Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets

Transcription

1 THE JOURNAL OF FINANCE VOL. LIX, NO. 1 FEBRUARY 2004 Optmal Consumpton and Investment wth Transacton Costs and Multple Rsky Assets HONG LIU ABSTRACT We consder the optmal ntertemporal consumpton and nvestment polcy of a constant absolute rsk averson CARA nvestor who faces fxed and proportonal transacton costs when tradng multple rsky assets. We show that when asset returns are uncorrelated, the optmal nvestment polcy s to keep the dollar amount nvested n each rsky asset between two constant levels and upon reachng ether of these thresholds, to trade to the correspondng optmal targets. An extensve analyss suggests that transacton cost s an mportant factor n affectng tradng volume and that t can sgnfcantly dmnsh the mportance of stock return predctablty as reported n the lterature. THIS PAPER STUDIES THE OPTIMAL INTERTEMPORAL CONSUMPTION and nvestment polcy of an nvestor wth a constant absolute rsk averson CARA preference and an nfnte horzon. The nvestor can trade n one rsk-free asset and n 1 rsky assets. In contrast to the standard settng, the nvestor faces both fxed and proportonal transacton costs n tradng any of these rsky assets. In the absence of transacton costs and when rsky asset prces follow geometrc Brownan motons, the optmal nvestment polcy s to keep a constant dollar amount n each rsky asset, as shown by Merton Ths tradng strategy requres contnuous tradng n all the rsky assets. In addton, the optmal consumpton s affne n the total wealth. In the presence of transacton costs, however, tradng contnuously n a rsky asset would ncur nfnte transacton costs. Therefore, rsky assets are traded only nfrequently n ths case. The lterature on optmal consumpton and nvestment wth multple rsky assets subject to transacton costs s lmted. Leland 2000 examnes a multasset nvestment fund that s subject to transacton costs and captal gans taxes. Under the assumpton that the fund has an exogenous target for each rsky asset, he develops a relatvely smple numercal procedure to compute the no-transacton regon. Akan, Menald, and Sulem 1996 consder an optmal consumpton and nvestment problem wth proportonal transacton costs for Lu s from the John M. Oln School of Busness, Washngton Unversty n St. Lous. I thank Domenco Cuoco, Sanjv Das, Phl Dybvg, Bob Goldsten, Zhongfe L, Mark Loewensten, Mark Schroder, Dmtr Vayanos, Guofu Zhou, and partcpants n the 2002 WFA Conference, the 2002 Internatonal Fnance Conference, the 2001 Internatonal Mathematcal Fnance Conference, and the Unversty of Kentucky semnar for helpful comments. I am especally ndebted to an anonymous referee, Rck Green the Edtor and Kerry Back for very useful suggestons. Any remanng errors are of course mne. 289

2 290 The Journal of Fnance a constant relatve rsk averson CRRA nvestor when asset returns are uncorrelated. They also use numercal smulatons to compute the no-transacton regon. Lynch and Tan 2002 numercally solve a smlar problem when stock returns are predctable n a dscrete tme settng. Deelstra, Pham, and Touz 2001 use the dual approach to obtan the suffcent condtons for the exstence of a soluton to the optmal nvestment problem for an nvestor who maxmzes expected utlty from her termnal wealth. Eastham and Hastngs 1988 address the optmal consumpton and portfolo choce problem wth transacton costs and multple stocks; however, they assume consumpton can only be changed at the same tme that stock holdngs are changed. Beleck and Plska 2000 analyze a smlar problem wth a general transacton cost structure and rsk-senstve crtera but exclude ntertemporal consumpton. None of these models obtan any analytcally explct shape for the no-transacton regon. Our frst contrbuton n ths paper s to derve the optmal transacton polcy n an explct form when the rsky asset returns are uncorrelated, up to some constants that can be solved numercally. In partcular, t s shown that the optmal nvestment polcy n each rsky asset s for the nvestor to keep the dollar amount nvested n the asset between two constant levels. Once the amount reaches one of these two thresholds, the nvestor trades to the correspondng optmal targets. To the best of our knowledge, ths s the frst paper to present such an explct form of tradng strategy n the case of multple rsky assets subject to fxed transacton costs. 1 The optmal tradng strategy mples that the no-transacton and target boundares have corners and only on reachng a corner does the nvestor trade n more than one rsky asset. Snce the corner s of measure zero relatve to the no-transacton boundary, wth probablty 1, the nvestor only trades n at most one rsky asset at any pont n tme. When there are only proportonal transacton costs for a rsky asset, we show that the optmal tradng polcy nvolves possbly an ntal dscrete change jump n the dollar amount nvested n the asset, followed by trades n the mnmal amount necessary to mantan the dollar amount wthn a constant nterval. The presence of fxed transacton costs mples that any optmal transacton nvolves a lump-sum trade. In the absence of proportonal transacton costs, the optmal tradng polcy for each rsky asset s to trade to the same target dollar amount as soon as the amount n a rsky asset goes beyond a constant range. If there are also proportonal transacton costs, the optmal nvestment polcy then nvolves buyng to a target amount as soon as the amount n the rsky asset falls below a lower bound and sellng to a dfferent target amount as soon as the amount n the rsky asset rses above an upper bound. Thus, the target amounts depend on the drecton of a trade. These results generalze the no-transacton-cost case the Merton case where the optmal polcy for a CARA nvestor s to mantan a constant dollar amount n a rsky asset. 1 In contrast to Leland 2000, the form and the magntude of the targets n ths paper are endogenously derved.

3 Optmal Consumpton and Investment 291 In the presence of transacton costs, the dependence of the optmal consumpton on total wealth s also dfferent from the standard results derved by Merton In partcular, the optmal consumpton s no longer affne n total wealth. Instead, t s affne only n the dollar amount nvested n the rsk-free asset but nonlnear n the dollar amounts n rsky assets. Our second contrbuton s that we conduct an extensve analyss of the optmal polcy n the lterature. We provde a smple way to compute the notransacton and target boundares. We analyze the mpact of rsk averson, rsk premum, and volatlty on the no-transacton regon, the target amounts and the tradng frequency. We also derve n closed-form the steady-state dstrbuton of the amount nvested n a rsky asset and examne the steady-state average amount nvested n the asset. Wth no explct form of tradng strategy derved, the exstng lterature provdes only a very lmted analyss of the tradng strategy, rarely gong beyond the computaton of the no-transacton regon and the target amounts. The explct form of the boundares up to some numercally computed constants allows us to conduct ths extensve analyss, whch enhances our understandng of the relatonshp between fundamental parameters and optmal nvestment polcy n the presence of transacton costs, and also yelds some nterestng results. Frst, we fnd that small transacton costs can nduce large devatons from the no-transacton-cost case. For example, wth $5 fxed cost and 1 percent proportonal cost whch ncludes the bd ask spread, the nvestor would purchase addtonal unts of a rsky asset to reach the buy target of $104,300 only when the actual amount fell below $93,500. On the other hand, only when the actual amount rose above $152,600, would the nvestor sell the rsky asset to reach the sell target of $138,300. In contrast, n the absence of transacton costs, the nvestor would trade contnuously to keep a constant amount of $121,900 n the rsky asset. Ths large devaton mples a very low frequency of tradng. For example, wth $5 fxed cost and 1 percent proportonal cost, the average tme between sales would be about 1.2 years and the average tme between purchases would be about 2.5 years. We show that tradng more frequently than the optmal strategy would result n sgnfcant utlty loss. Ths suggests that the gan from ncorporatng stock return predctablty see, e.g., Kandel and Stambaugh 1996 would be sgnfcantly decreased f transacton costs were consdered. Also, snce transacton costs have dramatc effects on both tradng frequency and tradng sze, to explan the observed tradng volume, t seems that one must also consder transacton costs along wth other standard factors consdered n the lterature such as nformaton asymmetres and heterogeneous belefs e.g., Admat and Pflederer 1988 and Wang Second, we show that condtonal on postve nvestment n a rsky asset, the steady-state average amount nvested n the asset ncreases as the transacton cost ncreases. Ths result suggests that the presence of transacton costs makes the nvestor less rsk averse overall. Intutvely, to compensate for the 2 I thank the referee for pontng out the relevance of transacton costs to the predctablty and tradng volume lterature.

4 292 The Journal of Fnance transacton costs, the nvestor overshoots by nvestng more than otherwse optmal n the rsky asset. Ths fndng, n partcular, mples that after an ncrease n transacton costs, to nduce an nvestor to hold the same average amount as before, one needs to lower the expected return of the rsky asset ceters parbus. In addton, we fnd that as the return volatlty of a rsky asset rses, the notransacton regon narrows, the expected tme to the next purchase after a trade decreases, but the expected tme to the next sale after a trade ncreases. Ths fndng seems counterntutve because as the volatlty ncreases the nvestor could be expected to wden the no-transacton regon to decrease the tradng frequency n order to save on transacton costs. However, savng transacton costs s not the nvestor s only concern. As volatlty ncreases, so does rsk, and hence, on average, the nvestor holds less n the rsky asset. Over tme then, the nvestor needs to sell the rsky asset less frequently to ncrease current consumpton, and actually buys more often to fnance future consumpton. A large body of lterature addresses the optmal transacton polcy for an agent facng a proportonal transacton cost n tradng a sngle rsky asset see, e.g., Constantndes 1986, Davs and Norman 1990, Dumas and Lucano 1991, Shreve and Soner 1994, Cuoco and Lu 2000, and Lu and Loewensten In contrast, ths paper consders multple rsky assets wth both proportonal costs and fxed costs. Closely related models of optmal consumpton and nvestment wth fxed costs and one rsky asset have been prevously analyzed by Schroder 1995, Øksendal and Sulem 1999, and Korn These papers do not provde explct forms for the no-transacton or target boundares and they use numercal procedures to drectly solve the Hamlton Jacob Bellman partal dfferental equatons HJB PDE wth free boundares. Lo, Mamaysky, and Wang 2001 study the effect of fxed transacton costs on asset prces and fnd that even small fxed costs can gve rse to a sgnfcant llqudty dscount on asset prces. Ths fndng s n sharp contrast to the proportonal transacton cost case consdered by Constantndes 1986 and shows the mportance of fxed transacton costs n a fnancal market. Also related are papers that assume quas-fxed transacton costs see, e.g., Duffe and Sun 1990, Morton and Plska 1995, and Grossman and Laroque Whle the assumpton of quas-fxed costs smplfes analyss e.g., wth power utlty functon, the homogenety of the value functon s preserved and hence the HJB PDE can be smplfed nto an ordnary dfferental equaton ODE, the soluton s at best an approxmaton for nvestors who face fxed costs such as those charged by brokers. In a dfferent context, Constantndes 1976 and Constantndes and Rchard 1978 study the optmal cash management polcy n the presence of fxed and proportonal transacton costs. Cadenllas and Zapatero 1999 examne the optmal nterventon of a central bank n the foregn exchange market where the bank drectly controls the exchange rate but ncurs fxed and proportonal nterventon costs. Korn 1997 nvestgates a one-dmensonal optmal mpulse control for a cost mnmzaton problem when there are both fxed and proportonal control costs.

5 Optmal Consumpton and Investment 293 The three man aspects of the model we present here make t more tractable and thus better able to yeld an extensve analyss than other models n the lterature. Frst, CARA preferences and the absence of borrowng constrants 3 mply the separablty of optmal polces for the rsk-free asset, rsky assets, and consumpton. Second, the assumpton of uncorrelated rsky asset returns enables us to further break down the analyss of multple rsky assets nto an analyss of ndvdual assets. Thrd, the standard assumpton of no transacton cost n lqudatng the rsk-free asset to buy the consumpton good s also mportant. Wthout ths feature, consumpton would only occur at optmal stoppng tmes, whch would n turn requre a more complcated analyss. The case of uncorrelated asset returns s of practcal nterest. Uncorrelated assets are commonly recommended to acheve effcent dversfcaton, and there exst asset classes wth nearly zero correlatons. Indeed, some nvestors e.g., funds of funds vew themselves as facng a menu of uncorrelated assets. In addton, other nvestors may also fnd t benefcal to lmt ther tradng to uncorrelated portfolos. The remander of the paper s organzed as follows. Secton I descrbes the model. Secton II solves the nvestor s optmal consumpton and nvestment problem n the absence of transacton costs, provdng a benchmark for the subsequent analyss. Secton III contans a heurstc dervaton of the optmal polces n the presence of only proportonal transacton costs. It also provdes suffcent condtons under whch the conjectured polces are ndeed optmal. Secton IV derves the optmal polcy n the presence of only fxed transacton costs. Secton V obtans the optmal polcy n the presence of both fxed and proportonal transacton costs. Secton VI addresses the correlated asset case. Secton VII contans an extensve analyss of the optmal polcy. Secton VIII concludes the paper and dscusses some possble extensons. In Appendx A, we provde the proofs for the man results and n Appendx B, we provde the soluton algorthms. I. The Model A. The Asset Market Throughout ths paper we assume a probablty space, F, P and a fltraton {F t }. Uncertanty n the model s generated by a standard n-dmensonal Brownan moton w a n 1 column vector. There are n + 1 assets our nvestor can trade. The frst asset the bond s a money market account growng at a constant, contnuously compounded rate of r > 0. The other n assets are rsky hereafter we wll use stocks and rsky assets nterchangeably. The nvestor can buy stock at the ask prce of S t and sell t at the bd prce of 1 α S t, where 0 α < 1 represents the proportonal transacton cost rate. 4 In addton, the nvestor has to pay a fxed 3 We do mpose portfolo constrants to rule out any arbtrage opportuntes. 4 It should also be noted that t s wthout loss of generalty to represent the proportonal transacton cost ths way nstead of havng proportonal costs for both sales and purchases because one can always normalze the latter representaton to obtan the former.

6 294 The Journal of Fnance brokerage fee F 0 for each transacton n ether drecton when tradng stock. 5 Let α = α 1, α 2,..., α n and F = F 1, F 2,..., F n. For smplcty, we assume no dvdend s pad by any stock. For = 1, 2,..., n, the ask prce S t s assumed to follow a geometrc Brownan moton ds t S t = µ dt + σ dw t, 1 where w s the th element of the n-dmensonal standard Brownan moton w, µ > r, and σ > 0. 6 B. The Investor s Problem There s a sngle pershable consumpton good the numerare. Followng Merton 1971, we assume that the nvestor derves her utlty from ntertemporal consumpton c of ths good. We use C to denote the nvestor s admssble consumpton space, whch conssts of progressvely measurable consumpton processes c t such that t 0 c s ds < for any t [0,. In addton, smlar to Merton 1971, Vayanos 1998, and Lo et al. 2001, we assume that the nvestor has a CARA preference wth tme dscountng, that s, uc, t = e δt e βc for some absolute rsk averson coeffcent β>0 and tme dscount parameter δ>0. We further assume that consumpton wthdrawals, stock trades, and transacton cost payments are all made through the money market account. Let x be the amount nvested n the money market account, y be the amount n the th stock, and y = y 1, y 2,..., y n. We then have the followng dynamcs for x t and y t : dx t = rx t dt c t dt n =1 dit 1 α dd t + F 1 {dit+dd t>0}, 2 dy t = µ y t dt + σ y t dw t + di t dd t, = 1, 2,..., n, 3 where the processes D and I represent the cumulatve dollar amount of sales and purchases of the th stock, respectvely. These processes are nondecreasng, rght-contnuous, and adapted, wth D0 = I0 = 0, where D = D 1, D 2,..., D n and I = I 1, I 2,..., I n. In addton, let n [ W t = x t + 1 α y + t y t F 1 { yt 0}] =1 4 denote the lqudated wealth at tme t. 5 It s straghtforward to extend ths analyss to the case where the fxed cost for a purchase s dfferent from the one for a sale. 6 When µ < r, the nvestor shorts the stock. Ths analyss s symmetrc to the case analyzed n ths paper. The fact that only one element of the Brownan moton appears n each stock return equaton mples that the stock returns are assumed to be uncorrelated. Some dscusson of the correlated return case wll be provded later.

7 Optmal Consumpton and Investment 295 To rule out any arbtrage opportunty such as doublng strateges or Ponz schemes, smlar to Lo et al. 2001, we restrct the set of tradng polces to be such that 7 lm t E[ e δt rβw t ] = 0 T and E y t e δt rβw t 2 dt <, T [0,. 5 0 Ths set of tradng polces s also the set wthn whch the Merton soluton see next secton n the no-transacton-cost case s optmal. We use x, y to denote the set of admssble tradng strateges I, D, c such that the mpled x t and y t from equatons 2 and 3 satsfy condton 5 startng from x 0 = x and y 0 = y. The nvestor s problem s then to choose admssble tradng strateges I, D, and c to maxmze E[ 0 uc t, t dt]. We defne the value functon at tme t to be vx, y = sup E I, D,c x, y [ t e δs t e βc s ds ] Ft, x t = x, y t = y. 6 II. Optmal Polces wth No Transacton Costs For the purpose of comparson we present n ths secton the man results for the no-transacton-cost case.e., α = 0 and F = 0 wthout proof see Merton In the absence of transacton costs, the cumulatve purchase and sale processes of the stocks can be of nfnte varaton and n ths case the lqudated wealth W t = x t + y t 1, where 1 s an n-element column vector of 1 s. The nvestor s problem can then be rewrtten as [ vw = sup E e δt e βc t dt ] W0 = w y,c 0 subject to dw t = rw t dt + n µ r y t dt + σ y t dw t c t dt. =1 THEOREM 1: Suppose α = F = 0. Let y M = µ r, = 1, 2,..., n. 7 rβσ 2 7 Mathematcally speakng, the second part of condton 5 s to ensure that T 0 y t e δt rβw t dw t s a martngale, whch s necessary for the Merton soluton to be optmal n the no-transactoncost case. As shown by Cox and Huang 1989, the optmal polces wth nonnegatve wealth and consumpton constrants converge to the polces wthout these constrants as the ntal wealth of the nvestor ncreases. We thus do not mpose these constrants to smplfy the analyss but focus accordngly on nvestors wth large ntal wealth such as mutual funds and hedge funds.

8 296 The Journal of Fnance The optmal consumpton and nvestment polces are c t = rw t + γ, y t = y M, = 1, 2,..., n, for all t > 0, respectvely, where Wt followng the above polces and γ = δ r rβ s the optmal wealth process derved from n + µ r 2. 2rβσ 2 =1 Moreover, the value functon s vw = 1 r e rβw βγ. Thus, wthout transacton costs, the optmal polcy nvolves nvestng a constant dollar amount n each stock, and the optmal consumpton s an affne functon of total wealth. Ths nvestment polcy requres contnuous tradng n every stock. We wll show later that none of these results hold n the presence of transacton costs. III. The Proportonal Transacton Cost Case We begn by addressng the case wth only proportonal transacton costs.e., α>0 and F = 0. In contrast to the no-transacton-cost case, stock tradng wll now become nfrequent. We provde a heurstc dervaton of the optmal polcy n ths secton. In the sngle-stock case, Davs and Norman 1990, Shreve and Soner 1994, and Lu and Loewensten 2002 show that n the presence of proportonal costs there exst a no-transacton regon and a transacton regon. Smlarly, n the multple stock case, we conjecture that there exsts a transacton regon wheren the nvestor trades at least one stock and a no-transacton regon NT where she does not trade any stock. Insde NT, the value functon must satsfy the HJB equaton, n 1 max c 2 σ 2 y 2 v y y + µ y v y + rxv x cv x δv e βc = 0. 8 =1 The optmal consumpton s thus c = 1 β log vx β whch mples that 8 becomes n 1 2 σ 2 y 2 v y y + µ y v y + rxv x + v x β log =1, vx β δv v x β = 0. 9

9 We conjecture that Optmal Consumpton and Investment 297 vx, y 1, y 2,..., y n = 1 r e rβx n =1 ϕ rβ y, 10 for some functons ϕ :IR IR. For expostonal convenence, we let z = rβy be the scaled amount n the th stock and ψ be the restrcton of ϕ n the no-transacton regon. Then equaton 9 becomes n =1 1 2 σ 2 z2 ψ 1 2 σ 2 z2 ψ 2 + µ z ψ rψ + δ r = For equaton 11 to hold, t s clear that the followng n ODEs must be satsfed: 1 2 σ 2 z2 ψ 1 2 σ 2 z2 ψ 2 + µ z ψ rψ + δ r n λ = 0, 12 for some constants λ such that n =1 λ = 0 and = 1, 2,..., n. We note that the above ODE system s not only ndependent of the amount x n the money market account but also completely separable n z s. Ths observaton suggests that f the boundary condtons are also separable n z s, then the optmal stock transacton polcy n stock would depend only on the amount n the stock, but not on the amount n the money market account or the amounts n other stocks. We wll show later that ths s ndeed the case. We thus further conjecture that there exst two crtcal numbers, ȳ and ȳ wth ȳ < ȳ, whch characterze the optmal tradng strategy for ths stock. To be specfc, we conjecture that the optmal polcy s to buy enough to reach the buy boundary ȳ f y t ȳ and sell enough to reach the sell boundary ȳ f y t ȳ. Accordng to the proposed transacton polcy, n a stock s transacton regon the margnal ndrect utlty from the bond holdng must be always equal to the margnal utlty from the stock holdng, net of transacton costs. Therefore, the dfferental equaton n a transacton regon where stock s purchased can be wrtten as v y x, y 1, y 2,..., y,..., y n = v x x, y 1, y 2,..., y,..., y n 13 and smlarly, n a transacton regon where stock s sold the dfferental equaton must be v y x, y 1, y 2,..., y,..., y n = 1 α v x x, y 1, y 2,..., y,..., y n. 14 In addton, the optmalty of ȳ and ȳ mples that v s C 2 n all ts arguments and n all regons cf. Dumas 1991.

10 298 The Journal of Fnance Usng equatons 10, 13, and 14 and lettng = rβȳ and z = rβ ȳ,we then obtan the followng forms for ϕ n the transacton regons: f z <, and f z > z, ϕ z = C 1 + z ϕ z = C α z, where C 1 and C 2 are two constants to be determned. The proposed transacton polcy and the C 2 property of the value functon then mply the followng sx boundary condtons n terms of ψ : ψ = C 1 +, 15 ψ = 1, 16 ψ = 0, 17 ψ z = C α z, 18 and ψ z = 1 α, 19 ψ z = Therefore, the boundary condtons are ndeed all ndependent of the holdngs n the bond and separable n z s. Thus, the above conjectures about the form of the no-transacton regon and the related optmal transacton polcy are justfed. Next, consder a varaton of the ODE 12 for stock : 1 2 σ 2 z2 ψ 1 2 σ 2 z2 ψ 2 + µ z ψ rψ + δ r n λ η = 0, 21 where η s a constant. Suppose ψ,, and z are the soluton to 12 subject to the boundary condtons 15 20, then f z = ψ z η /r and the same boundares and z are the soluton to equaton 21 subject to the correspondng sx boundary condtons derved from replacng ψ wth f n condtons Ths result holds because and z are ndependent of any constant term n ψ. Ths observaton also apples to the cases consdered n subsequent sectons

11 Optmal Consumpton and Investment 299 and mples n partcular that the boundares are ndependent of δ n all the cases consdered n ths paper. Ths shows that the undetermned λ n equaton 12 does not affect the optmal boundares or z. In addton, because of the condton n =1 λ = 0 and the property of the soluton, v s also ndependent of λ. Therefore, wthout loss of generalty, we can set λ = 0 for all = 1, 2,..., n. Consequently, we have 1 2 σ 2 z2 ψ 1 2 σ 2 z2 ψ 2 + µ z ψ rψ + δ r n = 0, 22 for = 1, 2,..., n. The above dscusson suggests that when there are multple rsky assets subject to proportonal costs and ther returns are uncorrelated, we can compute the optmal boundares separately for each stock. Ths greatly reduces the dmensonalty of the computaton problem, makng t feasble to compute the optmal tradng strategy for a large number of rsky assets. Defne C α z f z z ϕ z = ψ z f < z < z 23 C 1 + z f z. We next provde a verfcaton theorem whch shows the valdty of the aboveconjectured optmal polces and the form of the value functon. THEOREM 2: Assume α>0 and F = 0, and {1, 2,..., n}, let ϕ be as defned n 23. Consder any stock. Suppose there exst constants C 1, C 2,, and z such that ψ s a soluton of ODE 22 subject to condtons and n addton, 1 α <ψ z < 1, z, z. 24 Then ψ s the unque soluton to ODE 22 subject to condtons and 24, from whch the correspondng optmal consumpton polcy s c t = rx t + 1 β n ϕ rβ y t, =1 and the correspondng optmal rsky asset tradng polcy s to transact the mnmal amount necessary to mantan y t between ȳ and ȳ, where x t and y t are the bond holdng and rsky asset holdng processes derved from followng the above polces. Moreover, the value functon s vx, y = 1 r e rβx n =1 ϕ rβ y.

12 300 The Journal of Fnance Proof: The proof of ths theorem s only a slght varaton of the proof of Theorem 4 see below and s thus omtted. 8 Q.E.D. If equaton 22 has a closed form soluton, then we would have a soluton for ψ wth two ntegraton constants A and B. Usng the above sx boundary condtons, we would then solve for the sx unknowns, C 1, C 2,, z, A, and B. Unfortunately, equaton 22 belongs to a specal class of Abel dfferental equatons whose closed form soluton, f any, has not yet been obtaned see, e.g., Cheb-Terrab and Roche 1999 except for the specal case where µ = 1 2 σ 2. However, the above free-boundary problem can be numercally solved qute easly usng a smple algorthm Algorthm 1 as explaned n Appendx B. 9 To facltate understandng of the optmal polcy, we provde numercal llustratons below. Snce the optmal stock tradng strategy s separable n ndvdual stocks, most of the followng numercal analyss wll focus on the sngle stock case and for clarty, we wll suppress all subscrpts when there s only one stock consdered n a fgure. For all numercal llustratons, we use the followng default values for the parameters unless otherwse stated. Accordng to Ibbotson and Snquefeld 1982, we set the excess return µ r and the volatlty σ at 5.9 percent and 22 percent, respectvely; n addton, followng Grossman and Laroque 1990, we set the real rsk-free rate r at 1 percent and the tme dscount rate δ at 0.01; fnally, Lo et al examne cases n whch β les between and 5.000, and we set t to the low end, 0.001, to emphasze the effect of transacton costs. Of course, ths s by no means an attempt to calbrate our model for emprcal analyss. Fgure 1 dsplays the optmal no-transacton boundares and z as functons of the proportonal transacton cost rate. Wthout transacton costs α = 0, the nvestor would always keep $121,900 n the stock, as represented by the thn mddle lne. Note that ths s the actual amount that s equal to the scaled amount n the fgure dvded by rβ. In the presence of transacton costs, t s no longer optmal to always mantan a fxed amount n the stock. Instead, the nvestor allows the amount n the stock to fluctuate wthn a certan range. When α = 0.01, for example, the nvestor wll not adjust the amount she nvests n the stock untl t reaches the bounds of $99,400 or $144,700. Thus, the presence of transacton costs has a sgnfcant mpact on the optmal tradng strategy. It should also be noted that as the transacton cost rate ncreases, the buy boundary decreases and the sell boundary ncreases, makng the nvestor trade less frequently. 8 Interested readers may also see Shreve and Soner 1994 and Theorem VIII.4.1 n Flemng and Soner 1993 for smlar proofs for the CRRA case wth one stock. Although we have not been able to prove that condton 24 n ths theorem and smlar condtons n Theorems 3 4 are automatcally satsfed by the correspondng ϕ, we strongly suspect that ths s ndeed the case from checkng these condtons n all the cases we examned. 9 Although we cannot show the exstence of a soluton of the correspondng conjectured forms n Theorems 2 4, the numercal algorthms n Appendx B have always successfully found one n every numercal case consdered n ths paper.

13 Optmal Consumpton and Investment 301 Fgure 1. Boundares as functons of the proportonal cost. The graph plots the notransacton boundares and z aganst proportonal cost α for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, fxed cost F = 0, and absoluton rsk averson coeffcent β = The thn mddle lne s the Merton lne. IV. The Fxed Transacton Cost Case When there are fxed transacton costs, the nfntesmal transacton polcy proposed n the prevous secton s no longer optmal. In ths case, all transactons nvolve lump-sum trades, because cost s ndependent of the sze of a trade. In ths secton, we consder the case when the nvestor pays only fxed costs but not proportonal transacton costs.e., F > 0 and α = 0. In the presence of only fxed costs, we conjecture that the optmal polcy for any stock s characterzed by three nstead of two, as n the prevous secton crtcal numbers: ȳ, y, and ȳ. When the amount n the stock reaches the buy boundary, ȳ, or the sell boundary, ȳ, t s optmal to trade to y. For the form of the value functon, we conjecture that 10 s stll vald. In the no-transacton regon, the HJB ODE system 22 n the prevous secton stll holds. However, the condtons n the transacton regons.e., where y ȳ or y ȳ need to be changed. Accordng to the proposed transacton polcy, we have vx, y 1, y 2,..., y,..., y n = v x F y y, y1, y 2,..., y,..., y n 25 for any y ȳ and vx, y 1, y 2,..., y,..., y n = v x F + y y, y1, y 2,..., y,..., y n 26 for any y ȳ. In addton, the optmalty of y mples that v y x, y1, y 2,..., y,..., y n = vx x, y1, y 2,..., y,..., y n. 27

14 302 The Journal of Fnance Let ψ be the restrcton of ϕ n the no-transacton regon, = rβȳ, z = rβy, and z = rβ ȳ. To provde suffcent condtons for optmalty, we focus on the case where the value functon s C 1. Usng equatons 10, and the C 1 property, we obtan the followng seven boundary condtons: ψ = C 1 +, 28 ψ = 1, 29 ψ z = 1, 30 ψ z = C 2 + z, 31 ψ z = 1, 32 and ψ z = C1 + rβ F + z, 33 ψ z = C2 + rβ F + z, 34 where C 1 and C 2 are two constants to be determned. Comparng equatons 33 and 34, we have C 1 = C 2. Ths result mples that for any stock, we only need to solve sx equatons as n the prevous secton for sx unknowns: C 1,, z, z, and two ntegraton constants. We note that, n contrast to the case wth only proportonal costs, n the presence of fxed costs the above free boundary problem s no longer β free. In partcular, β enters the boundary condtons 33 and 34. However, gven values of r, F, and β that are of economcally meanngful magntudes,, z, and z are generally not senstve to changes n β. The followng theorem records results for the value functon and the optmal tradng strategy n ths case. THEOREM 3: Assume F > 0 and α = 0, and {1, 2,..., n}, let ϕ be as defned n 23. Consder any stock. Suppose there exst constants C 1, C 2,, z, and z such that ψ s a soluton of ODE 22 subject to condtons and n addton, and ψ z > 1, z, z, 35 0 <ψ z < 1, z z, z. 36

15 Optmal Consumpton and Investment z _ 1.2 z* 1 z_ Fxed Cost F Fgure 2. Boundares as functons of the fxed cost. The graph plots the optmal boundares, z, and z aganst fxed cost F for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0, and absoluton rsk averson coeffcent β = Then ψ s the unque soluton to ODE 22 subject to condtons 28 36, from whch the correspondng optmal consumpton polcy s c t = rx t + 1 β n ϕ rβ y t, =1 and the correspondng optmal rsky asset tradng polcy s to transact to y only when yt ȳ or y t ȳ, where x t and y t are the bond holdng and rsky asset holdng processes derved from followng the above polces. Moreover, the value functon s vx, y = 1 r e rβx n =1 ϕ rβ y. Proof: Ths theorem s a specal case of Theorem 4 see below. Q.E.D. Fgure 2 dsplays the optmal no-transacton boundares and z and the optmal target z as functons of the fxed cost. In the presence of fxed transacton costs, t s no longer optmal for the nvestor to transact an nfntesmal amount to keep the amount n the stock wthn a specfed range. When F = $5, for example, the nvestor wll allow the actual amount n the stock to fluctuate between $105,200 and $139,800. If the actual amount reaches $105,200, the nvestor wll buy $16,600 worth of the stock. On the other hand, f the actual amount reaches $139,800, the nvestor wll sell $18,000 worth of the stock. Thus, the presence of fxed transacton costs also has a sgnfcant mpact on tradng. The large sze of the no-transacton regon derves manly from the

16 304 The Journal of Fnance Fgure 3. Equvalent fxed costs as functons of the proportonal cost. The graph plots the equvalent fxed cost F aganst proportonal cost α for absolute rsk averson coeffcents β = 0.01, β = 0.1, β = 1, and other parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, and return volatlty σ = low rsk averson we used n the numercal llustraton. As the rsk averson β ncreases, the sze of the no-transacton regon shrnks, as wll be shown later. In addton, t should be noted that as n the prevous case, as transacton costs ncrease the buy boundary decreases and the sell boundary ncreases. However, the senstvty of the optmal target y to changes n transacton costs s very small. It only decreases from $121,900 to $121,500 as the fxed cost ncreases from $0 to $30, makng z ndstngushable from the Merton lne n the fgure. Ths fndng s consstent wth the ntuton that roughly speakng, the nvestor s better off beng around the Merton lne, on average, even n the presence of transacton costs. Based on the nsenstvty of the target amount to fxed costs, to obtan the optmal boundares, one can frst fx z to be the Merton lne, and then choose C 1 to satsfy all the condtons except 34. Ths one-dmensonal search s straghtforward. To measure the relatve effect of the proportonal and fxed costs on the welfare of the nvestor, we defne the equvalent fxed cost F for a gven proportonal cost α to be the fxed cost such that the nvestor s ndfferent between facng only the fxed cost and facng only the proportonal cost, that s, F such that vx, y; F = vx, y; α. For a gven α, f the fxed cost exceeds the equvalent F, then the nvestor prefers to face the proportonal transacton cost. Otherwse, the nvestor prefers to face the fxed transacton cost. Fgure 3 plots the equvalent fxed cost F aganst the proportonal cost α for several rsk averson levels. For β = 1, the nvestor s ndfferent between facng a proportonal cost of 5 percent and facng a fxed cost of $2. As the proportonal cost ncreases, the equvalent fxed cost ncreases at an ncreasng rate. In addton, as the rsk

17 Optmal Consumpton and Investment 305 averson decreases, the equvalent fxed cost ncreases sgnfcantly. For example, f β = 0.1, the equvalent fxed cost for a 5 percent proportonal cost becomes as hgh as $18. Intutvely, as the nvestor s rsk averson decreases, the amount the nvestor holds n a stock ncreases. Therefore, the relatve mpact of a gven fxed cost becomes smaller. V. The Fxed and Proportonal Cost Case When the nvestor s subject to both fxed and proportonal costs for each transacton, the problem becomes even more complcated. We conjecture that n ths case, there exst four nstead of three, as n the prevous secton crtcal numbers, ȳ, ȳ, ȳ, and ȳ ȳ < ȳ < ȳ < ȳ, characterzng the optmal tradng strategy. Specfcally, we conjecture that the optmal polcy s to transact mmedately to the buy-target ȳ f y t ȳ and to jump to the sell-target ȳ f y t ȳ. In addton, the value functon stll satsfes the HJB ODE system 22 n the no-transacton regon. Accordng to the proposed transacton polcy, we must have vx, y 1, y 2,..., y,..., y n = v x F ȳ y, y1, y 2,..., ȳ,..., y n for any y ȳ, and vx, y 1, y 2,..., y,..., y n = v x F + 1 α y ȳ, y1, y 2,..., ȳ,..., y n for any y ȳ, where = 1, 2,..., n. The optmalty of ȳ and ȳ mples that v y x, y1, y 2,..., ȳ,..., y n = vx x, y1, y 2,..., ȳ,..., y n and v y x, y1, y 2,..., ȳ,..., y n = 1 α v x x, y1, y 2,..., ȳ,..., y n, for any = 1, 2,..., n. Pluggng equaton 10 nto the boundary condtons and usng the C 1 property of v, we obtan the followng eght boundary condtons: ψ = C 1 +, 37 ψ = 1, 38 ψ = 1, 39 ψ z = 1 α, 40 ψ z = 1 α, 41

18 306 The Journal of Fnance and ψ z = C α z, 42 ψ = C1 + rβ F +, 43 ψ z = C2 + rβ F + 1 α z, 44 for = 1, 2,..., n, where = rβȳ, = rβȳ, z = rβ ȳ, and z = rβ ȳ. We then have the followng result for the value functon and the optmal tradng strategy. THEOREM 4: Assume F > 0 and α>0, and {1, 2,..., n}, let ϕ be as defned n 23. Consder any stock. Suppose there exst constants C 1, C 2,,, z, and z such that ψ s a soluton of ODE 22 subject to condtons 37 44, and n addton, ψ z > 1, z,, 45 1 α <ψ z < 1, z, z, 46 and 0 <ψ z < 1 α, z z, z. 47 Then ψ s the unque soluton to ODE 22 subject to condtons 37 47, from whch the correspondng optmal consumpton polcy s c t = rx t + 1 β n ϕ rβ y t, 48 =1 and the correspondng optmal rsky asset tradng polcy s to transact to ȳ only when yt ȳ, and transact to ȳ only when yt ȳ, where x t and y t are the bond holdng and rsky asset holdng processes derved from followng the above polces. Moreover, the value functon s vx, y = 1 r e rβx n =1 ϕ rβ y. Proof: See Appendx A. Q.E.D. To help us compute the optmal boundares and understand the boundary behavor, we present the followng proposton that provdes some bounds on the optmal boundares. PROPOSITION 1: For any = 1, 2,..., n, f ȳ α, F and ȳ α, F are, respectvely, the optmal buy and sell boundares as specfed n Theorem 4 for gven α and

19 Optmal Consumpton and Investment 307 Fgure 4. Frst dervatve of ϕ. The graph plots ϕ z aganst z for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0.01, fxed cost F = 5, and absolute rsk averson coeffcent β = F, wth α + F > 0, then y α, F < y M and ȳ α, F > y M, 49 1 α where y M s the Merton lne for stock as defned n 7. In addton, for F > 0, we have y α, F < ȳ α,0 and ȳ α, F > ȳ α,0. 50 Proof: See Appendx A. Q.E.D. Proposton 1 shows that the buy and sell boundares always bracket the Merton lne. In addton, as α 1, the sell boundary goes to nfnty and thus cannot be bounded from above. Moreover, the boundares wth fxed costs always bracket the correspondng boundares wth no fxed costs. Ths proposton makes the computaton of the optmal boundares more effcent by provdng better ntal values for the boundares and the drecton of changes as transacton costs change. Accordng to Theorem 4, we need to fnd,, z, z, C 1, and C 2 such that ψ solves ODE 22 and satsfes condtons Appendx B presents an algorthm that effectvely reduces the problem to a two-dmensonal search procedure. Fgure 4 shows the typcal shape of ϕ z wthn the no-transacton regon. Clearly, t satsfes condtons n the above theorem. Ths fgure also shows that the value functon s C 2 almost everywhere except at and z, where t s only C 1. In addton, ϕz s frst convex, then turns nto a concave functon, 10 Both C 1 and C 2 can be easly elmnated to reduce the number of equatons to sx. We choose not to do so to preserve clarty.

20 308 The Journal of Fnance Amount n Stock Buy 1 Sell 2 NT 1 Sell 2 Sell 1 Sell 2 A D a d Buy 1 NT 2 Sell 1 NT 2 B Buy 1 Buy 2 b NT 1 Buy Amount n Stock 1 c C Sell 1 Buy 2 Fgure 5. No-transacton and transacton regons for two stocks. The graph shows the no-transacton and transacton regons when there are two stocks subject to both fxed and proportonal costs for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, stock 1 expected return µ 1 = 0.069, stock 2 expected return µ 2 = 0.10, stock return volatltes σ 1 = σ 2 = 0.22, proportonal costs α 1 = α 2 = 0.01, fxed costs F 1 = F 2 = 5, and absolute rsk averson coeffcent β = then changes back nto a convex functon. Ths mples that the value functon v s not globally concave. Ths s because a convex combnaton of two polces does not always outperform these two polces due to the presence of fxed costs. Fgure 5 shows the no-transacton and transacton regons when there are two stocks subject to both fxed and proportonal costs. The nteror of ABCD represents the no-transacton regon; abcd and ts extensons nsde ABCD are the target boundares. There are eght transacton regons. The arrow lnes represent the transacton drectons n these transacton regons. For example, n the Sell 1 Buy 2 regon the quadrant startng at pont C, the nvestor sells stock 1 and buys stock 2 to reach the target pont c. Smlarly, n the NT 1 Sell 2 regon, the nvestor sells stock 2 but does not trade n stock 1 to reach the target pont on the segment ad. After the ntal trade, the nvestor always stays n ABCD. In addton, only when she reaches one of the four corners, A, B, C, or D, does she trade smultaneously n more than one stock. Ths event s obvously of probablty zero because the set of these corners s of measure zero relatve to the no-transacton boundary, and z 1t and z 2t follow geometrc Brownan motons nsde ABCD. In general, when there are n stocks, the nvestor trades n more than one stock only when these stocks smultaneously reach ther respectve transacton boundares. Ths mples that when there are multple rsky assets, wth probablty 1, the nvestor only trades n at most one stock at any pont n tme. Ths fgure s n contrast to that of Morton and Plska 1995 whose numercal computaton shows that the no-transacton regon approxmates an ellpse. It s generally suspected that the no-transacton regon boundary should be an ellpse and thus dfferentable everywhere. We show, however, that ths s not true n our case. In partcular, the boundary of the no-transacton regon n our

21 Optmal Consumpton and Investment z _ 1.4 _ z * Fxed Cost z* _ z_ F Fgure 6. Boundares as functons of the fxed cost. The graph plots the boundares,, z, and z aganst fxed cost F for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0, and absoluton rsk averson coeffcent β = The thn mddle lne s the Merton lne. model s not an ellpse, but rather does have corners n general, a set wth dmenson n 2, and thus s not dfferentable at these ponts. The assumpton of uncorrelated returns s not the reason for ths dfference. In the presence of correlatons among the stock returns, we conjecture the no-transacton boundares would also have corners as long as the correlatons were not perfect. The only dfference would be that the no-transacton boundares would be skewed one way or the other dependng on the sgns of the correlatons see next secton for an example. 11 Moreover, the assumpton of a CARA preference s not crtcal ether. For other utlty functons such as a CRRA preference, the notransacton and target boundares would also have these nonsmooth ponts. Intutvely, these corners arse because, to the nvestor, one stock s not a perfect substtute for another. Fgure 6 plots the optmal boundares,, z, and z as functons of the fxed cost for α = In the presence of both fxed and proportonal transacton costs, t s no longer optmal to trade to the same boundary as was suggested n the prevous secton. If F = $5, for example, the nvestor would buy $10,800 worth of the stock to reach the buy target of $104,300 when the actual amount of the nvestment decreases to $93,500. If, on the other hand, the market goes up and the actual amount of the nvestment ncreases to $152,600, the nvestor would sell $14,300 worth of the stock to reach the sell target of $138,300. In addton, as the fxed cost decreases toward zero, and z and z approaches the z for the case wth only proportonal costs. Furthermore, as the fxed cost F ncreases, and z converge to z n the fxed cost case. Ths convergence 11 In other words, the optmal dollar amount range for a stock would not be constant but rather would depend on the amounts n other stocks.

22 310 The Journal of Fnance Fgure 7. Boundares as functons of the proportonal cost. The graph plots the boundares,, z, and z aganst proportonal cost α for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, fxed cost F = 5, and absoluton rsk averson coeffcent β = The thn mddle lne s the Merton lne. occurs because as F becomes much larger than the proportonal cost, α, the mpact of transacton costs orgnates more and more from the fxed costs. Fgure 7 shows the optmal boundares,, z, and z as functons of the proportonal cost rate for F = $5. If α = 0.05, for example, the nvestor wll buy $8,200 worth of stock when the actual amount of the nvestment reaches $79,600. If the market goes up and the actual amount ncreases to $171,900, the nvestor wll sell $13,500 worth of stock. As the proportonal transacton cost ncreases, both the sze of a purchase after reachng the buy boundary and the sze of a sale after reachng the sell boundary z decrease. In addton, as the proportonal cost approaches zero, and z approach z for the case wth only fxed costs. VI. Fxed and Proportonal Costs wth Correlated Asset Returns In ths secton, we extend the analyss n the prevous sectons to the case wth correlated asset returns. We assume that the asset prces stll evolve as n 1. However, we allow the correlatons among the asset returns to be nonzero, that s, w t and w j t may have nonzero correlaton. We denote the correlaton between asset return and asset j return as ρ j, wth ρ = 1, = 1, 2,..., n. Whle we extend the logc of the prevous secton to conjecture the optmal polces n ths case, we cannot make the formal statement analogous to Theorem 4. Insde NT, the value functon must satsfy the HJB equaton: 1 2 n =1 j =1 n n ρj σ σ j y y j v y y j + =1 µ y v y + rxvx + v x β log vx β δv v x β = 0. 51

23 We conjecture that Optmal Consumpton and Investment 311 vx, y 1, y 2,..., y n = 1 r e rβx ϕrβ y 1,...,rβ y n, 52 for some functon ϕ :IR n IR. Let ψ be the restrcton of ϕ n the no-transacton regon. Then equaton 51 becomes 1 2 n =1 j =1 n [ ] n ρj σ σ j z z j ψz z j ψ z ψ z j + µ z ψ z rψ + δ r = We conjecture that n ths case, there exst four crtcal functons nstead of numbers, as n the prevous secton, ȳ y, ȳ y, ȳ y, and ȳ y, where y = y 1,..., y 1, y +1,..., y n, defnng the no-transacton regon and the optmal target boundares. Accordngly, we must have = 1, 2,..., n, vx, y 1, y 2,..., y,..., y n = v x F ȳ y y, for any y ȳ y, and =1 y 1, y 2,..., ȳ y,..., y n, vx, y 1, y 2,..., y,..., y n = v x F + 1 α y ȳ y, y 1, y 2,..., ȳ y,..., y n, for any y ȳ y. The optmalty of ȳ y and ȳ y mples that and v y x, y1, y 2,..., ȳ y,..., y n = vx x, y1, y 2,..., ȳ y,..., y n v y x, y1, y 2,..., ȳ y,..., y n = 1 α v x x, y1, y 2,..., ȳ y,..., y n, for any = 1, 2,..., n. Pluggng equaton 52 nto the boundary condtons and usng the C 1 property of v, we obtan the followng eght boundary condtons: ψz 1,..., z,..., z n = C 1 z + z, 54 ψ z z 1,..., z,..., z n = 1, 55 ψ z z1,..., z,..., z n = 1, 56

24 312 The Journal of Fnance and ψ z z 1,..., z z,..., z n = 1 α, 57 ψ z z1,..., z z,..., z n = 1 α, 58 ψz 1,..., z z,..., z n = C 2 z + 1 α z z, 59 ψ z 1,..., z,..., z n = C1 z + rβ F + z, 60 ψ z 1,..., z z,..., z n = C2 z + rβ F + 1 α z z, 61 for = 1, 2,..., n, where z = z 1,..., z 1, z +1,..., z n, = rβȳ, = rβȳ, z = rβ ȳ, and z = rβ ȳ. We then need to solve for,, z, z, C 1, and C 2 for all, whch are all functons of z. Ths n-dmensonal nonlnear PDE wth 4n free boundares s dffcult to solve even numercally, especally when n s large. To get some dea of how correlaton affects the no-transacton regon and to see f the uncorrelated return case provdes some useful nsghts nto the correlated case, we use Algorthm 3 descrbed n Appendx B, whch s essentally the projecton method ntroduced by Judd 1999, to numercally solve the two-asset case wth a correlaton of ρ 12 = 0.1. Smlar to Leland 2000, we assume that the four no-transacton boundares and the four target boundares see Fgure 5 are all straght lnes. 12 Although ths lnearty has already sgnfcantly smplfed the computaton, we stll need to optmally choose 16 + m + 1m + 2/2 constants to mnmze a test functon, where m s the order of the seres soluton n the projecton method. When the correlaton s small, an order of two s generally suffcent m = 2, whch means we need to mnmze over 22 constants. As n and m grow, the number of constants we need to mnmze over grows quckly. In general, one needs to mnmze over n2 n+1 + m n + j 1! j =0 whch s equal to 796 when n = 6, m = 2 constants. Ths j!n 1! llustrates the extreme dffculty of computng the optmal boundares n the correlated return case when n s large. Fortunately, when the correlaton s small, as Fgure 8 suggests, the soluton to the uncorrelated return case provdes a reasonable estmate of the optmal boundares. In Fgure 8, we present the no-transacton regon and the target boundares for a two-stock example wth 10 percent correlaton. 13 The dashed lnes show the boundares when the correlaton s zero. Ths fgure suggests that the boundares for the uncorrelated return case are close to those of the correlated case. In addton, all the boundares are negatvely sloped. Ths s because n the presence of postve correlaton, the two stocks have substtuton effects for each other. Furthermore, compared to the boundares n the uncorrelated return 12 We fnd that relaxng ths assumpton to allow all the boundares to be pecewse lnear does not yeld any notceable changes n the optmal boundares. 13 Other numercal examples we nvestgated yeld smlar qualtatve results.

25 Optmal Consumpton and Investment 313 Amount n Stock Amount n Stock 1 Fgure 8. No-transacton and target boundares for two correlated stocks. The graph shows the no-transacton and target boundares when there are two correlated stocks subject to both fxed and proportonal costs for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, stock expected returns µ 1 = µ 2 = 0.069, stock return volatltes σ 1 = σ 2 = 0.22, proportonal costs α 1 = α 2 = 0.01, fxed costs F 1 = F 2 = 5, absolute rsk averson coeffcent β = 0.001, and return correlaton ρ 12 = 0.1. The dashed lnes are correspondng boundares for the uncorrelated return case. case, all the boundares n the correlated return case move southwest, respectvely. Ths suggests that n the presence of postve correlaton, one tends to nvest less n each stock. Ths s because the dversfcaton beneft of a stock s smaller when ts return s correlated wth another stock. VII. Analyss of the Optmal Polcy One of the man reasons for nvestng n multple rsky assets s to reduce portfolo rsk through dversfcaton. There are asset classes that have nearly zero correlatons and for dversfcaton purposes nvestors may fnd t effcent to lmt ther tradng to these uncorrelated asset classes. Ths suggests that from an economc pont of vew the uncorrelated return case s an mportant case to study. Therefore, n ths secton, we provde some further analyss of the optmal tradng strategy n the uncorrelated return case. As shown n Sectons III V, analyss of the optmal polcy for multple stocks can be decomposed nto analyss for ndvdual stocks n ths case. We thus wthout loss of generalty pck one of the stocks and conduct the analyss on ths stock. The locaton of free boundares, the senstvty of these boundares to changes n rsk averson and volatlty, the frequency of transacton, and the optmal sze of a purchase and a sale are all examples of questons we wll address n ths secton.

26 314 The Journal of Fnance Fgure 9. Boundares as functons of the absolute rsk averson coeffcent. The graph plots the optmal boundares ȳ, ȳ, ȳ, and ȳ aganst absolute rsk averson coeffcent β for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0.01, and fxed cost F = 5. A. Optmal Boundares A.1. Changes n Rsk Averson Fgure 9 plots the optmal boundares ȳ, ȳ, ȳ, and ȳ the actual amount, nstead of the scaled amount aganst the absolute rsk averson coeffcent β. As rsk averson ncreases, ȳ, ȳ, ȳ, and ȳ all decrease. The amount of each purchase and sale also decreases. The target amounts ȳ and ȳ quckly converge to the Merton lne. The pattern of the boundares also suggests that, on average, the amount nvested n the stock decreases as the nvestor becomes more rsk averse. A.2. Changes n Volatlty Fgure 10 plots the optmal boundares,, z, and z aganst the stock return volatlty, σ. As the volatlty ncreases,,, z, and z all decrease. In contrast to the ntuton that as volatlty ncreases, to save transacton costs, the nvestor would wden the no-transacton regon, the NT regon actually shrnks see subsecton B for the mplcaton on the frequency of tradng. Furthermore, both and z move closer to the Merton lne, but the amount of each transacton s not very senstve to changes n the volatlty. A.3. Changes n Expected Return Fgure 11 plots the optmal boundares,, z, and z aganst the expected stock return µ. As the expected return ncreases,,, z, and z all ncrease.

27 Optmal Consumpton and Investment 315 Fgure 10. Boundares as functons of the return volatlty. The graph plots the optmal boundares,, z, and z aganst return volatlty σ for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = Fgure 11. Boundares as functons of the expected return. The graph plots the optmal boundares,, z, and z aganst expected return µ for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, return volatlty σ = 0.22, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = Both and ncrease at a lower rate than the Merton lne, whle z and z ncrease at a hgher rate. Ths mples that the no-transacton regon wdens as the expected return rses. In addton, the sze of each purchase and sale also ncreases. B. Frequency of Tradng To better understand the optmal transacton polcy n the stock, we now analyze the stochastc behavor of the nvestment n the stock n ths subsecton.

28 316 The Journal of Fnance Wthn the no-transacton regon, the scaled amount n stock z evolves as follows: Now let z 0 = z, z be fxed, dz t = µz t dt + σ z t dw t. τ = nf{t 0:z t /, z} denote the tme of the next transacton, P z τ< = Pτ < z 0 = z denote the condtonal probablty that τ s fnte, and E z [τ] = E[τ z 0 = z] denote the condtonal expectaton of τ. The followng proposton states that wth postve probablty the nvestor wll transact n the stock, and that the expected tme to the next transacton s always fnte. PROPOSITION 2: If 0 < < z <, then P z τ< = 1 and E z [τ] < for all z, z. Moreover, both boundares of the no-transacton regon, and z, can be reached wth postve probablty. Proof: Ths follows mmedately from the propostons n Secton 5.5 of Karatzas and Shreve Q.E.D. Snce for the case n whch 0 < < z < both boundares can be reached n fnte expected tme, we can compute a set of measures of tradng frequency; for example, expected tme to the next trade, expected tme to the next sale after a purchase, and so on. Let τ s = nf{t 0:z t = z} and τ b = nf{t 0:z t = } represent the frst tme z t reaches the sell boundary z and the buy boundary of the no-transacton regon, respectvely. Let E z [τ s ] = E[τ s z 0 = z] and E z [τ b ] = E[τ b z 0 = z] denote the condtonal expectatons of τ s and τ b, respectvely. Lettng Tz = E z [τ s ] and applyng Itô s lemma, we fnd that T satsfes the followng dfferental equaton cf. Karln and Taylor 1981, p. 192: 1 2 σ 2 z 2 T + µzt + 1 = 0. 62

29 Optmal Consumpton and Investment 317 For the boundary condtons, frst we note that obvously T z = 0. Snce the transacton polcy s to jump to from as soon as s reached, we must have the second boundary condton T = T. Solvng the above ODE 62 subject to these two boundary condtons and followng a smlar procedure for E z [τ b ], we have the followng result. PROPOSITION 3: Suppose 0 < < z <. Then E z [τ s ] = 1 σ log 2 log z/z k k log / z k z k µ 1 2 σ 2 k k z z log zz f µ 1 2 σ 2 f µ = 1 2 σ 2 and where log/z z k z k log z / z k z k µ E z [τ b ] = 1 2 σ 2 z k z k 1 σ z log z log 2 z z f µ 1 2 σ 2 f µ = 1 2 σ 2, k = 1 2µ σ Fgure 12 plots the expected tme to the next sale after a sale and the expected tme to the next purchase after a purchase aganst the proportonal transacton cost rate. When α = 0.01, on average, t takes about 1.2 years from sell to sell and about 2.5 years from buy to buy. As the transacton costs ncrease, the transacton frequency decreases and the dfference between the expected tme from buy to buy and the expected tme from sell to sell also becomes greater. A wealth of lterature exsts on stock return predctablty e.g., Kandel and Stambaugh 1996, Barbers 2000, Xa Generally, t s found that ncorporatng predctablty would sgnfcantly ncrease the welfare of an nvestor, even n the presence of parameter uncertanty. However, most of these studes do not take transacton costs nto account. The large devaton of tradng polcy n the presence of transacton costs from optmal polcy n the absence of transacton costs mples, as found n the above analyss, a very low frequency of tradng. Ths nfrequency of tradng seems to suggest that the gan from ncorporatng predctablty would be sgnfcantly decreased f transacton costs were consdered. We wll return to ths pont later. Ths fndng of low tradng frequency n the presence of transacton costs also has some mplcatons for models of tradng volume. Snce transacton costs have dramatc effects on both tradng frequency and tradng sze, to explan the observed tradng volume, t seems that one has to consder transacton costs n addton to other

30 318 The Journal of Fnance Fgure 12. Expected tme to the next transacton as functons of the proportonal cost. The graph plots the expected tme to the next transacton E z [τ s ] and E z [τ b ] startng from z and respectvely aganst proportonal cost α for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, fxed cost F = 5, and absoluton rsk averson coeffcent β = Fgure 13. Expected tme to the next transacton as functons of the absolute rsk averson coeffcent. The graph plots the expected tme to the next transacton E z [τ s ] and E z [τ b ] startng from z and, respectvely, aganst absolute rsk averson coeffcent β for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0.01, and fxed cost F = 5. standard factors consdered n the lterature e.g., Admat and Pflederer 1988 and Wang 1994 such as nformaton asymmetry and heterogeneous belefs. Fgures 13 to 15 plot the expected tme to the next sale after a sale and the expected tme to the next purchase after a purchase aganst the absolute

31 Optmal Consumpton and Investment 319 Fgure 14. Expected tme to the next transacton as functons of the return volatlty. The graph plots the expected tme to the next transacton E z [τ s ] and E z [τ b ] startng from z and, respectvely, aganst return volatlty σ for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = Fgure 15. Expected tme to the next transacton as functons of the expected return. The graph plots the expected tme to the next transacton E z [τ s ] and E z [τ b ] startng from z and, respectvely, aganst expected return µ for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, return volatlty σ = 0.22, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = rsk averson coeffcent β, stock return volatlty σ, and the expected return µ, respectvely. As the nvestor becomes more rsk averse, the frequency of tradng decreases. The expected tme between purchases ncreases faster than the expected tme between sales. It should be emphaszed that although the

32 320 The Journal of Fnance no-transacton regon narrows as β ncreases, the tradng frequency stll decreases. Ths nverse correlaton suggests that tradng frequency not only depends on the wdth of the no-transacton regon but also on the locaton of the NT regon. As stock return volatlty ncreases, whle the expected tme between sales ncreases, the expected tme between purchases decreases. Ths fndng seems counterntutve because as volatlty ncreases t seems probable that the nvestor would wden the no-transacton regon to decrease the tradng frequency so as to save on the transacton costs. However, the nvestor s response s more sophstcated than smply savng transacton costs. As volatlty ncreases, the rsk ncreases. So, on average, the nvestor holds less n the stock as suggested by Fgure 10 and can be verfed usng the measure developed n the next subsecton. Over tme then, the nvestor needs to sell less frequently to fnance current consumpton and actually needs to buy stock more often to fnance future consumpton. As the expected stock return ncreases, both the expected tme between purchases and the expected tme between sales decrease, but the expected tme between sales decreases faster. Agan, t should be noted that the frequency of transacton s not determned only by the wdth of the no-transacton regon Fgure 11 shows that the regon wdens as µ grows. Wth the rsk premum ncreasng from 5 percent to 9 percent, the expected tme between sales reduces from 1.4 years to about 7 months. C. Average Amount Invested n Stock In ths subsecton, we compute a measure of the average amount the nvestor would optmally hold n the rsky asset. When 0 < < z <, the expected tme to reach ether boundary s fnte; therefore, t follows that z s a postvely recurrent process. Let k be as defned n equaton 63 and { 2x k Gx, ξ = k z k ξ k ξ 1 k /[σ 2 z k k ] f x ξ z 2 z k x k ξ k k ξ 1 k /[σ 2 z k k ] f ξ x z be the Green functon of z nsde the no-transacton regon. We focus on the case wth both fxed and proportonal costs. Then f z = z k z k G, z + k k G z, z z z z k z k, η + k k G z G, η dη s the statonary or steady-state probablty densty functon cf. Karln and Taylor 1981, p Fgure 16 shows the typcal shape of the statonary densty functon. As expected, sgnfcant mass falls around the optmal targets and z, because these are the ponts to whch the nvestor must return after reachng the transacton boundares. Snce µ>0, there s greater mass around z than around.

33 Optmal Consumpton and Investment 321 Fgure 16. The statonary densty functon of the amount n a stock. The graph plots the statonary densty functon f z for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = Fgure 17. The average amount n a stock as a functon of the proportonal cost. The graph plots the average amount n stock aganst proportonal cost α for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, fxed cost F = 5, and absoluton rsk averson coeffcent β = Usng the statonary dstrbuton, we can compute the average amount nvested n the stock n the steady state as t approaches. Fgure 17 shows the steady-state average amount nvested n the stock as a functon of the proportonal transacton cost rate α. Surprsngly, the average amount nvested n the stock ncreases as the transacton costs ncrease. As transacton costs ncrease, to save on such costs, the nvestor wdens the no-transacton regon. The tenson occurs between nvestng more on average versus transactng more often to keep a lower average but payng hgher transacton costs. In ths case, savng transacton costs s domnant. Fgure 18 shows the steady-state average amount nvested n the stock as a functon of the fxed transacton cost F. Agan, as transacton costs ncrease, the average amount ncreases. However, the ncrease n the average amount as the fxed cost ncreases from $0

34 322 The Journal of Fnance Fgure 18. The average amount n stock aganst fxed cost. The graph plots the average amount n a stock as a functon of the fxed cost F for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, expected return µ = 0.069, return volatlty σ = 0.22, proportonal cost α = 0.01, and absoluton rsk averson coeffcent β = Fgure 19. The expected return that mples the same average amount n a stock as a functon of the proportonal cost. The graph plots an expected return that mples the same average amount n a stock aganst proportonal cost α for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, return volatlty σ = 0.22, fxed cost F = 5, and absoluton rsk averson coeffcent β = to $30 s small compared to that shown n Fgure 17 because the fxed cost s small compared to the actual transacton sze of $22,000 when F = $30. That the steady-state average amount nvested n the stock ncreases as transacton costs ncrease suggests that to nduce an nvestor to hold the same average amount as before, one needs to make the stock less attractve, for example, by lowerng the expected return of the stock. Fgure 19 shows the expected returns that nduce the nvestor to hold the same steady-state average amount as that n the absence of transacton costs as a functon of the proportonal cost rate α when the fxed cost s $5. Consstent wth the above analyss, the expected return of the stock that mples the same average amount n stock s nversely related to the transacton costs. In addton, ths relatonshp s almost lnear n ths range of the transacton costs.

35 Optmal Consumpton and Investment 323 Fgure 20. The requred extra rsk premum as a functon of the average tme between transactons. The graph plots the requred extra rsk premum aganst the average tme between transactons for the followng parameters: tme dscount rate δ = 0.01, rsk-free rate r = 0.01, return volatlty σ = 0.22, proportonal cost α = 0.01, fxed cost F = 5, and absoluton rsk averson coeffcent β = As already shown, n the presence of transacton costs, optmal tradng occurs nfrequently. To measure how much the nvestor loses from tradng at a hgher frequency than the optmal one and to be consstent wth the conventon of usng extra rsk premum to measure utlty gan from ncorporatng predctablty, we compute the extra rsk premum requred to compensate the nvestor for tradng more frequently than the optmal tradng strategy. Specfcally, we suppose that the nvestor shrnks the optmal buy boundary and sell boundary symmetrcally about the md-pont of the no-transacton regon, but then chooses optmally the buy and sell targets. Ths change would mply an ncrease n tradng frequency and a loss of utlty. Fgure 20 plots the extra rsk premum requred aganst the average tme between transactons n the steady state. Ths fgure shows that wth a monthly tradng frequency, the nvestor would need about 25 bass ponts extra premum. Wth a daly tradng frequency, the extra premum requred would be as hgh as 300 bass ponts. These numbers seem to suggest that the mportance of predctablty as reported n the lterature would be sgnfcantly dmnshed f transacton costs were taken nto account. VIII. Conclusons and Extensons In ths paper we consder the optmal ntertemporal consumpton and nvestment polcy of an nfnte-horzon CARA nvestor who faces both fxed and proportonal transacton costs n tradng multple rsky assets. We fnd that n the presence of even small transacton costs, tradng n rsky assets becomes nfrequent and ncreasng tradng frequency beyond the optmal frequency results n sgnfcant utlty loss. These fndngs suggest that transacton costs s an mportant factor n affectng tradng volume, and the mportance of stock return predctablty as reported n the lterature would be sgnfcantly dmnshed f transacton costs were taken nto account. In addton, we fnd that

36 324 The Journal of Fnance condtonal on nvestment, as transacton costs ncrease, the average amount nvested n each rsky asset ncreases. Compared to the exstng lterature, ths paper provdes a smple model that makes t feasble to compute the optmal tradng strateges when there are large numbers of rsky assets subject to both fxed and proportonal transacton costs. Incorporatng more realstc market features such as stochastc nvestment opportuntes, portfolo constrants, exogenous ncome, correlated asset returns, and ncomplete nformaton would be economcally nterestng but mathematcally challengng for future research. Appendx A In the frst part of Appendx A, we provde a proof of Theorem 4. The proofs of Theorems 2 and 3 are specal cases and are thus omtted. Snce the nvestor s problem nvolves contnuous consumpton and dscrete stock transactons at stoppng tmes, ths optmal control problem belongs to the class of combned stochastc control as studed by Brekke and Øksendal In contrast to Brekke and Øksendal 1998, the nvestor n ths model has an nfnte horzon. The proof n ths appendx s a varaton of the proofs n Brekke and Øksendal 1998 and Korn We frst ntroduce some notaton and termnology, then provde a modfed verson of the verfcaton theorems of Brekke and Øksendal 1998 and Korn 1998, and fnally show that the condtons provded n Theorem 4 satsfy the condtons n ths verfcaton theorem. DEFINITION 1: An mpulse control χ ={τ j, ζ j, j IN} s a sequence of tradng tmes τ j and tradng amounts ζ j = di τ j dd τ j IR n such that j IN, 1. 0 τ j τ j+1 a.s., 2. τ j s a stoppng tme and ζ j s F τ j measurable, and 3. Plm n τ n K = 0, K 0, where IN denotes the set of natural numbers, and I and D are the cumulatve purchase and sale processes, respectvely. DEFINITION 2: For a gven mpulse control χ and a consumpton polcy c, the par ξ = χ,c s called a combned stochastc control. DEFINITION 3: A combned stochastc control ξ = χ, c s admssble f the mpled processes I, D and c form an admssble strategy as defned n the text; that s, the mpled x t and y t from 2 and 3 satsfy 5. Let W denote the set of admssble combned stochastc controls. Next, we let H denote the space of all measurable functons h :IR n+1 IR. We defne the maxmum operator M : H H by Mhx, y sup ζ IR n \{0} h x n =1 14 I thank the referee for pontng ths out. F 1 {ζ 0} + ζ + 1 α ζ, y + ζ,

37 Optmal Consumpton and Investment 325 where ζ s the th element of ζ. For each x, y IR n+1, let ˆζ h x, y be such that Mhx, y = h x n F 1 {ˆζ h x, y 0} + ˆζ h x, y+ =1 1 α ˆζ h x, y, y + ˆζ h x, y. A1 For a gven consumpton polcy c, we next defne the dfferental operator L c by L c gx, y 1 2 σ 2 y 2 g yy + µ yg y + rxg x cg x δ g, for all functons g :IR n+1 IR for whch the dervatves nvolved exst at x, y. We now provde a lemma whch serves as a verfcaton theorem for solvng the nvestor s problem. It provdes suffcent condtons under whch a combned stochastc control ξ = χ, c solves the nvestor s optmal consumpton and nvestment problem, and a gven functon V s the value functon. LEMMA 1. Verfcaton Theorem a. Suppose there exsts a C 1 functon V :IR n+1 IR, whch s C 2 except over a Lebesgue measure zero subset of IR n+1, such that 1. L c V x, y + uc 0, c C; 2. V x, y MV x, y; 3. T T [0,, E e δs y s V Y x s, y s 2 ds < 0 A2 and lm T E[ e δt V x T, y T ] = 0, A3 for any process x t, y t correspondng to an admssble combned stochastc control, where V Y s the n n dagonal matrx wth V y = 1, 2,..., n as ts dagonal elements and represents the Eucldan norm; and, 4. {e δt Vx t, y t } t 0 s unformly ntegrable. Then V x, y v ξ x, y, ξ W,x, y IR n+1, A4 where v ξ x, y s the value functon from followng ξ.

38 326 The Journal of Fnance b. Defne NT ={x, y :V x, y > MV x, y}. Suppose n addton to the condtons n part a, there exsts a functon ĉ : NT IR such that Lĉx, y V x, y + uĉx, y = 0, A5 for all x, y NT. Defne the mpulse control ˆχ ˆτ 1, ˆτ 2,...; ˆζ 1, ˆζ 2,... nductvely as follows: ˆτ 0 = 0 and k = 0, 1, 2,..., and where x k t ˆτ k+1 = nf { t > ˆτ k : x k t ˆζ k+1 = ˆζ V x k t, y k } / NT t, y k,, y k s the result of applyng the combned stochastc control t ˆξ k ˆτ 1,..., ˆτ k ; ˆζ 1,..., ˆζ k, ĉ t and ˆζ V s as defned n A1 forv.if ˆξ ˆχ, ĉ s admssble, then V x, y = vx, y and the combned stochastc control ξ = ˆξ s optmal, where vx, y s the value functon defned n 6. Proof: a Assumng that V satsfes the condtons n part a, we let ξ = χ, c W be any admssble combned stochastc control, where χ = τ 1, τ 2,...; ζ 1, ζ 2,... Let T [0, be fxed. For all k 0, defne θ k = τ k T,

39 Optmal Consumpton and Investment 327 wth τ 0 = 0 and let x t, y t = x ξ t, y ξ t. We can then wrte for every n IN, e δθ n V x θn, y θn V x, y n [ = e δθ V x θ, y θ e δθ 1 V x, y θ 1 θ 1] =1 + n =1 1 {τ <T}e δθ [ V x θ, y θ V xθ ], y θ. A6 Snce y t s a contnuous sem-martngale n the stochastc nterval [θ k, θ k+1 and V s C 2 except over a Lebesgue measure zero subset of IR n+1 and C 1 n IR n+1, Lemma 45.9 a generalzed verson of Itô s lemma of Rogers and Wllams 2000 apples see also Korn Therefore, IN, we have e δθ V x θ = θ, y θ e δθ 1 V x, y θ 1 θ 1 θ 1 e δs L c V x s, y s ds + θ θ 1 e δs y s V Y x s, y s σ dw s, A7 where σ s a n n dagonal matrx wth σ = 1, 2,..., n as ts elements. By condton 1, we have e δθ V x θ θ, y θ e δθ 1 V x, y θ 1 θ 1 θ 1 uc s, s ds + θ θ 1 e δs y s V Y x s, y s σ dw s. A8 By condton 2, we have V x θ, y θ V xθ Combnng A6 A9 and takng expectatons, we then get [ V x, y E e δθ n V x θn, y θn + n =0 By A2, for any fxed n, wehave θ θ 1 uc s, s ds, y θ 0. A9 θ [ θn ] E e δs y s V Y x s, y s σ dw s = 0. 0 θ 1 e δs y s V Y x s, y s σ dw s From condton 3 n Defnton 1 and condton 4 n ths lemma, we have lm n E[ e δθ n V x θn, y θn ] = E [ e δt V x T, y T ]. ]. A10

40 328 The Journal of Fnance Therefore, takng the lmt as n n A10 and usng the monotone convergence theorem, we have V x, y E [ e δt V x T, y T ] [ T ] + E uc s, s ds. 0 Takng the lmt as T and usng A3 and the monotone convergence theorem, we obtan [ ] V x, y E uc s, s ds, 0 for all ξ W and thus Vx, y v ξ x, y. b By A5, we have equalty rather than nequalty n A8. Gven the defnton of ˆξ we also have equalty n A9. Combnng ths wth A4, we then get V x, y sup v ξ x, y vˆξ x, y = V x, y. ξ W Hence Vx, y = vx, y and ξ = ˆξ s optmal. Q.E.D. Snce one of the condtons n the above verfcaton theorem s that ˆξ s an admssble combned stochastc control, we next show that the combned stochastc control mpled by the consumpton polcy and tradng strategy specfed n Theorem 4 s ndeed admssble. LEMMA 2: Let ˆξ = ˆχ, ĉ represent the stock tradng strategy specfed n Theorem 4. Then ˆξ s an admssble combned stochastc control. Proof: Let τ j, j IN denote the tme when the nvestor trades accordng to the polcy specfed n Theorem 4. Snce the prescrbed stock tradng strategy s to trade stock whenever y t s outsde ȳ, ȳ, the tradng tme s clearly a stoppng tme wth 0 τ j τ j+1 a.s., j IN. For all j IN, defne ˆζ j = y y τ j f y τ j ȳ ȳ y τ j f y τ j ȳ 0 otherwse. Obvously, ˆζ j s adapted to F τ j. Because t 0,, P{z t [, z ]}=1, and z t s postvely recurrent by Proposton 2 for = 1, 2,..., n, we fnd that Plm m τ m K = 0, K 0 and thus condton 3 n Defnton 1 s also satsfed. To complete the proof we now show that 5 s also satsfed. For all

41 Optmal Consumpton and Investment 329 t > 0 and m IN, by 2 and 48, we have rβx t τm = rβx 0 + n n =1 m =1 j =0 t τm 0 1 {τ j <t} rψ z s ds rβ F 1 { ˆζ j >0} rβ ˆζ j α rβ ˆζ j, where ˆζ j s the th element of ˆζ j. By 37 and 42 44, we fnd that rβ F 1 { ˆζ j >0} rβ ˆζ j α rβ ˆζ j = ψ zτ j ψ zτ j, where at tme τ j > 0: f t s a purchase n th stock, then z τ j = and z τ j = ; f t s a sale then z τ j = z and z τ j = z ; f there s no trade n the stock.e., ˆζ j = 0 then z τ j = z τ j. We then have m 1 {τ j <t} ψ zτ j ψ zτ j = ψ z,0 ψ z,t τm j =0 + m [ψ z,t τ j ψ z,t τ j 1 ]. Snce ψ s a soluton of 22 subject to and 22 satsfes the condtons of Corollary 4.1 of Hartman 1964, ϕ z as defned n 23 s C 2 except at {, z } a Lebesgue measure zero set and C 1 at these ponts. Usng the generalzed verson of Itô s lemma, we then obtan ψ z,t τ j ψ z,t τ j 1 = t τ j Therefore, rβx t τm = rβx t τm 0 t τm 0 + j =1 t τ j σ 2 z2 s ψ 1 2 σ 2 z2 s ψ 2 + µ z s ψ ds t τ j t τ j 1 [ 1 2 σ 2 z2 s ψ 2 ds + σ z s ψ dw s]. n ψ z,0 ψ z,t τm =1 1 2 σ 2 z2 s ψ 1 2 σ 2 z2 s ψ 2 + µ z s ψ rψ ds [ 1 2 σ 2 z2 s ψ 2 ds + σ z s ψ dw s].

42 330 The Journal of Fnance By 22, we then have rβx t τm = rβx n =1 n ψ z,0 ψ z,t τm δ rt τ m =1 t τm 0 [ 1 2 σ 2 z2 s ψ 2 ds + σ z s ψ dw s]. A11 Takng the lmt as m on both sdes of A11, by condton 3 n Defnton 1 shown above, we get rβx t = rβx By 4 we have t [0,, n =1 n ψ z,0 ψ z,t δ rt =1 t 0 [ 1 2 σ 2 z2 s ψ 2 ds + σ z s ψ dw s]. e δt rβw t = e rt rβx 0 n =1 [ψ z,0 ψ z,t +1 α z + t z t F 1 {zt 0}] Nt, where Nt = e n t =1 0 [ ] 1 2 σ 2z2 s ψ 2 ds+σ z s ψ dw s. Snce t [0,, z t, ψ z t, and ψ z t are all bounded and E[Nt] = 1, we obtan 0 lm t E[e δt rβw t ] lm t [Ke rt ENt] = 0, where K s some fnte constant. Ths shows that the frst part of 5 holds,.e., lm t E[e δt rβw t ] = 0. In addton, snce t [0,, z t and ψ z t are both bounded, we have for some fnte constant K 1, e 2δt 2rβW t = e 2rt 2rβx 0 2 n =1 [ψ z,0 ψ z,t +1 α z + t z t F 1 {zt 0}] Nt 2 < K 1 e 2rt Nt 2. Snce t [0,, z t and ψ z t are also bounded, we obtan E[Nt 2 ] < e K 2t, for some fnte constant K 2. Therefore, we have = 1, 2,..., n and T [0,,

43 Optmal Consumpton and Investment 331 [ T ] E y t e δt rβw t 2 dt = 1 [ T ] 0 rβ E z t e δt rβw t 2 dt <. 0 Ths shows that the second part of 5 s also satsfed. Q.E.D. We are now ready to prove Theorem 4. Proof of Theorem 4: In ths proof, we show that all the condtons n the verfcaton theorem Lemma 1 are satsfed. Frst, by Lemma 2, the combned stochastc control proposed n Theorem 4 s admssble. Also, as explaned n the proof of Lemma 2, the ϕ s are C 2 except over a Lebesgue measure zero subset of IR and C 1 n IR and thus vx, y are C 2 except over a Lebesgue measure zero subset of IR n+1 and C 1 n IR n+1. Next, we show that A2 and A3 hold wth the proposed value functon. Frst recall that and vx, y 1,..., y n = 1 r e rβx n =1 ϕ rβ y v y x, y 1,..., y n = βϕ e rβx n =1 ϕ rβ y, = 1, 2,..., n. By 4 we have t [0,, 0 e δt rβx t rβ n =1 y + t y t e δt rβx t rβ n =1 [1 α y + t y t F 1 { yt 0}] = e δt rβw t. A12 Takng the expectaton and the lmt, 5 then drectly mples that lm E[ e δt rβx n t+ =1 y t ] = 0. t A13 Snce z <, ϕ z = C 1 + z and z, ϕ z > 0 accordng to the condtons n the theorem, we then have 0 e δt rβx T n =1 ϕ rβ y T e δt rβx T + n =1 y T n =1 C 1, A14 and thus takng the expectaton and the lmt as T we have A3 by A13; that s, lm E[ e δt vx T, y T ] = lm 1 T T r E[ e δt rβx T n =1 ϕ rβ y T ] = 0. A15 The above expresson mples that for any fxed t 0, E [ e δt vx t, y t ] < A16 and thus e δt vx t, y t snl 1. In addton, A15 also mples that e δt vx t, y t converges to 0 n L 1. By Theorem 13.7 n Wllams 1994, we have that condton 4 n Lemma 1 holds.

44 332 The Journal of Fnance For all T [0,, we then have for some fnte constants K 1, K 2 > 0, T E e δt y t v y x t, y t [ T n 2 2 dt = E rβ y t ϕ e δt vx t, y t ] dt 0 0 =1 n T < E K 1 0 T < E K 2 0 <, =1 yt 2 e δt vx t, y t 2 dt n y t e δt rβw t 2 dt where the frst nequalty stems from the fact that ϕ s bounded, the second nequalty follows from A12 and A14, and the last nequalty follows from 5. Therefore, A2 also holds. Next, defnng =1 =1 vx Gvx, y 1 2 σ 2 y 2 v yy + µ yv y + rxv x + v x β log β n 1 = 2 σ 2 z2 ϕ 1 2 σ 2 z2 ϕ 2 + µ z ϕ rϕ we then have for an arbtrary consumpton polcy c, δv v x β + δ r vx, y, L c vx, y + uc maxl c vx, y + uc = L c vx, y + uc = Gvx, y, c where the frst equalty follows from the optmalty of c defned n 48 for a gven vx, y, whch s straghtforward to verfy. By 22 and summng up over, we then have Gvx, y = 0 n NT and thus A5 s satsfed n NT wth ĉ = c. By 38 and 45, we must have ψ > 0. By 40 and 47, we must have ψ z > 0. By 22 and pluggng n 37, 38, 40, and 42, we then have and 1 2 σ µ rc 1 + r δ r n 0 A σ 2 1 α z 2 + µ r1 α z rc 2 + δ r n 0, A18 for = 1, 2,..., n. Equatons A17 and A18 then, respectvely, mply that z, 1 2 σ 2 z2 + µ rz rc 1 + δ r n 0

45 Optmal Consumpton and Investment 333 and z z, 1 2 σ 2 1 α z 2 + µ r1 α z rc 2 + δ r n 0, for = 1, 2,..., n. Summng over and notng the fact that ϕ z = 0 outsde NT, we then have L c vx, y + uc Gvx, y 0 outsde the NT regon. Therefore, condton 1 n part a also holds. Next, we show that condton 2 n part a s true. Frst, Mvx, y = 1 r e rβx n =1 ν y, A19 where ν y sup ϕ rβ y + ζ + 1 α rβζ rβζ + rβ F 1 {ζ 0}, ζ where ζ 0 for at least one. Condtonal on a trade, by 45 47, we have ψ rβȳ rβ ȳ y rβ F f y < ȳ ν y = ψ rβ y rβ F f ȳ y ȳ ψ rβ ȳ + rβ1 α A20 y ȳ rβ F f y > ȳ. By 43 and 44, we fnd that A20 becomes C 1 + rβ y f y < ȳ ν y = ψ rβ y rβ F f ȳ y ȳ A21 C α rβ y f y > ȳ. By 45 and ψ rβȳ = C 1 + rβȳ,wehave y ȳ, ȳ, ψ rβ y > C 1 + rβ y. Smlarly, by 47 and ψ rβ ȳ = C α rβ ȳ, we have y ȳ, ȳ, ψ rβ y > C α rβ y. Combned wth A19 and A21, ths mples that n NT vx, y > Mvx, y. For any stock that s n the buy regon of ths stock, that s, y ȳ, ϕ rβy = C 1 + rβy by 23. Smlarly, for any stock that s n the sell regon of ths stock, that s, y ȳ, ϕ rβy = C α rβy. Thus, outsde NT, we have vx, y = Mvx, y. Therefore, condton 2 n part a also holds. Fnally, we show that f there s a soluton to 22 subject to condtons 37 44, then t s unque. We prove by contradcton.

46 334 The Journal of Fnance Suppose there are two dfferent optmal combned controls ξ and ˆξ. Clearly, the value functons assocated wth these two controls must be dentcal n IR IR n for both to be optmal. The separablty of the value functon n ϕ then mples that = 1, 2,..., n, ϕ s dentcal to ˆϕ and thus C 1 = Ĉ 1 and C 2 = Ĉ 2, where ϕ and ˆϕ are assocated wth ξ and ˆξ, respectvely. Snce the optmal consumpton polcy s completely determned by the value functon, t must also be dentcal for any gven x t and y t. Therefore, the dfference n ξ and ˆξ must come from the dfference n the optmal stock tradng polcy for at least one stock. Wthout loss of generalty, we suppose for stock k between 1 and n, there are two dfferent optmal polces k, k {, z k, z k} and k, k {ẑ ẑ, ˆ z k, ˆ z k }. Wthout loss of generalty, we suppose k > a > k, where a s a constant such that ẑ k < a < k ẑ ẑ. By 37, we have ψ k a = 1. On the other hand, by 45, we have ˆψ k a > 1, whch contradcts the fact that ϕ k s dentcal to ˆϕ k. Therefore, the soluton of the conjectured form s unque. Ths completes the proof of Theorem 4. Q.E.D. Proof of Proposton 1: Dfferentatng 22 once, we obtan 1 2 σ 2 z2 ψ + σ 2 z σ 2 z2 ψ + µ z ψ σ 2 z ψ 2 + µ rψ = 0. By 38 and 39, we have ψ = ψ = 1. Ths mples that there must exst a ẑ, such that ψ ẑ = 0 and ψ ẑ < 0. Otherwse, at any pont z such that ψ z = 0 we would have ψ z < 1, contradctng 45. We therefore have Ths mples σ 2 ẑψ 2 + µ rψ > 0. ẑ rβ < µ r rβσ 2 ψ < µ r rβσ 2 = y M. Snce < ẑ and ȳ α, F = rβ, the frst nequalty n 49 must hold. Smlarly, by 40, 41, and 47, the second nequalty n 49 must also hold. Next, we show that 50 holds. We let = α, F and = α, 0. Smlar to 15 17, we have ψ = C 1 +, ψ = 1, and ψ = 0. By 22, ths mples that 1 2 σ µ rc 1 + r δ r n = 0. A22 By 37 and 38, we have ψ = 1 and ψ = C 1 +. By 45, we then have ψ > 0. By 22, we then have

47 Optmal Consumpton and Investment σ µ r C 1 + r δ r n < 0. A23 Gven a z < mn,, we have ψ z = C 1 + z and ψ z = C 1 + z by the boundary condtons. Because an ncrease n the fxed cost from zero to F > 0 decreases the value functon for any gven z, we then must have C 1 > C 1. Combnng ths observaton wth A22 and A23, we then have 0 < 1 2 σ µ rc 1 C 1 r < 1 2 σ µ r. A24 By the frst nequalty of 49, we have < µ r σ 2 A24 then mples the frst nequalty of 50. Smlarly, usng the boundary and < µ r. Inequalty σ 2 condtons at the sell boundary and the second nequalty of 49, we fnd that the second nequalty of 50 holds. Q.E.D. Appendx B In ths appendx, we provde the soluton algorthms for solvng the freeboundary problems. Algorthm 1: When there are only proportonal costs. 1. Defne a test functon q :IR + IR + as follows: for a gven canddate, solve the ODE 22 subject to equaton 16 and ψ = 1 2 σ µ + δ r r whch s obtaned from equaton 22 evaluated at usng equatons 16 and 17; then solve equaton 20 for z. If there s no z satsfyng equaton 20, set q equal to an arbtrarly large postve number, such as ten. If there s a z satsfyng equaton 20, set q equal to 1 α ψ z Use a standard mnmzaton algorthm to fnd the optmal [0, rβ y M ] that mnmzes q. 15 Algorthm 2: When there are both fxed and proportonal costs. 1. Defne a test functon q :IR 2 + IR + as follows: for a canddate and a canddate d for ψ, solve the ODE 22 subject to condton 38 and, 15 Accordng to Theorems 2 to 4, the mnmum of q s theoretcally zero. Alternatve numercal procedures proposed n an earler verson of ths paper also work well and obtan the same solutons. However, ths procedure and Algorthm 2 offer the advantage that they need vrtually no nterventon on the startng ponts and are thus more robust for a wde range of parameters.

48 336 The Journal of Fnance ψ = 1 2 σ 2 2 d 1 2 σ µ + δ r, r whch s obtaned from ODE 22 evaluated at usng condton 38; then solve condtons for, z and z, respectvely. If there s no soluton for, z,or z, set q equal to an arbtrarly large postve number, such as ten. Otherwse, set C 1 = ψ, C 2 = ψ z 1 α z, and q equal to [ψ C 1 + rβ F + ]2 + [ψ z C 2 + rβ F + 1 α z ]2. 2. Use a standard mnmzaton algorthm to fnd the optmal [0, rβ y M ] and d > 0 that mnmzes q, whose mnmum theoretcal value s zero at the optmal soluton. Algorthm 3: When asset returns are correlated. To save space, we only descrbe the algorthm for the two-stock case. For the general case of n stocks, the procedure s smlar. Ths algorthm s an applcaton of the projecton method proposed by Judd 1999 to our problem. We thus only provde the man steps n applyng ths method here. For detals and ts theoretcal foundaton, we refer readers to Judd Let m = 0 and ẑ, = 1, 2,..., 16, denote the coordnates of the eght corners of the no-transacton and target boundares e.g., ponts A, B, C, D, a, b, c, and d n Fgure Set m = m + 1. Let the approxmaton functon be ψ m z 1, z 2 = m m a j H z 1 H j z 2, =0 j =0 where the H. s the Hermte functon of order and coeffcents a j are to be determned. 2. Integrate the left-hand sde of the PDE 53 over the no-transacton regon, NT usng ψ m z 1, z 2 n place of ψz 1, z 2. Denote ths value as d Next, reduce the four boundary condtons 54 and to two condtons by elmnatng C 1 and C 2. Use ψ m z 1, z 2 n place of ψz 1, z 2 to compute the dfference between the left-hand sde value and the rghthand sde value of each of the resultng sx boundary condtons plus the newly obtaned two condtons for each stock. Denote these dfferences sx for each stock as d j, j = 1, 2,..., Then defne the sum-of-squares test functon q m = 13 =1 d Fnally, use a standard optmzaton procedure to mnmze q m over a j, + j m,0, j and ẑ, = 1, 2,..., 16. Note that q m s a polynomal functon of a j s. 6. Repeat Steps 1 5 untl both q m and q m+1 q m are smaller than a preset approxmaton error tolerance level.

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