ S ¯ and an admissible control ν = (C, L, M ) ∈ A 0 (z, y), the controlled state process reduces to the pair (Z ν , Y ν ) which evolves according to the dynamics : ν dZt ν = (rZt − Ct ) dt + Ytν [(µ − r)dt + σdWt ] − dYtν = Ytν [(µ − r)dt + σdWt + dLt − dMt ] − together with the initial condition (Z, Y )0− = (z, y). This model presents some minor diﬀerences with the classical Merton model of Section 3.1. First, the investment strategies are constrained to have bounded variation. We shall see that this induces a non-existence of an optimal control for the problem V 0 (z, y), but ¯ does not entail any diﬀerence between V 0 and V . Second, the above dynamics imply that zero is an absorbing boundary for the Y variable which describes the holdings in stock. From the solution of the classical Merton model reported in Theorem 3.1, notice that the investment in stock is always positive, except the case of zero initial capital z = 0. We ¯ therefore expect that the value functions V and V 0 do coincide except on the boundary ¯ ∂ yS := {(z, y) ∈ S : z > 0 and y = 0}. ¯ Observe that the analysis of both problems V and V 0 is trivial on the boundary ¯ ∂ zS := {(z, y) ∈ S : z = 0}, since there is no possibility neither for consumption nor for investment. The following result characterizes the value function V 0 on the boundary of S which, according to the previous notations, is partitioned into ∂S = ∂ zS ∪ ∂ yS . Proposition 3.1 The solution of the problem V 0 on the boundary ∂S is given by : (i) For (z, y) ∈ ∂ zS, ¯ ˆ ˆ ˆ V 0 (z, y) = V (z) = 0 with optimal controls (C, L, M )t = (0, 0, 1), t ≥ 0 , ˆ ˆ and optimal state process (Z, Y )t = 0 for t ≥ 0. β−rp (ii) Set γ0 := 1−p and assume β > r whenever T = +∞. Then, for (z, y) ∈ ∂ yS, 1−p zp 1 1 0 VT (z, y) = + 1− e−γ0 T , (3.4) p γ0 γ0 with optimal controls −1 1 1 ˆ ˆ ˆ c ˆ (C, L, M )t = (ˆ0 (t)Zt , 0, 0), ˆ c0 (t) := + 1− e−γ0 (T −t) (3.5) γ0 γ0 ˆ for 0 ≤ t ≤ T ; the optimal state processes are Y = 0 and t ˆ Zt := z exp rt − c0 (s)ds , ˆ 0≤t≤T . 0 11 ˆ ˆ ˆ Proof. For item (i), it is suﬃcient to observe that the investment strategy {(C, L, M )t = (0, 0, 1), t ≥ 0} is the only admissible strategy. We now concentrate on item (ii). Since ∂ yS is an absorbing boundary, we are reduced to the (deterministic) control problem : T 0 VT (t, z, 0) = sup e−βt U (Ct ) dt + e−β(T −t) U (ZT ) 1{T <∞} , t where the state dynamics are given by dZt = (rZt − Ct )dt . 1. We ﬁrst solve the ﬁnite horizon problem T < ∞. We shall use a veriﬁcation argument by guessing a solution to the Hamilton-Jacobi equation of this problem : ∂VT 0 ∂V 0 ∂V 0 0 0 = −βVT (t, z) (t, z) + rz T (t, z) + sup U (ξ) − ξ T (t, z) ∂t ∂z ξ≥0 ∂z 0 p/p−1 0 ∂VT ∂V 0 p ∂VT0 = −βVT (t, z) (t, z) + rz T (t, z) + (t, z) ; ∂t ∂z 1−p ∂z the argument y = 0 has been omitted for notational simplicity. We guess a solution to the above ﬁrst order partial diﬀerential equation in the separable form z p h(t), and determine h so that the terminal condition 0 1 VT (T, z) = U (z) or, equivalently, h(T ) = p is satisﬁed. This leads to the candidate solution deﬁned in (3.4). By usual veriﬁcation arguments, this candidate is then shown to be the solution of the problem, and the optimal controls are identiﬁed. 2. We now concentrate on the inﬁnite horizon case T = +∞. It is clear that the optimal state process Z should be set to zero at inﬁnity. In terms of optimal control, this is a natural transversality condition for the problem. In order to take advantage of this information, we solve the problem by the calculus of variation approach. Direct calculation from the local Euler equation of the problem leads to the following characterization of the optimal state : ˙ ¨ (p − 1) rZ − Z ˙ = (β − r)(rZ − Z) , ˙ where Z = dZ/dt denotes the time derivative of the state Z. . this ordinary diﬀerential equation can be solved explicitly by the technique of variation of the constant. In view of the boundary conditions Z0 = z and Z∞ = 0, this provides the unique solution to the local Euler equation : β−r ˙ Zt = z exp − t with optimal consumption Ct = rZt − Zt = γ0 Zt . 1−p Notice that the condition β > r is here necessary in order to ensure that Z∞ = 0. 2 To conclude our analysis of the tax-free model, we now focus on the value function in the interior of the domain S. in contrast with the situation on the boundary, trading in the 12 stock is now possible. The following result shows that the value function V 0 coincides with ¯ V , the maximal utility in the classical Merton model. The price for the control restriction to the class of bounded variation processes is that existence does not hold any more for the problem V 0 . 0 ¯ Proposition 3.2 For all (z, y) ∈ S, we have VT (z, y) = VT (z). 0 In view of (3.3), the only non-trivial inequality in the above result is that VT (z, y) ≥ ¯ VT (z). This follows directly from our main Theorem 4.1, by considering the case f ≡ 0. 4 Optimal consumption-investment under capital gain taxes In this section, we consider the case of a ﬁnancial market with no tax credits, i.e. f (b) ≥ 0 for all b ≥ 0 . (4.1) The general case will be studied in the subsequent section. In the context of (4.1), we shall prove the ﬁrst main result of the paper which states that the maximal utility in the ﬁnancial market is not altered by the capital gains taxation rule, i.e. V f = V 0 . This result is of course trivial when the initial holding in stock is zero, Y0− = 0, since ∂ yS is an absorbing boundary. For a non-zero initial holding y in stock, we shall prove this result by forcing the relative tax basis Bt to be as close as desired to unity, and tracking Merton’s optimal strategy, i.e. keep the proportion of wealth invested in the risky asset Yt πt := 1 , 0 ≤ t ≤ T, Zt {Zt =0} and the proportion of wealth dedicated for consumption Ct ct := 1 , 0 ≤ t ≤ T, Zt {Zt =0} π ¯ close to the pair (¯ , c(t)) deﬁned in Theorem 3.1. To do this, we ﬁrst ﬁx some t > 0, and deﬁne a convenient sequence (ν t,n )n≥1 := (C t,n , Lt,n , M t,n )n≥1 for all (z, y) ∈ S ∪ ∂ zS. We shall denote by Z t,n , Y t,n , B t,n = t,n t,n t,n Zν ,Y ν , Bν the corresponding state processes. For each integer n ≥ 1, the consumption- investment strategy ν t,n is deﬁned as follows. t,n 1. At time 0 choose the transfers (∆Lt,n , ∆M0 ) so as to adjust the proportion of wealth 0 ¯ to π : z z ∆Lt,n := 0 π ¯ t,n − 1 1{¯ z≥y} and ∆M0 := π 1−π ¯ 1{¯ z<y} , π y y so that t,n Y0t,n π0 := t,n ¯ = π; Z0 13 recall that B0δ − = B0− = 1. 2. At the ﬁnal time t, ﬁx the jumps ∆Lt,n , ∆M (t,n) t so that all the wealth is transferred to the bank : (t,n) ∆Lt,n := 0 and ∆Mt t = 1. This implies that (t,n) t,n Ytt,n = 0 and Zt = Xt . t,n 3. In Step 3 below, we shall construct a sequence of stopping times (τk )k≥1 . Our con- sumption strategy is deﬁned by t,n t,n Cs := c(t)Zs ¯ for 0 ≤ s ≤ T . The investments strategy is piecewise constant : t,n t,n dLt,n = dMs = 0 for all s ∈ [0, T ] \ {τk , k ≥ 1} . s t,n 4. We now introduce the sequence of stopping times τk as the hitting times of the pair t,n process (π, B) of some barrier close to (¯ , 1). Set τ0 := 0, and deﬁne the sequence of π stopping times t,n π B τk := T ∧ τk ∧ τk , where t,n τk := inf s ≥ τk−1 : |πs − π | > n−1 , π t,n ¯ B τk := inf s ≥ τk−1 : t,n 1 − Bsδ > n−1 λk , where λ is a parameter in (0, 1) to be ﬁxed later on. 5. To conclude the deﬁnition of ν t,n , it remains to specify the jumps ∆Lt,n , ∆M t,n at t,n each time τk . The idea here is to re-set the proportion π t,n to the constant π , and to ¯ push-back the relative tax basis B to unity. To do this, we ﬁrst consider some parameter λ ∈ (0, 1) such that : 1 + λ(1 − π ) > 0 . ¯ t,n We then deﬁne for all s ∈ {τk , k ≥ 1} : t,n π ¯ 1 − πs− f Bsδ − ∆Lt,n s := t,n n and ∆Ms := 1 − λ∆Ln . s t,n πs− 1 + λ 1 − π f Bs − ¯ δ Using the dynamics of (Z, Y, B), we have : t,n Yst,n t,n t,n πs− (1 + ∆Lt,n − ∆Ms ) s πs = t,n = Zs t,n t,n t,n 1 − πs− ∆Ms f Bsδ − 14 and t,n t,n t,n t,n t,n t,n Bs Yst,n − Bs− Ys− = Ys− ∆Lt,n − Bs− ∆Ms s , so that with the above deﬁnition of the jumps (∆Lt,n , ∆M t,n ), we have t,n t,n t,n 1 + λBs− t,n πs = π and Bs = ¯ for s ∈ {τk , k ≥ 0} . 1+λ Remark 4.1 Since f (1) = 0 and f is continuous, it is immediately checked from the above deﬁnitions that, for suﬃciently large n : t,n t,n 0 < ∆Lt,n < 1 and 0 < ∆Ms < 1 for s ∈ {τk , k ≥ 0} . s This guarantees that the process of holdings in risky assets Y t,n is positive P−a.s. 2 t,n Remark 4.2 The sequence τk is strictly increasing, and converges to T . To see k≥0 t,n π this, we ﬁrst make the trivial observation that τk < τk+1 P−a.s. On the other hand, since t,n t,n L and M are constant in the stochastic interval [τk−1 , τk ), we have 1 − B t,n ≤ λk /n. t,n τk − Then : λ λk+1 λk+1 1 − B t,n t,n = 1 − B t,n t,n ≤ < . τk 1+λ τk − n(1 + λ) n t,n B t,n t,n This guarantees that τk < τk+1 P−a.s. In particular τk −→ τ∞ ≤ T . The proof of our t,n claim is completed by observing that the limit τ∞ is necessarily equal to T . 2 The main property of the sequence ν t,n n is the following. Lemma 4.1 Let t > 0 be some ﬁxed time horizon. Then for any map f satisfying (2.5), we have 2 E t,n ¯ sup Zs − Zs ≤ n−2 αeαt , 0≤s≤t for some constant α depending on t. t,n Proof. By deﬁnition of the sequence of stopping times τk , we have k t,n 1 t,n λk sup πs − π ≤ ¯ and sup 1 − Bs ≤ for all k ≥ 1 . (4.2) 0≤s≤t n τk−1 ≤s<τk n ¯ Set D := Z t,n − Z. Since D0 = 0, we decompose D into : Ds = Fs + Gs + Hs , where s t,n Fs := Du (r − c(u))du + πu ((µ − r)du + σdWu ) , ¯ 0 s Gs := ¯ t,n Zu πs − π ((µ − r)du + σdWu ) ¯ 0 s t,n t,n ¯ t,n t,n t,n ¯ t,n Hs := − πu− f Buδ − Du− + Zu− dMu = πu− f Buδ − Du− + Zu− ∆Mu . 0 u≤s 15 In the subsequent calculation, A will denote a generic (t−dependent) constant whose value may change from line to line. We shall also denote by Vs∗ := sup0≤u≤s |Vu | for all process (Vs )s . ¯ We ﬁrst start by estimating the ﬁrst component F . Observe that c(.) is bounded and the process π t,n is bounded by 2¯ for large n. Then π s 2 s 2 2 t,n t,n |Fs | ≤ 2 Du (r − c(u) + ¯ πu (µ − r))du +2 Du πu σdWu 0 0 s s 2 ∗ ≤ A |Du |2 du + 2 t,n Du πu σdWu . 0 0 By the Buckholder-Davis-Gundy inequality, this provides s s ∗ ∗ E|Fs |2 ≤ A E|Du |2 du + E t,n |Du |2 |πu |2 σ 2 du 0 0 s ∗ ≤ A E|Du |2 du . (4.3) 0 Similarly, it follows from (4.2) that : s 2 s 2 2 ¯ ¯ |Gs |2 ≤ (µ − r)2 Zu du + σ2 Zu dWu . n2 0 0 Using again the Buckholder-Davis-Gundy inequality, this provides A E|G∗ |2 ≤ s . (4.4) n2 Finally, since the jumps of M t,n are bounded by 1, we estimate the component H by : 2 2 t,n t,n t,n ¯ |Hs |2 ≤ 2 |πu− |f Buδ − |Du− | + 2 f Buδ − Zu− u≤s u≤s 2 ∗ ¯∗ t,n ≤ A |Ds |2 + |Zt |2 f Buδ − u≤s 2 ∗ ¯∗ t,n ≤ A |Ds |2 + |Zt |2 f Buδ − u≤T 2 ∗ ¯∗ λk ≤ A |Ds |2 + |Zt |2 n k≥0 A ∗ = 1 + |Ds |2 , (4.5) n2 where the last inequality follows from (4.2) and (2.5) which implies that f is locally Lipschitz at b = 1. We now collect the estimates from (4.3), (4.4) and (4.5) to see that : s A ∗ A ∗ 1− E|Ds |2 ≤ +K E|Du |2 du for all s ≤ t . n2 n2 0 16 The required result follows from the Gronwall inequality. 2 We are now ready for the ﬁrst main result of this paper which states that the value function of the consumption investment problem is not altered by the capital gain taxes rule. Notice that the proof produces a precise description of optimal consumption-investment behavior : in the ﬁnite horizon case T < ∞, (ν T,n )n is a maximizing consumption-investment strategy, in the inﬁnite horizon case, a maximizing consumption-investment strategy is obtained by means of a diagonal extraction argument from the sequence (ν t,n ). Theorem 4.1 Let T ∈ R+ ∪ {+∞} be some given maturity. Assume the the function f f 0 ¯ deﬁning the taxation rule satisﬁes (2.5) and (4.1). Then VT = VT on S. In particular, in S∪∂ zS, the value function of the consumption investment problem under taxes coincides with the value function of the classical Merton problem. Proof. 1. We ﬁrst show that t lim Jtf (z, y; ν t,n ) = Jt (z) := E ¯ e−βs U (Cs Zs )ds + e−βT U (Zt ) ¯ ¯ ¯ n→∞ 0 for all (z, y) be in S and t ∈ R+ . Indeed, since the utility function U is p-holder continuous t,n c ¯ t,n ¯ |U (¯(s)Zs ) − U (¯(s)Zs )| ≤ c(s)|Zs − Zs |p c ¯ p Then using the Jensen inequality with the concave function x :→ x 2 : p p t,n ¯ t,n E|Zs − Zs |p = E |Zs − Zs |2 ¯ 2 ≤ t,n ¯ E|Zs − Zs |2 2 Now, using the estimate provided by lemma (4.1) p p p E|Zs − Zs |2 t,n ¯ 2 ≤ n−2 αeαt 2 = n−p αe 2 αt It follows that : t 2 p 2 p Jt (z) − Jtf (z, y; ν t,n ) ¯ ≤ n−p c(0)p + α e 2 αt − e 2 αs ds . p p 0 ¯ 2. Combining Proposition 2.2 together with (3.3), we see that V f ≤ V 0 ≤ V on S ∪ ∂ zS. ¯ In order to prove that equality holds, it suﬃces to show that V f ≥ V . In the ﬁnite horizon case, the proof is completed by taking t = T in Step 1. We next concentrate on the inﬁnite horizon case T = +∞. Fix some positive integer k. By Remark 3.1, we have : ¯ VT (z) = ¯ lim Jt (z) . t→∞ Then ¯ ¯ 1 Jtk (z) ≥ VT (z) − , k for some tk > 0. By the ﬁrst step of this proof : lim J f (z, y; ν tk ,n ) ¯ = Jtk (z) . n→∞ tk 17 Then, there exists some integer nk 1 1 Jtfk (z, y; ν tk ,nk ) ≥ Jtk (z) − ¯ ¯ ≥ VT (z) − . 2k k 3. Finally, we deﬁne the consumption-investment strategies ν k consisting in following ν tk ,nk ˆ up to tk , then liquidating at tk the risky asset position and making a null consumption on the time interval (tk , T ). Then: f 1 JT (z, y; ν k ) ˆ ≥ Jtfk (z, y; ν tk ,nk ) ≥ VT (z) − for all k ≥ 0 . ¯ k f ¯ 0 This proves that VT (z, y) ≥ lim supk→∞ JT (z, y; ν k ) ≥ VT (z). ˆ 2 5 Extension to Taxation rules with possible tax credits In this section we consider the case where the ﬁnancial market allows for tax credits, and we restrict our analysis to the case δ = 0. Our main purpose is to extend Theorem 4.1 to this context. Theorem 5.1 Consider a taxation rule deﬁned by the map f satisfying (2.5), and let δ = 0. Assume further that f (b) inf > −∞ . (5.1) b≥0 1−b ¯ f 0 Then, for all T ∈ R ∪ {+∞} and (z, y) ∈ S, we have VT (z, y) = VT (z, y). f f + Proof. 1. Since f ≤ f + , VT ≥ VT by Proposition 2.2. Now f + deﬁnes a taxation rule without tax credits as required by Condition (4.1). It then follows from Theorem 4.1 that f 0 V T ≥ VT . f 2. From (5.1) it follows that VT ≤ VT , where is the linear map f (b) (b) := α (1 − b) with α := inf . b≥0 1−b 0 f 0 We shall prove in Proposition 5.2 that VT = VT , hence VT ≤ VT . 2 In the above proof, we reduced the problem to the linear taxation rule of Example 2.3. The rest of this section is specialized to this context. 5.1 Immediate realization of capital losses for the linear taxation rule Consider the taxation rule deﬁned by f (b) := α(1 − b) for all b ≥ 0 . (5.2) 18 We ﬁrst intend to prove that it is always worth realizing capital losses whenever the tax ¯ basis exceeds unity. In other words, for each (z, y) in S, any admissible consumption- investment strategy for which the relative tax basis exceeds 1 at some stopping time τ can be improved strictly by increasing the sales of the risky asset at τ . We shall refer to this property as the optimality of the immediate realization of capital losses. In a discrete-time framework, this property was stated without proof by [4]. Proposition 5.1 Let f be the linear taxation rule of (5.2), δ = 0, and consider some ¯ initial holdings (z, y) in S. Consider some consumption-investment strategy ν := (C, L, M ) ∈ Af (z, y), and suppose that there is a ﬁnite stopping time τ ≤ T with P [τ < T ] > 0 and ν Bτ > 1 a.s. on {τ < T }. Then there exists an admissible strategy ν = V(ν, τ ) such that ˜ ˜ ˜ ˜ Y ν = Y ν , Z ν ≥ Z ν , B ν ≤ B ν , C ≥ C, ˜ and f f JT (z, y; ν ) > JT (z, y; ν) . ˜ We start by proving the following lemma which shows how to take advantage of the tax credit at time τ . Lemma 5.1 Let f be as in (5.2), δ = 0, and consider some initial holdings (z, y) be in S.¯ Consider some consumption-investment strategy ν := (C, L, M ) ∈ A f (z, y), and suppose ν that there is a ﬁnite stopping time τ ≤ T with P [τ < T ] > 0 and Bτ > 1 a.s. on {τ < T }. ¯ ¯ ¯ ¯ Deﬁne ν = (C, L, M ) by ¯ ¯ ¯ C := C and (L, M ) := (L, M ) + (1, 1)(1 − ∆Mτ )1t≥τ . (5.3) Then ν ∈ Af (z, y) and the resulting state processes are such that ¯ T ¯ ¯ Y ν = Y ν , Z ν ≥ Z ν , Bν ≤ Bν , ¯ and, almost surely, ¯ ν ¯ ν ν Bτ = 1, Zτ > Zτ on {τ < T } . ¯ ¯ ¯ Proof. 1. Since ν and ν diﬀer only by the jump at the stopping time τ , and ∆Lτ = ∆Mτ , we have ¯ Yν = Yν , and ν ν Zt¯ , Bt¯ ν ν = (Zt , Bt ) for all t < τ . ¯ By the deﬁnition of ν , it is easily seen that ¯ ν Bτ = 1 19 and ¯ ν ν Zτ − Zτ ν ν = −f Bτ − Yτν− (1 − ∆Mτ ) = αYτν− Bτ − − 1 (1 − ∆Mτ ) . (5.4) Since ν ν 1 − ∆Mτ 0 > (1 − Bτ ) = (1 − Bτ − ) on {τ < T } , (5.5) 1 + ∆Lτ − ∆Mτ it follows that Bτ − > 1 and 1 − ∆Mτ > 0 a.s. on {τ < T }. We therefore deduce from (5.4) that ¯ ν ν Zτ − Zτ > 0 on {τ < T } . ν ¯ ¯ 2. We next examine the state variable Bt¯ for t > τ . Since Y ν = Y ν , we have K ν − K ν = ¯ Y ν (B ν − B ν ), and ν ν ν ¯ d(Kt¯ − Kt ) = −(Kt− − Kt− ) dMt , with Kτ − Kτ = Yτν (1 − Bτ ) . ν ¯ ν ν ν (5.6) This linear stochastic diﬀerential equation can be solve explicitly : c c Kt¯ − Kt = (Kτ − Kτ ) e−Mt +Mτ ν ν ¯ ν ν (1 − ∆Mu ), for all t ≥ τ , τ <u≤t ν ¯ ν ν where M c denotes the continuous part of M . Since Kτ −Kτ = Yτν (1−Bτ ) < 0 on {τ < T }, this shows that ν ν Kt¯ ≤ Kt ν ν and therefore Bt¯ ≤ Bt for t ≥ τ . (5.7) ν ν 3. In this step, we intend to prove that Zt¯ ≥ Zt for t ≥ τ . From the dynamics of the processes Z ¯ ν and Z ν we have, for t > τ : t e−r(t−τ ) Zt¯ − Zt ν ν ν ¯ ν = Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu ν ¯ ν ν τ t ν ¯ ν = Zτ − Zτ + α e−r(u−τ ) Yu− Bu− − Bu− ) dMu ν ¯ ν ν τ t ν ¯ ν = Zτ − Zτ + α e−r(u−τ ) Ku− − Ku− ) dMu , ¯ ν ν τ ¯ where the last equality follows from the fact that Y ν = Y ν . We next use (5.7) and (5.6) to see that, for t ≥ τ : t e−r(t−τ ) Zt¯ − Zt ν ν ¯ ν ν ≥ Zτ − Zτ + α ¯ ν ν Ku− − Ku− ) dMu τ ν ¯ ν = Zτ − Zτ + α Kτ − Kτ − Kt¯ + Kt . ν ¯ ν ν ν ¯ ν ν Now, since Yτ = Yτ − (1 + ∆Lτ − ∆Mτ ), it follows from (5.4) and (5.5) that Zτ − Zτ = ¯ ν ν −α (Kτ − Kτ ). Hence : e−r(t−τ ) Zt¯ − Zt ν ν ν ν ≥ −α Kt¯ − Kt ≥ 0 for t ≥ τ , 20 by (5.7). ¯ 4. Clearly, ν satisﬁes Conditions A1-A2-A3, and Z ν ≥ Z ν by the previous steps of this ¯ proof. Hence ν ∈ Af (z, y). ¯ 2 ˜ ˜ ˜ ˜ Proof of Proposition 5.1. Consider the slight modiﬁcation ν = (C, L, M ) of the consumption- investment strategy introduced in Lemma 5.1 : ˜ ¯ ν ν Ct := Ct + ξ Zt˜ − Zt 1t≥τ and ˜ ˜ L, M := ¯ ¯ L, M , (5.8) ˜ ˜ ¯ ¯ ν ν where ξ is an arbitrary positive constant. Observe that (Y ν , B ν ) = (Y ν , B ν ), and Zt˜ = Zt¯ ˜ ˜ ˜ for t ≤ τ . In order to check the admissibility of the triple (C, L, M ), we repeat Step 3 of the proof of Lemma 5.1 : t e−r(t−τ ) Zt˜ − Zt ν ν = ˜ ν ν Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu ν ν ˜ ν τ t +ξ e−r(u−τ ) Zu − Zu du ˜ ν ν τ t = ¯ ν ν Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu ν ν ¯ ν τ t +ξ e−r(u−τ ) Zu − Zu du ˜ ν ν τ t ··· ≥ ξ e−r(u−τ ) Zu − Zu du . ˜ ν ν τ ν ˜ ν ν ¯ ν Since Zτ − Zτ = Zτ − Zτ ≥ 0 by Lemma 5.1, it follows from the Gronwall lemma that Zt˜ ν ≥ Ztν a.s. Hence ν ∈ Af (z, y). ˜ T ν ˜ ν ¯ Recall from Lemma 5.1 that Zτ > Zτ on {τ < T }. Since the process (Z ν − Z ν ) is right- continuous, the strict inequality holds on some nontrivial time interval almost surely on ¯ {τ < T }. Hence C > C with positive Lebesgue⊗P measure, and f f JT (z, y; ν ) > JT (z, y; ν) . ˜ 2 5.2 Reduction to the tax-free ﬁnancial market In view of the optimality of the immediate realization of capital losses stated in Proposition ¯ 5.1, we expect that, given (z, y) ∈ S, and for all constant ε > 0, the problem of maximizing f JT (z, y; ν) can be restricted to those admissible control processes ν inducing a cumulated tax credit bounded by ε. This result, stated in Lemma 5.2, will allow to prove that V f = V 0 in the context of a linear taxation rule, hence completing the proof of Theorem 5.1. Lemma 5.2 Let f be the linear taxation rule deﬁned in (5.2), δ = 0, and let t > 0 be some ﬁnite maturity, ε > 0, and ν in Af (z, y). Then, there exists ν ε = (C ε , Lε , M ε ) in Af (z, y) such that t ε ε ε Jtf (z, y; ν ε ) ≥ Jtf (z, y; ν) and ν ν ν Bu− − 1 Yu− dMu ≤ ε a.s. 0 21 Proof. Let θ0 := 0, ν 0 := ν, n n ν ν θn+1 := t ∧ inf{s > θn : (Bs − 1) 1 ∨ Hs > ε} , with n n νn c n Hs ν := Ys− eMs ν (1 − ∆Mu )−1 , ν u≤s and ν n+1 := ν n 1{θn+1 =t} + V(ν n , θn+1 )1{θn+1 <t} . nc n where M ν denotes the continuous part of M ν , and V is deﬁned in Proposition 5.1 so as to take advantage of the tax credit while decreasing the relative tax basis. We shall simply n n n denote (Z n , Y n , B n ) := (Z ν , Y ν , B ν ). Then : C n+1 ≥ C n , Y n = Y 0 , Z n+1 ≥ Z n , B n+1 ≤ B n , (5.9) and n ν n (Bs − 1) 1 ∨ Hs ≤ ε for t ≤ θn+1 . (5.10) Clearly, for a.e. ω ∈ Ω, θn (ω) = t for n ≥ N (ω), where N (ω) is some suﬃciently large integer. Therefore the sequence ν n (ω) is constant for n ≥ N (ω) and ν n −→ ν ε a.s. for some ν ε ∈ Af (z, y). Also, by construction of the sequence ν n , we have : ε n νs = νs for s ≤ θn+1 . (5.11) By (5.9), it is immediately checked that Jtf (z, y; ν ε ) ≥ Jtf (z, y; ν). We ﬁnally use (5.10) o and (5.11), together with Itˆ’s lemma, to compute that t t ε ε ε ε ε ε ν ν ν Bu− − 1 Yu− dMu ≤ ε Yu− (Hs )−1 dMu ν ν ν 0 0 t νε c ε ε = ε e−Ms ν ν (1 − ∆Mu )dMu 0 u≤s νε −Mt νε = ε 1 − e (1 − ∆Mu ) u≤t ≤ ε, ε by the fact that 1 − ∆M ν ≤ 1. 2 We are now ready to state the extension of Theorem 4.1 to the linear taxation rule case with tax credits. 22 Proposition 5.2 Consider the linear taxation rule of (5.2), and let δ = 0. Then, for all ¯ f 0 T ∈ R ∪ {+∞} and (z, y) ∈ S, we have VT (z, y) = VT (z, y). Proof. The result is trivial for (z, y) ∈ ∂ yS. We then assume in the sequel that (z, y) ∈ S ∪ ∂ zS. f f+ 1. Since f ≤ f + := max{f, 0}, we have VT ≥ VT . Now f + deﬁnes a taxation rule without f + tax credits as required by Condition (4.1). It then follows from Theorem 4.1 that VT = 0 f 0 VT and therefore VT ≥ VT . 2. We next concentrate on the reverse inequality. Notice that J∞ (z, y, ν) ≤ lim inf Jtf (z, y, ν). f t→∞ This follows by the monotone convergence theorem and the fact that the utility function is non-negative. Therefore, in order to prove the required inequality, it is suﬃcient to show that, for any ﬁxed ﬁnite maturity 0 < t ≤ T : Jtf (z, y; ν) ≤ VT (z, y) for all ν ∈ Af (z, y) . 0 (5.12) Let ε > 0, and consider the consumption-investment strategy ν ε ∈ Af (z, y) deﬁned in ¯ Lemma 5.2. Let Z ε be the wealth process induced by the strategy ν ε in a tax-free market, i.e. with f ≡ 0. Since the taxation rule f allows for tax credits, there is no reason for the ¯ process Z ε to be non-negative. Using the dynamics of Z ν , it follows from Lemma 5.2 that u νε ε ε ε e−ru Zu − Zu ¯ε = α ν ν ν (1 − Bs− )Ys− dMs ≥ −αε . (5.13) 0 Now, from Lemma 5.2 and the increase of the utility function U and its concavity, we have Jtf (z, y; ν) ≤ Jtf (z, y; ν ε ) ≤ Jtf (z + 2αε, y; ν ε ) = Jt0 (z + 2αε, y; ν ε ) νε +e−βt E U (2αεert + Zt ) − U (2αεert + Zt ) . ¯ε Using (5.13) together with the increase and the concavity of U , this provides : ε ε Jtf (z, y; ν) ≤ Jt0 (z + 2αε, y; ν ε ) + e−βt E U (2αεert + Zt ) − U (αεert + Zt ) ν ν ε ν ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t E U αεert + Zt ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t U αεert = Jt0 (z + 2αε, y; ν ε ) + pe−βt U αεert . Now, observe from (5.13) that ν ε ∈ Af (z + 2αε, y). Then : Jtf (z, y; ν) ≤ Vt0 (z + 2αε, y) + e−βt U αεert = Vt (z + 2αε) + e−βt U αεert , ¯ where the last equality follows from Propositions 3.1 and 3.2. The required inequality (5.12) is obtained by sending ε to zero and using the continuity of the Merton value function V . ¯ 2 23 References [1] J. Cox and C.F. Huang (1989). Optimal consumption and portfolio policies when asset prices follow a diﬀusion process, Journal of Economic Theory 49, 33-83. c [2] J. Cvitani´ and I. Karatzas (1992). Convex duality in constrained portfolio opti- mization. Annals of Applied Probability 2, 767-818. [3] M.H.A. Davis and A.R. Norman (1990). Portfolio selection with transaction costs. Mathematics of Operations Research 15, 676-713. [4] R.M. Dammon, C.S. Spatt and H.H. Zhang (2001). Optimal consumption and in- vestment with capital gains taxes. The Review of Financial Studies 14, 583-616. [5] I. Karatzas, J.P. Lehoczky and S.E. Shreve (1987). Optimal portfolio and consump- tion decisions for a ”small investor” on a ﬁnite horizon. SIAM Journal on Control and Optimization 25, 1557-1586. [6] R.C. Merton (1969). Lifetime portfolio selection under uncertainty: the continuous- time model. Review of Economic Statistics 51, 247-257. [7] R.C. Merton (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3, 373-413. [8] S.R. Pliska (1986). A stochastic calsulus model of continuous trading: optimal port- folios, Mathematics of Operations Research 11, 371-382. [9] E. Jouini, P.-F. Koehl and N. Touzi (1997). Optimal investment with taxes : an opti- mal control problem with endogeneous delay, Nonlinear Analysis : Theory, Methods and Applications 37, 31-56. [10] E. Jouini, P.-F. Koehl and N. Touzi (1999). Optimal investment with taxes: an existence result, Journal of Mathematical Economics 33, 373-388. [11] M. Gallmeyer, R. Kaniel and S. Tompaidis (2002). Tax Management Strategies with Multiple Risky Assets, preprint. 24

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