Optimal Tax-Timing and Asset Allocation when Tax Rebates on Capital Losses are Limited

Transcription

1 Opimal Tax-Timing and Asse Allocaion when Tax Rebaes on Capial Losses are Limied Marcel Marekwica This version: January 15, 2007 Absrac Since Consaninides (1983) i is well known ha in a marke where capial losses qualify for ax rebaes indefiniely i is opimal o realize losses immediaely. However, he US-ax code as well as many oher ax-codes around he world resrics ax rebaes on capial losses. This paper shows ha Consaninides resul generalizes o markes in which capial loss deducion is limied. Neverheless, in such markes an invesor who is no locked-in will hold subsanially less risky asses han in markes wih unlimied capial loss deducion. The less capial losses qualify for ax rebaes, he lower he equiy exposure. JEL Classificaion Codes: G11, H21, H24 Key Words: ax-iming, asse allocaion, capial losses I would like o hank Alexander Schaefer and Michael Samos for helpful commens and discussion. Financial suppor from Friedrich Naumann Foundaion is graefully acknowledged. Opinions and errors are solely hose of he auhor and no of he insiuions wih whom he auhor is affiliaed. c 2007 by Marcel Marekwica. All righs reserved. Johann Wolfgang Goehe Universiy, Deparmen of Finance, Posfach (Uni-Pf. 58), D Frankfur am Main, Germany, Phone: , Fax: , marekwica@finance.unifrankfur.de

2 1 Inroducion The ax rules privae invesors are facing are a poenially imporan facor influencing he household porfolio srucure. Even hough pre-ax reurns are usually repored in newspapers, on elevision or on he inerne, only afer-ax reurns should have an impac on invesmen decisions of privae invesors as only hese reurns have an impac on consumpion. In many counries around he world capial gains are axable a a consan capial gains axrae. On he oher hand capial losses only qualify for ax rebaes i hey do no exceed a cerain upper bound (limied capial loss deducion). Under curren US-ax code his upper bound is $ 3,000. Boh hese limiaions and he opporuniy o defer he axaion of capial gains have an impac on he risk-reurn-profile of an asse and hus have an impac on he invesor s asse allocaion decision. According o he seminal work of Consaninides (1983), i is opimal o realize losses immediaely if capial loss deducion is unlimied and if wash-sale rules do no apply. 1 This paper generalizes his resul o ax-sysems in which capial loss deducion is limied and he remaining losses can be carried over indefiniely as a loss carryforward. I shows ha in such ax-sysems Consaninides resul o realize a loss immediaely (and poenially building up a loss carryforward) remains an opimal ax-iming sraegy. I furher shows ha compared o ax-sysems wih unlimied capial loss deducion invesors usually canno aain he same level of wealh as in ax-sysems wih limied capial loss deducion. This is due o he fac ha in sysems wih unlimied capial loss deducion he invesor effecively receives some ax-refund in cash which he can inves o earn some reurns on he ax rebae. In a ax-sysem wih limied capial loss deducion, however, he migh end up wih some loss carryforward remaining ha does no pay any ineres. This in urn implies ha in ax-sysems wih limied capial loss deducion invesors will hold less risky asses han in ax-sysems wih unlimied capial loss deducion. The remainder of his paper is organized as follows. Secion 2 reviews he relaed lieraure, secion 3 inroduces he model and shows ha selling all asses facing a loss is an opimal aximing sraegy in ax-sysems wih limied capial loss deducion as well. Secion 4 analyzes he impac of capial gains axaion in ax-sysems wih limied and unlimied capial loss deducion and concludes ha in ax sysems wih limied capial loss deducion invesors will hold significanly less risky asses. Secion 5 concludes. 1 A ransacion is ermed a wash sale if a sock is sold o realize a capial loss and repurchased immediaely. Under curren US ax-rules wash sales do no qualify for he capial loss deducion if he same sock is repurchased wihin hiry days before or afer he sale. 1

3 2 Relaed Sudies The axaion of capial gains has several impacs on he ax-iming decisions of privae invesors. Firs, i reduces he expeced afer-ax reurn which migh lead some invesors o decide no o inves heir funds bu o consume hem. Second, invesors having high unrealized capial gains in some asses migh no wan o sell hem o avoid paying he capial gains ax and hus ge locked-in. This ax-saving behavior can resul in porfolios ha are no well diversified. Especially for older invesors he disadvanages of badly diversified porfolios migh be ouweighed by he forgiveness of capial gains under curren US ax-law when bequesed. Third, he deferral of capial gains axaion resuls in compound reurns on he posponed axes and hus decreases he effecive capial gains ax-rae (Chay, Choi, and Poniff (2006)). Fourh, if capial loss deducions are limied, he risk-reurn profile of he asses changes in a disaracing manner compared o he case wih unlimied capial loss deducion which resuls in a lower exposure o he risky asse. According o Poerba (1987) such loss-offse consrains are binding o abou weny percen of US axpayers. Many asses face boh profis from capial gains and dividends which are axed a differen raes. These wo ypes of profis differ in wo ways. On he one hand dividends are axable he year hey are obained while capial gains are axable he year when he asse is sold and he gains are realized. In case of a beques capial gains are even enirely forgiven under curren US ax-law. On he oher hand dividends are usually subjec o a higher ax-rae han capial gains. On he assumpion ha he reurns of each asse are subjec o he same ax-rae Auerbach and King (1983) show ha an opimal porfolio is a weighed average of a marke porfolio and a porfolio ha is chosen on he basis of ax consideraions ignoring risk. Thus, expeced changes in he capial gains ax-rae resul in vas realizaions of capial gains (Auerbach (1988)). While according o Shefrin and Saman (1985), Odean (1998), Barber and Odean (2000), and Barber and Odean (2003) invesors end o hold asses incurring losses oo long and end o sell asses wih unrealized capial gains oo early which clearly violaes efficien ax-planning, he sudy by Jin (2006) finds selling decisions by insiuions serving ax-sensiive cliens o be sensiive o cumulaive capial gains. Barclay, Pearson, and Weisbach (1998) show ha fund managers manage ax liabiliies o arac new invesors. Bergsresser and Poerba (2002) find fund flows o be sensiive o ax burdens. This finding suggess ha privae invesors migh be more subjec o he disposiion effec han insiuional invesors. Ivkovich, Poerba, and Weisbenner (2005) compare he realizaions of capial gains in axable and ax-deferred 2

4 accouns and find a srong lock-in effec for capial gains in he former ype of accouns. This finding in urn suggess ha privae invesors migh ride efficien ax-iming sraegies. On he oher hand he empirical evidence in Seyhun and Skinner (1994) suggess ha he $ 3,000 limi on capial loss deducions represens an imporan consrain on ax-reducion sraegies and invesors end o follow very simple ax-iming sraegies like realizing losses early and posponing gains. However, in heir sudy, abou 90% of he invesors seem o follow simple buy-and-hold sraegies. Oher sudies on he relaion beween axaion and invesor behavior include Blouin, Radey, and Shackelford (2003) and he survey of Poerba (2002). Under curren US ax laws i is no allowed o shor a securiy in which one has a long posiion o avoid realizing capial gains. Invesors realizing such a shoring-he-box-sraegy are reaed as if hey had sold he long posiion and hence heir capial gains are axed. Gallmeyer, Kaniel, and Tompaidis (2006) address his issue and propose ax-managemen sraegies o circumven he capial gains ax. Their so-called rading flexibiliy sraegy minimizes fuure ax-induced rading coss by shoring one of several socks even if none of he socks has an unrealized gain. This sraegy is in paricular useful if he benefis from holding a well-diversified porfolio are ouweighed by he expeced fuure rebalancing coss. This is in paricular he case if wo asses are highly correlaed. To circumven he shoring-he-box-sraegy Sigliz (1983) suggess selling (or shoring if necessary) highly correlaed asses insead of realizing capial gains. Neverheless his shoring-sraegy can be subjec o significan coss ha have o be aken ino accoun. Dammon, Spa, and Zhang (2001) on he oher hand show ha selling an asse wih an unrealized capial gain can be an opimal ax-iming sraegy. According o heir sudy he diversificaion benefi of reducing a volaile posiion can significanly ouweighed he ax cos of selling he asse wih an unrealized capial gain. The resuls of Dammon, Dunn, and Spa (1989) sugges ha he value of he opion o realize long-erm gains o regain he opporuniy of realizing shor-erm losses is negaively relaed o he socks price volailiy. Consaninides (1984) shows ha if shor-erm capial gains are axed a a higher ax-rae han long-erm capial gains i can be opimal o sell asses wih an unrealized capial gain as soon as hey qualify for long-erm reamen in order o regain he opporuniy of producing shor-erm losses. Dammon and Spa (1996) exend his approach by allowing he number of rading periods before a shor erm posiion becomes a long erm posiion o be greaer han one. In paricular hey show ha conrary o inuiion, i can be opimal o defer small shor-erm losses even in he absence of ransacion coss. This finding is due o he fac ha realizing hese losses and repurchasing he asse resars he shor-erm holding period and hus 3

5 he ime he invesor has o wai unil poenial fuure gains qualify for long-erm reamen. Under plausible parameer values, hey find ha i can be opimal for invesors o defer realizing shor-erm capial losses of abou 10% in he absence of ransacion coss. Consaninides and Scholes (1980) argue, ha even when an invesor sells an asse wih an unrealized capial gain, he can defer his gain by hedging. Neverheless, all of hese sudies assume ha capial loss deducion is unlimied. This paper akes he limied capial loss deducion explicily ino accoun, generalizes he finding of Consaninides (1983) ha losses should be realized immediaely o a ax-sysem wih limied capial loss deducion in secion 3 and shows he impac of such limiaions on opimal asse allocaion in secion 4. 3 Analyical Resuls Following Consaninides (1983), a marke is considered, in which invesors are price akers and only rade a equilibrium prices, here are no ransacion coss and he ax-sysem allows for wash-sales. Unrealized gains remain unaxed, while realized capial gains reduced by realized capial losses are axable. The difference beween a realized capial gain and a realized capial loss in period k is called a ne capial gain G k if G k 0 or a ne capial loss if G k < 0. A ne capial gain is axable a ax-rae τ. As in Consaninides (1983) i is assumed ha here is only one capial gains ax-rae and i is no disinguished beween long-erm and shor-erm capial gains. In conras o Consaninides i is assumed ha capial losses only qualiy for a ax-rebae up o a cerain amoun. For a ne capial loss an invesor only receives a ax rebae in period k for ha par of he ne capial loss G k no exceeding a maximum amoun of M k 0 dollars in period k in absolue value. This rebae is paid a he end of he period, when he invesor s ne capial gain is known. Even hough in ax-law he maximum loss deducion M k is usually a consan and hus independen from he ime index k, he heoreical analysis wih ime dependen M k allows for a deeper undersanding of he problem. The ne capial gain (or loss) T k in period k ha is subjec o he capial gains ax is given by T k := max (G k + L k 1 ; M k ) (1) Realized ne losses ha exend he amoun of M k can be indefiniely carried forward o he following periods. Thus, he loss L k ha can be carried forward from period k o period k + 1 4

6 is given by L k := min (G k T k + L k 1 ; 0) (2) In a one-period model he evoluion of an invesor s wealh in a ax-sysem wih limied capial loss deducion is hus equal o ha of an invesor in a ax-sysem wih unlimied capial loss deducion who has sold a pu-opion a ime zero wih srike P 1 P 0 M 1 and paricipaion rae of τ wihou receiving a premium for ha pu-opion. However, in a ax-sysem wih limied capial loss deducion he invesor receives he loss carryforward according o Equaion (2) as a kind of compensaion when he pu-opion is in he money a ime of mauriy. The Case wih One Risky Asse In he following he invesmen decision of an invesor a ime is considered who has a loss carryforward of L 1. He has he opporuniy o inves ino a risk-free asse ha pays an afer-ax reurn of r > 0 per period. Le R := 1 + r denoe he gross afer-ax reurn of he risk-free asse. Furhermore, he invesor has he opporuniy o inves ino a risky asse. He wans o hold one uni of he risky asse unil ime T > ha he aains alive wih probabiliy one. Le P denoe he price of he risky asse a ime and assume he reurn of his asse only consiss of capial gains, i.e. i does no pay any dividend or ineres. Le P denoe he purchase price of ha asse a he end of period. Thus, P 1 is he purchase price before rading. Le P [inf i [,+1) P i, sup i [,+1) P i ] be some price in ime [, + 1). For simpliciy i is assumed ha he invesor holds one single uni of he risky asse which is infiniely divisible for he enire invesmen horizon. In case P < P 1 he invesor could realize a ne capial loss in period. If ha loss does no exceed M, ha is ( P P 1) M he classical resul of Consaninides (1983) applies and he invesor should sell he asse o realize ha loss and earn he ax-rebae on i. If, however, he ne capial loss exceeds M, i.e. P P 1 < M, he precondiions under which his resul holds, are no longer full-filled. In he following i is shown ha i remains opimal o sell he asse even hough he invesor does no necessarily aain he same wealh-level as in a ax-sysem wih unlimied capial loss deducion. The Relaion beween Wealh, Unrealized Gains and he Loss Carryforward In ax-sysems wih limied capial loss deducion, an opimal asse allocaion decision depends on he invesor s oal wealh before rading, his unrealized capial gains, his loss carryforward 5

7 and he lengh of he remaining invesmen horizon. The key in undersanding opimal aximing in such a ax-sysem is o undersand he relaion beween he firs hree facors. A loss carryforward L 1 of one dollar in period can be used in wo ways. Firs, i can be subraced from a realized capial gain o reduce capial gains axes. Second, in he absence of a realized capial gain, he loss carryforward can be claimed as a ne capial loss ha is subjec o he capial gains ax if M > 0 according o Equaion (1). Thus, one dollar of loss carryforward can be shifed o τ dollars of wealh if M τ. Shifing he loss carryforward o wealh is a dominaing sraegy, as each dollar of loss carryforward can reduce fuure ax burden by no more han τ dollars. Furhermore, in conras wih he loss carryforward, he τ dollars of ax rebae can be reinvesed. By invesing he ax rebae in he risk-free asse, is value is always a leas as high as he fuure ax burden of he unrealized capial gain. Thus, if wo invesmen sraegies resul in he same unrealized capial gains a some poin in ime before rading, bu one of hem resuls in a higher pre ax wealh W before rading and he oher in a higher loss carryforward L 1, he sraegy wih he higher pre ax wealh is a leas as good as he sraegy wih he higher loss carryforward, if for every τ exra dollars of wealh W, he second sraegy does no have more han one dollar of exra loss carryforward. Le A B denoe he fac ha A is a leas as good as B, han his finding can also be expressed as W = τ W = 0 (3) L 1 = 0 L 1 = 1 An invesor endowed wih one dollar of unrealized capial gains U a he beginning of period before rading and one dollar of loss carryforward can use his loss carryforward in wo ways. I can eiher be used o realize he capial gain or i can be used o generae a ne capial loss a ime and hus o earn a ax rebae of τ dollars if M τ. As shown above, he value of he ax rebae is a leas as high as he fuure ax burden of he unrealized capial gain when invesed in he risk-free asse. Realizing he ne capial loss hus is a dominaing sraegy. An invesor who is neiher endowed wih ha dollar of unrealized capial gain nor ha dollar of loss carryforward can be considered an invesor who has realized ha capial gain and used his loss carryforward o avoid he capial gains ax. He hen lacks he dominaing opporuniy 6

8 of realizing he ne capial loss and keeping he unrealized capial gain. Hence: U = 1 U = 0 (4) L 1 = 1 L 1 = 0 The relaion beween wealh and unrealized capial gains follows from he relaion beween wealh and losses and he relaion beween unrealized gains and losses. If wo invesmen sraegies resul in a loss carryforward of zero a some poin in ime before rading, bu he firs of hem resuls in a higher pre ax wealh and in higher capial gains han he oher, he firs sraegy is a leas as good as he second sraegy, if for ever τ exra dollars of pre ax wealh, he unrealized capial gains of he firs sraegy does no exend one dollar. Then his is due o he fac ha according o Equaions (3) and (4) i holds ha W = 0 W = τ W = 0 W = 0 U = 1 + U = 0 U = 0 + U = 0 L 1 = 1 L 1 = 0 L 1 = 0 L 1 = 1 (5) W = τ W = 0 U = 1 U = 0 L 1 = 0 L 1 = 0 The economic inuiion for his resul is ha each dollar of unrealized capial gains resuls in a ax burden of τ dollars. Furhermore, he τ dollars of wealh allow for earning he risk-free ineres rae in forhcoming periods. These hree findings on he relaion beween wealh, unrealized capial gains and he loss carryforward have a nice inuiive inerpreaion. Wealh allows for earning he risk-free rae in forhcoming periods. This is why wealh is preferred o a loss carryforward ha does no earn any ineres. Furhermore, a loss carryforward can be convered ino wealh easier han a lower unrealized capial gain. Thus, a loss carryforward is closer o being convered ino wealh han a lower unrealized capial gain. Hence, one can expec a loss carryforward o conver o wealh earlier han he lower unrealized capial gain and hus, he loss carryforward allows earning he risk-free ineres rae earlier han he lower unrealized capial gain. This explains why wealh is preferred o a loss carryforward, which in urn is preferred o lower unrealized capial gains. 7

9 The Opimal Tax-Timing Sraegy To derive he opimal ax-iming sraegy of an invesor who holds one uni of he risky asse plus some risk-free bonds a ime, i suffices o analyze he case of an invesor who holds only one uni of he risky asse. Even more, i suffices o analyze hree ax-iming sraegies in period as all oher sraegies are linear combinaions of hese hree sraegies. Firs, he invesor can sell he risky asse o realize he unrealized ne capial loss, and immediaely repurchase one uni of he risky asse (sraegy one). Second, he invesor can hold he asse and do no ransacions in period (sraegy wo). Third, he invesor can sell ha much of he risky asse ha he realizes he maximum loss ha can be offse in period, and repurchase ha much of he risky asse as he has sold (sraegy hree). In case ha his loss carryforward L 1 exceeds he limi on he loss deducion, he does no even have o sell any asses o realize he desired capial loss and sraegies wo and hree coincide. All oher ax-iming sraegies are linear combinaions of hese hree sraegies. Any sraegy selling some fracion of he risky asse which is more han ha of sraegy hree, bu less ha ha of sraegy one resuls in a porfolio and a loss carryforward ha is a linear combinaion of hose of sraegy one and hree. Accordingly any sraegy selling some fracion of he risky asse which is less ha ha of sraegy hree, bu more ha ha of sraegy wo resuls in a porfolio and a loss carryforward ha is a linear combinaion of hose of sraegy wo and hree. To prove ha sraegy one is an opimal ax-iming sraegy, i hus suffices o show ha sraegy one does a leas as good as sraegies wo and hree. The hree sraegies only differ in heir purchase prices of he risky asse, he loss carryforward and he invesor s wealh afer rading in period. When he invesor follows sraegy one and sells he risky asse he realizes a ne capial loss of P P 1 and increases his purchase price o P = P. As P P 1 < M P P 1 + L 1 < M his axable ne capial loss is T (1) = max ( P P 1 + L 1 ; M ) = M (6) Thus, his ax refund is M τ dollars. His remaining loss carryforward is given by L (1) = min ( P P 1 + M + L 1 ; 0 ) = P P 1 + M + L 1 (7) If he invesor follows sraegy wo and does no do any ransacions in period, his purchase 8

10 Table 1: Comparison of ax-iming sraegies sraegy 1 sraegies 2,3a sraegy 3b P P P 1 mixed L P P 1 + M + L 1 min (L 1 max (L 1 ; M ) ; 0) 0 W +1 P +1 + M τ P +1 max (L 1 ; M ) τ P +1 + M τ U +1 P +1 P P +1 P 1 P +1 P 1 + M + L 1 price remains a P = P 1, his ne capial gain is T (2) = max (0 + L 1 ; M ) = max (L 1 ; M ) (8) Thus, his ax refund is max (L 1 ; M ) τ. His remaining loss carryforward is L (2) = min (0 max (L 1 ; M ) + L 1 ; 0) = min (L 1 max (L 1 ; M ) ; 0) (9) If he invesor follows sraegy hree, he chooses his invesmen sraegy such ha his ne capial loss is given by T (3) = M (10) and hus his ax refund under sraegy hree is M τ. His remaining loss carryforward is L (3) = 0 (11) Le W (i) denoe he pre ax wealh in period of sraegy i (i {1, 2, 3}) before rading. Then W (1) +1 =P +1 + M τ (12) W (2) +1 =P +1 max (L 1 ; M ) τ (13) W (3) +1 =P +1 + M τ (14) If he invesor follows ax-iming sraegy hree wo cases have o be disinguished. Firs, if max (L 1 ; M ) = M, hen he loss carryforward L 1 from period 1 suffices o realize he desired ne capial loss in period. This case will be referred o as sraegy hree, case a), or jus hree a). In his case he invesor does no have o do any ransacions. Thus, if max (L 1 ; M ) = M sraegies wo and hree coincide. Second, if L 1 > M, he invesor sill has o sell some of his risky asses. The amoun 9

11 of he risky asses he has o sell is equivalen o a fracion f of he risky asse, such ha M = f ( ) P P 1 + L 1 f = M L 1. This case will be referred o as sraegy hree, case b), or jus hree b). P P 1 Le U (i) denoe he unrealized capial gains (or losses) in period of sraegy i (i {1, 2, 3}) before rading. Then U (1) +1 =P +1 P (15) U (2) +1 = U (3a) +1 =P +1 P 1 (16) U (3b) +1 =P +1 ( ) fp + (1 f) P 1 ( L 1 M =P +1 P P P 1 ( L 1 M =P +1 P P 1 =P +1 P 1 + L 1 + M Table 1 summarizes he properies of he hree sraegies. ( 1 + L 1 M P P 1 ) ( ) P P 1 P 1 ) ) P 1 As for max (L 1 ; M ) = M sraegies wo and hree resul in he same invesmen behavior in period, sraegies wo and hree a) do no differ. To show ha sraegy one is he dominaing ax-iming sraegy i hus suffices o show ha sraegy one dominaes sraegy wo and sraegy hree b). Wih Equaion (4) i holds for he relaion beween sraegies one and hree b) ha W (1) +1 U (1) +1 L (1) = P +1 + M τ P +1 P P P 1 + M + L 1 P +1 + M τ P +1 P 1 + M + L 1 = 0 W (3b) +1 U (3b) +1 L (3b) (17) (18) Thus, sraegy one is a leas as good as sraegy hree b). This is due o he higher flexibiliy of he loss carryforward ha can be easier ransferred o wealh and hen earn he risk-free 10

12 ineres rae. Wih Equaion (3) i holds for he relaion beween sraegies wo and one ha W (2) +1 U (2) +1 L (2) P +1 max (L 1 ; M ) τ = P +1 P 1 min (L 1 max (L 1 ; M ) ; 0) P +1 + M τ P +1 P 1 min (L 1 max (L 1 ; M ) ; 0) + M + max (L 1 ; M ) (19) For boh L 1 M and L 1 < M i holds wih P P 1 < 0 and Equaion (4) ha W (2) +1 U (2) +1 L (2) P +1 + M τ P +1 P 1 L 1 + M P +1 + M τ P +1 P P P 1 + M + L 1 = W (1) +1 U (1) +1 Thus, sraegy one is a leas as good as sraegy wo, which shows ha independen from he realizaion of fuure prices of equiy and he relaion of he maximum loss deducion M and he iniial loss carryforward L 1, sraegy one always does a leas as good as sraegies wo and hree. Furhermore, sraegy one someimes resuls in higher wealh han sraegy wo, by allowing o earn he risk-free ineres rae in fuure periods on he loss carryforward convered o wealh. Hence, sraegy one is an opimal ax-iming sraegy and unrealized losses should be realized immediaely. Even more, if P inf i [,+1) P i, he invesor can sill increase his realized loss in ime [, + 1) by rading whenever he price of he asse is below his purchase price. In his case he above resuls wih P = inf i [,+1) P i apply. In he proof i has hus far been assumed, ha he risky asse does no pay any dividend or ineres. If, however, he risky asse does pay some dividend or ineres, boh sraegies are affeced from hese paymens in he same way as in boh ax-sysems he invesor holds one uni of he risky asse and hence receives he same amoun of dividend or ineres. Thus, an asse paying dividends affecs invesors in ax-sysems wih limied and unlimied capial loss deducion he same way. L (1) (20) Hence, he resuls derived above also hold for risky asses whose reurns conain of boh capial gains and dividend or ineres paymens. 11

13 The Case wih Muliple Risky Asses In case he invesor only holds one risky asse, i is opimal for he invesor o realize losses a ime immediaely in order o earn he ineres on he ax rebae if M > 0. In case ha M = 0, here is no ax rebae and he invesor can only use a loss carryforward o reduce fuure realized capial gains. Thus, for he one-asse case when M = 0, boh sraegies one and hree and any combinaion of hem is opimal as here are no ax rebaes ha allow for earning a reurn. In he one-asse case he invesor never faces he siuaion in which one of his asses faces a capial gain and anoher faces a capial loss. Thus, in his case, a ne capial loss ha exceeds he amoun of M can only be carried forward. If however he invesor holds more han one risky asse he can use a loss carryforward L 1 realized in some asse S 1 in period o reduce his ne capial gains from some oher asse S 2 realized in period k > if he wans o reallocae his porfolio. As in conras o sraegy one, sraegy hree does no allow for his ransfer of realized losses of some asse S 1 o some oher asse S 2, sraegy one dominaes sraegy hree in he muliple-asse case. To make sraegy one preferable o sraegy hree i is no even necessary o assume ha he invesor holds more han one risky asse in period. I suffices if he holds more han one risky asse in some period k ( < k T ) wih posiive probabiliy. Aainable Wealh As shown in Consaninides (1983) and above in boh a ax sysem wih limied and unlimied capial loss deducion i is opimal o realize losses immediaely. Le W (l) denoe he beginning of period wealh before rading an invesor can aain in a ax-sysem wih limied capial loss deducion following he opimal ax-iming sraegy, i.e. W (l) := W (1). Le furhermore W (u) denoe he corresponding wealh he can aain in a ax-sysem wih unlimied capial loss deducion following he opimal ax-iming sraegy o realize losses in he period hey occur. As here are no loss carryovers in ax-sysems wih unlimied capial loss deducion, i is assumed, ha L 1 = 0 in he ax-sysem wih limied capial loss deducion o make he wo ax-sysems comparable. In case an invesor realizes a capial loss in period ha does no exceed M he evoluion of his wealh from o + 1 is he same in boh ax sysems. If 12

14 however he capial loss exceeds M, i.e. P P 1 < M, hen W (l) +1 =P +1 + M τ (21) U (l) +1 =P +1 P (22) L (l) =P P 1 + M (23) and W (u) +1 =P +1 ( P P 1) τ (24) U (u) +1 =P +1 P (25) L (u) =0 (26) Wih Equaion (3) i holds ha W (u) +1 U (u) +1 L (u) P +1 + M τ ( ) P P 1 + M τ = P +1 P 0 P +1 + M τ P +1 P P P 1 + M = W (l) +1 U (l) +1 L (l) (27) Thus, no very surprisingly, an invesmen opporuniy se wih unlimied capial loss deducion is preferable o an invesmen opporuniy se wih limied capial loss deducion. The advanage of he invesmen opporuniy se wih unlimied capial loss deducion is he opporuniy o ge an unlimied ax rebae on capial losses and earn he ineres on hese losses, while in a ax-sysem wih limied capial loss deducion no ineres is paid on he loss carryforward. Furhermore, one dollar of cash a hand can be used a lo more flexible, han one dollar of loss carryforward, especially when he limis on he maximum loss deducions M 1,..., M T are small. 4 Capial Gains Taxaion and Asse Allocaion This secion analyzes he impac of capial gains axaion on asse allocaion decisions. As above invesmens for a given invesmen horizon in he absence of exogenous increases or decreases of wealh like income or consumpion are considered. If here was no ax on capial gains, he classical resul of Meron (1969) and Samuelson (1969) would apply and he invesor s asse allocaion would be he same in each period. Thus, each deviaion of his asse allocaion 13

15 from his benchmark mus be due o he axaion of capial gains. This secion shows, how axaion of capial gains affecs he invesor s asse allocaion and in paricular how limiaion of capial loss deducion does. In a ax-sysem wih unlimied capial loss deducion, capial gains and capial losses are reaed symmerically. Tha is, no maer if a capial gain or a capial loss is realized, he invesor is confroned wih he same axable reamen. In ax-sysems wih limied capial loss deducion, however, here is an asymmeric axaion of capial gains and losses. While capial gains are axed a he capial gains ax-rae wihou any limis, capial losses qualify for ax rebaes only up o a cerain amoun. Thus, he invesor receives he fracion 1 τ of poenial capial gains, bu bears he enire risk for losses exceeding he maximum loss deducion when he has no loss carryforward. The compensaion for his risk comes as a loss carryforward. However, in conras o ax rebaes, he invesor canno earn any ineres on he loss carryforward. Furhermore, he also bears he risk ha he has no use for he enire amoun of loss carryforward in forhcoming periods. This risk is especially imporan, if he remaining invesmen horizon is shor. Thus, compared o a ax-sysem wih unlimied capial loss deducion, an invesmen ino a risky asse offers he same opporuniies o he invesor when reurns are posiive, bu bears higher risks when reurns are negaive. Thus, in a ax-sysem wih limied capial loss deducion, invesors will hold less risky asses han in a ax-sysem wih unlimied capial loss deducion when hey are no endowed wih a loss carryforward. The size of he advanage ha resuls from he opporuniy o inves in a ax-sysem wih unlimied capial loss deducion insead of a ax-sysem wih limied capial loss deducion depends on five facors. Firs, i depends on M 1,..., M T, he amouns up o which realized losses qualify for ax rebaes. The higher hese values he lower he advanage of he axsysem wih unlimied capial loss deducion. Second, i depends on he capial gains ax-rae τ. The higher τ, he higher he ax rebaes and hus he more advanageous he ax-sysem wih unlimied capial loss deducion. Third, i depends on he evoluion of he price of he risky asse, P 1,..., P T. The earlier and he higher capial losses ha exceed M 1,..., M T, he bigger he advanage of he opporuniy o inves in a ax-sysem wih unlimied capial loss deducion. Thus, he more volaile he risky asse, he bigger he disadvanage of being confroned wih a ax-sysem wih limied capial loss deducion. Hence, in ax-sysems wih limied capial loss, invesors will decrease heir holdings in risky asses he sronger, he more volaile hey are. Fourh, he lower he risk-aversion of an invesor he higher his exposure o risky asses. Thus he higher his disadvanage when being confroned wih a ax-sysem wih limied capial loss deducion. Fifh, he higher he risk-free rae he higher he disadvanage 14

16 of he loss carryforward no paying any ineres and hus, he higher he advanage of he ax-sysem wih unlimied capial loss deducion. If, however, he invesor is endowed wih a loss carryforward, his has a posiive impac on his risk-reurn profile. In his case, an invesmen ino a risky asse is more preferable han wihou a loss carryforward as i allows he invesor o earn an amoun of capial gains no exceeding he loss carryforward ax-free. Thus, if he invesor is endowed wih some loss carryforward, here is no longer a dominaing relaion beween he wo ax-sysems from he invesor s poin of view. In he ax-sysem wih limied capial loss deducion, on he one hand he bears he risk ha he reasury does no paricipae in high losses via ax rebaes. On he oher hand, he has he opporuniy of earning some capial gains ax-free. Numerical Evidence Following Dammon, Spa, and Zhang (2001) i was assumed, ha he invesor can only rade a ime 0, 1,..., T and ha he purchase price used o compue capial gains is he average weighed hisorical purchase price. If P denoes he ax basis of a risky asse (also referred o as equiy in he following) afer rading a ime, hen his ax basis is defined by P = q 1 P 1 +max(q q 1,0)P q 1 +max(q q 1,0) P if P 1 < P if P 1 P. (28) This specificaion akes he fac ino accoun, ha in i is opimal o realize capial losses (i.e. P 1 P ) immediaely, which decreases he average purchase price from P 1 o P. If, however, he invesor is endowed wih an unrealized capial gain (i.e. P 1 < P ) he change in his ax-basis depends on his rading in period. If he invesor sells some asses (i.e. q < q 1 ), his ax basis remains unchanged. If insead, he invesor buys some asses (i.e. q > q 1 ) his ax basis is a weighed average of he previous ax basis and he purchase price P of he new asse. As according o he opimal ax-iming sraegy losses shall be realized immediaely, he invesor s realized capial gains in period is given by G = ( ) 1 P 1 >P q P P 1 max (q 1 q, 0) (P ) P 1 (29) In he ax-sysem wih unlimied capial loss deducion, he opimizaion problem is he same as in Dammon, Spa, and Zhang (2001). In he ax-sysem wih limied capial loss deducion, 15

17 he invesor s opimizing problem is s.. max q E [U [W T ]] (30) W = q 1 (1 + d ) P + b 1 R, = 0,..., T (31) W = τt + q P + b = 0,..., T 1 (32) W T = W T τt T (33) q 0 = 0,..., T 1 (34) q T = 0 (35) where d is he afer-ax dividend of equiy, given he iniial holding of bonds b 1, socks q 1, he iniial ax-basis P 1, he price of one uni of he sock P 0 and his iniial loss carryforward L 1. According o Equaion (30) he invesor maximized his uiliy he derives from erminal wealh afer axes. To have he classical resul of Meron (1969) and Samuelson (1969) as a benchmark, he invesor is assumed o have no beques moive. Equaion (31) defines he invesor s beginning of period wealh as he sum of his wealh in socks and his wealh in bonds before rading a ime, including he afer-ax ineres and dividend income, bu before any capial gains axes resuling from rading a ime. Equaion (32) is he invesor s budge consrain a ime. If he invesor rades equiy he possibly has o pay some capial gains ax or ges some ax-refund on his ne capial gain T as defined in Equaion (1). By leing X denoe he vecor of he invesor s sae variables a ime, he Bellmann equaion for he maximizaion problem can be wrien as follows: V (X ) = max q E [V +1 (X +1 )] (36) for = 0,..., T 1 subjec o Equaions (1), (2), (28), (29) and (31) o (35). The sae variables needed o solve ha problem a ime are he invesors beginning-of-period-wealh W before rading, his iniial loss carryforward L 1, his ax basis P 1 and he number of socks q 1 he is holding a he beginning of period before rading. Thus, his vecor of sae variables can be represened as X = [P, W, L 1, P 1, q 1 ] (37) 16

18 For he numerical analysis i is assumed, ha M is a negaive muliple of W, i.e. m := M W some negaive real value and consan in ime, i.e. m := m ( T ). This assumpion, which is made o reduce he number of sae-variables, is in conras wih he US-ax code, ha allows a consan amoun up o $ 3,000 o qualify for a ax-rebae. In his case, he impac of he is limiaion of capial loss deducion depends on he absolue wealh level as well. However, his relaion can also be capured by varying m. Neverheless, excep for M = 0, m canno capure he impac of an increase in oal wealh on he relaion beween M and W. In axsysems where M > 0 is consan, invesors should c.p. hold a lower proporion of equiy wih increasing wealh as oherwise hey run a higher risk o end up wih capial losses exceeding M. Wih he assumpion, ha m is a consan, he above opimizaion problem can be simplified by normalizing wih beginning-of-period-wealh W. Le s := q 1P W he invesor s beginning-of-period-wealh before rading invesed ino equiy, α denoe he fracion of := qp W invesor s fracion of beginning-of-period-wealh allocaed o equiy afer rading, b := b W fracion of he beginning-of-period-wealh allocaed o risk-free bonds afer rading, p 1 := P 1 P he invesors basis-price raio, := T W ha is axable a he capial gains ax-rae, l 1 he he he fracion of he invesor s beginning-of-period-wealh := L 1 W carryforward o his beginning-of-period-wealh, g = P +1 P period, and he fracion of he invesors loss 1 he capial gain on he sock in R := α (1 + d ) (1 + g ) + b R, (38) α + b he gross nominal reurn on he invesor s porfolio afer rading in period afer paymen of axes on dividends and ineres, bu before paymen of capial gains axes. Assuming he invesor o have CRRA-uiliy wih γ 1 U [W T ] = W 1 γ 1 γ (39) wih parameer of risk-aversion γ 0, and defining v (x ) = V(X) W 1 γ and w +1 = W +1 W, he 17

19 invesor s opimizaion problem can be rewrien as v (x ) = max α E [ ] v +1 (x +1 ) w 1 γ +1 (40) s.. 1 = τ + α + b = 0,..., T 1 (41) w +1 = (1 τ ) R = 0,..., T 1 (42) α 0 = 0,..., T 1 (43) in which he fracion of realized gains o beginning-of-period-wealh is given by δ := G ( ) = 1 p W 1 >1s + 1 p 1 1 max (α s, 0) (1 ) p 1 (44) and p is given by s p 1 +max(α s;0) p (s = +max(α s ;0))(g +1) p 1 < 1 1 oherwise g +1 (45) A ime T, he invesor s value funcion akes he value v T = (1 τ T ) 1 γ. (46) This problem can be solved numerically using backward inducion wih sae variables x = [s, p 1, l 1 ]. To do so, a ( ) grid is spanned. The opimizaion problem wih unlimied capial loss deducion wih sae variables x = [s, p 1] is solved wih a grid. For values beween he grid, cubic spline inerpolaion is performed. To expedie compuaion, he spline funcion for each of he wo problems is compued symbolically for each period. For he numerical analysis, i is assumed, ha he risk-free rae is 6%, he reurn of equiy is binomially disribued, comes wih an expeced capial gain of 7%, a sandard deviaion of 20% and a consan dividend rae of 2%. The ax-rae on ineres and dividends is assumed o be 36%. The ax-rae on realized capial gains is assumed o be 20%. This choice of parameers follows Dammon, Spa, and Zhang (2001) and Gallmeyer, Kaniel, and Tompaidis (2006). The lengh of he invesmen horizon is T = 10 years. These parameer values are referred o as he base case scenario. For he base-case i is assumed, ha he relaive risk-aversion of he invesor is γ = 3. In he absence of he ax-iming opion, i.e. for an asse whose reurns are 18

20 reduced by 20% bu no longer subjec o he capial gains ax-rae, bu whose volailiy is kep consan, he opimal fracion of socks is 20.1% in each period. All deviaions from his value resul from he ax-iming opion as well as he limiaion of capial loss deducion. Limied versus Unlimied Capial Loss Deducion Unlimied, =0, γ=3 Limied, =0, γ=3, l 1 =0, m=0 Opimal equiy proporion Opimal equiy proporion Basis price raio Iniial equiy proporion Basis price raio Iniial equiy proporion Figure 1: This figure shows he opimal equiy proporion afer rading α of an invesor wih risk-aversion of γ = 3 in a ax-sysem wih unlimied capial loss deducion (lef graph) and in a ax-sysem wih limied capial loss deducion (righ graph), when he invesor is endowed wih no iniial loss carryforward (l 1 = 0) and here is no ax rebae on capial losses (m = 0) as a funcion of he invesor s iniial equiy exposure s before rading a ime and he basisprice-raio p 1 when he invesmen horizon is 10 years. The lef graph in Figure 1 shows he opimal asse allocaion afer rading α of an invesor wih risk-aversion γ = 3 in a ax-sysem wih unlimied capial loss deducion. If he invesor is neiher endowed wih an unrealized capial gain nor a loss (i.e. p 1 = 1), he invesor holds 41.5% of his beginning-of-period wealh in equiy which is significanly more han in he benchmark case wihou capial gains axes, which reflecs he value of he ax-iming opion. In paricular, his equiy proporion afer rading does no depend on his fracion of socks relaive o beginning-of-period-wealh s, as for p 1 = 1 he invesor does no have o pay any capial gains axes when rearranging his asse allocaion. If he invesor is endowed wih a capial loss (i.e. p 1 > 1), he receives some ax rebae on his capial loss, which increases his wealh afer rading. Wih increasing beginning-of-period-wealh afer rading and a consan fracion of socks relaive o ha wealh, he fracion of socks afer rading relaive o beginning-ofperiod wealh α increases. The higher he capial loss and he higher he fracion of socks before rading s he higher he ax rebae and hus he higher he fracion of socks relaive o beginning-of-period-wealh before rading α. 19

21 If, however, he invesor is facing an unrealized capial gain (i.e. p 1 < 1), he has o decide, wheher o realize ha gain or o pospone i. Concerning his decision, here are wo opposing effecs. On he one hand, posponing he realizaion of he capial gain allows for earning compound reurns on he axes ha have been posponed. On he oher hand, posponing he realizaion of he capial gains can resul in unbalanced porfolios. Especially if some asse has been performing well in he pas, is fracion relaive o he invesor s oal wealh can be subsanial and so is is impac on he reurn of he porfolio. However, selling his asse o rebalance he porfolio would resul in a subsanial capial gains ax. In case he invesor is endowed wih a low fracion of socks before rading s, he invesor s fracion of socks afer rading is decreasing in his fracion of socks before rading s and increasing in his basis-price-raio p 1. Even hough his oal fracion of socks afer rading is higher han his fracion of socks before rading, i is no as high as if he invesor was no facing a capial gain (i.e. p 1 = 1). This is due o he reason, ha he new basis-price-raio p is defined as a weighed average of he beginning-of-period basis-price-raio p 1 and one. The lower p 1 and he higher s, he lower p and hus he higher he risk of geing locked-in in wih a significan amoun in fuure periods. Wih increasing fracion of socks before rading, he invesor someime aains a level of socks before rading s ha exceeds his desired fracion of socks afer rading α. As long as his desired fracion of socks afer rading does no deviae oo far from he fracion of socks afer rading α he has o hold, o avoid paying he capial gains ax, he rebalancing moive is ouweighed by he opporuniy o earn compound reurns on he unrealized capial gains. In case he invesor is endowed wih an even higher fracion of socks before rading s, he rebalancing moive ouweighs he impac of he posponing of capial gains, and he invesor s wealh afer rading is reduced by he capial gains axes, which is why his fracion of socks afer rading relaive o beginning-of-period-wealh is he lower, he higher his fracion of socks before rading. The righ graph in Figure 1 shows he opimal asse allocaion of an invesor wih risk-aversion γ = 3 and no iniial loss carryforward (i.e. l 1 = 0) in a ax-sysem wih limied capial loss deducion where realized capial losses can only be carried forward and deduced from fuure realized capial gains, bu do no qualify for ax rebaes (i.e. m = 0). Such a axable reamen of realized capial losses is e.g. implemened in he German ax-code. If he invesor is neiher endowed wih an unrealized capial gain nor wih an unrealized capial loss (i.e. p 1 = 1), his equiy exposure is abou 33.7%, which is significanly above he benchmark case wihou he ax-iming opion, bu well below he 41.5% in he ax-sysem 20

22 wih unlimied capial loss deducion. This is due o he fac, ha on he one hand in he ax-sysem wih limied capial loss deducion, he invesor bears he enire risk when having ne capial losses, as he does no ge any ax rebaes on hem. On he oher hand, capial gains are reaed he same way as in a ax-sysem wih unlimied capial loss deducion. Hence, equiy is less aracive in he ax-sysem wih limied capial loss deducion which is why he invesor holds a subsanially lower fracion of his wealh in equiy. If he invesor is facing an unrealized capial loss (i.e. p 1 > 1), i is opimal o realize ha loss immediaely as shown above. The invesor is hen endowed wih a loss carryforward which is he higher, he higher he ne capial loss p 1 and he higher he equiy proporion before rading s. As his loss carryforward allows for earning some fuure capial gains ax-free, he invesor chooses a higher equiy proporion afer rading compared o he case of p 1 = 1. While in he ax-sysem wih unlimied capial loss deducion he invesor receives he loss carryforward in cash as a ax rebae, in he ax-sysem wih limied capial loss deducion, i only has a value o he invesor, if i can be subraced from forhcoming capial gains. This is why in he ax-sysem wih limied capial loss deducion he increase in he fracion of socks is sronger han in he ax-sysem wih unlimied capial loss deducion bu remains a a cerain level wih increasing realized losses for no ending up wih a porfolio ha is oo heavily invesed ino equiy. If, however, he invesor is facing an unrealized capial gain (i.e. p 1 < 1) and his fracion of socks before rading is low, his fracion of socks afer rading will be abou he same as wihou he capial gain and his basis-price-raio afer rading will decline. Compared o he ax-sysem wih unlimied capial loss deducion, he danger of geing locked-in wih a higher amoun of wealh seems o be neglecable. The reason for his is ha in he ax-sysem wih limied capial loss deducion, each capial gain in equiy increases his fuure capial gains ax, while each loss in equiy decreases his fuure capial gains ax. This decrease in he fuure capial gains ax resuls in fuure capial gains and capial losses being reaed equally and hus has a higher value for he invesor han a loss carryforward, as he loss carryforward carries he risk of poenially remaining unused. The numerical resul suggess ha he opporuniy of reducing he embedded capial gain ouweighs he risk of geing locked in. If he invesor is facing an unrealized capial gain and his fracion of socks before rading is high, he rebalancing moive ouweighs he opporuniy of earning compound reurns on he unrealized capial gains and he resuls in he ax-sysem do no differ from hose in he ax-sysem wih unlimied capial loss deducion. In general, he fracion of equiy is a leas as high in he ax-sysem wih unlimied capial 21

23 loss deducion as in he ax-sysem wih limied capial loss deducion. The reason for his is ha equiy is more aracive in he laer ax-sysem due o he ax rebae on realized capial losses. The Impac of a Loss Carryforward and Tax Rebaes Limied, =0, γ=3, l 1 = 0.2, m=0 Opimal equiy proporion Basis price raio Iniial equiy proporion Limied, =0, γ=3, l 1 =0, m=0.02 Limied, =0, γ=3, l 1 = 0.2, m=0.02 Opimal equiy proporion Basis price raio Iniial equiy proporion Opimal equiy proporion Basis price raio Iniial equiy proporion Figure 2: This figure shows he opimal equiy proporion afer rading α of an invesor wih risk-aversion of γ = 3 in a ax-sysem wih limied capial loss deducion. The upper graph shows he asse allocaion when he invesor is endowed wih an iniial loss carryforward of l 1 = 0.2 and here is no ax rebae on capial losses (m = 0) as a funcion of he invesor s iniial equiy proporion s before rading a ime and he basis-price-raio p 1 when he invesmen horizon is 10 years. The lower graphs show his asse allocaion if he ax sysem allows for a rebae on capial losses for up o 2% of presen wealh (m = 0.02) for an invesor wih no iniial loss carryforward (l 1 = 0, lef graph) and an iniial forward of l 1 = 0.2 (righ graph). Figure 2 shows he impac of variaions in he iniial loss carryforward l 1 and he maximum capial loss deducion m in a ax-sysem wih limied capial loss deducion for an invesmen horizon of en years. In he upper graph he invesor is assumed o have an iniial loss car- 22

24 ryforward of 20% of his curren wealh (i.e. l 1 = 0.2) and he ax-sysem is assumed no o provide any ax rebaes on capial losses (i.e. m = 0). Compared o he case in which he invesor has no loss carryforward (righ graph in Figure 1) he invesor increases his equiy proporion o 36.7% when he is neiher endowed wih an unrealized capial gain nor an unrealized capial loss (i.e. p 1 = 1). This is due o he fac ha due o he loss carryforward he invesor can realize some capial gains in equiy wihou having o pay he capial gains ax. Thus, he risk-reurn profile in ha asse becomes more aracive for him, which is why he increases his exposure o i. Wih decreasing basis-price-raio p 1 and increasing fracion of socks before rading s, he invesor realizes more of his unrealized capial gains han wihou a loss carryforward o ge a porfolio ha is beer diversified. However, if his capial gains exceed his loss carryforward, similar o he case wihou an iniial loss carryforward, he invesor does no realize ha much of his capial gains as o come up wih he same fracion of socks afer rading as wihou such a high unrealized capial gain. In he lower lef graph of Figure 2, he opimal asse allocaion of an invesor wih no iniial loss carryforward (i.e. l 1 = 0) is shown who is rading in a ax-sysem ha allows for ax rebaes on capial losses of up o 2% of he invesor s beginning-of-period wealh (i.e. m = 0.02). If such an invesor is neiher endowed wih an unrealized capial gain nor an unrealized capial loss (i.e. p 1 = 1), his fracion of socks afer rading is abou 36.6%, which is well above he level in a ax-sysem wih no ax rebaes (i.e. m = 0) and significanly below he level in a ax-sysem wih unlimied capial loss deducion. If he invesor is endowed wih a capial loss, his exposure o equiy is slighly decreasing as soon as he capial loss exceeds he maximum amoun qualifying for a ax rebae. This is due o he reason ha he invesor wans o avoid increasing his loss carryforward even more. If, however, his ne capial loss is of significan heigh, he risk of no using his loss carryforward ouweighs he risk of increasing i even furher, which is why in his case; his opimal equiy exposure is increasing again. If he invesor is endowed wih an unrealized capial gain (i.e. p 1 < 1) and his fracion of socks before rading is small, he invesor increases his equiy exposure. If his unrealized capial gain is small, he increases his equiy exposure even above he level when no being endowed wih a capial gain o decrease p ha much, ha he regains he opporuniy of geing ax rebaes on poenial fuure capial losses. If, however, his unrealized capial gains are large, he decreases his equiy exposure even below he fracion of socks for p 1 = 1 o reduce o risk of geing locked-in wih a higher amoun of wealh. If he invesor faces a significan capial loss (i.e. p 1 > 1) and ends up wih a loss carryforward, he does no change his equiy exposure afer rading α significanly compared o he case of p 1 = 1. Due 23