Investors preference for a positive tax rate depends on the level of the interest rate

Transcription

1 Math Finan Econ (27) 1: DOI 117/s x Investors preference for a positive tax rate depends on the level of the interest rate Cristin Buescu Abel Cadenillas Received: 4 February 27 / Accepted: 9 October 27 / Published online: 1 November 27 Springer-Verlag 27 Abstract In a financial market with only one stock, Cadenillas and Pliska (Financ Stoch 3: , 1999) showed that sometimes investors can take advantage of a positive tax rate to maximize their portfolio return Buescu et al (Math Finance 17: , 27) generalized this surprising result to a market with one stock and one bank account with zero interest rate We consider instead a financial market with one stock and one bank account with positive interest rate As in the papers above, we assume that there are taxes and transaction costs in the financial market We succeed in solving the problem of an investor who wants to maximize the long-run growth rate of his investment, even though the positivity of the interest rate increases the dimensionality of the problem and the difficulty of the computations We characterize how the investors preference for a positive tax rate depends on the interest rate level: investors prefer a positive tax rate when the level of the interest rate is low, and the opposite occurs when the level of the interest rate is high Keywords Portfolio management Taxes Interest rate Transaction costs Optimal stopping JEL Classification G11 H2 C63 E44 Most of the contributions of C Buescu were made during his doctoral studies at the University of Alberta The research of C Buescu and A Cadenillas was supported by the Social Sciences and Humanities Research Council of Canada grants and We are grateful to Stanley R Pliska for comments and suggestions to a previous version of the paper, and to the associate editor and referees for constructive remarks Existing errors are our sole responsibility C Buescu Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA A Cadenillas (B) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada acadenil@mathualbertaca

2 164 C Buescu, A Cadenillas 1 Introduction We consider a financial market consisting of one risk-free asset with positive interest rate and several stocks modeled by geometric Brownian motions An investor wants to maximize the long-run growth rate of his portfolio in the presence of taxes and transaction costs Taxation on capital gains plays a crucial role in investment decisions, and this importance has been well acknowledged in the mainstream finance literature The incorporation of taxes not only makes financial models more realistic, but also creates challenging mathematical and computational problems (surprisingly, taxation has received little attention in the financial mathematics literature) Important papers on portfolio management with taxes include the classical works of Constantinides [9,1], and the more recent works of Ben Tahar et al [3,4], Buescu et al [7], Cadenillas and Pliska [8], Dammon and Spatt [12], Dammon et al [13,14], DeMiguel and Uppal [15], Gallmeyer et al [18], Hur [19], Jouini et al [2], and Leland [23] The works by Buescu et al [7] and Cadenillas and Pliska [8] are the closest to the present paper Cadenillas and Pliska [8] were the first to provide a rigorous mathematical proof that it can be optimal to make a transaction not only when there is a loss, but also when there is a gain Another surprising result was that sometimes investors are better off with a positive tax rate Those two results were obtained for investors maximizing the long-run growth-rate of the portfolio in a market with only one stock, whose price was modeled by a geometric Brownian motion Those results were later generalized by Buescu et al [7] to a more realistic financial market which consisted not only of one stock, but also of one bank account with zero interest rate The main difference between our paper and that of Buescu et al [7] is in the interest rate of the risk-free asset: they assume that the interest rate is zero, while we assume that it is positive A positive interest rate makes the model more realistic, but increases the dimensionality of the problem and the difficulty of the analysis (we shall see the details below) After presenting the model with multiple stocks and one bank account, we use numerical methods to obtain the solution in the case of one stock and one bank account We show that the two main economic results of Buescu et al [7] are also valid in our financial market: it can be optimal to make a transaction when there is a gain, and sometimes investors prefer a positive tax rate In addition, our model enables us to investigate how the level of the interest rate influences investors preference for a positive tax rate This analysis could not be done previously by Buescu et al [7] because they assumed that the level of interest rate was zero We show the existence of a threshold level of the interest rate beyond which a positive tax rate is no longer advantageous to the investor s portfolio growth In other words, investors prefer a positive tax rate when the level of the interest rate is low, and the opposite occurs when the level of the interest rate is high The stochastic model of the financial market is presented in Sect 2 The stock prices are modeled by geometric Brownian motions Transaction costs are paid as a fraction of the entire wealth, while gains are taxed at a fixed rate with symmetric taxation In the case of losses, we assume that the tax credits are realized as actual cash payments The objective of the investor is to maximize the long-run growth rate of the portfolio wealth Applying renewal theory arguments, we reduce the problem to that of optimally managing the portfolio in the first transaction cycle An application of the dynamic programming principle yields the equivalent moving boundary problem While in Buescu et al [7] this problem involved an ordinary differential equation with explicit solution, in our case the moving boundary problem contains a partial differential equation without explicit solution In Sect 3, we apply the Markov chain approximating method of Kushner and Dupuis [22] to solve numerically the moving boundary problem in the case of one stock and one risk-free asset We show

3 Investors preference for a positive tax rate depends on the level of the interest rate 165 the interplay between the interest rate and the preference of the investor for a positive tax rate We also provide a sensitivity analysis of the results to various model parameters The conclusions of this paper are presented in the last section 2 The financial market We consider a financial market modeled by a probability space {,F, P} with a filtration {F t } t that is the P-augmentation of the filtration generated by an n-dimensional standard Brownian motion {Wt 1, W t 2,,W t n} t with W 1 = W 2 = = W n = In this financial market we consider one risk-free asset (bank account) and n risky assets (stocks) At time t the price of the risk-free asset is St,andthepricesofthen stocks are St 1, S2 t,,sn t The dynamics of the asset prices are given, for every t, by dst = rst dt, (21) dst i /Si t = µ i dt + σ ij dw j t, 1 i n, (22) where µ i and σ ij are positive constants representing the drift coefficient and, respectively, the volatilities of the ith stock Here r is a positive constant representing the interest rate of the risk-free asset For each i {1, 2,,n} we denote the long-run growth rate of the ith stock by σij 2 λ i := µ i > (23) 2 This means that we are assuming that each stock has a positive long-run growth rate At initial time τ =, for each i {1, 2,,n} a proportion πτ i [, 1] of the wealth is invested in the ith stock, and the remaining proportion πτ := 1 n πτ i of wealth is invested in the risk-free asset The portfolio at time τ = isdefinedby π τ := (πτ,πτ 1,πτ 2,,πτ n ) Generally, the portfolio at time t isdefinedbyπ t := (πt,π1 t,π2 t,,πn t ), where for each i {1, 2,,n} πi t is the proportion of wealth investedintheith stock at time t,andπt := 1 n πt i is the proportion of wealth invested in the risk-free asset at time t If V t denotes the value of the portfolio at time t, the value of the investment just before the time τ 1 of the first transaction can be written as V τ1 = i= πτ i V S i Sτ i 1 (24) Suppose that at time τ 1 the investor makes a transaction in the market, which incurs a cost equal to a fixed fraction α [, 1) of the portfolio value 1 The remaining proceedings $(1 α)v τ1 generate a profit (or a loss) (1 α)v τ1 V 1 Atkinson et al [1], Atkinson and Wilmott [2], Bielecki and Pliska [5], Buescu et al [7], Cadenillas and Pliska [8], Duffie and Sun [16], Morton and Pliska [24], and Pliska and Selby [26] make the same assumptions about the structure of transaction costs

4 166 C Buescu, A Cadenillas which is taxed at a constant rate β [, 1) The value of the investment after paying transaction costs and taxes is V τ1 = V τ1 αv τ1 β[(1 α)v τ1 V ], so the overall factor by which wealth is increased over the first cycle is [ ] M 1 := V τ 1 = β + (1 α)(1 β) π i Sτ i 1 τ V τ S i (25) i= If, after paying transaction costs, the profit (1 α)v τ1 V is positive, then the investor pays a tax equal to β times that profit, whereas if the sale incurs a loss, then the investor receives a tax credit equal to β times the absolute value of the loss We assume that the tax credits are realized as actual cash payments As in Cadenillas and Pliska [8] and Buescu et al [7], we ignore wash rules and other tax limitations, and assume that tax payments and credits are made at transaction times, and not at the end of the tax year We observe that at time t [,τ 1 ) the portfolio π t := (πt,π1 t,π2 t,,πn t ) is given by π t = and, for every k {1, 2,,n}, by V π S t /S ni= V π i Si t /S i = π S t /S ni= π i Si t /S i πt k V π k = Sk t /Sk ni= V π i Si t /S i = π k Sk t /Sk ni= π i Si t /S i This shows that the portfolio depends on the prices of the assets during [,τ 1 )However,it cannot be optimal to rebalance the portfolio continuously in time because of the transaction costs (this will be confirmed in the numerical examples of Sect 3, which show that E[τ 1 ] > for the optimal τ 1 ) At time τ 1, for each i {1, 2,,n} a proportion πτ i 1 [, 1] of the current wealth is invested in the ith stock, and the remaining proportion 1 n πτ i 1 in the risk-free asset This transaction cycle is repeated at times τ 2,τ 3, and so on Definition 21 An admissible strategy is a sequence of pairs {(τ m,π τm )} m N, where each τ m is a stopping time (with respect to the filtration {F t } t ) such that = τ τ 1 τ 2,andtheF τm -measurable random vector π τm = (πτ m,,πτ n m ) is chosen at time τ m such that { } P πτ i m = 1 = 1, and i= That is, for every m N ={, 1, 2,}: P{ i {, 1,,n} : π i τ m [, 1]} = 1 P { π τm P } = 1,

5 Investors preference for a positive tax rate depends on the level of the interest rate 167 where { P := x R n+1 : i {, 1,,n} :x i [, 1] and } x i = 1 The cash value of the investment at the end of each transaction cycle [τ m,τ m+1 ) is given by V τm+1 = V τm M m+1 = = V τ M 1 M m+1, (26) where M m+1 is the factor by which wealth was increased over the (m + 1)-st cycle [ M m+1 := β + (1 α)(1 β) π i S i ] τ m+1 τ m S i (27) i= τ m Generally, for t [τ m,τ m+1 ) we have [ ] V t = V τm πτ i St i m S i i= τ m Definition 22 A measure of the growth of the investor s wealth is the long-run growth rate of the investment portfolio, defined by 1 lim inf t t E[log V t] (28) Since the long-run growth rate depends on the particular investment strategy chosen, different strategies will result in different growth rates We assume an infinite time horizon Problem 21 The investor wants to determine the admissible strategy that maximizes the long-run growth rate of the investment The strategy that maximizes this criterion will be called the optimal strategy, and the corresponding long-run growth rate will be denoted by R This problem generalizes the problem studied by Cadenillas and Pliska [8] by incorporating one bank account, and the problem of Morton and Pliska [24] by incorporating taxes Problem 21 was solved analytically by Buescu et al [7] in the case of one stock (n = 1) and one bank account with zero interest rate (r = ) We want to find the strategy that maximizes the long-run growth rate of the portfolio when the risk-free asset has a positive interest rate (r > ) In this case, however, we cannot find explicit analytic solutions, and we have to apply numerical methods In the following we use renewal theory to reduce Problem 21 to finding the optimal strategy for the first transaction cycle After choosing the vector of optimal proportions π τ = π P at time τ = insome optimal fashion, and then optimally stopping at time τ 1, the after-tax wealth is invested in the same bank account (St = Sτ 1 e r(t τ1), t τ 1 ) and in the same stocks The price of the ith stock (1 i n) St i = Sτ i 1 exp λ i (t τ 1 ) + has identical distribution to Sτ i 1 exp λ i (t τ 1 ) + σ ij (W j t W j σ ij W j t τ 1 τ 1 ) i=, t τ 1,,

6 168 C Buescu, A Cadenillas so we are in the same probabilistic context as before [see (21), (22)] Therefore we choose the same vector of optimal proportions π τ2 π, and the optimal stopping time τ 2 such that τ 2 τ 1,τ 1 τ are independent and identically distributed (iid) random variables Generally, at the beginning of the transaction cycle [τ m 1,τ m ) we choose the optimal vector of proportions π τm 1 π, and we stop the process at time τ m following the same policy employed in the previous transaction cycles, thus having τ m τ m 1,τ m 1 τ m 2,,τ 1 τ iid In this case it follows from (24) and the discussion above that V τ1, V τ 2, V V τ1 are also iid This implies by (25) that the pairs (τ m τ m 1, M m ), m N, are iid in the optimal strategy Let us denote by S the class of positive stopping times, and by S := {τ S : E[τ] (, )} (29) the subclass of stopping times with positive and finite expectation Define the function g :[, ) R by g(x) := log {β + (1 β)(1 α)x} (21) Suppose at initial time τ = the investor allocates the wealth using the vector of proportions π τ = π P For this fixed π P we define, for t ( ) I t (π) = 1 π i S t S + π i Si t S i ( ) = 1 π i e rt + π i exp λ i t + σ ij Wt j (211) According to (24), this models the factor by which wealth is increased before paying taxes and transaction costs Lemma 21 For every η (, ), τ S, and π P E[g(I τ (π)) ητ] =g(1) + E Here the stochastic process Y = Y (η) is defined by [ ] ( ) (1 α)(1 β) Y t := 1 π i re rt β + (1 α)(1 β)i t (π) + µ i π i exp λ i t + σ ij W j t 1 [ ] (1 α)(1 β) 2 σ ik π i exp 2 β + (1 α)(1 β)i t (π) λ i t + k=1 τ Y t dt (212) σ ij W j t 2 η (213)

7 Investors preference for a positive tax rate depends on the level of the interest rate 169 Proof Applying the multidimensional Itô s formula to I t (π) as a function of t and Wt 1, W t 2,, Wt n,wegetforeveryτ S and π P: τ ( ) I τ (π) = π i re rt + µ i π i exp λ i t + σ ij W j t dt τ + σ ik π i exp λ i t + σ ij Wt j dwt k k=1 Using the above semimartingale form of I (π) to apply Itô s formula to the function g(x) ηt gives Since τ τ (1 α)(1 β) g(i τ (π)) ητ = g(1) + Y t dt + β + (1 α)(1 β)i k=1 t (π) σ ik π i exp λ i t + σ ij Wt j dwt k (214) σ ik π i exp λ i t + ( ) σ ij Wt j σ ul π i exp λ i t + u=1 l=1 ( ) σ ul I t (π), u=1 l=1 σ ij W j t we observe that the integrands of the above stochastic integrals are bounded Hence, those stochastic integrals have expected value equal to zero Therefore, taking the expected value in (214), we obtain (212) Proposition 21 If the pairs (τ m τ m 1, M m ),m N, are independent and identically distributed with E[τ 1 ] <, then lim inf t with V τ1 /V given in (25) 1 t E[log V t]= E[log M 1] E[τ 1 ] = E[log{V τ 1 /V }], E[τ 1 ] Proof We check that the stochastic process Y of Lemma 21 is bounded The first term of the right hand side of (213) is positive and bounded by r + µ j,

8 17 C Buescu, A Cadenillas because ( ) 1 π i re rt + µ i π i exp λ i t + σ ij W j t ( ) 1 π i r + µ j e rt + r + µ j π i exp λ i t + = r + I t (π) µ j The term multiplying the negative half in (213) is positive and bounded by ( ) 2 σ ik, because k=1 2 σ ik π i exp λ i t + σ ij Wt j k=1 ( ) 2 2 σ ik π i exp λ i t + σ ij W j t k=1 ( ) 2 [It σ ik (π) ] 2 k=1 σ ij W j t Hence the process Y is bounded Using Lemma 21 with a bounded process Y and a stopping time τ with E[τ] <, we obtain E[g(I τ (π))] < The definition of the function g given in (21) and that of I t (π) givenin(211) on one hand, and the overall factor M 1 givenin(27) on the other hand, show that g(i τ1 (π)) = log M 1 = log V τ 1 V This implies E[log M 1 ] <, and we can apply renewal theory (see Theorem 361 of [27]) to get the conclusion of this Proposition Remark 21 In terms of Definition 21, the pairs that define the admissible strategies have the form: (τ =,π), (τ 1,π), (τ 2,π), (τ 3,π), Here {τ m τ m 1 ; m N} is the set of times between transactions, and π P is the vector of proportions of wealth to be invested in each of the n + 1 assets We observe that the optimal strategy is characterized by the optimal stopping time ˆτ of the first transaction cycle, and by the vector of optimal proportions ˆπ P For fixed π P we denote the maximum long-run growth rate of the portfolio by R π := sup τ S E[log{V τ /V }] (215) E[τ]

9 Investors preference for a positive tax rate depends on the level of the interest rate 171 Thus R = sup R π, (216) π P and for every π P : R π R, with equality for π =ˆπ Problem 21 can be rewritten in the following equivalent form (see Chap 7 of [6] for details) Problem 22 For each fixed portfolio π = (π,π 1,π 2,,π n ) P, determine the value R π for which the following optimal stopping problem has value zero [ { } ] Vτ sup E log R π τ =, (217) τ S V where V τ /V is given in (25) That is, for each fixed π P and each fixed θ (, ) solve the optimal stopping problem with value τ { } H(θ) = sup E Vτ ( θ)du + log (218) τ S V Then, for that fixed π P, determine the value R π such that H(R π ) = (219) Finally, select the value of π that maximizes {R π ; π P} That is, find ˆπ P such that R ˆπ = sup R π (22) π P We note that (218) is the problem of optimal stopping when there is a continuation fee paid at the rate θ per unit of time, and a reward-for-stopping equal to the log-term We observe that problem (215) is equivalent to problem (217), and problem (217) is equivalent to problem (218) (219) Since (215) definesr π uniquely, then problem (219) has a unique solution According to (25), the log-term in (218) depends not only on the stock prices, but also on time via the price of the risk-free asset, making this a non-homogeneous problem To transform (218) into a homogeneous problem we follow the method of Dynkin [17, Sect 46], and presented with adjustments in Krylov [21, p 14], Shiryayev [28, p 23], and Øksendal [25, Sect 12] We consider the homogenized (n + 1)-dimensional process {X t } t having time as the first component and the natural logarithm of the normalized prices of the stocks as the other n components The dynamics of the log-prices of the stocks are given, for i {1,,n} and t, by d log S i t /Si = λ i dt + σ ij dw j t The dynamics of the process X are given then, for each t, by dx t = dt, dx i t = λ i dt + σ ij dw j t, i {1, 2,,n}

10 172 C Buescu, A Cadenillas Define for our fixed π P the function G : R n+1 R by { [ G(y, y 1,,y n ) := log β + (1 α)(1 β) π exp (ry ) + ]} π i exp (y i ) (221) Applying the principle of dynamic programming, we have that, with initial condition X = (y, y 1, y 2,,y n ) =: y R + R n, the value function u : R + R n R defined by u(y) y + u(y) := sup τ S satisfies the moving boundary problem ( ) σ ik σ jk λ i u(y) y i E y i, τ ( θ)dw + G(X τ ) (222) k=1 2 u(y) y i y j = θ, if y C (223) u(y) = G(y), if y, (224) where the continuation and stopping regions are given, respectively, by C := {y R + R n : u(y) >G(y)}, (225) := {y R + R n : u(y) = G(y)} (226) Thesolutionto(218) forthefixedπ P is then H(θ) = u(,,,) }{{} n+1 times As mentioned above, this solution cannot be derived explicitly because the problem involves the partial differential equation (223) Instead, we will use the Markov chain approximation method of Kushner and Dupuis [22] to obtain numerically the value function u, and thus H(θ) This method works when their condition (5312) is satisfied: 1 i n, y : a ii (y) a ij (y), (227) where a ij (y) = j: j =i σ ik σ jk k=1 Once we have a numerical approximation of H(θ) andhave solved Eq (219), the solution to our initial problem will be obtained using Eq (22) 3 Numerical results for the financial market with one stock and one bank account We consider in this section the case n = 1andr >, when the market consists of one stock and one bank account with positive interest rate We compute numerically the optimal strategy, and use it to show that sometimes the investor prefers a positive tax rate We also investigate, among other things, the effects of the interest rate on this preference

11 Investors preference for a positive tax rate depends on the level of the interest rate 173 Since we are assuming n = 1, the vector of proportions is simply π = (1 π, π), with π [, 1] Notation 31 Each combination of the parameters (µ, σ, r) of the two assets, transaction cost rate α [, 1], taxrateβ [, 1], and proportion of money invested in the stock π [, 1], will result in the best long-run growth-rate denoted by R (r,µ,σ ) := sup τ S E[log{V τ /V }] (31) E[τ] Remark 31 If all the money is invested in the bank account (π ), then the long-run growth rate is simply R (r,µ,σ ),α,β = r Proposition 31 For fixed π [, 1], we have the following bounds for the long-run growth rate R (,µ,σ ) R (r,µ,σ ) r + R (,µ r,σ ) (32) Proof Using (25), the growth rate R (r,µ,σ ) can be written as [ { ]}] E log β + (1 α)(1 β) [(1 π)e rτ + πe (µ σ 2 /2)τ+σ W τ R (r,µ,σ ) = sup τ S E[τ] Since e τ = 1 e rτ, the inequality involving the lower bound follows For the inequality involving the upper bound, use β βe rτ : [ { ]}] E log β + (1 α)(1 β) [(1 π)e rτ + πe (µ σ 2 /2)τ+σ W τ E[τ] [ { ]}] E log βe rτ + (1 α)(1 β) [(1 π)e rτ + πe (µ σ 2 /2)τ+σ W τ E[τ] [ { { ]}}] E log e rτ β + (1 α)(1 β) [(1 π)+ πe (µ r σ 2 /2)τ+σ W τ = E[τ] [ { ]}] E log β + (1 α)(1 β) [(1 π)+ πe (µ r σ 2 /2)τ+σ W τ = r + E[τ] This gives, by taking sup τ S, R (r,µ,σ ) r + R (,µ r,σ ) For fixed tax rate β (, 1) and transaction cost rate α (, 1), the above bounds involve three values for the triplet (interest rate, drift, volatility): (,µ,σ),(r,µ,σ)and (,µ r,σ) It is unlikely that the corresponding optimal proportions are all equal, so let us assume them to be π 1,π 2 and π 3, respectively That is, for every π [, 1] we have R (,µ,σ ) R (,µ,σ ) π 1,α,β, R(r,µ,σ ) R (r,µ,σ ) π 2,α,β, R(,µ r,σ ) R (,µ r,σ ) π 3,α,β

12 174 C Buescu, A Cadenillas In particular we have R (r,µ,σ ) π 1,α,β R(r,µ,σ ) π 2,α,β, R(,µ r,σ ) π 2,α,β R (,µ r,σ ) π 3,α,β By Proposition 31 we also have R (,µ,σ ) π 1,α,β R(r,µ,σ ) π 1,α,β, R(r,µ,σ ) π 2,α,β r + R(,µ r,σ ) π 2,α,β Combining these inequalities gives the range of the optimal growth rate for the triplet (r,µ,σ), without restricting it to a particular proportion Corollary 31 The range of the long-run growth rate when r > is R (,µ,σ ) π 1,α,β R(r,µ,σ ) π 2,α,β r + R(,µ r,σ ) π 3,α,β Since both R (,µ,σ ) ) π 1,α,β and R(,µ r,σ π 3,α,β correspond to cases with zero interest rates, they can be computed using the method of Buescu et al [7] This yields the expected range for every with r > R (r,µ,σ ) π 2,α,β Example 31 If we take r = 15, µ = 65, σ = 3, α = 2 and β = 3, then we get the following range: R (,65,3) 1,2,3 = R (15,65,3) 15 + R (,5,3) ˆπ,2,3 8,2,3 = Here, ˆπ is the unknown optimal proportion to be invested in the stock We now summarize in Table 1 the case µ = 65, σ = 3, α = 2 When π =, all the money is invested in the risk-free asset, hence the optimal long-run growth rate is given by the interest rate of the risk-free asset, say r = r (Remark 31) The case π = 1(allthe money goes into the stock, so the interest rate of the risk-free asset is irrelevant), was solved by Cadenillas and Pliska [8]for β = (R = λ = 2) and β = 3 (their Example 51) For π (, 1) and β = we use the approach of Morton and Pliska [24] to obtain the long-run growth rates corresponding to r = andr = 15 The case β = 3 andr = can be solved by the method of Buescu et al [7], and the case β = 3 andr = 15 is presented in Example 31 above Example 32 Consider the parameters values µ = 65, σ = 3 and α = 2 Allowing π (, 1), but using only the lower bound of Corollary 31 for the case β = 3 and r = 75, we obtain Table 2 in a similar way to Table 1 Based on this we conclude that, when r >, an investor is sometimes better off with taxes than without taxes! Table 1 Optimal long-run growth rate R R π = π (, 1) and r = π (, 1) and r = 15 π = 1 β = r β = 3 r [22311, 27449] 22311

13 Investors preference for a positive tax rate depends on the level of the interest rate 175 Table 2 In this example the investor is better off with a positive tax rate R r = r = 75 β = R = R = β = 3 R = R The same qualitative result was obtained previously by Cadenillas and Pliska [8] ina financial market consisting of only one stock, and by Buescu et al [7] in a financial market consisting of one stock and one bank account with zero interest rate We obtain the same qualitative result, but in a more realistic financial market This example allows us to state the following result Theorem 31 In a financial market with a risky asset and a risk-free asset with positive interest rate, sometimes investors prefer a positive tax rate Now, we want to find a solution to Problem 22 In the first step, we want to find [see (218)], for each π [, 1] and θ (, ): τ { } H(θ) = sup E Vτ ( θ)du + log τ S V This is a problem of maximizing the average reward when costs are incurred at a rate θ per unit of time and a reward log{v τ /V } is collected at time τ In the second step, for fixed value of π [, 1], we will choose ˆθ = R π such that H( ˆθ) = In the third step, the best long-run growth rate R will then be obtained using (22) We derive the equivalent moving boundary problem (223) (224) for the case of one stock and one risk-free asset with positive interest rate Recall that (24)gives V τ V = (1 π) S τ S + π S1 τ S 1 = (1 π) e rτ + π exp {(µ 12 σ 2 ) τ + σ W τ } The explicit dependence on time via the exponential price of the risk-free asset shows that this is a non-homogeneous Markov process To transform this into a homogeneous case we use (25) to express the increase V t /V, and consider the two dimensional process {X t } t that has as components time and the homogeneous process {S 1 t /S1 } t We observe that the positivity of the interest rate r (as opposed to the case r = studied by Buescu et al [7]), makes it necessary to increase the dimensionality of the problem Define the function G :[, ) (, ) R by G(x 1, x 2 ) := log { β + (1 β)(1 α) [ (1 π)exp (rx 1 ) + π x 2 ]} (33) The principle of dynamic programming gives that, with initial condition X = x := (s, p) [, ) [, ), the value function v :[, ) [, ) R defined by τ v(s, p) := sup E (s,p) ( θ)du + G(X τ (s,p) ) (34) τ S

14 176 C Buescu, A Cadenillas satisfies the moving boundary problem (see also [11]) v v (s, p) + µp s p (s, p) + σ 2 p 2 2 v 2 p 2 (s, p) = θ, if (s, p) C, (35) v(s, p) = G(s, p), if (s, p), (36) where the continuation and stopping regions are given, respectively, by C ={(s, p) [, ) [, ) : v(s, p) >G(s, p)}, ={(s, p) [, ) [, ) : v(s, p) = G(s, p)} This problem is easier to implement numerically when written in the equivalent form given by Eqs (221) (226) involving the logarithm of the stock price In that equivalent form, the problem is solved for each fixed θ in a bounded interval (given by Corollary 31) using the Markov chain approximation method of Kushner and Dupuis [22] (it is easy to verify that condition (227) is satisfied) In addition, we develop a method similar to that of Morton and Pliska [24] to update the parameter θ so that it converges to R (see [6] for details) The solution is then expressed in terms of the two-dimensional process {(t, St 1/S t ); t [, )} of the original problem (33) (36) If we take in this model the interest rate and the tax rate to be equal to zero, we can check that the results are consistent with those obtained using existing approaches Example 33 In the case of no taxes and a bank account with zero interest rate (β =, r = ), consider the parameter values µ = 65, σ = 3, α = 2 1 The approach of Morton and Pliska [24], which uses β =, yields R = , ˆπ = , ˆτ = inf{t : St 1 /S1 / (22848, )} 2 The method of Buescu et al [7], which uses r =, results in R = , ˆπ = 73, ˆτ = inf{t : St 1 /S1 / (22932, )} 3 The present Markov chain approximation method gives R = 22532, ˆπ = 73, ˆτ = inf{t : St 1 /S1 / (229, 7576)} The results are consistent no matter which method is used to derive them To go one step further in checking this consistency, we consider one example with a positive interest rate Example 34 Consider the parameter values of Example 33, but with a positive interest rate: µ = 65, σ = 3, α = 2, r = 15 We are going to consider two tax rates: β = andβ = 3 1 For β = the method of Morton and Pliska [24]gives R = , ˆπ = , while the Markov chain approximation method gives R = , ˆπ = 557

15 Investors preference for a positive tax rate depends on the level of the interest rate STOPPING REGION S 1 (t) S1 () 1 5 CONTINUATION REGION 1 STOPPING REGION E(tau*)= t Fig 1 Continuation region for µ = 65, σ = 3, α = 2, r = 15, β = 3 2 For β = 3 the Markov chain approximation method gives R = , ˆπ = 64 We note that the growth rate is within the range obtained in Example 31 The continuation region is presented in Fig 1 Example32 showed that the investor can prefer a positive tax rate Considering from now on the parameter values µ = 65, σ = 3, α = 2, it turns out that for r = 75 the best tax rate for the investor is ˆβ = 3 (seefig2) Now, we investigate the effect that the interest rate has on the tax rate that is best for the investor Figure 3 shows the growth rate versus the interest rate in two cases: β = and β = 3 It is clear that the optimal growth rate is an increasing function of the interest rate It is also obvious that if r is small, then the investor should allocate a higher proportion of the money in the stock Although this is a risky investment, it is especially interesting to remark that when the interest rate is small the investor obtains a higher long-run growth rate of the investment R Best Beta= tax rate (beta) Fig 2 Determining the best tax rate when µ = 65, σ = 3, α = 2, r = 75

16 178 C Buescu, A Cadenillas R no taxes tax rate 3% r Fig 3 Growth rate versus interest rate for different tax rates portfolio in the presence of a positive tax rate However, as the interest rate increases this is no longer the case We summarize this discussion in the following result Theorem 32 For small values of the interest rate investors prefer a positive tax rate, while for high values of the interest rate a positive tax rate is detrimental to the investment The intuitive explanation is the following: if the interest rate is low, then a large proportion of money is invested in the stock, and therefore a positive tax rate is advantageous to reduce the risk of the investment On the other hand, if the interest rate is high, then a small proportion of money is invested in the stock, and therefore a positive tax rate has small potential to reduce the risk of the investment (in comparison to the investment in the risk-free asset) In practice, the tax rate is not selected by the investor Instead, the tax rate is decided by the current fiscal policy of the government, and is known to the investor before he enters the financial market Theorem 32 says that, for small values of the interest rate, investors are better-off with a positive tax rate; while for large values of the interest rate, investors are better-off with a null tax rate This result would be interesting and useful to the economists and politicians who decide the fiscal policy of a government, since it provides a link between the level of taxation and the level of the prime interest rate We conclude with an analysis of the dependence of the optimal proportion on some of the parameters Remark 32 The optimal proportion of initial wealth invested in the stock is a decreasing function of the interest rate r (see Fig 4), and an increasing function of the tax rate β (see Fig 5) It is obvious that the larger the interest rate, the smaller the proportion of money to be invested in the stock However, we observe in Fig 4 that there is a dramatic difference between the case of no taxes and the case of a positive tax rate We observe that, for a fixed positive value of the interest rate, the larger the tax rate the larger the proportion of money to be invested in the risky asset The intuition is that, for a fixed positive value of the interest rate, the positive tax rate reduces the risk of the investment, and therefore the investor can afford to invest a larger proportion of money in the risky asset We observe in Fig 5 that the optimal proportion of money invested in the stock increases with respect to the tax rate until it reaches 1 (put all the money in the stock) The intuitive

17 Investors preference for a positive tax rate depends on the level of the interest rate 179 pi no taxes tax rate 3% r Fig 4 Optimal proportion versus interest rate for different tax rates pi tax rate (beta) Fig 5 Optimal proportion versus tax rate when r = 75 explanation is that the larger the tax rate, the better the cushion against volatility, so the investor is willing to allocate an increasing proportion of the initial investment in the stock We observe in Fig 5 that, for the best tax rate (in this example, the value of the best rate is 3), the optimal strategy is to invest all the money in the stock, so the interest rate of the risk-free asset makes no difference (provided r is small enough) 4 Conclusions We succeeded in solving a portfolio management problem with taxes and transaction costs Due to the computational complexity, we had to apply a numerical method: the Markov chain approximation method In addition, we obtained two interesting economic results The first one is a generalization to a market with one stock and one bank account with positive interest rate of the result that sometimes investors prefer a positive tax rate The second result presents the interplay between taxes and interest rates in portfolio optimization Namely, when the interest rate is small the investors can take advantage of the

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