SIAM J. FINANCIAL MATH. Vol. 1, pp. 366 395 c 21 Sociey for Indusrial and Applied Mahemaics Meron Problem wih Taxes: Characerizaion, Compuaion, and Approximaion Imen Ben Tahar, H. Mee Soner, and Nizar Touzi Absrac. We formulae a compuaionally racable exension of he classical Meron opimal consumpioninvesmen problem o include he capial gains axes. This is he coninuous-ime version of he model inroduced by Dammon, Spa, and Zhang [Rev. Financ. Sud., 14 (21), pp. 583 616]. In his model he ax basis is compued as he average cos of he socks in he invesor s porfolio. This average rule inroduces only one addiional sae variable, namely he ax basis. Since he oher ax rules such as he firs in firs ou rule require he knowledge of all pas ransacions, he average model is compuaionally much easier. We emphasize he linear axaion rule, which allows for ax credis when capial gains losses are experienced. In his conex wash sales are opimal, and we prove i rigorously. Our main conribuions are a firs order explici approximaion of he value funcion of he problem and a unique characerizaion by means of he corresponding dynamic programming equaion. The laer characerizaion builds on echnical resuls isolaed in he accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Conrol Opim., 46 (27), pp. 1779 181]. We also sugges a numerical compuaion echnique based on a combinaion of finie differences and he Howard ieraion algorihm. Finally, we provide some numerical resuls on he welfare consequences of axes and he qualiy of he firs order approximaion. Key words. opimal consumpion and invesmen in coninuous ime, ransacion coss, capial gains axes, finie differences AMS subjec classificaions. 91B28, 35B37, 35K2 DOI. 1.1137/8742178 1. Inroducion. Since he seminal papers of Meron [26, 27], here has been exensive lieraure on he problem of opimal consumpion and invesmen decision in financial markes subjec o imperfecions. We refer he reader o Cox and Huang [1] and Karazas, Lehoczky, and Shreve [21] for he case of incomplee markes, Cvianić andkarazas[11] for he case of porfolio consrains, and Consaninides and Magill [9], Davis and Norman [13], Shreve and Soner [28], Barles and Soner [3], and Duffie and Sun [15] for he case of ransacion coss. However, he problem of axes on capial gains received limied aenion, alhough axes represen a much higher percenage han ransacion coss in real securiies markes. Compared o ordinary income, capial gains are axed only when he invesor sells he securiy, allowing for a deferral opion. One may hink ha he axes on capial gains have an appreciable impac on an individual s consumpion and invesmen decisions. Indeed, under axaion Received by he ediors December 1, 28; acceped for publicaion (in revised form) March 8, 21; published elecronically May 26, 21. This research was par of he Chair Financial Risks of he Risk Foundaion sponsored by Sociéé Générale, he Chair Derivaives of he Fuure sponsored by he Fédéraion Bancaire Française, and he Chair Finance and Susainable Developmen sponsored by EDF and Calyon. hp://www.siam.org/journals/sifin/1/74217.hml CEREMADE, Universié Paris Dauphine, 75775 Cedex 16, Paris, France (imen@ceremade.dauphine.fr). Sabanci Universiy, Isanbul 34956, Turkey (hmsoner@ehz.ch). This auhor s research was parly suppored by he King Saud Universiy D.S.F.P and by he Turkish Technical and Scienific Council, TÜBITAK. CMAP, Ecole Polyechnique, 91128 Palaiseau Cedex, Paris, France (nizar.ouzi@polyechnique.edu). 366
MERTON PROBLEM WITH TAXES 367 of capial gains, he porfolio rebalancing implies addiional charges, herefore alering he available wealh for fuure consumpion. This possibly induces a depreciaion of consumpion opporuniies compared o a ax-free marke. On he oher hand, since axes are paid only when embedded capial gains are acually realized, he invesor may choose o defer he realizaion of capial gains and liquidae his/her posiion in case of a capial loss, paricularly when he ax code allows for ax credis. The firs relevan work in he previous lieraure is due o Consaninides [8], who shows ha he invesmen and consumpion decisions are separable and ha he opimal sraegy consiss in realizing losses and deferring gains. These resuls rely heavily on he possibiliy of shor selling he risky asse. Since capial gains realizaions are observed in real securiies markes, he subsequen lieraure considers he problem under he no-shor-sales consrain. In a muliperiod conex, many challenging difficulies appear because of he pah dependency of he problem. The axaion code specifies he basis o which he price of a securiy has o be compared in order o evaluae he capial gains (or losses). The ax basis is defined as eiher (i) he specific purchase price of he asse o be sold, (ii) he purchase price of a freely chosen share held in he porfolio (of course he number of chosen shares mus be more han he ones o be sold), or (iii) he weighed average of pas purchase prices. In some counries, invesors can choose any one of he above definiions of he ax basis. A deerminisic model wih he above definiion (i) of he ax basis, ogeher wih he firs in firs ou prioriy rule for he sock o be sold, is inroduced by Jouini, Koehl, and Touzi [2, 19]. An exisence resul is proved, and he firs order condiions of opimaliy are derived under some condiions. However, he numerical complexiy due o he pah dependency of he problem is no solved in he conex of his model. A financial model wih he above definiion (ii) of he axaion rule was considered by Dybvig and Koo [16] in he conex of a four-period binomial model. Some numerical progress was achieved laer by DeMiguel and Uppal [14],whowereableoconsidermoreperiodsin he binomial model and/or more socks. This numerical progress is limied, as hese auhors werenoableogobeyond1periodsinhesingle-asseframework. The axaion rule (iii), where he ax basis is he weighed average of pas purchase prices, was firs considered by Dammon, Spa, and Zhang [12] in he conex of a binomial model wih shor-sales consrains and he linear axaion rule. The average ax basis is acually used in Canada. Dammon, Spa, and Zhang [12] considered he problem of maximizing he expeced discouned uiliy from fuure consumpion and provided a numerical analysis of his model based on he dynamic programming principle. The imporan echnical feaure of his model is ha he pah dependency of he problem is seriously reduced, as he dynamics of he ax basis is Markov. This implies a significan advanage of his model in comparison o [16]. This advanage was furher jusified by DeMiguel and Uppal [14], who provided numerical evidence ha he cerainy equivalen loss from using he average ax basis (iii) insead of he exac ax basis (ii) is ypically less han 1% for a large choice of parameer values. The analysis of [12] was furher exended o he muliasse framework by Gallmeyer, Kaniel, and Tompaidis [18]. In his paper, we formulae a coninuous-ime version of he Dammon Spa Zhang uiliy maximizaion problem under capial gains axes. Our model is similar o ha of Leland [24], who insead considered he problem of minimizing he racking error o some benchmark index.
368 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI The financial marke consiss of a ax-free riskless asse and a risky one. The holdings in he risky asse are subjec o he no-shor-sales consrain, and he oal wealh is resriced by he no-bankrupcy condiion. The risky asse is subjec o axes on capial gains. As in [12], he ax basis is defined as he weighed average of pas purchase prices, and he axaion rule is linear, hus allowing for ax credis. However, we differ from [12] by considering an infinie horizon problem, as our main goal is o provide analyical ools for his class of problems. In paricular, our model does no allow for ax forgiveness a deah. Clearly, one should keep his difference in mind when inerpreing resuls for invesors wih a shor horizon. The invesor preferences are described by a power uiliy wih a consan relaive risk aversion facor. This assumpion is needed only o reduce he compuaional complexiy of he problem. However, wih his reducion explici descripions of he value funcion and he opimal sraegy are sill no available. This model enables us o rigorously prove several ineresing properies observed in pracice. Alhough hese resuls are someimes inuiively clear, heir proofs require careful analysis and he use of he racabiliy of he model. The firs of hese resuls is he opimaliy of he wash sales. Namely, in Proposiion 3.5 we prove ha i is always opimal o realize capial losses whenever he ax basis exceeds he spo price. This propery is observed in pracice and is saed and embedded direcly in he definiion of he ax basis in [12]. We also prove he coninuiy of he value funcion (and even Lipschiz coninuiy, up o a change of variables). We recall ha, in he ax-free models of [26, 9, 13, 28], his propery follows from he obvious concaviy of he value funcion. Under capial gains axes, he concaviy argumen fails, and he numerical resuls of secion 6 sugges ha he value funcion is indeed no concave! The firs main resul of his paper is o provide an explici approximaion of he value funcion which follows from an upper and a lower bound proved in secion 4. In view of he absence of closed form soluions, such an approximaion is useful for undersanding he model beer. Alhough his explici approximaion holds for small ineres rae and ax parameers, ournumerical experimens indicae ha hisapproximaion is saisfacory wih realisic values of ineres rae and ax parameers, as i leads o a relaive error wihin 1%. These findings are repored in secion 6. This firs order approximaion allows one o draw he following observaions: The lower bound is derived as he limi of he value implied by a sequence of sraegies which mimics he Meron opimal sraegy in a Meron-ype ficiious fricionless financial marke wih ax-deflaed drif and volailiy coefficiens. The risk premium of his ficiious financial marke is smaller han ha of he original marke. So, even if he opimal sraegy in our problem is no available in explici form, our firs order expansion is accompanied by an explici sraegy which achieves he firs order maximal uiliy value. In a siuaion of a capial loss, our firs order approximaion is increasing in he ax rae. For small ineres rae and ax parameers, he advanage aken from an iniial ax credi is never compensaed by he increase of ax over he lifeime horizon. This is in agreemen wih Cadenillas and Pliska [7], who found ha someimes invesors are beer off wih a posiive ax rae. Finally, he invesmen componen of his approximaion sequence exhibis a smaller exposiion o he risky asse. This is in line wih he risk premium puzzle highlighed
MERTON PROBLEM WITH TAXES 369 by Mehra and Presco [25]. However, one should noe ha his model is only a parial equilibrium model and ha he level of he equiy premium is deermined by general equilibrium consideraions. Our analysis of he opimal consumpion-invesmen problem relies on a numerical approach based on dynamic programming and parial differenial equaions. Therefore, he second main resul of his paper is a characerizaion of he value funcion as he limi (uniformly on compac subses) of an approximaing sequence defined by a sligh perurbaion of he naural dynamic programming equaion of our problem. The financial inerpreaion of our perurbaion is, on he one hand, o inroduce a small ransacion cos parameer and, on he oher hand, o modify simulaneously he axaion rule when he ax basis approaches he criical poin zero. Our analysis relies on he echnical resuls in our accompanying paper [5], which shows ha he perurbed dynamic programming equaion has a unique coninuous viscosiy soluion wihin he class of polynomially growing funcions. Finally, based on our dynamic programming characerizaion, we sugges a numerical approximaion mehod combining finie differences wih he Howard ieraions. Unforunaely, we have no heoreical convergence resul for our algorihm. Indeed, esablishing such convergence resuls for Hamilon Jacobi Bellman equaions corresponding o singular conrol problems is an open quesion in numerical analysis, and he exising resuls, based on he monoone scheme mehod of Barles and Souganidis [4], are resriced o he bounded conrol conex; see Bonnans and Zidani [6], Krylov [22, 23], Barles and Jakobsen [2], and Fahim, Touzi, and Warin [17]. This difficuly was already observed in he relaed lieraure on ransacion coss; see Akian, Menaldi, and Sulem [1] and Tourin and Zariphopoulou [31]. Following he laer papers, we herefore concenrae our effor on realizing he empirical convergence of he algorihm. The numerical scheme is implemened o obain he qualiaive behavior of he soluion and o undersand he welfare consequences of he axaion. In paricular, he numerical approximaion of he opimal sraegy displays a bang-bang behavior, as expeced in our singular conrol problem. As in he ransacion cos conex of [9, 13, 28], he sae space is pariioned ino hree regions: he no-ransacion region NT, he buy region B, and he sell region S; bu in conras wih he ransacion cos framework hese regions are no cones. Noaion. For a domain D in R n,wedenoebyusc(d) (resp., LSC(D)) he collecion of all upper semiconinuous (resp., lower semiconinuous) funcions from D o R. The se of coninuous funcions from D o R is denoed by C (D) :=USC(D) LSC(D). For a parameer δ>, we say ha a funcion f : D R has δ-polynomial growh if sup x D f(x) 1+ x δ <. We finally denoe USC δ (D) :={f USC(D) :f has δ-polynomial growh}. The ses LSC δ (D) and C δ (D) are defined similarly. 2. Consumpion-invesmen models wih capial gains axes. 2.1. The financial asses. Throughou his paper, we consider a complee probabiliy space (Ω, F, P), endowed wih a sandard scalar Brownian moion W = {W, }, andwe denoe by F he P-compleion of he naural filraion of he Brownian moion. We consider
37 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI a financial marke consising of one bank accoun wih consan ineres rae r>andone risky asse whose price process evolves according o he Black Scholes model: (2.1) dp = P [(r + θσ)d + σdw ], where θ> is a consan risk premium, and σ> is a consan volailiy parameer. The posiiviy resricion on he risk premium coefficien ensures ha posiive invesmen in he risky asse is ineresing. 2.2. Taxaion rule on capial gains. The sales of he sock are subjec o axesoncapial gains. The amoun of ax o be paid for each sale of risky asse, a ime, is compued by comparison of he curren price P o an index B defined as he weighed average price of he shares purchased by he invesor up o ime. When P B, i.e., he curren price of he risky asse is greaer han he weighed average price, he invesor would realize a capial gain by selling he risky asse. Similarly, when P B, he sale of he risky asse corresponds o he realizaion of a capial loss. In order o beer explain he definiion of he ax basis B, we provide he following example derived from he official Canadian ax code. Table 1 repors ransacions performed by an individual on shares of STU Ld and how he ax basis of he individual changes over ime. Table 1 Exraced from Capial Gains 27; hp://www.cra.gc.ca. Transacion Price P Number of shares Porfolio composiion Tax basis B (dollars) (uniless) (uniless) (dollars) Purchase a 1 15. 1 1 : $15./share 15. Purchase a 2 2. 15 1 : $15./share 15 : $2./share 18. Sale a 3-2 2 : $15./share = 4 (1 + 15) 3 : $2./share 18. 5 Purchase a 4 21. 35 2 : $15./share 3 : $2./share 35 : $21./share 2.625 Jus afer a sale ransacion, he ax basis is no changed. However, sales do aler he ax basis saring from he dae of he nex purchase. Noice, however, ha he ax basis is affeced only by he number of shares sold and no by he sale price. The sale of a uni share of sock a some ime is subjec o he paymen of an amoun of ax compued according o he ax basis of he porfolio a ime. In his paper, we consider a linear axaion rule, i.e., his amoun of ax is given by (2.2) l(p B ):=α (P B ), where α [, 1) is a consan ax rae coefficien. Our ineres is of course in he case α>. When he ax basis is smaller han he spo price, he invesor realizes a capial gain. Then, by selling one uni of risky asse a he spo price P, he amoun of ax o be paid is α(p B ).
MERTON PROBLEM WITH TAXES 371 When he ax basis is larger han he spo price, he invesor receives he ax credi α(b P ) for each uni of asse sold a ime. Remark 2.1. In pracice, he realized capial losses are deduced from he oal amoun of axes ha he invesor has o pay, and he annual deducible capial losses amoun may be limied by he ax code. In our model, we follow Dammon, Spa, and Zhang [12] by adoping he simplifying assumpion ha capial losses are credied immediaely wihou any limi. Remark 2.2. Our definiion of he ax basis B is slighly differen from ha of Dammon, Spa, and Zhang [12], who se he ax basis o be equal o he spo price whenever he average purchase price exceeds he curren price. This does no affec he resuls, as Proposiion 3.5 shows ha wash sales are opimal. 2.3. Consumpion-invesmen sraegies. We denoe by X he (cash) posiion on he bank, Y he amoun invesed in he risky asses, and (2.3) K := B Y P,, he posiion on he risky asse accoun evaluaed a he basis price. The rading in risky asses is subjec he no-shor-sales consrain (2.4) Y P-a.s. for all, and he posiion of he invesor is required o saisfy he solvency condiion (2.5) Z := X + Y l (P B ) Y P = X +(1 α)y + αk P-a.s.; i.e., he afer-ax liquidaion value of he porfolio is nonnegaive a any poin in ime. Trading on he financial marke is described by means of he ransfers beween he wo invesmen opporuniies defined by wo F-adaped, righ-coninuous, and nondecreasing processes L = {L, } and M = {M, } wih L = M =. The amoun ransferred from he bank o he nonrisky asse accoun a ime is given by dl and corresponds o a purchase of risky asse. The amoun ransferred from he risky asse accoun o he bank a ime, corresponding o a sale of risky asse, is given by Y dm andisexpressedinermsof proporions of he oal holdings in risky asse as in he example of Table 1. To force he shor-sales consrain (2.4) o hold, we resric he jumps of M by (2.6) ΔM 1 for P-a.s. Wih hese noaions, he evoluion of he wealh on he risky asse accoun is given by (2.7) dy = Y dp P + dl Y dm, and, by definiion of he ax basis B and (2.3), we have (2.8) dk = dl K dm.
372 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI Observe ha he conribuion of he sales in he dynamics of K is evaluaed a he basis price. For any given iniial condiion (Y,K ), (2.7) (2.8) define a unique F-adaped process (Y,K) aking values in R 2 +, he nonnegaive orhan of R 2. In addiion o he rading aciviies, he invesor consumes in coninuous ime a he rae C = {C, }. Here, C is an F-progressively measurable process wih (2.9) C and T C d < P-a.s. for all T>. Then, he bank componen of he wealh process saisfies he dynamics (2.1) dx =(rx C ) d dl + Y dm l (P B ) Y dm P =(rx C ) d dl +[(1 α)y + αk ] dm. Since he processes Y and K have been defined previously, he above dynamics uniquely defines an F-adaped process X valued in R for any given iniial condiion X. Remark 2.3. In Dammon, Spa, and Zhang [12], he nonrisky asse is also subjec o a consan proporional axaion rule. This is obviously caugh by our model by inerpreing r as he afer-ax insananeous ineres rae. For laer use, we repor he dynamics of he corresponding liquidaion value process defined in (2.5), which follows from (2.7), (2.8), (2.9), and (2.1): (2.11) dz =(rz C ) d +(1 α) Y ( dp P ) rd rαk d. Definiion 2.1. (i) A consumpion invesmen sraegy is a riple of F-adaped processes ν =(C, L, M), wherec saisfies (2.9), L and M are nondecreasing and righ coninuous, L = M =,andhejumpsofm saisfy (2.6). (ii) Given an iniial condiion s =(x, y, k) R R + R + and a consumpion-invesmen sraegy ν, wedenoebys s,ν =(X s,ν,y s,ν,k s,ν ) he unique srong soluion of (2.7), (2.8), (2.9), and(2.1) wih iniial condiion S s,ν = s. (iii) Given an iniial condiion s =(x, y, k) R R + R +, a consumpion-invesmen sraegy ν is said o be s-admissible if he corresponding sae process S s,ν saisfies he nobankrupcy consrain (2.5). We shall denoe by A(s) he collecion of all s-admissible consumpion-invesmen sraegies. The admissibiliy condiions imply ha he process S s,ν is valued in he closure S of (2.12) S = { (x, y, k) R 3 : x +(1 α)y + αk >,y>,k> }. We pariion he boundary of S ino S = z S y S k S wih y S := { (x, y, k) S : y = }, k S := { (x, y, k) S : k = } and z S = { (x, y, k) S : z := x +(1 α)y + αk = }.
MERTON PROBLEM WITH TAXES 373 2.4. The consumpion-invesmen problem. The invesor preferences are characerized by a power uiliy funcion wih consan relaive risk aversion coefficien 1 p (, 1): U(c) := cp, c, for some p (, 1). p The resricion on he relaive risk aversion coefficien o (, 1) allows us o simplify he analysis of his paper, as he boundary condiion on z S is easily obained; see Proposiion 3.2. However, several of our resuls hold for a general parameer p <, and we will indicae whenever i is he case. For every iniial daa s S and any admissible sraegy ν A(s), we inroduce he consumpion-invesmen crierion [ T ] (2.13) J T (s, ν) :=E e β U(C )d + e βt U(Z s,ν T )1 {T< }, T R + {+ }. The consumpion-invesmen problem is defined by (2.14) V (s) := sup J (s, ν), s S. ν A(s) We shall assume ha he parameers r, θ, σ, p, andβ saisfy he condiion (2.15) c(r, θ) := β pr 1 p pθ 2 2(1 p) 2 >, which has been poined ou as a sufficien condiion for he finieness of he value funcion in he conex of a financial marke wihou axes in [26] and[28]. 2.5. Review of he ax-free model. In his secion, we briefly review he soluion of he consumpion-invesmen problem when he financial marke is free from axes on capial gains. The properies of he corresponding value funcion are going o be useful o sae relevan bounds for he maximal uiliy achieved in a financial marke wih axes. In he classical formulaion of he ax-free consumpion-invesmen problem [26], he invesmen conrol variable is described by means of a unique process π which represens he proporion of wealh invesed in risky asses a each ime, and he consumpion process C is expressed as a proporion c of he oal wealh: (2.16) d Z = Z [(r c )d + π σ(θd + dw )]. In his conex, a consumpion-invesmen admissible sraegy is a pair of adaped processes (c, π) such ha c is nonnegaive and T c d + T π 2 d < P-a.s. for all T >. We shall denoe by Ā he collecion of all such consumpion-invesmen sraegies. For every iniial condiion z andsraegy(c, π) Ā, here is a unique srong soluion o (2.16) ha we denoe by Z z,c,π. The fricionless consumpion-invesmen problem is [ (2.17) V (z) := sup E e β U ( ] ) c Zz,c,π d. (c,π) Ā
374 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI Theorem 2.1 (see [26]). Le condiion (2.15) hold. Then, for all z, p 1 zp V (z) = c(r, θ) p, and he consan consumpion-invesmen sraegy c(r, θ), π(σ, θ) := θ (1 p)σ is opimal. Remark 2.4. The reducion of he model of secion 2 o he fricionless case, i.e., α =, does no aler he value funcion. This can be proved by approximaing any invesmen sraegy by a sequence of bounded variaion sraegies. However, he invesmen sraegies in our formulaion are consrained o have bounded variaion. This is needed because sales and purchases have differen impacs on he bank componen of he wealh process (2.1). Since he Meron opimal sraegy is well known o be unique and has unbounded variaion, i follows ha exisence fails o hold in our formulaion. 3. Firs properies of he value funcion. We firs show ha he opimal consumpioninvesmen problem under axes reduces o he Meron problem when he ineres rae is zero. Le c(r, θ) beasintheorem2.1. Proposiion 3.1. For r =, V (s) = c(,θ)p 1 p (x +(1 α)y + αk) p, s =(x, y, k) S. Proof. Noice ha he opimal consumpion-invesmen problem (2.14) can be expressed equivalenly in erms of he sae processes (Z, Y, K) insead of (X, Y, K), and observe ha he ineres rae parameer r is involved only in he dynamics of Z. Whenr =, he dynamics of he process Z in (2.11) dp dz = C d +(1 α) Y P is independen of he ax basis K. Since he dynamics of Y in (2.7) is independen of K, i follows ha he value funcion does no depend on he variable k. Nex,forZ >, defining c := C /Z and π := (1 α)y /Z, we see ha he soluion Z of he above equaion is he same as he soluion Z z,c,π of (2.16). In view of he sae consrain Z, our sae dynamics in he conex r = is hen equivalen o (2.16). Then, he only difference beween he conrol problem V and he corresponding Meron problem V is he class of admissible rading sraegies, which does no induce any difference on he value funcion; see Remark 2.4. The argumen in he above proof clearly does no involve he specific naure of he uiliy funcion. Therefore an analogous resul holds for any uiliy funcion. Since he ax basis is no inflaed by he ineres rae r, for nonzero values of r he ax basis plays a role in he soluion. This explains he imporance of he poin r =. We nex discuss he value funcion on he boundary of he sae space S. Observe ha here is no a priori informaion on he boundary componens y S and k S. This is one source of difficuly in he numerical par of his paper, as his sae consrain problem needs special reamen; see [5]. Proposiion 3.2. For every s z S, we have V (s) =. Proof. Le s be in z S,andleν be in A(s). By he definiion of he se admissible conrols, he process Z s,ν is nonnegaive. By Iô s lemma, ogeher wih he nonnegaiviy of C and K and he nondecrease of L, his provides e r Z s,ν (1 α) e ru Y s,ν u σ [θdu + dw u].
MERTON PROBLEM WITH TAXES 375 Le Q be he probabiliy measure equivalen o P under which he process {θu + W u,u } is a Brownian moion. The process appearing on he righ-hand side of he las inequaliy is a Q-supermaringale because i is a nonnegaive Q-local maringale. By aking expeced values under Q, i hen follows from he las inequaliies ha Z s,ν = Y s,ν = K s,ν = C = L. We have hen proved ha, for s z S, any admissible sraegy ν =(C, L, M) A(s) issuch ha C = L, implying ha V (s) =. Proposiion 3.3. The value funcion V is nondecreasing wih respec o each of he variables x, y, andk. Moreover, for (x, y, k) S wih z := x +(1 α)y + αk >, ( y (3.1) V (x, y, k) = z p V z, k ), where V(ξ,ζ) :=V (1 (1 α)ξ αζ,ξ,ζ). z Proof. 1. The monooniciy propery wih respec o x, y, andk follows immediaely from he dynamics of he problem and he bound ΔM 1. 2. Le ν =(C, L, M) be an arbirary sraegy in A(s), and define he sraegy ν := (δc, δl, M). We easily verify ha S δs,ν = δs s,ν S, which implies ha ν is in A(δs), and herefore [ ] V (δs) E e βu U(δC u )du = δ p J (δ, ν), where he las equaliy follows from he homogeneiy propery of he uiliy funcion U. By he arbirariness of ν in A(s), his shows ha V (δs) δ p V (s). 3. By wriing V (s) =V ( δ 1 δs ) δ p V (δs), i follows from he previous sep ha we in fac have V (δs) =δ p V (s), and he required resul follows immediaely from his homoheiciy propery. In he absence of axes on capial gains, i.e., α =, i is easy o deduce from he concaviy of U ha he value funcion V is concave and herefore coninuous. The numerical resuls exhibied in secion 6 reveal ha his propery is no longer valid when α>. The proof of he following coninuiy resul is obained by firs reducing he coninuiy problem o he ray {(x,, ), x R + }. This is achieved by means of a comparison resul in he sense of viscosiy soluions. Then he coninuiy on he laer ray is proved by a direc argumen. Proposiion 3.4. The funcion V of (3.1) is Lipschiz coninuous on S. Proof. See Appendix A. We now show ha i is always worh realizing capial losses whenever he ax basis exceeds he spo price of he risky asse. In oher words, given s =(x, y, k) S, every admissible sraegy ν A(s), wih Kτ s,ν >Yτ s,ν (i.e., Bτ s,ν >P τ ) for some sopping ime τ, canbe improved sricly by realizing he capial loss on he enire porfolio a ime τ. This propery is observed in pracice and is known as a wash sale. I was saed in [12] and embedded direcly in he definiion of he ax basis. This resul is independen of he choice of he uiliy funcion. Proposiion 3.5. Consider some s S and ν =(C, L, M) A(s). Assume ha K s,ν Y s,ν τ > ( τ a.s. for some finie sopping ime τ. Then, here exiss an admissible sraegy ν = C, L, M) A(s) such ha, for any uiliy funcion, i.e., a wash sale is opimal. Y ν = Y ν, Δ M ΔM = 1 {τ}, and J (s, ν) > J (s, ν);
376 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI Proof. We organize he proof in wo seps. 1. Se (L,M ):=(L, M)+(Y τ, 1)(1 ΔM τ )1 τ. We shall prove ha ν =(C, L,M ) A(s) and ha he resuling sae process saisfies Y s,ν = Y s,ν, Z s,ν Z s,ν, K s,ν K s,ν a.s. and Z ν >Z ν a.s. on { >τ}. To see his, observe ha since ν and ν differ only by he jump a he sopping ime τ, and ΔYτ ν =ΔY ν τ,wehave Y s,ν = Y s,ν, (Z s,ν ),K s,ν =(Z s,ν,k s,ν ) for <τ and Zτ s,ν = Zτ s,ν by he coninuiy of he process Z. Observe ha he newly defined sraegy ν consiss in selling ou he whole porfolio a ime τ as ΔM τ = 1. Hence Kτ s,ν = Yτ s,ν = Yτ s,ν,andwe compue direcly from (2.8) ha K s,ν =(Yτ s,ν Kτ s,ν ) e M c+m τ c (1 ΔM u ) > for τ, since Y s,ν τ K s,ν K s,ν τ e r ( Z s,ν <. By (2.11), his provides ) Z s,ν = rα τ τ<u e ru ( K s,ν u ) Ku s,ν du > for >τ, which shows ha Z s,ν andν A(s). 2. Define he sraegy ν =( C, L, M) by ( ) (3.2) C := C + ξ Z s, ν Z s,ν 1 τ and ( L, M) := ( L,M ), where ξ is an arbirary posiive consan. Observe ha (Y s, ν,k s, ν )=(Y s,ν,k s,ν ), and Z s, ν = Z s,ν = Z s,ν for τ. In paricular, K s, ν K s,ν = K s,ν K s,ν. Se ΔK := K s, ν K s,ν and ΔZ := Z s, ν Z s,ν. In order o check he admissibiliy of he sraegy ν, we direcly compue ha e r( τ) ΔZ =ΔZ τ rα ξ τ τ e r(u τ) ΔZ u du. e r(u τ) ΔK u du + ξ τ e r(u τ) ΔZ u du By he Gronwall inequaliy, his implies ha Z ν >Z ν on { >τ}, and herefore C >Con { >τ} wih posiive Lebesgue P measure. Hence J (s; ν) >J (s; ν). Remark 3.1. I follows from he previous proposiion ha V (x, y, k) =V (x + y + α(k y),, ) whenever k>y. Then, we may resric our analysis of he value funcion o he se {(x, y, k) S : k y}. We could no find any benefi from his reducion. Even he numerical implemenaion is no simplified by his domain resricion because we have no naural boundary condiion on {k = y}. We herefore coninue our analysis of he value funcion V on he oal domain S.
MERTON PROBLEM WITH TAXES 377 Remark 3.2. The previous proposiion highlighs he difficuly in characerizing a soluion of he opimal consumpion-invesmen problem under axes. The opimaliy of wash sales suggess ha he opimal rading sraegy has a local ime ype of behavior. We have no heoreical resul o suppor his inuiion. Our very limied informaion on he regulariy of he value funcion is he main obsacle in developing a verificaion argumen similar o ha of Davis and Norman [13]. 4. The firs order approximaion. In his secion, we provide upper and lower bounds for he value funcion. The upper bound expresses ha here is no way for he invesor o ake advanage of ax credis in order o do beer han in he ax-free financial marke, and his holds for any uiliy funcion. Our derivaion of he lower bound explicily uses he power uiliy bu wihou any resricion on he risk aversion facor p. These bounds will be used in order o obain a firs order approximaion resul for p<1. Proposiion 4.1. For any s =(x, y, k) S, we have V (s) V (x +(1 α)y + αk). Proof. Le s =(x, y, k) bein S. Consider some consumpion-invesmen sraegy ν = (C, L, M) ina(s). Define a consumpion-invesmen sraegy ν =(C, (1 α)l, M), and denoe by ( X,Ỹ ) he corresponding ax-free bank and risky asse accoun processes wih he iniial endowmen (x+αk, (1 α)y). Clearly, Ỹ =(1 α)y s,ν. To see ha ν is admissible in he ax-free financial marke, observe ha he process Z := X + Ỹ saisfies Z Z s,ν = and Z Z s,ν e r e ru rαk s,ν u du, so ha Z s,ν Z s,ν. Hence, V (x +(1 α)y + αk) J (s, ν); see Remark 2.4. The required resul follows from he arbirariness of ν A(s). Proposiion 4.2. For s =(x, y, k) in S and z = x +(1 α)y + αk, here exiss a sequence of admissible sraegies (ν n ) n 1 A(s) such ha V (s) c (r, θ α) p 1 z p p = lim J (s, ν n ), where θα rα := θ n σ(1 α) ; i.e., he value funcion of he Meron fricionless problem wih he smaller risk premium θ α can be approached as close as possible in he conex of he financial marke wih axes. This resul is proved by producing a sequence of admissible sraegies (C n,l n,m n ) n 1 A(s) which approximaes Meron s value funcion wih he smaller risk premium θ α.togive an inuiive jusificaion of his resul, we rewrie (2.11) as ( (4.1) dz =(rz C ) d + Y σ α dw + θ ) α d + rα (Y K ) d, where θ α is defined as in he saemen of Proposiion 4.2 and σ α := (1 α)σ. Compare he above Z dynamics o (2.16) wih modified parameers ( σ α, θ α )andwihc = C / Z, π = Y / Z. The only difference is he erm rα(y K). However, in view of Proposiion 3.5, we expec his erm o be nonnegaive for he opimal sraegy (if i exiss). This hins ha he liquidaion value process Z (wih he above choices C and Y ) is larger han he wealh process in he ficiious ax-free financial marke wih a modified risk premium. This formally jusifies he inequaliy of Proposiion 4.2.
378 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI The proof repored in Appendix B exhibis an explici sequence of sraegies which mimics he opimal consumpion-invesmen sraegy in he Meron fricionless model while keeping he difference Y K small or, equivalenly, he ax basis close o he spo price of he risky asse. Remark 4.1. Le b := r + θσ be he insananeous mean reurn coefficien in our financial marke. Then, he modified risk premium θ α can be easily inerpreed in erms of he modified volailiy coefficien σ α =(1 α)σ and a similarly modified insananeous mean reurn coefficien b α := (1 α)b as θ α =(b α r)/σ α. This ficiious financial marke wih such modified coefficiens corresponds o he siuaion where he invesor is forced o realize he capial gains or losses, a each ime, before adjusing he porfolio. Proposiions 4.1 and 4.2 provide he following bounds on he value funcion V : (4.2) c(r, θ α ) p 1 (x +(1 α)y + αk) p V (x, y, k) p c (r, θ)p 1 (x +(1 α)y + αk) p, p where θ α is defined as in he saemen of Proposiion 4.2, and c is defined as in Theorem 2.1. Observe ha θ α = θ whenever α =orr =. Therefore, we migh expec ha hese bounds are igh for small ineres rae or ax parameers. Corollary 4.1. For s =(x, y, k) S, we have V (s) =V app (s)+o(α + r), where o(ξ) is a funcion on R wih o(ξ)/ξ as ξ, and ( V app (s) := c(,θ)p 1 1+ rp ) (x + y) p + α c(,θ) p 1 (k y)(x + y) p 1. p c(,θ) Proof. I is sufficien o observe ha he bounds on he value funcion V in (4.2) are smooh funcions wih he idenical parial gradien wih respec o (r, α) aheorigin.this follows from he fac ha ( θ α / α) =( θ α / r) =a(r, α) =(, ). Remark 4.2. Observe ha he funcion c defined in Theorem 2.1 is increasing in he r variable. Then, he above firs order expansion shows ha he value funcion V is also increasing in he ineres rae variable (for small ineres rae and ax parameers). This is inuiively clear, as he larger ineres rae provides he invesor a beer opporuniy se. The dependence of he value funcion on he ax rae α is more complex, and i depends on he iniial posiion of he ax basis. If he iniial ax basis is larger han he spo price, i.e., in a siuaion of capial gain loss, he invesor akes immediae advanage of he ax credi, as saed in Proposiion 3.5, and he value funcion V is increasing in α (for small α). In he opposie siuaion, i.e., when he iniial ax basis is smaller han he spo price, he value funcion is decreasing in α. Finally, when he iniial ax basis coincides wih he spo price, he value funcion is no sensiive o he ax rae in he firs order. This variaion of he value funcion (up o he firs order) in erms of he ax rae α is somehow surprising. Indeed, in a capial loss siuaion, an increase of he ax parameer implies wo opposing resuls: an increase of he ax credi is received iniially by he agen; a larger amoun of ax is paid during he infinie lifeime of he agen.
MERTON PROBLEM WITH TAXES 379 Our firs order expansion shows ha, for small ineres rae and ax parameers, he increase of iniial ax credi is never compensaed by he increase of ax over he infinie lifeime. This is in agreemen wih Cadenillas and Pliska [7], who found ha someimes invesors are beer off wih a posiive ax rae. The same reasoning also shows ha when here are no iniial embedded capial gains (i.e., when y = k = ) he effec of he ax parameer is only second order. Remark 4.3. Since he lower bound in (4.2) has he same firs order Taylor expansion as he value funcion V, we can view he corresponding sraegy as nearly opimal. From he discussion following Proposiion 4.2, he porfolio allocaion defining he lower bound is by definiion an approximaion of he consan porfolio allocaion π( σ α, θ α ) = 1 (1 p)σ 2 [ ] b 1 α r (1 α) 2, where b := σθ + r is he insananeous mean reurn of he risky asse. Direc compuaion shows ha π( σ α, θ α ) π(σ, θ) if and only if r (1 α)(b (1 α)(b r)). Using he daa se of Dammon, Spa, and Zhang [12] (r = 6%, b = 9%, α = 36%), we see ha π( σ α, θ α ) π(σ, θ). Since his nearly opimal sraegy exhibis a smaller exposiion o he risky asse, he presence of axes on capial gains conribues o explaining he equiy premium puzzle highlighed by Mehra and Presco [25]. Noice ha his observaion is in conradicion wih he numerical resuls of Dammon, Spa, and Zhang [12], who found ha he exposiion o he risky asse is increased by he presence of axes. This is due o he fac ha he bank accoun in heir model is also subjec o axes wih he same ax rae as for he risky asse, which implies ha he opimal porfolio sraegy in he forced realizaion case is given by ˆπ α = b(1 α) r(1 α) (1 p)σ 2 (1 α) 2 = π(σ, θ) 1 α, which is increasing in α. 5. Characerizaion by he dynamic programming equaion. The chief goal of his secion is o provide a characerizaion of V by means of a second order parial differenial equaion for which we shall provide a numerical soluion in he subsequen secion. Unforunaely, we are unable o obain a characerizaion of V by he corresponding dynamic programming equaion. Therefore, we shall exhibi a consisen approximaion V ε as he unique soluion of an approximaing second order parial differenial equaion. For s in S and ν =(C, L, M) ina, he jumps of he sae processes S are given by ΔS s,ν = ΔL g b ΔM [ (1 α)y s,ν + ] αks,ν g s ( S s,ν ), where he vecor fields g b and g s (x, y, k) are defined by g b := 1 1 1 and g s (s) := 1 1 1 α + α 1 α 1 k (1 α) y + αk 1 (y,k).
38 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI The value funcion of our consumpion-invesmen problem is formally expeced o solve he corresponding dynamic programming equaion: { } (5.1) min Lv, g b Dv, g s Dv = on S \ z S and v = on z S, where L is he second order differenial operaor defined by Lv = βv + rxv x + byv y + 1 2 σ2 y 2 v yy + Ũ(v x), wih Ũ(q) = sup(u(c) cq). c Observe ha we have no informaion on he regulariy of he value funcion V ; hence we canno prove ha V is a classical soluion o (5.1). Moreover, he value funcion V is known only on he boundary z S (see Proposiion 3.2), bu here is no possible knowledge of V on y S k S. We hen need o use he noion of viscosiy soluions which allows for a weak formulaion of soluions o parial differenial equaions and boundary condiions. Because g s is no locally Lipschiz coninuous, i is no clear ha here is a unique characerizaion of he value funcion V as he consrained viscosiy soluion of (5.1). Due o his echnical difficuly, we isolaed in he accompanying paper [5] some viscosiy resuls for a slighly modified equaion. The objecive of his secion is o build on hese resuls in order o characerize he value funcion V as a limi of an approximaing funcion defined as he unique viscosiy soluion of a convenienly perurbed equaion. For every ε>, we define he funcion ( ) k + f ε (x, y, k) := 1 εz 1, wih z := x +(1 α)y + αk, ogeher wih he approximaion of g b and g s : 1+ε gε b := 1 and gε(x, s y, k) :=g s (x, y, kf ε (s)) 1 for s S\ z S.Noicehagε s is locally Lipschiz coninuous on S \ z S,andgε(s) s =g s (s) whenever k 2εz. The main resul of his secion provides a characerizaion of he value funcion V by means of he approximaing equaion: } (5.2) min { Lv, gε b Dv, gs ε Dv = on S \ z S and v = on z S. We firs recall he noion of a consrained viscosiy soluion firs inroduced in [29, 3]. For a locally bounded funcion u : S R, we shall use he classical noaion in viscosiy heory for he corresponding upper semiconinuous and lower semiconinuous envelopes: u (s) := lim sup S s s u(s ) and u (s) := lim inf S s s u(s ). Definiion 5.1. (i) A locally bounded funcion u is a consrained viscosiy subsoluion of (5.2) if u on z S,andforalls S \ z S and ϕ C 2 ( S) wih =(u ϕ)(s) = max S\ z S (u ϕ) we have min { Lϕ, g b ε Dϕ, g s ε Dϕ }.
MERTON PROBLEM WITH TAXES 381 (ii) A locally bounded funcion u is a consrained viscosiy supersoluion of (5.2) if u on z S,andforalls Sand ϕ C 2 (S) wih =(u ϕ)(s) =min S (u ϕ) we have min { Lϕ, gε b Dϕ, gε s Dϕ }. (iii) A locally bounded funcion u is a consrained viscosiy soluion of (5.2) if i is a consrained viscosiy subsoluion and supersoluion. In he above definiion, here is no boundary value assigned o he value funcion on y S k S. Insead, he subsoluion propery holds on his boundary. Noice ha he supersoluion propery is saisfied only in he inerior of he domain S. Theorem 5.1. For each ε>, he boundary value problem (5.2) has a unique consrained viscosiy soluion V ε in he class Cp ( S). Moreover, he following hold: (i) he family (V ε ) ε> is nonincreasing and converges o he value funcion V uniformly on compac subses of S as ε ; (ii) for every s S and δ, we have V ε (δs) =δ p V ε (s). Proof. The exisence of V ε as he unique consrained viscosiy soluion of (5.2) inhe class Cp ( S) isshownintheorem3.2of[5], where we inroduced he value funcion v ε,λ of a consumpion-invesmen problem wih ransacion coss λ>andanε-modified axaion rule near he ray {(x,, ), x R + }.HereV ε = v ε,ε. We nex use Theorem 5.1 of [5], which provides a sochasic conrol represenaion of v ε,λ. In paricular, i follows direcly from his represenaion ha v ε,λ (δs) =δ p v ε,λ (s) for all δ. This homoheiciy propery is obviously inheried by V ε = v ε,ε, as announced in (ii). We now prove (i) in he nex wo seps. Sep 1. The monooniciy of he sequence (V ε ) ε> is inheried from he nonincrease of he sequence ( v ε,λ), proved in Proposiion 6.2 of [5], ogeher wih he decrease of he sequence ( ε> v ε,λ ) λ>. I follows from his monooniciy propery ha V := lim ε V ε is well defined. Now observe ha v ε,λ V ε v,ε V for every λ ε. Then lim λ lim ε v ε,λ V V. By Proposiion 6.3 of [5], his implies ha (5.3) lim λ v λ v,ε V, where v λ := lim ε v ε,λ is he value funcion of he opimal consumpion-invesmen problem wih axes and proporional ransacion cos λ>. In order o conclude ha V = V,iremainsoshow ha lim λ vλ = V. Sep 2. For fixed s S, leν n =(C n,l n,m n ) be such ha V (s) 1 n J (s, ν n ) for all n 1. Denoe by Z λ,n he afer-ax liquidaion value process wih consumpion-invesmen sraegy ν n in a financial marke subjec o axes and consan proporional ransacion cos parameer λ. See Appendix A for he precise formulaion. We also denoe by Z n he corresponding aferax liquidaion value process wihou ransacion coss. Then, i follows immediaely from he dynamics of hese processes ha Z λ,n = Z n λl n for all.
382 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI Then he sopping imes { τ λ := inf >: Z λ,n 1 } 2 Zn saisfy τ λ as λ P-a.s. Define he sraegies ν n by ν n := ν n 1 [,τ λ ) + (,L n τ λ,mn τ λ +1) 1 [τ λ, ). Clearly, ν n is admissible for he problem wih ransacion coss, i.e., ν n A λ (s) in he noaion of Appendix A. Then [ ] τ λ [ ] v λ (s) J (s, ν n ) = E U(C n )d E U(C n )d V (s) 1 n, where we use he monoone convergence heorem. By he arbirariness of n and (5.3), his shows ha V = V. Finally he convergence holds uniformly on compac subses by he monooniciy of (V ε ) and he coninuiy of he limi V. 6. Numerical esimae for V. We have saed in he previous secion ha he value funcion V is approximaed by he funcions (V ε ) ε>,where,foreachε>, V ε can be compued as he unique viscosiy soluion of he boundary value problem (5.2). In his secion, we provide a numerical esimae for V, based on a numerical scheme for (5.2). Unforunaely, we have no heoreical convergence resul for our algorihm. Indeed, as discussed in he inroducion, esablishing convergence resuls for Hamilon Jacobi Bellman equaions corresponding o singular conrol problems is an open quesion in numerical analysis. We herefore follow previous relaed works such as [1] and[31] by concenraing our effor on realizing he empirical convergence of he algorihm. Following Akian, Menaldi, and Sulem [1], we adop a numerical scheme based on he finie difference discreizaion and he classical Howard algorihm. For he convenience of he reader we briefly describe i hereafer, and we refer he reader o [1] for a deailed discussion. 6.1. Change of variables and reducion of he sae dimension. By he homoheiciy propery of V ε (Theorem 5.1(ii)) we have for s =(x, y, k) S\ z S and z := x+[(1 α)y+αk] ( y V ε (s) = z p V ε z, k ), where V ε (ξ 1,ξ 2 ) := V ε (1 (1 α)ξ 1 αξ 2,ξ 1,ξ 2 ). z Nex, for a vecor ξ R 2 +, we define he vecor ζ [, 1) 2 by ζ i := ξ i /(1 + ξ i ), i =1, 2, and Ψ ε (ζ) := V ε (ξ). This reduces he domain of V ε from R 2 + o he bounded domain [, 1) 2. By changing variables, i is immediaely checked ha Ψ ε is a coninuous consrained viscosiy soluion on [, 1) [, 1) of { } 2 (6.1) min β(a)ψ ε (ζ) b i (a, ζ) D i Ψ ε (ζ) 1 2 η ij (a, ζ)d 2 a A 2 ijψ ε (ζ) g(a) =, i=1 where he conrol se A and he expressions of β, (b i ) i=1,2,(η i,j ) i,j=1,2,andg are obained by immediae calculaion. i,j=1
MERTON PROBLEM WITH TAXES 383 6.2. Finie differences for (6.1). We adop a classical finie difference discreizaion in order o obain a numerical scheme for (6.1). Le N be a posiive ineger, and se h := 1 N, he finie difference sep; we se e 1 := (1, ) and e 2 := (, 1), and we define he uniform grid S h := [, 1] 2 (hz) 2. We denoe by ζ h := (ζ1 h,ζh 2 ) a poin of he grid S h,andwese S h := (, 1) [, 1) (hz) 2. In order o define a discreizaion of (6.1), we approximae he parial derivaives of Ψ ε by he corresponding backward and forward finie differences { b i (a, ζ) i Ψ ε bi (a, ζ)d + (ζ) i Ψε (ζ) if b i (a, ζ), b i (a, ζ)di Ψε (ζ) if b i (a, ζ) <, ii Ψ ε (ζ) Di 2 Ψ ε (ζ), { η ij (a, ζ) ij Ψ ε ηij (a, ζ)d ij + (ζ) Ψε (ζ) if η ij (a, ζ), η ij (a, ζ)dij Ψε (ζ) if η ij (a, ζ) <, where he finie difference operaors are defined for i j {1, 2} by D + i Ψε (ζ) = Ψε (ζ + he i ) Ψ ε (ζ) h, Di Ψε (ζ) = Ψε (ζ) Ψ ε (ζ he i ), h D 2 i Ψ ε (ζ) = Ψε (ζ + he i ) 2Ψ ε (ζ)+ψ ε (ζ he i ) h 2, D ± ij Ψε (ζ) = 1 2h 2 {2Ψε (ζ)+ψ ε (ζ + he i ± he j )+Ψ ε (ζ he i he j ) Ψ ε (ζ + he i ) Ψ ε (ζ he i ) Ψ ε (ζ + he j ) Ψ ε (ζ he j )}. In order o compue hese differences a every poin of S h,weexendψ ε as follows: ( ) ( ) ( ) ( ) Ψ ε ζ h = Ψ ε ζ h + he 1, Ψ ε ζ1 h = Ψ ε ζ1 h he 1 for ζ h {} [, 1] and ζh 1 {1} [, 1] and ( ) ( ) ( ) Ψ ε ζ h he 2 = Ψ ε ζ h, Ψ ε ζ1 h ( ) = Ψ ε ζ1 h he 2 for ζ h [, 1] {} and ζh 1 [, 1] {1}. This provides a sysem of (N 1)N nonlinear equaions wih he (N 1)N unknowns Ψ ε h (ζh ), ζ h S h : (6.2) min a A {Aa h Ψε h g(a)} =. 6.3. The classical Howard algorihm. To solve (6.2) we adop he classical Howard algorihm, which can be described as follows: Sep : sar from an iniial value for he conrol a A, Ψ h soluion of Aa h ϕ g(a ) =, { } Sep k +1,k : find a k+1 argmin a A A a h Ψk h g(a), Ψ k+1 h soluion of A ak+1 h ϕ g(a k+1 ) =.
384 IMEN BEN TAHAR, H. METE SONER, AND NIZAR TOUZI 7. Numerical resuls. We implemen he above numerical algorihm wih he following parameers: p =.3, σ =.3, and β =.1. We also fix he insananeous mean reurn of he risky asse o b := θσ + r =.11. Our numerical experimens showed ha by aking a finie difference sep 1/2 h 1/4 our algorihm converges wihin a reasonable compuaion ime: convergence error Ψ k+1 h Ψ k h 1 6, compuaion ime 1 5 minues. 7.1. Accuracy of he firs order Taylor expansion. I is clear ha one should no expec his algorihm o be reliable near he boundary of he grid. However, realisic iniial poins are far from he boundary, and we expec he error o be small for hese poins. The main purpose of his subsecion is o examine he accuracy of he firs order approximaion for differen ses of parameers r and α: r {.1,.1,.7} and α {.1,.1,.5,.1,.2,.3,.36}. Figure 1 plos he mean relaive error beween he resuls of he firs order expansion and he numerical algorihm over all poins of he grid: ( ) ( 1 Vε h ζij h V app ζ h ij) ), N(N 1) i,j V (ζ app ij h where N(N 1) is he oal number of poins in he grid, Vε h obained by our numerical scheme, and is he approximaion of V ε V app (ξ 1,ξ 2 ):=V app (1 (1 α)ξ 1 αξ 2,ξ 1,ξ 2 ). As expeced, he relaive error is zero a he origin and increases when he values of he parameers r and α increase. The error size is large due o he boundary effecs. Indeed, in Figure 2 we focus our aenion on a region away from he boundary and concenrae on (y, k) [, 1] 2 (i.e., a region wih small iniial embedded capial gains). We observe ha he average relaive error is remarkably small and is of he order of 4% for realisic values of r and α. This figure is our main numerical resul, as i shows he reasonable accuracy of he firs order Taylor approximaion V app of he value funcion V. 7.2. Welfare analysis. In view of Remark 4.3, anε-maximizing sraegy is given by he consan porfolio allocaion π α and he consan consumpion-wealh raio c(r, θ α ). The expeced uiliy realized by following his approximaing sraegy corresponds o he lower bound Ṽ (z) = c(r, θ α )z p /p of Proposiion 4.2. In order o compare his approximaing sraegy o he opimal one, we repor in Figures 3 and 4 he welfare cos, z such ha V (1 (1 α)ξ 1 αξ 2,ξ 1,ξ 2 )=Ṽ (1 + z ), wih he following parameers: p =.3, β =.1, b := r + θσ =.11, σ =.3, and r =.7.
MERTON PROBLEM WITH TAXES 385 Figure 1. Mean relaive error on [, 1] 2. Figure 2. Mean relaive error on [, 1] 2. Figure 3. Welfare cos for α =.2. Figure 4. Welfare cos for α =.36. The welfare cos is nonincreasing wih respec o he ax basis and remains relaively small for reasonable values of he parameers α: i reaches a maximum of 8% for α =.2 andof 12% for α =.36. 7.3. Opimal consumpion-invesmen sraegies. Throughou his subsecion we implemen our numerical algorihm wih he following parameers: p =.3, β =.1, b := r + θσ =.11, σ =.3, and r =.7. The ax-free model. For α =., our algorihm produces he well-known resuls of he Meron fricionless model. Given he above values of he parameers, Meron s opimal sraegy is given by π =.6349 and c =.174. Figure 5 repors he numerical soluion for he funcion Vε h. We verify ha he funcion Vε h in his ax-free conex does no depend on he variable ξ 2, so ha he value funcion Vε h does no depend on he k componen. We also see ha he value funcion is concave. Figure 6 repors he opimal invesmen sraegy and produces he expeced pariion of he sae space ino hree regions: The region of no ransacion (NT) corresponds o posiions such ha he proporion of wealh allocaed o he risky asse y/(x + y) isequalo π. In his region no posiion adjusmen is considered by he invesor. The Sell region is where he invesor immediaely sells risky asses so as o aain he region NT by moving along he ray (1, 1).