OPTIMAL PORTFOLIO MANAGEMENT WITH TRANSACTIONS COSTS AND CAPITAL GAINS TAXES

Transcription

1 OPTIMAL PORTFOLIO MANAGEMENT WITH TRANSACTIONS COSTS AND CAPITAL GAINS TAXES Hayne E. Leland Haas School of Business Universiy of California, Berkeley Curren Version: December, 1999 Absrac We examine he opimal rading sraegy for an invesmen fund which in he absence of ransacions coss would like o mainain asses in exogenously fixed proporions, e.g. 60/30/10 in socks, bonds and cash. Transacions coss are assumed o be proporional, bu may differ wih buying and selling, and may include a posiive capial gains ax componen. We show ha he opimal policy involves a no-rade region abou he arge sock proporions. As long as he acual proporions remain inside his region, no rading should occur. When proporions are ouside he region, rading should be underaken o move he raio o he region s boundary. We compue he opimal muli-asse no-rade region and resuling annual urnover and racking error of he opimal sraegy. Almos surely, he sraegy will require rading jus one risky asse a any momen, alhough which asse is raded varies sochasically hrough ime. Compared o he curren pracice of periodic rebalancing of all asses o heir arge proporions, he opimal sraegy wih he same degree of racking error will reduce urnover by almos 50%. The opimal response o a capial gains ax is o allow proporions o subsanially exceed heir arge levels before selling. When an asse s proporion exceeds a criical level, selling should occur o bring i back o ha criical level. Capial gains axes lead o lower opimal iniial invesmen levels. Similarly, saring from a zero-invesmen posiion, i is opimal o inves less iniially in asse classes ha have high ransacions coss, such as emerging markes. Our analysis makes precise he effecs of ransacions coss on opimal iniial invesmen and subsequen rading. The auhor graefully acknowledges suppor from BARRA and he BSI Gamma Foundaion. Andrea Belrai, Greg Connor, Avinash Dixi, Ron Kahn, and Dan Sefek have provided valuable commens. Hui Ou-Yang provided insighs ino he soluion o he muli-asse problem. Klaus Tof correced an error in an earlier version. The auhor bears sole responsibiliy for errors ha may remain. This paper significanly exends he resuls of wo earlier working papers, "Opimal Asse Rebalancing in he Presence of Transacions Coss," IBER Working Paper RPF-61, Augus 1996, and Muliple Asse Rebalancing in he Presence of Transacions Coss and Capial gains Taxes, Sepember 1997.

2 OPTIMAL PORTFOLIO MANAGEMENT WITH TRANSACTIONS COSTS AND CAPITAL GAINS TAXES I. Inroducion Many invesors, boh insiuional and privae, sae heir invesmen sraegy in erms of desired asse proporions, such as a 60/40 raio of socks o bonds, or 40/40/0 proporions of domesic asses, foreign asses, and cash. As asse values move randomly, asse raios diverge from heir arges. Bu when asse reurns follow a diffusion process, i is well known e.g. Leland [1985] ha an infinie amoun of rading is required o keep asses coninuously a heir arge proporions. This creaes a problem: frequen readjusmens o keep asses close o heir arge levels will incur high rading coss. Bu infrequen revision will creae racking error relaive o he ideal porfolio. Typical curren pracice is o "rebalance" o desired arge proporions on a periodic basis, e.g. quarerly or annually. Less frequen rebalancing lowers he expeced amoun of rading, bu creaes higher average deviaions from he desired asse raios. Bu he convenional sraegy of periodic rebalancing o he arge raios will no be opimal when ransacions coss are proporional o he dollar amouns raded. Work by Magill and Consaninides [1976], Taksar, Klass, and Assaf [1988], and Davis and Norman [1990] on he single risky asse case shows ha he opimal sraegy is characerized by a "no rade" inerval abou he arge risky asse proporion. When he proporion varies randomly wihin his inerval, no rading is needed. When he risky asse raio moves ouside he no-rade inerval, i should be adjused back o he

3 neares edge of he inerval--no o he arge proporion. 1 Dixi [1991], Dumas [1991], and Shreve and Sohner [1994] provide furher mahemaical resuls for his and relaed problems wih a single risky asse, based on work by Harrison and Taksar [1983], Harrison [1985] and ohers on regulaed Brownian moion. Akian, Menaldi, and Sulem [1996] consider a muli-dimensional version of Davis and Norman [1990]. Dixi [1997] and Eberly and Van Mieghem [1997] examine he relaed problem of a profimaximizing firm facing parially-irreversible invesmen decisions in muliple facors of producion. Our work differs from previous work in several ways. Firs, our focus is on managing porfolios ha have a given se of arge or ideal asse raios. Many large invesors, including pension funds, formulae heir invesmen sraegies in erms of desired long-run asse allocaion raios. These raios are based on perceived risks and reurns, for example using a mean-variance opimizaion approach, and ypically do no adjus for ransacions coss. 3 Raher han assuming a specific uiliy funcion over wealh which invesmen managers can rarely specify, we posulae a loss funcion ha is naural o many porfolio managers: he sum of rading coss and he coss associaed wih racking error--divergences from he desired arge raios. This permis a possible disincion beween risk aversion for asse selecion and risk aversion owards racking error, a disincion ha many praciioners consider imporan. 4 1 The inuiion behind his resul is as follows. The loss L from diverging from he opimal raio is approximaely U-shaped. Because i is fla a he boom, very lile loss reducion resuls from moving he las small amoun o he opimum raio: he gain is of second order, and insufficien o jusify he firs order rading coss. Relaed problems include he opimal cash managemen problem examined by Connor and Leland [1995] and he opion replicaion problem in he presence of ransacions coss see, e.g., Leland [1985] and Hodges and Neuberger [1989]. 3 An ad hoc approach has been o adjus he mean reurn of an asse or asse class downwards o reflec rading coss. This approach is erroneous, as discussed in Secion X. 4 See, for example, Grinold and Kahn

4 Second, we develop a echnique for esimaing he expeced urnover and he expeced racking error of arbirary policies. This allows an invesor o assess he radeoff beween he required volume of rading and he racking accuracy of alernaive efficien policies, and o compue he loss associaed wih following sub-opimal sraegies. There are imporan cos savings o be realized from following opimal rebalancing sraegies raher han radiional periodic rebalancing sraegies. For he same average racking error, he opimal sraegy will reduce rading coss by almos 50%. Third, we examine he effecs of capial gains axes on opimal rading sraegies. Capial gains axes can be deferred by no selling bu no selling an asse may lead he porfolio o become dangerously over-invesed in ha asse. We deermine an invesor s opimal sraegy when facing a capial gains ax. Fourh, we can use our resuls o examine imporan ax and regulaory policy quesions. For example, how would a ransacions ax on porfolio rading, or a capial gains ax on sales, affec he long-erm average volume of rading? We show below ha, under reasonable assumpions, a % ransacions ax could cu rading by over 40%. Finally, we consider how rading coss affecs he opimal iniial invesmens in differen asse classes. Popular wisdom holds ha rading coss should be modeled as reducing he expeced reurn of an asse, wih a consequen scaling back of he amoun invesed in ha asse. Bu we show ha his will no be a correc approach. Iniial invesmens ha exceed ideal levels may be opimal for an asse ha incurs rading coss, if hose coss are smaller han hose of oher correlaed asses. Furhermore, if an invesor should iniially hold a larger posiion han he arge amoun, i may well be opimal o reain ha 4

5 posiion. Our resuls permi a rigorous examinaion of he impac of rading coss on he opimal holding of asses. II. Asse Price and Proporion Dynamics Consider dollar holdings S i of asse i which evolve as a regulaed logarihmic Brownian moion, τ : 1 ds i τ = µi S i dτ + σi S i dz i τ + dl i τ dm i τ i = 0,..., N, wih iniial values S i - = S i i = 0,, N. where he dz i are he incremens o a join Wiener process wih correlaions ρij. S i - denoes he lefhand limi of he process S i a ime. By assumpion, asse i = 0 is riskfree, wih σ0 = 0 and µ0 = r. L i τ and M i τ are righ-coninuous and nondecreasing cumulaive dollar purchases and sales of asse i on [, τ], respecively, wih L i - = M i - = 0. Noe ha iniial rades purchases or sales of asse i are given by L i or M i. For simpliciy, i is assumed ha any ransacions coss incurred will be paid by addiional conribuions o he fund. Wih his excepion, here are no ne conribuions or wihdrawals from he invesor s holdings, implying a self-financing consrain Σi 0 dl i τ dm i τ = 0, for all τ. 5

6 Le µ denoe he column vecor µ1,...,µn of insananeous expeced raes of reurn of he risky asses, and V denoe he insananeous variance covariance marix of risky raes of reurn, wih elemens ρij σi σj, i, j = 1,, N. Le Σi 0 denoe he summaion operaor over all asses, i = 0,..., N, and Σi denoe he summaion operaor over risky asses i = 1,...,N. Define he following: Wτ = Σi 0 S i τ: invesor wealh a ime τ, assumed sricly posiive for all τ. 5 w i τ = S i τ/wτ : he proporion of wealh held in risky asse i a ime τ, i = 0,,N. Noe Σi 0 w i τ = 1. wτ: he vecor of he risky asses proporions w i τ, i = 1,, N. x: he vecor of iniial asse proporions w i - = S i /W, i = 1,, N. w i * : he given arge proporion of wealh in asse i, i = 0,, N. Noe Σi 0 w i * = 1. Then w * : he vecor of arge risky asse proporions w i *, i = 1,..., N. 5 Sufficien condiions for nonnegaive wealh are ha no shor posiions or borrowing be allowed. While saisfied by he examples we consruc below, hese are no necessary condiions. 6

7 dwτ/wτ = Σi 0 ds i τ/wτ = Σi 0 µi S i τ/wτdτ + Σi 0 σi S i τ/wτdz i + Σi 0 dl i τ dm i τ = Σi 0 µi w i τdτ + Σi 0 σi w i τdz i, = r + Σi µi - rw i τdτ + Σi σi w i τdz i where he firs line follows from he definiion of Wτ, he second line uses equaion 1, he hird uilizes he definiion of w i τ and he self-financing consrain, and he las follows from w 0 τ = 1 - Σi w i τ. The weighs w i τ will be changing hrough ime, bu ypically will be close o w i * when asses are opimally raded. Thus o a close order of approximaion, dwτ/wτ * * = Σi0 µi w i dτ + Σi0 σi w i dz i. = r + Σi µi - rw * i dτ + Σi σi w * i dz i. Define µw = * r + Σi µi - rw i σw = Σi Σj w i * w j * ρijσiσj σiw = σi Σj w * j ρij σj Observe µw dτ = E[dW/W] σw dτ = E[dW/W ] and, when here is no rading dl i = dm i = 0, σiw dτ = E[dS i /S i dw/w] 7

8 Since w i τ = S i τ/wτ, i follows from Io s Lemma ha 3 dw i τ = µi - µw + σw - σiw w i dτ + σi w i dz i - Σj σj w j * dz j w i + δw i τ, where δw i τ = dl i τ dm i τ/wτ. The nonnegaive process wτ is hus righ coninuous wih lef-hand limi. Define a i = µi - µw + σw - σiw q ij = σi σj ρij - σiw - σjw + σw, and le a and Q represen he 1xN vecor and NxN marix wih elemens a i and q ij, respecively. 6 Noe ha, when here is no rading dl i τ = dm i τ = 0, a i w i dτ = E[dw i ] q ij w i w j dτ = E[dw i dw j ]. III. The Invesor s Objecive The invesor wishes o hold risky asses in arge proporions w *. Divergence beween he acual raios wτ and w * creaes an expeced uiliy loss resuling from racking error. This loss can be reduced by rading more frequenly bu more frequen rading will lead o greaer ransacions coss. The opimal invesmen sraegy will minimize he sum of racking error coss and rading coss. 6 Observe ha he dynamics of he w i τ can vary significanly from he dynamics of he S i τ. For example, if here are wo posiively correlaed risky asses whose weighs sum o one implying w 0 τ = 0, hen dw i τ/w i τ will be perfecly negaively correlaed, i = 1,. 8

9 III.a. Tracking Error The incremenal loss duτ of uiliy a ime τ, measured in dollar erms, is assumed proporional o he variance of racking error he variance of rae of reurn from holding asses in proporion wτ raher han w * over he inerval dτ: 4 duτ = λ w * - wτ' V w * - wτdτ, where primes denoe ransposes and λ is he invesor's "price of racking error." 7 III.b. Trading Coss Over an infinie horizon, he invesor wishes o minimize he discouned inegral of racking error losses duτ, plus he discouned rading coss associaed wih adjusing asse proporions wτ. Trading coss are assumed o be proporional o he dollar amoun of rades. The analysis immediaely below assumes ha buying and selling coss are he same. Laer, coss are allowed o vary wih he ype of ransacion. 7 Equaion 4 can be jusified by assuming mean-variance preferences over raes of reurn o wealh. In ha case, 4 can be expressed as he difference beween mean-variance uiliy a w * and a wτ: Lτ = [µ - r w * dτ - λ w * V w * dτ] - [µ -r wτ dτ - λ wτ V wτ dτ] = λ w * - wτ V w * - wτdτ, where λ is he price of porfolio risk variance, and he second line uses he mean-variance opimal porfolio condiion ha w * = 1/λV -1 µ - r. Noe ha his raionale would imply ha he same "λ" is used o choose he opimal porfolio and o price racking error. More generally, however, we can separae he wo. Revealed behavior by porfolio managers suggess ha racking error is ofen accorded a higher cos han he λ implied by he selecion of asse proporions. 9

10 Define k i : he ransacions cos per dollar of asse i raded 8, k: he vecor of coss k 1,..., k N. We assume he riskless asse i = 0 is cosless o rade. Recalling δw i τ = dl i τ dm i τ/wτ, he incremenal rading coss dt i τ of asse i a ime will be dt i τ = k i Wτ δw i τ. Since he loss 4 is expressed in erms of he racking error on porfolio raes of reurn i.e. dollar reurn divided by wealh, we also normalize dollar ransacions cos by wealh Wτ. Define dtc i τ = k i δw i τ. Therefore, a any ime, he change dcτ in cos per uni wealh of racking error plus rading cos will be given by 5 dcτ = duτ + Σi dtc i τ III.c. Toal Discouned Coss = λ wτ - w * ' V wτ - w * dτ + k' δwτ. 8 Alernaively, proporional ransacions coss which vary wih buying and selling could be represened by k i+ for purchases dl i, and k i- for sales dm i. See, for example, Secion VIIIb below. 10

11 The presen value of fuure expeced coss Jx; β a ime depends on he chosen rading sraegy β defined more specifically below and on he iniial asse proporions x before rade, where x = w -. J is he discouned inegral of expeced fuure coss dcτ, τ : J x; β = E[ = E[{ e r τ e r τ dc τ x, β ] λ w τ w* V w τ w* d + e r τ k δw τ } x, β ] Jx; β is convex in x as he insananeous cos funcion dc is sricly convex and he cos of rading is proporional o rade size see Harrison and Taksar [1983] and Dumas [1991]. 9 Following he insighs of Magill and Consaninides [1976] and he resuls of Davis and Norman [1990] for a single conrol variable, and he resuls of Akian, Menaldi, and Sulem [1996] and Eberly and Van Mieghem [1997] for N-dimensional conrol, he opimal rading sraegy will describe a conneced compac no-rade or coninuaion region χ ε R N when J is convex in x. In he case of a single risky asse, where w is a scalar, he no-rade region is an inerval: χ = [w min, w max ]. When x ε χ, no iniial rading akes place and w = w - = x. If x χ, iniial rading is required and 9 The ineresed reader may wonder wheher our approach is consisen wih expeced uiliy maximizaion, and wheher w* should iself be a funcion of ransacions coss. Our approach generaes opimal sraegies for an invesor seeking o maximize he expeced inegral of discouned uiliy, wih a mean-variance uiliy funcion over insananeous raes of reurn and a price of risk variance = λ: A each momen in ime, insananeous meanvariance uiliy Uwτ can from foonoe 6 be expressed as Uwτ = Uw* - λw*- wτ Vw*- wτ. Since Uw* is a consan given our saionariy assumpions, maximizing he discouned inegral of Uw less ransacions coss is he same as minimizing he discouned inegral of λw*- wτ Vw*- wτ plus ransacions coss which is precisely our J. 11

12 w = x + δw ε χ. 10 Similar o he resuls of he previously cied papers, when ransacions coss are proporional, rading will always move asse raios o a poin on he boundary of he no-rade region χ. Afer a poenially large iniial rade, subsequen rades will be infiniesimal in size as he coninuous diffusion process governing he movemen of asse raios wτ will no carry hese raios far ouside he boundary before rading back o he boundary occurs. Harrison and Taksar [1983] label his siuaion as one of insananeous conrol. Our objecive is o characerize χ and deail he naure of porfolio rading in he presence of proporional ransacions coss and capial gains axes. The boundary of χ will be denoed βχ, a subse of χ wih elemens "poins" denoed {w β }. Wih a single risky asse, βχ = {w min, w max }. Deermining he boundary β compleely deermines χ, given he assumpions above. Hereafer he boundary of he norade region, β = {w β }, is reaed as he invesor's choice variable. Thus he invesor's objecive is o find he boundary β = β * of he no-rade region ha minimizes discouned expeced coss Jx; β. IV. Deermining he Funcion Jx; β for a given Boundary β Firs consider he case where he no-rade region χ and is boundary β are given, bu are no necessarily opimal. When iniial asse proporions x ε χ, here is no rading a, and w = x. The only cos incurred over d is he cos of diverging from he opimal w *. From he definiion of Jx; β, 10 The opimal soluion will specify how he rades δwτ are deermined. 1

13 7 J x; β = E[ + d τ = λ w τ w* V w τ w* dτ + e = λ x w* V x w* d + E[ e rd rd J x + dw, β ] J x + dw, β w = x] when x ε χ. Expanding he expecaion erm of 7, and simplifying gives he parial differenial equaion 8 x a J x; β +.5 x [ Q J x; β ] x + λ x w* V x w* rj x; β = 0 x where J x = J 1,..., J N is he vecor of firs parial derivaives of Jx; β w.r.. x; J xx is he N x N marix of cross parials wih elemens J i j ; 1 is he N-dimensional uni vecor; a J x is he vecor wih elemens {a i J i }; and [Q J xx ] is he N x N marix wih elemens {q i j J i j }. xx As J is convex, J xx x; β is posiive semi-definie and posiive definie for x in he inerior of χ. For x χ, he sraegy chooses δw o insanly move he asse raios o a boundary poin w β x β. For x ouside he no-rade region χ, i follows immediaely ha Jx, β = Jw β x; β + k x - w β x. Associaing a boundary poin w β x β wih arbirary poins x ouside he boundary is examined in Appendix A. From A.1a A.1c, for poins x χ, 13

14 9 J i w β x; β = k i when x i > w i β x = -k i when x i < w i β x J i w β x, β < k i only if x i = w i β x. Condiions 9 hold for arbirary no-rade regions χ and heir associaed boundaries βχ = {w β }. Dumas 1991 erms hese value maching condiions. As will be seen below, mos boundary poins will be characerized by 9 holding for a single i. Bu he N corner poins on he boundary for which J i w β ; β = ±k i, for all i, will be of considerable compuaional imporance. Le Mβ ε β denoe such poins. Now consider he problem of deermining he opimal boundary β. The case of a single risky asse N = 1 builds from he analysis of Magill and Consaninides [1979], Davis and Norman [1990], and Dumas [1991]. 11 The general case wih an arbirary number of risky asses N can subsequenly be examined. V. Deermining he Opimal No-Trade Region in he Single Risky Asse Case N = 1 When here is a single risky asse, he opimal sraegy moves x o he boundary poin w min if x < w min, and o w max if x > w max. We drop he subscrip "1" for he single risky asse case. Therefore w β x = w max for all x > w max, and w β x = w min for all x < w min. Observe ha Mβ = β = {w min, w max }: every boundary poin belongs o Mβ, which is no rue for N. A he boundary poins when w min and w max, i follows from 9 ha 11 Our one-dimensional case differs from Dumas in ha he dynamics 3 of he asse weigh w follows a logarihmic Brownian moion. I differs from Consaninides in he form of he loss funcion. 14

15 10 J 1 w min ; w min,w max = -k 11 J 1 w max ; w min,w max = k where J n ;, is he derivaive of J wih respec o he nh argumen. 1 In he single risky asse case, equaion 8 is an ordinary differenial equaion wih soluion x xw* w* c11 c1 1 J x; wmin, wmax = λσ + + C1x + C x r a Q r a r where c 11 and c 1 are uniquely deermined as 13 c 11 = -a + Q/ + [a-q/ + Qr].5 /Q; c 1 = -a + Q/ - [a-q/ + Qr].5 /Q, where from 3 a = 1 - w * µ - r - σ w * ; Q = σ 1 - w * For a given β = {w min, w max }, he consans C 1 and C are deermined by he boundary condiions 10 and 11. The opimal boundary β * = {w * min, w * max} is deermined by he "super conac" condiions, which require ha Jx; w min, w max be minimized w.r.. w min and w max. This provides he wo final condiions needed for opimizaion, ha a he opimal w min and w max 14 J w min ; w min, w max = 0, 15 J 3 w max ; w min, w max = 0. 1 Dumas [1991] shows ha hese condiions are no smooh-pasing opimaliy condiions, bu raher he limi of a "value maching" condiion. The condiions someimes ermed super conac associaed wih he opimal boundary β * are given by 16 and

16 Following Dumas [1991], i can in urn be shown ha hese condiions imply 16 J 11 w min ; w min, w max = 0, 17 J 11 w max ; w min, w max = 0. Solving 8 subjec o he condiions 10, 11, 16, and 17 generaes soluions for he opimal sraegy parameers w * min and w * max, and for he consans C 1 and C of equaion 1, hereby uniquely deermining Jx; β * = Jx; w * min, w * max. VI. Deermining he Opimal No-Trade Region wih Muliple Risky Asses N The soluion o he parial differenial equaion 8 is much more difficul when N, since he boundary se β is now described by an infinie number of poins raher han he wo poins {w min, w max. From equaions A.1a A.1c, recall ha a every boundary poin w β, 18 J i w β, β k i, i = 1,..., N, wih equaliy holding for a leas one i. In addiion, he maximizing condiions equivalen o 16 and 17 are ha whenever J i w β ; β = k i, hen 19 J i i w β ; β = 0. 16

17 When N, we are unaware of closed form soluions o equaion 8 ha saisfy condiions 18 and 19 a all poins of he boundary β = {w β } of he no-rade region. So we urn now o finding an approximaion of he opimal sraegy, which we erm he quasi-opimal sraegy. VIa. Deermining A Quasi-Opimal No-Trade Region Our sraegy is o find a quasi-opimal soluion JAx; B which saisfies he p.d.e. equaion 8, and which saisfies condiions 18 and 19, wih JA replacing J, a a finie se of boundary poins B. Given JA, we can hen consruc he remaining boundary poins β beween hese poins B by an algorihm described in Appendix D. 13 In general, hese in-beween boundary poins will be consruced o saisfy eiher 18 or 19, bu will no excep by chance saisfy boh. We develop a measure of how well his quasi-opimal soluion approximaes an exac soluion, and show ha i will be highly accurae for realisic choices of parameers. Clearly he quasi-opimal funcion JAx; B will depend on he choice of he poins B a which boh condiions 18 and 19 will be saisfied. The soluion o equaion 8 can be represened by he sum of a homogeneous soluion and a paricular soluion. From Appendix B, here exiss a homogeneous soluion in he form H x; B = K { C k B N k = 1 i= 1 x cik i } where he coefficiens {c ik } saisfy equaion B5 and he coefficiens {C k } depend upon he boundary 13 Since here are only wo boundary poins when N = 1 B = {w min, w max }, i follows immediaely ha he quasiopimal soluion is he fully opimal soluion in his case. 17

18 condiions. 14 The acual number of erms K needed will depend upon he number of boundary poins a which boh condiions 18 and 19 are presumed o be saisfied. A naural se of boundary poins B are he N corner poins where condiions 18 hold wih equaliy for all i, as well as condiions 19. Each corner poin has N dimensions, so here are N N variables characerizing he poins w B ε B o be deermined. In addiion, we mus deermine he K consans {C k } in he homogeneous soluion. The oal number of variables o be deermined herefore is N N + K. The equaions o be saisfied a each of he N corner poins are he N equaions 18--which hold wih equaliy a w B ε B--and he N equaions 19. Thus N condiions mus hold a each of he N poins, for a oal of N+1 N equaions. For he number of equaions o equal he number of unknowns, i follows ha K = N N. We inroduce his number of power funcions in he homogeneous erm B., where he exponens {c i k } of each funcion k = 1,, K saisfy equaion B.5. A paricular soluion exiss of form B.9 in Appendix B. The quasi-opimal funcion JAx; B is given by he sum of he homogeneous soluion and he paricular soluion, saisfying he equaions 18 wih equaliy and 19 a he boundary corner poins w B ε B. These boundary condiions joinly deermine he boundary corner poins and he consans {C k }, k = 1,, K. Figure 1, discussed in deail in Secion IX, locaes corner poins {X,Y,Z,V}. We can use wo alernaive echniques o consruc he remaining boundary poins connecing adjacen corner poins w B ε B, given he funcion JAx; B. The firs assures ha condiions 18 for JA are me 14 In he one risky asse case N = 1, he coefficiens c ik are uniquely deermined. When N > 1, an infinie number of possible coefficiens exis, subjec o consrain B5 being saisfied. We can always find a soluion for he corner poins when he c ik are all possible combinaions of zero or one. 18

19 along his boundary; he second assures ha condiions 19 are saisfied. Appendix D develops hese algorihms. Le β1* denoe he se of boundary poins {w β1* } deermined by he firs echnique, and β* denoe he se of boundary poins {w β* } deermined by he second echnique. Because he second echnique proved more racable compuaionally, we used i o perform all calculaions ha follow. The calculaion of he boundary β* using 19 or D.3 also offers a measure of accuracy of our approximaion. Equaion 0 will in general no be saisfied: JA j w β*, B - k j 0. Bu for each poin w β* on he boundary we can compue he ransacions cos k j w β* which would make his erm zero, i.e. k j w β* JA j w β*, B. For such ransacions coss, he quasi-opimal boundary would be fully opimal, since i would saisfy boh condiions 18 and 19. The maximal error over he enire boundary β *, E = Max [ Max [ k j - k j w β* ] {w β*, j} indicaes he maximal amoun by which ransacions coss would have o vary for he quasi-opimal soluion o be he opimal soluion. For realisic parameers and modes asse correlaions <.30, his number is usually small: less han Thus if he quasi-opimal soluion is based on ransacions coss of say 1%, hen he quasi-opimal soluion is exac when ransacions coss range appropriaely as w β* varies beween.95% and 1.05%. Since i is rare o have exac esimaes of acual ransacions coss, his range seems olerable for mos pracical siuaions. Bu as asse correlaions increase, we observe from he examples in Secion IX below ha errors become larger reaching a maximum of.003 when ransacions coss are 1 percen. The soluion in his case is 19

20 exacly opimal only if ransacions coss were o vary appropriaely beween 0.70% and 1.30%, a fairly broad range. To furher reduce errors, a sraighforward exension of he approximaion mehod oulined above is now considered. VIb. Greaer Accuracy of he Quasi-Opimal Soluion The accuracy of he quasi-opimal soluion can be improved by finding a JA soluion ha saisfies he appropriae condiions 18 and 19 a more han jus he corner poins. 15 In wo dimensions, for example, we could require maching he appropriae opimaliy condiions a a finie number of poins along he boundary segmens XY, YZ, ZV, VX as well as a he corner poins X, Y, Z, V. An obvious se of addiional poins would be he four midpoins of he boundary segmens. We would need an addiional four erms in he homogeneous sum B. wih exponens saisfying B.5. Thus K = 1 in his siuaion. Maching he boundary condiions a more and more poins B leads o greaer and greaer accuracy as measured by, a he cos of he exra compuaional requiremens as K rises. In Secion IX, a se of examples examine he differences beween soluions wih K = 8 and soluions wih K = 1. In he laer soluions, opimaliy condiions are me a he midpoins of he boundary segmens as well as a he corners. For asse reurn correlaions of 0. or less, he no-rade regions are virually indisinguishable, wih he exreme poins {X,Y,Z,V} differing by less han 1% in boh dimensions and he segmens joining hese poins coninuing o appear as virually "sraigh" lines. However, he maximal 15 A subse of coordinaes for hese addiional poins will be fixed e.g., he midpoin of a boundary segmen 0

21 error E defined by falls o abou half is previous level. When asse correlaion rises o 0.7, he norade regions differ percepively, and he error of he corners only soluion K = 8 rises o.003. The maximal error falls o.0008 for he K = 1 soluion. VII. Coss, Turnover, and Tracking Error of Opimal Trading Sraegies We have developed echniques for deermining he quasi-opimal rading sraegies, as expressed by a no rade region wih boundary β* ε R N, where β* = β1* or β*, depending on wheher echnique 1 or is used o deermine he boundary. Now consider a mehod, firs inroduced by Leland and Connor [1995] in he single risky asse case, o esimae he expeced presen value of ransacions coss and annual expeced urnover from following he opimal sraegy. We also develop a measure of he expeced racking error of he opimal sraegy. Consider he funcion Tx; β Jx; β λ = 0 ha saisfies he value-maching condiions 18 on he boundary β. This is he expeced presen cos of he rading sraegy wih no-rade region deermined by boundary β when here is no cos of racking error, since λ = 0. Therefore Jx; β λ = 0 measures he expeced ne presen value of rading coss when he no-rade region has boundary β. By seing β = β*, he boundary of he opimal no-rade region for he original problem, he resuling funcion Tx; β* will measure he expeced ne presen value of rading coss for he opimal sraegy. When N > 1, here is no known closed form soluion for T, so we mus use he approximaion mehods we used previously. Le TAx; B denoe he approximae cos funcion, given he boundary deermined by joining wo corner poins. A hese poins, a subse of condiions 18 and 19 will hold: See Appendix D. 1

22 opimizing JAx; B. I will saisfy he differenial equaion 8 wih value-maching boundary condiions as in 18 holding wih equaliy a w β ε B. I will no saisfy condiions 19, since he boundary poins are opimal for JA, no TA. The soluion TAx; B will be he sum of a homogeneous soluion o he p.d.e. 8; plus a paricular soluion whose coefficiens are all zero and hence can be ignored. The homogeneous equaion will again be of he form B., wih exponens saisfying B.5. The N N coefficiens C k, k = 1,...,K mus be chosen such ha he N N value-maching condiions 18 hold wih equaliy a w β ε B. TAx; B herefore gives he discouned expeced oal rading coss, from τ = o infiniy, of he quasiopimal rading sraegy associaed wih JAx; B. The annualized expeced rading coss ATC are simply 3 ATC = r TAx; B. If all asses have he same rading coss k i = k, i = 1,..., N, hen he annualized expeced one-way urnover is 4 Turnover = ATC / k. 16 The approximae expeced discouned racking error cos TE comprises he residual cos: TE = JA - TA. 16 When k i varies across asses, i is sill possible o esimae urnover associaed wih he quasi-opimal sraegy JAx; B. We consruc a funcion TA*x; B as above, bu which saisfies he value-maching condiions TA i *x; w B = k for all i, where is k is an arbirary consan across all asses i. The soluion is he expeced discouned ransacions coss associaed wih B, when k is he common cos of rading each asse. Annualized urnover is given by rta*/k.

23 The annualized racking error is rte, and he annualized variance AV associaed wih his cos is simply 5 AV = r TE / λ = r JA - TA / λ. VIII. Opimal Policies wih a Single Risky Asse: Some Examples Consider he following base parameers for asse reurns: Risky asse: µ =.15 σ =.040 Riskless ineres rae: r =.075 Targe proporion of risky asse: w* =.60. We firs assume ha he cos of selling and buying are idenical, and equal o k. Table I liss he opimal no-rade boundaries {w min, w max }, wih percen urnover and percen sandard deviaion of racking error in parenheses below, for a range of ransacions coss k and cos per uni of racking error variance λ. Table I examines a range of rading coss, from a low of 0.1% o a high of 10%. For comparison, wo values for he price of racking error λ are considered: 1, he value which would also lead a meanvariance invesor o choose a arge proporion of approximaely 60% in he risky asse, given is reurn 3

24 and risk; and 10, a larger value which ofen seems o characerize he acions of invesors rying o rack a arge raio. The upper numbers in each cell are w min and w max. The lower numbers in parenheses are annual urnover in percen from equaion 4 and annualized sandard deviaion of racking error in percen he square roo of AV from equaion 5. TABLE I k λ = 1.56/ / / / / / / / / /1.9 λ = / /.67.56/ / / / /.3 3.4/ / /.88 Noe ha he able cells are idenical when he raio k/λ is he same. 17 Therefore higher ransacions coss or greaer aversion o racking error play an inverse role in deermining he opimal sraegy. For small k and/or large λ, he size of he opimal no-rade inerval is approximaely proporional o he cube roo of k. 18 Thus doubling ransacions coss will increase he no-rade inerval by a facor of abou 1/3 = This resul also holds when N : The quasi-opimal boundary, urnover, and racking accuracy are homogeneous of degree zero in he vecor of rading coss and λ. 18 A heurisic demonsraion of his resul is as follows. For small k, Jw;β* is approximaely quadraic wih a minimum a w*, implying J 1 w* = J 111 w* = 0. Using a Taylor series expansion, J 1 w max J 1 w* = J 11 w max w max -w* + 1/J 111 w max w max -w*. Using 11 and 17, J 1 w max = k and J 11 w max = 0, implying k = 1/J 111 w max w max -w*. From a Taylor series expansion, J 111 w max = J 111 w* + J 1111 w*w max -w* = 4

25 Similarly, urnover will fall by his same facor. Tracking error a he opimum also rises by a facor k 1/3. We may summarize our resuls as follows: for k/λ small e.g..1, hen o a close order of approximaion he following proposiions hold: i The size of he opimal no-rade inerval w max - w min is proporional o he cube roo of ransacions coss. ii Turnover and he size of he opimal no-rade inerval are inversely proporional, implying ha urnover is inversely proporional o he cube roo of ransacions coss. iii The sandard deviaion of racking error and he size of he opimal no-rade inerval are proporional, implying ha he sandard deviaion of rading error is inversely proporional o he cube roo of ransacions coss. Proposiions iii and iv immediaely sugges iv The sandard deviaion of racking error and urnover are inversely proporional. Sensiiviy o changes in racking error aversion λ follows immediaely from he invariance of he opimal sraegy for consan k/λ., implying a change in λ has he same effec as a change in 1/k. J 1111 w*w max -w*, implying k = 1/ J 111 w max w max -w* 3. Thus w max -w* = Zk 1/3, where Z = /J 1111 w* 1/3. By symmery, w*- w min = w max w*. So he size of he no-rade inerval w max w min is approximaely Zk 1/3. 5

26 VIIIa. The Effec of a Turnover Tax on Opimal Trading Volume The resuls above can be used o assess he impac of proposed axes on rading by funds seeking o keep asses in given proporions. If oher rading coss averaged 0.5%, a % ax on porfolio rades, raising oal coss o.5%, would reduce rading volume from opimal rebalancing by /.05 1/3, or 4%. VIIIb. Asymmeric Buying/Selling Coss and Opimal Sraegies in he Presence of Capial Gains Taxes We now examine he case where he cos of selling k s differs from he cos of buying k b. These coss may differ simply because of marke condiions, or because here are capial gains axes associaed wih selling. How does he presence of capial gains axes, and herefore of asymmeric selling and buying coss, affec he opimal rading sraegy? 19 Assume ha selling incurs capial gains axes e.g. k s =.10, i.e. half he sales price is axable a a 0% rae, whereas buying incurs only a rading cos e.g. k b =.01. Consider he following conjecure: he opimal sraegy is o choose w max as in he case when boh buying and selling coss are.10, and w min as in he case when boh are.01. If λ = 10, from Table 1 his would imply 19 Since he ime horizon is infinie, capial gains axes can be avoided if asses are never sold. Some cauion is needed in exending our resuls o capial gains. Firs, we assume ha capial losses are immediaely realized and asses are replaced by repurchasing he equivalen of he curren posiion. This ignores wash sale rules. Second, our saionary model assumes ha capial gains ax raes will remain consan for fuure asse sales. Noe ha he dynamics of buying and selling could change he average cos basis and hence he effecive ax rae on sales revenue. The assumpion of consan selling coss also precludes a difference beween shor- and long-erm capial gains which Consaninides [1984] showed could creae an addiional reason for realizing long-erm gains. 6

27 w max =.669, w min =.56. Bu his conjecure is incorrec. When we solve for he opimal policy wih he boundary condiions J 1 w min ; w min, w max = -k b, J 1 w max ; w min, w max = k s, plus he opimizing condiions 16 and 17, we find w max =.661, w min =.534. While he upper rade raio w max is similar o ha when boh rading coss are 10%, he lower rade raio w min is subsanially lower han ha when boh rading coss are 1%. The reason for his becomes clear: he probabiliy of incurring capial gains in he long run depends roughly on he disance beween he midpoin of he no-rade inerval and he upper boundary. By lowering he lower boundary of he no-rade inerval, his disance can be increased while keeping he average exposure closer o w *. 0 Because w min is a decreasing funcion of he capial gains ax rae, an invesor commencing wih a cashonly posiion should iniially inves less in he risky asse o a level w min as he ax rae rises. In he long run, however, he average risky asse proporion will approximae w* when he opimal sraegy is followed. 0 Even if he cos of buying were zero, i would sill no be opimal o buy whenever w < w *, if he cos of selling is posiive. If k b = 0, and k s =.10, he opimal rading sraegy for he example above would be w min =.536. w max =

28 Our resuls should be conrased wih hose of Consaninides [1983], who argues ha capial gains never should be realized. Bu his conclusion presumes ha invesors can sell shor o offse long posiions "shoring he box" in order o avoid realizing capial gains. When his is no possible, he diversificaion argumens ha are imporan here canno be ignored: some capial gains mus be realized o keep racking error wihin bounds. Even if shoring he box or a highly correlaed securiy is possible, i is generally expensive. Synheically "selling" he original securiy using hese echniques can be incorporaed in our analysis, using heir high ransacions cos raher han he even higher capial gains coss. VIIIc. Turnover of he Opimal Sraegy vs. Periodic Rebalancing Consider now rebalancing periodically a a ime inerval δ. A he end of each rebalancing period, he random asse proporions w + δ are readjused back o he desired proporions w *. For equal racking error, we are ineresed in he urnover associaed wih his sraegy relaive o he opimal sraegy considered above. Appendix C gives formulae for he average annual racking error and urnover associaed wih a periodic rebalancing sraegy, when here is a single risky asse. Consider he example from above wih w * =.60 and σ =.04. From Table I, wih λ = 10 and k =.01 or equivalenly, λ = 1 and k =.001, annual racking error sandard deviaion is 0.41%. Using formula C5, seing δ = rebalancing approximaely hree imes per year gives an idenical racking error. Bu 8

29 plugging his value of δ ino C7 gives an expeced annual urnover of 6.36%, in comparison wih he opimal sraegy s urnover of 3.4%. Thus opimal rading reduces urnover by 49%. Comparisons wih oher cells in Table I give similar savings of abou 50%. IX. Opimal Policies wih a Muliple Risky Asses: Some Examples We now find quasi-opimal rading sraegies for he case wih wo risky asses plus a riskless asse. While higher-dimensional examples could be consruced wih he echniques of Secion VII above, hey are numerically inensive and mos salien poins can be seen in he wo risky asse case. We consider firs a symmeric base case, wih Risky asses: µ1 = µ =.15 σ1 = σ =.040 ρ =.00 Riskless ineres rae: r =.075 Targe proporions of risky asses: w 1 * = w * =.40 Transacions coss: k 1 = k =.01 Tracking error aversion: λ = 1.30 The racking error aversion facor of 1.30 is chosen o equal he variance aversion of he invesor choosing opimally o inves 40% in each risky asse, given he disribuion of reurns in he base case above. Alernaively, we laer allow he racking error aversion facor o exceed he invesor s variance aversion 9

30 see foonoe 5. Figure 1 shows he no-rade region for his λ using he corners only K = 8 soluion. The corner poin coordinaes are X = {.46,.46}, Y = {.478,.3}, Z = {.33,.33}, and V = {.3,.478}. Turnover is 3.% and he sandard deviaion of racking error is 1.13% per year. The K = 1 soluion differs by no more han.001 in any coordinae, and is no shown here. Alhough he segmens XY, YZ, ec. appear o be linear, hey are no exacly. This would be he case only if JAx; B were a quadraic funcion in he inerior of χ. Noneheless, once he boundary poins {X,Y,Z,V } have been found, "connecing he dos" seems a reasonable approximaion for mos parameer choices. The quasi-opimal policy is no fully opimal, as discussed above. Since we consruced he boundary segmens XY, YZ, ec. using he condiions JA 11 w β, B = 0, JA w β, B = 0, ec., we can measure he error using equaion, which deermines he maximal absolue error JA i - k i along he enire boundary. Figure 1A shows he error along he boundary XY, where w varies from Y =.33 o X =.46. Figure 1B shows he error along he boundary ZY, where w 1 varies from Z 1 =.33 o Y 1 =.478. The maximal error is approximaely.0003, or abou 3% of he 1% ransacions cos. Maximal errors along he oher segmens ZV and VX are similar. We conclude ha he quasi-opimal sraegy would be fully opimal if ransacions coss ranged appropriaely beween 0.97% and 1.03% along he boundary β *. The K = 1 soluion gives even igher bounds. When λ = 10, he no-rade region shrinks o x = {.43,.43}, y = {.438,.361}, z = { }, and v = 30

31 {.361,.438}. 1 Turnover rises o 6.8% and he sandard deviaion of racking error falls o 0.56% per year. Increasing he correlaion beween he asses o 0.7 increases he "skewness" of he no-rade region, as indicaed in he K = 8 soluion in Figure. Maximal errors, however, are subsanially larger using he K = 8 soluion. As can be seen in Figures A and B, hese errors reach.003, roughly a hird of he acual ransacions coss of 1%. Figure i illusraes he no-rade region using he K = 1 soluion. I is even more skewed han he K = 8 soluion. Figures Ai and Bi show ha maximal errors are reduced o.0008, or 8% of he acual ransacions coss. Noe ha even more accuracy could be achieved by maching he opimaliy condiions 18 and 19 a even more poins along he boundaries a he cos of increasing K and compuaional complexiy. When ransacions coss for all risky asses are scaled by a facor φ, he size of he no-rade region is approximaely proporional o φ 1/3, as in Proposiion i for he single risky asse case. Turnover and racking error also behave in he fashion oulined by Proposiions ii-iv above. No surprisingly, reducing ransacions coss of a paricular risky asse will shrink he size of he no-rade region in ha dimension. Less obvious is he fac ha he no-rade inerval will shif for asses whose 1 We coninue o assume ha w* =.40,.40. Thus he aversion o racking error in his case is higher han he aversion o risk which deermined he opimal invesmen proporions. See foonoe 5. We keep he invesor s aversion o variance of racking error a λ = 1.30 for purposes of comparison. This racking error aversion parameer was iniially seleced because i was also consisen wih he risk aversion owards porfolio variance of a mean-variance invesor choosing w* =.40,.40, given he base case disribuion. Noe, however, ha a differen risk aversion o porfolio variance would be required for he invesor o keep w* =.40,.40 given he new disribuion of asse reurns, wih higher correlaion. See also foonoe 5. 31

32 ransacions coss remain unchanged. Figure 3 illusraes he case where he rading cos of asse falls o 0.1 percen. 3 Oher parameers remain as in he base case. The opimal no-rade region has corner poins X = {.471,.41}, Y = {.473,.360}, Z = { }, and V = {.33,.443}. Conras he poin Z here wih Z =.367,.367, opimal when ransacions coss are boh 1 percen. As expeced, from a sar wih zero holdings of boh risky asses, more of asse should be bough iniially since is ransacions coss are lower. Bu less of asse 1 should be bough, reflecing he posiive correlaion of he wo asses. Posiive correlaion implies ha hey are parial subsiues. Buying more of he less-cosly asse permis a smaller purchase of he more-cosly asse. This illusraes he dependency of he no-rade region in one dimension upon ransacions coss in oher dimensions. X. The Effec of Transacions Coss on Iniial Asse Invesmens Consider an invesmen fund which iniially has cash holdings only. I wishes o inves in wo asses wih means and variances as given in he base case example of Secion IX. The opimal no-rade region is illusraed in Figure 1. How do ransacions coss affec iniial asse invesmens? Because he iniial allocaion x = 0, 0 is in Region V, he iniial invesmens will move asse proporions o he poin Z =.33,.33. When ransacions coss are equal, equal invesmen will occur in boh asses, bu less han he desired proporions.40,.40. Since iniial invesmen sars a a lower level han desired, he average asse 3 We use K = 1 for greaer accuracy. 3

33 exposure over a shor period of ime will be less han 40 percen o each asse. Only as he ime horizon becomes infinie will average invesmen proporions approach 40 percen each. Figure 3 illusraes he case wih asse correlaion 0.7, and ransacions coss of he wo asses are 1 percen and 0.1 percen, respecively. The aversion o racking error remains a λ = The no-rade region is compued wih K = 1, and errors are given in Figures 3A and 3B. Saring from an all-cash posiion, he opimal sraegy is o purchase iniial fracions given by he poin Z =.331,.41. No surprisingly, he amoun of asse 1, he high-cos asse, is scaled back subsanially from he ideal 40% proporion. Bu he iniial invesmen in asse, he low-rading-cos asse, is 41.%, which exceeds he desired proporion of 40%. The presence of rading coss can lead o a higher iniial purchase of an asse. In he example above, i is opimal o purchase more of he low-cos asse 1 o replace he lesser amoun of asse because he asses are highly correlaed and herefore ac as subsiues. In he long run, of course, he average asse proporions will approximae heir desired proporions when he opimal rading sraegy is followed. A currenly-popular echnique for dealing wih ransacions coss is o decrease he mean rae of reurn on asses, in proporion o heir ransacions coss. This sraegy yields allocaion fracions ha are permanenly less han arge levels. 4 And an invesor would never make an iniial invesmen ha exceeds he ideal arge proporion alhough ha is required by he opimal policy in Figure 3. The 4 The acual rading sraegy, afer hese adjused arge allocaions are iniially esablished, is no ariculaed. If acual proporions are reurned o he adjused arges on a periodic basis, furher inefficiencies resul as in Secion VIIIc. 33

34 mean-adjusmen echnique clearly has imporan shorcomings, and will be cosly o follow relaive o he opimal sraegy. XI. Exensions XIa. Nonconsan Targe Proporions Our analysis has examined he case where dw/w moves relaive o a fixed arge w *. Bu he analysis could be exended o cases where he arge w * iself follows a sochasic process. For example, say w * i = w * i Zτ, i = 1,...,N, τ, where Zτ is a vecor of variables following a join diffusion process. Zτ may include oher variables e.g. macroeconomic facors whose levels follow a diffusion as well as sock prices. The vecor wτ - w * Zτ will iself follow a diffusion process wih mean and volailiy deermined by Io s Lemma. If he process for Zτ has ime independen parameers, wτ - w * Zτ will also have ime-independen parameers, and our analysis will be direcly applicable wih he vecor a and he marix Q of equaion 3 being appropriaely modified. Examples of such w * Zτ processes include he case when he arge raios w i * are funcions of macroeconomic variables such as aggregae consumpion or ineres raes which follow a imeindependen diffusion, or when he arge raios are funcions of relaive sock price levels. Timeindependen sraegies such as consan proporion porfolio insurance e.g. Black and Jones [1988] are also included, bu opion-replicaion sraegies are no, due o he ime dependence of he arge dela. Our analysis could also be exended o alernaive ime-independen sochasic processes for he risky 34

35 asse values S i, for example an Ornsein-Uhlenbeck diffusion process exhibiing mean reversion. XIb. Fixed Componens of Transacions Coss Consider now a fixed cos componen in addiion o a proporional componen of ransacions coss. Work by Dixi [1991] suggess ha he opimal rading sraegy will be characerized by wo regions surrounding w *, one nesed wihin he oher. The ouer boundary will define he no-rade region. The inner boundary will be he asse levels opimal o rade o, when asse proporions move ouside he larger no-rade boundary. Dixi [1991] shows how o compue ouer and inner regions inervals in he single risky asse case. Even in one dimension he deerminaion of ouer and inner inervals is no easy, and we do no pursue he muli-dimensional exension here. XII. Conclusions We have considered opimal rebalancing sraegies in he presence of proporional ransacions coss. For boh single and muliple risky asses, he opimal sraegy is characerized by a no-rade region. When asse proporions lie wihin his no-rade region, he opimal policy will be o do nohing. When asse proporions move ouside he region, rades should be underaken o move he asse proporions o an appropriae poin on he no-rade region s boundary. I is no opimal o rade periodically o he arge asse proporions, despie he populariy of such a sraegy. The opimal sraegy wih idenical racking error reduces expeced urnover by almos 50%. 35