Modeling continuous-time financial markets with capital gains taxes ∗ Imen Ben Tahar

Nizar Touzi

CEREMADE, Universit´e Paris Dauphine

Centre de Recherche

and Universit´e L´eonard De Vinci

en Economie et Statistique

imen.ben tah[email protected]

and CEREMADE, Universit´e Paris Dauphine [email protected]

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 continuoustime, 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] for incomplete markets, Cvitani´c 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. Compared 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 depreciating 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 linear 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 particular, 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 2.1

The Model 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 Bt2 =

5St0 + 2St2 7St2

and Bt3 =

5St0 + 2St2 + 2St3 . 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 non-increasing, f (1) = 0, and lim inf b↑1

4

f (b) > −∞ . b−1

(2.5)

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. ˜ := (L ˜ t , t ≥ 0) the process of cumulative transfers form the bank account We denote by L ˜ := (M ˜ t , t ≥ 0) the process of cumulative transfers from the to the risky assets one, and M ˜ and M ˜ are two F−adapted, right-continuous, risky assets account to the bank. Here, L ˜ 0− = M ˜ 0− = 0. non-decreasing processes with L 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. ˜ M ˜ ), we denote by Xt the position on the Given a consumption-investment strategy (C, L, 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 ,

5

(2.6)

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 formulation of our problem. ˜ M ˜ ) is said to be admissible if the resulting The consumption-investment strategy (C, L, 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)

˜ M ˜ ), we introduce the stopping time : Finally, given an admissible strategy (C, L, τ

:= inf{t ≥ 0 : Yt 6∈ (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 strategy can be described by means of the non-decreasing right-continuous processes (Lt , Mt )t≥0 ˜ t, M ˜ t )t≥0 by which are related to (L Z t Z t −1 ˜ ˜ t, t < τ . Lt := Yt dLt and Mt := Yt−1 dM 0

0

Here, dLt and dMt represent the proportion of transfers of risky assets.

2.6

Controlled dynamics

˜ M ˜ ) be an admissible strategy, and define (L, M ) as in the previous paragraph. Let (C, L, 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 dXtν = (rXtν − Ct )dt − Yt− dLt + Yt− 1 − f (Btνδ − ) 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 ν dYt = Yt− + dLt − dMt and Y0− = y . (2.11) St This implies that the total wealth evolves according to ν [(µ − r)dt + σdW ] − Y ν f (B ν )dM dZtν = r(Ztν − Ct )dt + Yt− t t t− tδ − and Z0− = y + x .

6

(2.12)

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 dKtν = Yt− dLt − Kt− dMt and K0− = y,

(2.13)

ν = 1. Observe that the contribution of the sales in the dynamics of K is evaluated since B0− t ν at the basis price. We then define the relative basis process B by

Btν = 1{Ytν =0} +

Ktν 1 ν . Ytν {Yt 6=0}

(2.14)

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 Rt 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 6= 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 +∞. ¯ We Definition 2.1 Let ν = (C, L, M ) be a triple of F−adapted processes, and (z, y) ∈ S. 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 ˜t ˜t and dMt = dYM , instead of the volume of strategy by means of the proportions dLt = YdL t− t− ˜ t and dM ˜ t . This modification was needed for the specification of the tax basis transfers, dL

7

˜ M ˜ ), the dynamics of the by means of the process K defined above. Indeed, in terms of (L, state variables X and Y are given by : dYt = Yt−

dSt ˜ t − dM ˜ t and dXt = r(Xt − ct )dt − dL ˜ t + (1 − f (Bt − )) dM ˜t , + dL δ St

but the relative basis is defined by means of the process K whose dynamics are given by ˜t − dKt = dL

Kt− ˜ 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 p

U (c) :=

for all c ≥ 0,

where 0 < p < 1 is a given parameter. We next consider the investment-consumption criterion Z t f −βt ν Jt (z, y; ν) := E e U (Ct )dt + U (Xt )1{t 0. By the first step of this proof : lim J f (z, y; ν tk ,n ) n→∞ tk 17

= J¯tk (z) .

Then, there exists some integer nk 1 Jtfk (z, y; ν tk ,nk ) ≥ J¯tk (z) − 2k

1 ≥ V¯T (z) − . k

3. Finally, we define the consumption-investment strategies νˆk consisting in following ν tk ,nk up to tk , then liquidating at tk the risky asset position and making a null consumption on the time interval (tk , T ). Then: JTf (z, y; νˆk )

1 ≥ Jtfk (z, y; ν tk ,nk ) ≥ V¯T (z) − for all k ≥ 0 . k

This proves that VT0 (z, y) ≥ lim supk→∞ JTf (z, y; νˆk ) ≥ V¯T (z).

5

2

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 inf

b≥0

f (b) 1−b

> −∞ .

(5.1)

¯ we have V f (z, y) = V 0 (z, y). Then, for all T ∈ R ∪ {+∞} and (z, y) ∈ S, T T +

Proof. 1. Since f ≤ f + , VTf ≥ VTf 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 VTf ≥ VT0 . 2. From (5.1) it follows that VTf ≤ VT` , where ` is the linear map `(b) := α (1 − b) with α := inf

b≥0

f (b) . 1−b

We shall prove in Proposition 5.2 that VT` = VT0 , hence VTf ≤ VT0 .

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 . 18

(5.2)

We first intend to prove that it is always worth realizing capital losses whenever the tax ¯ any admissible consumptionbasis exceeds unity. In other words, for each (z, y) in S, 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 ¯ Consider some consumption-investment strategy ν := (C, L, M ) initial holdings (z, y) in S. ∈ Af (z, y), and suppose that there is a finite stopping time τ ≤ T with P [τ < T ] > 0 and Bτν > 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 JTf (z, y; ν˜) > JTf (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. f Consider some consumption-investment strategy ν := (C, L, M ) ∈ A (z, y), and suppose that there is a finite stopping time τ ≤ T with P [τ < T ] > 0 and Bτν > 1 a.s. on {τ < T }. ¯ L, ¯ M ¯ ) by Define ν¯ = (C, ¯ M ¯ ) := (L, M ) + (1, 1)(1 − ∆Mτ )1t≥τ . C¯ := C and (L,

(5.3)

Then ν¯ ∈ AfT (z, y) and the resulting state processes are such that Y ν¯ = Y ν , Z ν¯ ≥ Z ν , B ν¯ ≤ B ν , and, almost surely, Bτν¯ = 1, Zτν¯ > Zτν

on {τ < T } .

¯ τ = ∆M ¯τ , Proof. 1. Since ν and ν¯ differ only by the jump at the stopping time τ , and ∆L we have Y ν¯ = Y ν , and Ztν¯ , Btν¯

= (Ztν , Btν ) for all t < τ .

By the definition of ν¯, it is easily seen that Bτν¯ = 1 19

and Zτν¯ − Zτν

= −f Bτν− Yτν− (1 − ∆Mτ ) = αYτν− Bτν− − 1 (1 − ∆Mτ ) .

(5.4)

1 − ∆Mτ 1 + ∆Lτ − ∆Mτ

(5.5)

Since 0 > (1 − Bτν ) = (1 − Bτν− )

on {τ < T } ,

it follows that Bτ − > 1 and 1 − ∆Mτ > 0 a.s. on {τ < T }. We therefore deduce from (5.4) that Zτν¯ − Zτν > 0 on {τ < T } . 2. We next examine the state variable Btν¯ for t > τ . Since Y ν¯ = Y ν , we have K ν¯ − K ν = Y ν (B ν¯ − B ν ), and ν¯ ν d(Ktν¯ − Ktν ) = −(Kt− − Kt− ) dMt , with Kτν¯ − Kτν = Yτν (1 − Bτν ) .

(5.6)

This linear stochastic differential equation can be solve explicitly : Y c c Ktν¯ − Ktν = (Kτν¯ − Kτν ) e−Mt +Mτ (1 − ∆Mu ), for all t ≥ τ , τ τ : Z t −r(t−τ ) ν¯ ν ν¯ ν ν ν¯ ν e Zt − Zt = Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t ν ν¯ ν = Zτν¯ − Zτν + α e−r(u−τ ) Yu− Bu− − Bu− ) dMu τ Z t ν¯ ν = Zτν¯ − Zτν + α e−r(u−τ ) Ku− − Ku− ) dMu , 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 : Z t −r(t−τ ) ν¯ ν ν¯ ν ν¯ ν e Zt − Zt ≥ Zτ − Zτ + α Ku− − Ku− ) dMu τ ν¯ ν = Zτ − Zτ + α Kτν¯ − Kτν − Ktν¯ + Ktν . Now, since Yτ = Yτ − (1 + ∆Lτ − ∆Mτ ), it follows from (5.4) and (5.5) that Zτν¯ − Zτν = −α (Kτν¯ − Kτν ). Hence : e−r(t−τ ) Ztν¯ − Ztν

≥ −α Ktν¯ − Ktν 20

≥ 0 for t ≥ τ ,

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 ˜ L, ˜ M ˜ ) of the consumptionProof of Proposition 5.1. Consider the slight modification ν˜ = (C, investment strategy introduced in Lemma 5.1 : ˜ M ˜ := L, ¯ M ¯ , C˜t := C¯t + ξ Ztν˜ − Ztν 1t≥τ and L, (5.8) where ξ is an arbitrary positive constant. Observe that (Y ν˜ , B ν˜ ) = (Y ν¯ , B ν¯ ), and Ztν˜ = Ztν¯ ˜ L, ˜ M ˜ ), we repeat Step 3 of for t ≤ τ . In order to check the admissibility of the triple (C, the proof of Lemma 5.1 : Z t −r(t−τ ) ν˜ ν ν˜ ν ν ν˜ ν e Zt − Zt = Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t +ξ e−r(u−τ ) Zuν˜ − Zuν du τ Z t ν ν¯ ν = Zτν¯ − Zτν − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t +ξ e−r(u−τ ) Zuν˜ − Zuν du τ Z t ··· ≥ ξ e−r(u−τ ) Zuν˜ − Zuν du . τ

Since Zτν˜ − Zτν = Zτν¯ − Zτν ≥ 0 by Lemma 5.1, it follows from the Gronwall lemma that Ztν˜ ≥ Ztν a.s. Hence ν˜ ∈ AfT (z, y). Recall from Lemma 5.1 that Zτν˜ > Zτν on {τ < T }. Since the process (Z ν¯ − Z ν ) is rightcontinuous, the strict inequality holds on some nontrivial time interval almost surely on {τ < T }. Hence C¯ > C with positive Lebesgue⊗P measure, and JTf (z, y; ν˜) > JTf (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 ¯ and for all constant ε > 0, the problem of maximizing 5.1, we expect that, given (z, y) ∈ S, 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 Z t νε ε f f νε ε Bu− − 1 Yu− dMuν ≤ ε a.s. Jt (z, y; ν ) ≥ Jt (z, y; ν) and 0

21

Proof. Let θ0 := 0, ν 0 := ν, n

θn+1 := t ∧ inf{s > θn : (Bsν − 1) 1 ∨ Hsν

n

> ε} ,

with Hs ν

n

n

νn c

Y

ν Ms := Ys− e

n

(1 − ∆Muν )−1 ,

u≤s

and ν n+1 := ν n 1{θn+1 =t} + V(ν n , θn+1 )1{θn+1 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 Z u ε νε νε −ru ¯ ε νε (1 − Bs− )Ys− dMsν ≥ −αε . (5.13) e Zu − Zu = α 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 ) ε +e−βt E U (2αεert + Ztν ) − U (2αεert + Z¯tε ) . 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 + Ztν ) − U (αεert + Ztν ) ε ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t E U 0 αεert + Ztν ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t U 0 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

= V¯t (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. [2] J. Cvitani´c and I. Karatzas (1992). Convex duality in constrained portfolio optimization. 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 investment 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 consumption 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 continuoustime 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 portfolios, Mathematics of Operations Research 11, 371-382. [9] E. Jouini, P.-F. Koehl and N. Touzi (1997). Optimal investment with taxes : an optimal 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

Nizar Touzi

CEREMADE, Universit´e Paris Dauphine

Centre de Recherche

and Universit´e L´eonard De Vinci

en Economie et Statistique

imen.ben tah[email protected]

and CEREMADE, Universit´e Paris Dauphine [email protected]

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 continuoustime, 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] for incomplete markets, Cvitani´c 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. Compared 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 depreciating 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 linear 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 particular, 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 2.1

The Model 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 Bt2 =

5St0 + 2St2 7St2

and Bt3 =

5St0 + 2St2 + 2St3 . 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 non-increasing, f (1) = 0, and lim inf b↑1

4

f (b) > −∞ . b−1

(2.5)

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. ˜ := (L ˜ t , t ≥ 0) the process of cumulative transfers form the bank account We denote by L ˜ := (M ˜ t , t ≥ 0) the process of cumulative transfers from the to the risky assets one, and M ˜ and M ˜ are two F−adapted, right-continuous, risky assets account to the bank. Here, L ˜ 0− = M ˜ 0− = 0. non-decreasing processes with L 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. ˜ M ˜ ), we denote by Xt the position on the Given a consumption-investment strategy (C, L, 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 ,

5

(2.6)

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 formulation of our problem. ˜ M ˜ ) is said to be admissible if the resulting The consumption-investment strategy (C, L, 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)

˜ M ˜ ), we introduce the stopping time : Finally, given an admissible strategy (C, L, τ

:= inf{t ≥ 0 : Yt 6∈ (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 strategy can be described by means of the non-decreasing right-continuous processes (Lt , Mt )t≥0 ˜ t, M ˜ t )t≥0 by which are related to (L Z t Z t −1 ˜ ˜ t, t < τ . Lt := Yt dLt and Mt := Yt−1 dM 0

0

Here, dLt and dMt represent the proportion of transfers of risky assets.

2.6

Controlled dynamics

˜ M ˜ ) be an admissible strategy, and define (L, M ) as in the previous paragraph. Let (C, L, 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 dXtν = (rXtν − Ct )dt − Yt− dLt + Yt− 1 − f (Btνδ − ) 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 ν dYt = Yt− + dLt − dMt and Y0− = y . (2.11) St This implies that the total wealth evolves according to ν [(µ − r)dt + σdW ] − Y ν f (B ν )dM dZtν = r(Ztν − Ct )dt + Yt− t t t− tδ − and Z0− = y + x .

6

(2.12)

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 dKtν = Yt− dLt − Kt− dMt and K0− = y,

(2.13)

ν = 1. Observe that the contribution of the sales in the dynamics of K is evaluated since B0− t ν at the basis price. We then define the relative basis process B by

Btν = 1{Ytν =0} +

Ktν 1 ν . Ytν {Yt 6=0}

(2.14)

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 Rt 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 6= 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 +∞. ¯ We Definition 2.1 Let ν = (C, L, M ) be a triple of F−adapted processes, and (z, y) ∈ S. 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 ˜t ˜t and dMt = dYM , instead of the volume of strategy by means of the proportions dLt = YdL t− t− ˜ t and dM ˜ t . This modification was needed for the specification of the tax basis transfers, dL

7

˜ M ˜ ), the dynamics of the by means of the process K defined above. Indeed, in terms of (L, state variables X and Y are given by : dYt = Yt−

dSt ˜ t − dM ˜ t and dXt = r(Xt − ct )dt − dL ˜ t + (1 − f (Bt − )) dM ˜t , + dL δ St

but the relative basis is defined by means of the process K whose dynamics are given by ˜t − dKt = dL

Kt− ˜ 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 p

U (c) :=

for all c ≥ 0,

where 0 < p < 1 is a given parameter. We next consider the investment-consumption criterion Z t f −βt ν Jt (z, y; ν) := E e U (Ct )dt + U (Xt )1{t 0. By the first step of this proof : lim J f (z, y; ν tk ,n ) n→∞ tk 17

= J¯tk (z) .

Then, there exists some integer nk 1 Jtfk (z, y; ν tk ,nk ) ≥ J¯tk (z) − 2k

1 ≥ V¯T (z) − . k

3. Finally, we define the consumption-investment strategies νˆk consisting in following ν tk ,nk up to tk , then liquidating at tk the risky asset position and making a null consumption on the time interval (tk , T ). Then: JTf (z, y; νˆk )

1 ≥ Jtfk (z, y; ν tk ,nk ) ≥ V¯T (z) − for all k ≥ 0 . k

This proves that VT0 (z, y) ≥ lim supk→∞ JTf (z, y; νˆk ) ≥ V¯T (z).

5

2

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 inf

b≥0

f (b) 1−b

> −∞ .

(5.1)

¯ we have V f (z, y) = V 0 (z, y). Then, for all T ∈ R ∪ {+∞} and (z, y) ∈ S, T T +

Proof. 1. Since f ≤ f + , VTf ≥ VTf 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 VTf ≥ VT0 . 2. From (5.1) it follows that VTf ≤ VT` , where ` is the linear map `(b) := α (1 − b) with α := inf

b≥0

f (b) . 1−b

We shall prove in Proposition 5.2 that VT` = VT0 , hence VTf ≤ VT0 .

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 . 18

(5.2)

We first intend to prove that it is always worth realizing capital losses whenever the tax ¯ any admissible consumptionbasis exceeds unity. In other words, for each (z, y) in S, 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 ¯ Consider some consumption-investment strategy ν := (C, L, M ) initial holdings (z, y) in S. ∈ Af (z, y), and suppose that there is a finite stopping time τ ≤ T with P [τ < T ] > 0 and Bτν > 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 JTf (z, y; ν˜) > JTf (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. f Consider some consumption-investment strategy ν := (C, L, M ) ∈ A (z, y), and suppose that there is a finite stopping time τ ≤ T with P [τ < T ] > 0 and Bτν > 1 a.s. on {τ < T }. ¯ L, ¯ M ¯ ) by Define ν¯ = (C, ¯ M ¯ ) := (L, M ) + (1, 1)(1 − ∆Mτ )1t≥τ . C¯ := C and (L,

(5.3)

Then ν¯ ∈ AfT (z, y) and the resulting state processes are such that Y ν¯ = Y ν , Z ν¯ ≥ Z ν , B ν¯ ≤ B ν , and, almost surely, Bτν¯ = 1, Zτν¯ > Zτν

on {τ < T } .

¯ τ = ∆M ¯τ , Proof. 1. Since ν and ν¯ differ only by the jump at the stopping time τ , and ∆L we have Y ν¯ = Y ν , and Ztν¯ , Btν¯

= (Ztν , Btν ) for all t < τ .

By the definition of ν¯, it is easily seen that Bτν¯ = 1 19

and Zτν¯ − Zτν

= −f Bτν− Yτν− (1 − ∆Mτ ) = αYτν− Bτν− − 1 (1 − ∆Mτ ) .

(5.4)

1 − ∆Mτ 1 + ∆Lτ − ∆Mτ

(5.5)

Since 0 > (1 − Bτν ) = (1 − Bτν− )

on {τ < T } ,

it follows that Bτ − > 1 and 1 − ∆Mτ > 0 a.s. on {τ < T }. We therefore deduce from (5.4) that Zτν¯ − Zτν > 0 on {τ < T } . 2. We next examine the state variable Btν¯ for t > τ . Since Y ν¯ = Y ν , we have K ν¯ − K ν = Y ν (B ν¯ − B ν ), and ν¯ ν d(Ktν¯ − Ktν ) = −(Kt− − Kt− ) dMt , with Kτν¯ − Kτν = Yτν (1 − Bτν ) .

(5.6)

This linear stochastic differential equation can be solve explicitly : Y c c Ktν¯ − Ktν = (Kτν¯ − Kτν ) e−Mt +Mτ (1 − ∆Mu ), for all t ≥ τ , τ τ : Z t −r(t−τ ) ν¯ ν ν¯ ν ν ν¯ ν e Zt − Zt = Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t ν ν¯ ν = Zτν¯ − Zτν + α e−r(u−τ ) Yu− Bu− − Bu− ) dMu τ Z t ν¯ ν = Zτν¯ − Zτν + α e−r(u−τ ) Ku− − Ku− ) dMu , 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 : Z t −r(t−τ ) ν¯ ν ν¯ ν ν¯ ν e Zt − Zt ≥ Zτ − Zτ + α Ku− − Ku− ) dMu τ ν¯ ν = Zτ − Zτ + α Kτν¯ − Kτν − Ktν¯ + Ktν . Now, since Yτ = Yτ − (1 + ∆Lτ − ∆Mτ ), it follows from (5.4) and (5.5) that Zτν¯ − Zτν = −α (Kτν¯ − Kτν ). Hence : e−r(t−τ ) Ztν¯ − Ztν

≥ −α Ktν¯ − Ktν 20

≥ 0 for t ≥ τ ,

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 ˜ L, ˜ M ˜ ) of the consumptionProof of Proposition 5.1. Consider the slight modification ν˜ = (C, investment strategy introduced in Lemma 5.1 : ˜ M ˜ := L, ¯ M ¯ , C˜t := C¯t + ξ Ztν˜ − Ztν 1t≥τ and L, (5.8) where ξ is an arbitrary positive constant. Observe that (Y ν˜ , B ν˜ ) = (Y ν¯ , B ν¯ ), and Ztν˜ = Ztν¯ ˜ L, ˜ M ˜ ), we repeat Step 3 of for t ≤ τ . In order to check the admissibility of the triple (C, the proof of Lemma 5.1 : Z t −r(t−τ ) ν˜ ν ν˜ ν ν ν˜ ν e Zt − Zt = Zτ − Zτ − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t +ξ e−r(u−τ ) Zuν˜ − Zuν du τ Z t ν ν¯ ν = Zτν¯ − Zτν − e−r(u−τ ) Yu− f (Bu− ) − f (Bu− ) dMu τ Z t +ξ e−r(u−τ ) Zuν˜ − Zuν du τ Z t ··· ≥ ξ e−r(u−τ ) Zuν˜ − Zuν du . τ

Since Zτν˜ − Zτν = Zτν¯ − Zτν ≥ 0 by Lemma 5.1, it follows from the Gronwall lemma that Ztν˜ ≥ Ztν a.s. Hence ν˜ ∈ AfT (z, y). Recall from Lemma 5.1 that Zτν˜ > Zτν on {τ < T }. Since the process (Z ν¯ − Z ν ) is rightcontinuous, the strict inequality holds on some nontrivial time interval almost surely on {τ < T }. Hence C¯ > C with positive Lebesgue⊗P measure, and JTf (z, y; ν˜) > JTf (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 ¯ and for all constant ε > 0, the problem of maximizing 5.1, we expect that, given (z, y) ∈ S, 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 Z t νε ε f f νε ε Bu− − 1 Yu− dMuν ≤ ε a.s. Jt (z, y; ν ) ≥ Jt (z, y; ν) and 0

21

Proof. Let θ0 := 0, ν 0 := ν, n

θn+1 := t ∧ inf{s > θn : (Bsν − 1) 1 ∨ Hsν

n

> ε} ,

with Hs ν

n

n

νn c

Y

ν Ms := Ys− e

n

(1 − ∆Muν )−1 ,

u≤s

and ν n+1 := ν n 1{θn+1 =t} + V(ν n , θn+1 )1{θn+1 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 Z u ε νε νε −ru ¯ ε νε (1 − Bs− )Ys− dMsν ≥ −αε . (5.13) e Zu − Zu = α 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 ) ε +e−βt E U (2αεert + Ztν ) − U (2αεert + Z¯tε ) . 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 + Ztν ) − U (αεert + Ztν ) ε ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t E U 0 αεert + Ztν ≤ Jt0 (z + 2αε, y; ν ε ) + αεe(r−β)t U 0 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

= V¯t (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

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