Most investors focus on the management

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1 Long-Short Portfolo Management: An Integrated Approach The real benefts of long-short are released only by an ntegrated portfolo optmzaton. Bruce I. Jacobs, Kenneth. Levy, and Davd Starer BRUCE I. JACOBS and KEETH. LEVY are prncpals and DAVID STARER s a senor quanttatve analyst at Jacobs Levy Equty Management n Roseland (J 07068) Most nvestors focus on the management of long portfolos and the selecton of wnnng securtes. Yet the dentfcaton of wnnng securtes gnores by defnton a whole class of losng securtes. The ablty to sell short frees the nvestor to take advantage of the full array of securtes and the full complement of nvestment nsghts by holdng expected wnners long and sellng expected losers short. A long-short portfolo, by expandng the scope of the nvestor s sphere of actvty, can be expected to result n mproved performance from actve securty selecton vs-à-vs a long-only portfolo. But the benefts of long-short are to a large extent dependent on proper portfolo constructon. Only an ntegrated optmzaton of long and short postons has the potental to maxmze the value of nvestors nsghts. The benefts that emerge from ntegrated optmzaton encompass not only freedom from the short-sellng constrant but also freedom from the restrctons mposed by ndvdual securtes benchmark weghts. Of course, these benefts do not come wthout some cost. Much of the ncremental cost assocated wth a gven long-short portfolo reflects the strategy s degree of leverage. evertheless, as we wll see, longshort s not necessarly much costler or, ndeed, much rsker than long-only. Although most exstng long-short portfolos are constructed to be neutral to systematc rsk, we wll see that neutralty s nether necessary nor, n most cases, WITER 1999 THE JOURAL OF PORTFOLIO MAAGEMET 3

2 optmal. Furthermore, we show that long-short portfolos do not consttute a separate asset class; they can, however, be constructed to nclude a desred exposure to the return (and rsk) of vrtually any exstng asset class. LOG-SHORT: BEEFITS AD COSTS Consder a long-only nvestor who has an extremely negatve vew about a typcal stock. The nvestor s ablty to beneft from ths nsght s very lmted. The most the nvestor can do s exclude the stock from the portfolo, n whch case the portfolo wll have about a 0.01% underweght n the stock, relatve to the underlyng market. 1 Those who do not consder ths to be a materal constrant should consder what ts effect would be on the nvestor s ablty to overweght a typcal stock. It would mean the nvestor could hold no more than a 0.0% long poston n the stock a 0.01% overweght no matter how attractve ts expected return. The ablty to short, by ncreasng the nvestor s leeway to act on nsghts, has the potental to enhance returns from actve securty selecton. The scope of the mprovement, however, wll depend crtcally on the way the long-short portfolo s constructed. In partcular, an ntegrated optmzaton that consders both long and short postons smultaneously not only frees the nvestor from the non-negatvty constrant mposed on long-only portfolos, but also frees the long-short portfolo from the restrctons mposed by securtes benchmark weghts. To see ths, t s useful to examne one obvous (f suboptmal) way of constructng a longshort portfolo. Long-short portfolos are sometmes constructed by combnng a long-only portfolo, perhaps a preexstng one, wth a short-only portfolo. Ths results n a long-plus-short portfolo, not a true long-short portfolo. The long sde of ths portfolo s dentcal to a longonly portfolo; hence t offers no benefts n terms of ncremental return or reduced rsk. In long-plus-short, the short sde s statstcally equvalent to the long sde, hence to the long-only portfolo. 3 In effect: α L = α S = α LO ω L = ω S = ω LO That s, the excess return or alpha, α, of the long sde of the long-plus-short portfolo wll equal the alpha of the short sde, whch wll equal the alpha of the longonly portfolo. Furthermore, the resdual rsk of the long sde of the long-plus-short portfolo, ω, wll equal the resdual rsk of the short sde, whch wll equal the resdual rsk of the long-only portfolo. These equvalences reflect the fact that all the portfolos, the long-only portfolo and the long and short components of the long-plus-short portfolo, are constructed relatve to a benchmark ndex. Each portfolo s actve n pursung excess return relatve to the underlyng ndex only nsofar as t holds securtes n weghts that depart from ther ndex weghts. The ablty to pursue such excess returns may be lmted by the need to control the portfolo s resdual rsk by mantanng portfolo weghts that are close to ndex weghts. Portfolo constructon s ndex-constraned. Consder, for example, an nvestor who does not have the ablty to dscrmnate between good and bad ol stocks, or who beleves that no ol stock wll sgnfcantly outperform or underperform the underlyng benchmark n the near future. In long-plus-short, ths nvestor may have to hold some ol stocks n the long portfolo and short some ol stocks n the short portfolo, f only to control each portfolo s resdual rsk. The rato of the performance of the long-plusshort portfolo to that of the long-only portfolo can be expressed as follows: 4 IR IR L+ S LO = 1 + ρ L + S where IR s the nformaton rato, or the rato of excess return to resdual rsk, α/ω, and ρ L+S s the correlaton between the alphas of the long and short sdes of the long-plus-short portfolo. In long-plus-short, the advantage offered by the flexblty to short s curtaled by the need to control rsk by holdng or shortng securtes n ndex-lke weghts. A long-plus-short portfolo thus offers a beneft over a long-only portfolo only f there s a lessthan-one correlaton between the alphas of ts long and short sdes. In that case, the long-plus-short portfolo wll enjoy greater dversfcaton and reduced rsk relatve to a long-only portfolo. A long-only portfolo can derve a smlar beneft by addng a less than fully correlated asset wth comparable rsk and return, however, so ths s not a beneft unque to long-short. 4 LOG-SHORT PORTFOLIO MAAGEMET: A ITEGRATED APPROACH WITER 1999

3 The Real Benefts of Long-Short The real benefts of long-short emerge only when the portfolo s conceved of and constructed as a sngle, ntegrated portfolo of long and short postons. In ths framework, long-short s not a two-portfolo strategy. It s a one-portfolo strategy n whch the long and short postons are determned jontly wthn an optmzaton that takes nto account the expected returns of the ndvdual securtes, the standard devatons of those returns, and the correlatons between them, as well as the nvestor s tolerance for rsk. Wthn an ntegrated optmzaton, there s no need to converge to securtes benchmark weghts n order to control rsk. Rather, offsettng long and short postons can be used to control portfolo rsk. Ths allows the nvestor greater flexblty to take actve postons. Suppose, for example, that an nvestor s strongest nsghts are about ol stocks, some of whch are expected to do especally well and some especally poorly. The nvestor does not have to restrct the portfolo s weghtngs of ol stocks to ndex-lke weghts n order to control the portfolo s exposure to ol sector rsk. The nvestor can allocate much of the portfolo to ol stocks, held long and sold short. The offsettng long and short postons control the portfolo s exposure to the ol factor. Conversely, suppose the nvestor has no nsghts nto ol stock behavor. Unlke the long-only and longplus-short nvestors dscussed above, the long-short nvestor can totally exclude ol stocks from the portfolo. The excluson of ol stocks does not ncrease portfolo rsk, because the long-short portfolo s rsk s ndependent of any securty s benchmark weght. The flexblty afforded by the absence of the restrctons mposed by securtes benchmark weghts enhances the long-short nvestor s ablty to mplement nvestment nsghts. Costs: Percepton versus Realty Long-short constructon maxmzes the beneft obtaned from potentally valuable nvestment nsghts by elmnatng long-only s constrant on short-sellng and the need to converge to securtes ndex weghts n order to control portfolo rsk. Whle long-short offers advantages over long-only, however, t also nvolves complcatons not encountered n long-only management. Many of these complcatons are related to the use of short-sellng. Costs Related to Shortng. To engage n shortsellng, an nvestor must establsh an account wth a prme broker. The broker clears all trades for the longshort portfolo and arranges to borrow stock for shortng. For some shares, especally those of the smallestcaptalzaton companes, borrowablty may be problematc. Even when such shares are avalable for borrowng, they may pose a problem for the short-seller f they are later called back by the stock lender. In that case, the broker may not be able to fnd replacement shares, and the long-short nvestor wll be subject to a buy-n and have to cover the short postons. The fnancal ntermedaton cost of borrowng, whch ncludes the costs assocated wth securng and admnsterng lendable stocks, averages 5 to 30 bass ponts and may be hgher for harder-to-borrow names. Ths cost s ncurred as a harcut on the short rebate receved from the nterest earned on the short sale proceeds. Short-sellers may also ncur tradng opportunty costs because exchange rules delay or prevent short sales. Securtes and Exchange Commsson Rule 10a- 1, for example, states that exchange-traded shares can be shorted only at a prce that s hgher than the last trade prce (an uptck) or the same as the last trade prce f that prce was hgher than the prevous trade (zero-plus-tck). Such tck tests can be crcumvented by the use of prncpal packages (traded outsde U.S. markets) or the sale of call optons, but the costs nvolved may be hgher than the costs exacted by the rules themselves. For a long-short strategy that engages n patent tradng, where the plan s to sell short only after a prce rse, the ncremental mpact of uptck rules wll be mnmal. Tradng Costs. Some other costs of long-short may seem as though they should be hgh relatve to long-only and are often portrayed as such. For example, a long-short portfolo that takes full advantage of the leverage allowed by Federal Reserve Board Regulaton T (two-to-one leverage) wll engage n about twce as much tradng actvty as a comparable unlevered longonly strategy. The dfferental, however, s largely a functon of the portfolo s leverage. Long-short management does not requre leverage. Gven captal of $10 mllon, for example, the nvestor can choose to nvest $5 mllon long and sell $5 mllon short; tradng actvty for the resultng long-short portfolo wll be roughly equvalent to that for a $10 mllon long-only portfolo. 5 Asde from the tradng related to the sheer sze of WITER 1999 THE JOURAL OF PORTFOLIO MAAGEMET 5

4 the nvestment n long-short versus long-only, the mechancs of long-short management may requre some ncremental tradng not encountered n long-only. As securty prces change, for example, long and short postons may have to be adjusted n order to mantan the desred degree of portfolo leverage and to meet collateralzaton requrements (ncludng margn requrements and marks to market on the shorts). When a long-short portfolo s equtzed by a poston n stock ndex futures contracts, the need for such tradng s reduced because prce changes n the long futures postons wll tend to offset marks to market on the short stock postons. (For some examples, see Jacobs [1998].) Management Fees. Management fees for a longshort portfolo may appear to be hgher than those for a comparable long-only portfolo. Agan, the dfferental s largely a reflecton of the degree to whch leverage s used n the former and not n the latter. If one consders management fees per dollar of securtes postons, rather than per dollar of captal, there should not be much dfference between long-short and long-only. Furthermore, nvestors should consder the amount of actve management provded per dollar of fees. As noted, long-only portfolos must be managed wth an eye to the underlyng benchmark, as departures from benchmark weghts ntroduce resdual rsk. In general, long-only portfolos have a szable hdden passve component; only ther overweghts and underweghts relatve to the benchmark are truly actve. By contrast, vrtually the entre long-short portfolo s actve. In terms of management fees per actve dollars, then, long-short may be substantally less costly than long-only. Furthermore, long-short management s almost always offered on a performance-fee bass. Rsk. Long-short s often portrayed as nherently rsker than long-only. In part, ths vew reflects a concern for potentally unlmted losses on short postons. Although t s true that the rsk of a short poston s theoretcally unlmted because there s no bound on a rse n the prce of the shorted securty, ths source of rsk s consderably mtgated n practce. It s unlkely, for example, that the prces of all the securtes sold short wll rse dramatcally at the same tme, wth no offsettng ncreases n the prces of the securtes held long. And the nvestor can guard aganst precptous rses n the prces of ndvdual shorted stocks by holdng small postons n a large number of stocks, both long and short. In general, a long-short portfolo wll ncur more rsk than a long-only portfolo to the extent that t engages n leverage and/or takes more actve postons. A long-short portfolo that takes full advantage of the leverage avalable to t wll have at rsk roughly double the amount of assets nvested n a comparable unlevered long-only strategy. And, because t does not have to converge to securtes benchmark weghts n order to control rsk, a long-short strategy may take larger postons n securtes wth hgher (and lower) expected returns compared wth an ndex-constraned long-only portfolo. But both the portfolo s degree of leverage and ts actveness are wthn the explct control of the nvestor. Furthermore, proper optmzaton should ensure that ncremental rsks, and costs, are compensated by ncremental returns. THE OPTIMAL PORTFOLIO Here we consder what proper optmzaton nvolves, and what the resultng long-short portfolo looks lke. There are some surprses. In partcular, a rgorous look at long-short optmalty calls nto queston the goals of dollar- and beta-neutralty common practces n tradtonal long-short management. We use the utlty functon: 6 U = r P 1 σ P / τ where r P s the expected return of the portfolo over the nvestor s horzon, σ P s the varance of the portfolo s return, and τ s the nvestor s rsk tolerance. Ths utlty functon, favored by Markowtz [195] and Sharpe [1991], provdes a good approxmaton of other, more general, functons and has the agreeable characterstcs of provdng more utlty as expected return ncreases and less utlty as rsk ncreases. Portfolo constructon conssts of two nterrelated tasks: 1) an asset allocaton task for choosng how to allocate the nvestor s wealth between a rsk-free securty and a set of rsky securtes, and ) a rsky portfolo constructon task for choosng how to dstrbute wealth among the rsky securtes. Let h R represent the fracton of wealth that the nvestor specfcally allocates to the rsky portfolo, and let h represent the fracton of wealth nvested n the th rsky securty. There are three components of (1) 6 LOG-SHORT PORTFOLIO MAAGEMET: A ITEGRATED APPROACH WITER 1999

5 captal that earn nterest at the rsk-free rate. The frst s the wealth that the nvestor specfcally allocates to the rsk-free securty, and ths has a magntude of 1 h R. The second s the balance of the depost made wth the broker after payng for the purchase of shares long, and ths has a magntude of h R Σ L h, where L s the set of securtes held long. The thrd s the proceeds of the short sales, and ths has a magntude of Σ S h = Σ S h, where S s the set of securtes sold short. (For smplcty, we assume no harcut on the short rebate.) Summng these three components gves the total amount of captal h F that earns nterest at the rsk-free rate as h A number of nterestng observatons can be made about h F. Frst, note that t s ndependent of h R. Second, observe that, n the case of short-only management n whch = 1h = 1, the quantty h F s equal to two; that s, the nvestor earns the rsk-free rate twce. Thrd, n the case of dollar-balanced long-short management n whch = 1h = 0, the nvestor earns the rsk-free rate only once. Let r F represent the return on the rsk-free securty, and let R represent the expected return on the th rsky securty. The expected return on the nvestor s total portfolo s r = h r + hr Substtutng the expresson derved above for h F nto ths equaton gves the total portfolo return as the sum of a rsk-free return component and a rsky return component, expressed as r P = r F + r R. The rsky return component s r F P F F =1 R = 1 h = hr =1 = 1 (-A) where r = R r F s the expected return on the th rsky securty n excess of the rsk-free rate. The rsky return component can also be expressed n matrx notaton as r R = h T r (-B) where h = [h 1, h,..., h ] T and r = [r 1, r,..., r ] T. It can be shown that the varance of the rsky return component,, s σ R σ R T = h Qh where Q s the covarance matrx of the rsky securtes returns. The varance of the overall portfolo s σp = σr. Wth these expressons, the utlty functon n Equaton (1) can be expressed n terms of controllable varables. We determne the optmal portfolo by maxmzaton of the utlty functon through approprate choce of these varables. Ths maxmzaton s performed subject to the approprate constrants. A mnmal set of approprate constrants conssts of 1) the Regulaton T margn requrement, and ) the requrement that all the wealth allocated to the rsky securtes s fully utlzed. The soluton (provdng Q s non-sngular) gves the optmal rsky portfolo as (3) h = τq 1 r (4) where Q 1 s the nverse of the covarance matrx. We refer to the portfolo n Equaton (4) as the mnmally constraned portfolo. The optmal portfolo weghts depend on predcted statstcal propertes of the securtes. Specfcally, the expected returns and ther covarances must be quanttes that the nvestor expects to be realzed over the portfolo s holdng perod. As no nvestor knows the true statstcal dstrbuton of the returns, expected returns and covarances are lkely to dffer between nvestors. Optmal portfolo holdngs wll thus dffer from nvestor to nvestor, even f all nvestors possess the same utlty functon. The optmal holdngs gven n Equaton (4) have a number of mportant propertes. Frst, they defne a portfolo that permts short postons because no non-negatvty constrants are mposed durng ts constructon. Second, they defne a sngle portfolo that explots the characterstcs of ndvdual securtes n a sngle ntegrated optmzaton. Even though the sngle portfolo can be parttoned artfcally nto one subportfolo of only stocks held long and another subportfolo of only stocks sold short, there s no beneft WITER 1999 THE JOURAL OF PORTFOLIO MAAGEMET 7

6 to dong so. Thrd, the holdngs need not satsfy any arbtrary balance condtons; dollar- or beta-neutralty s not requred. Because optmal portfolo weghts are determned n a sngle ntegrated optmzaton, wthout regard to any ndex or benchmark weghts, the portfolo has no nherent benchmark. Ths means that there exsts no nherent measure of portfolo excess return or resdual rsk; rather, the portfolo wll exhbt an absolute return and an absolute varance of return. Ths return can be calculated as the weghted spread between the returns to the securtes held long and the returns to the securtes sold short. Performance attrbuton cannot dstngush between the contrbutons of the securtes held long and those sold short; the contrbutons of the long and short postons are nextrcably lnked. Separate long and short alphas (and ther correlaton) are meanngless. eutral Portfolos The flexblty afforded by the ablty to short stocks allows nvestors to construct long-short portfolos that are nsenstve to chosen exogenous factors. In practce, for example, most long-short portfolos are desgned to be nsenstve to the return of the equty market. Ths may be accomplshed by constructng the portfolo so that the beta of the short postons equals and offsets the beta of the long postons, or (more problematcally) the dollar amount of securtes sold short equals the dollar amount of securtes held long. 7 Market neutralty, whether acheved through a balance of dollars or betas, may exact costs n terms of forgone utlty. If more opportuntes exst on the short than the long sde of the market, for example, one mght expect some return sacrfce from a portfolo that s requred to hold equal-dollar or equal-beta postons long and short. Market neutralty could be acheved by usng the approprate amount of stock ndex futures, wthout requrng that long and short securty postons be balanced. Investors may nevertheless prefer long-short balances for mental accountng reasons. That s, nvestors may prefer to hold long-short portfolos that have no systematc rsk, wthout requrng seemngly separate management of dervatves overlays. Even f separate managers are used for long-short and for dervatves, however, there s no necessty for longshort balance; the dervatves manager can be nstructed to augment or offset the long-short portfolo s market exposure. Imposng the condton that the portfolo be nsenstve to the equty market return (or to any other factor) consttutes an addtonal constrant on the portfolo. The optmal neutral portfolo s the one that maxmzes the nvestor s utlty subject to all constrants, ncludng that of neutralty. Ths optmal neutral portfolo need not be, and generally s not, the same as the portfolo gven by Equaton (4) that maxmzes the mnmally constraned utlty functon. To the extent that the optmal neutral portfolo dffers from the mnmally constraned optmal portfolo, t wll nvolve a sacrfce n nvestor utlty. In fact, a neutral long-short portfolo wll maxmze the nvestor s mnmally constraned utlty functon only under the very lmted condtons dscussed below. Dollar-eutral Portfolos. We consder frst the condtons under whch a dollar-neutral portfolo maxmzes the mnmally constraned utlty functon. By defnton, the rsky portfolo s dollar-neutral f the net holdng H of rsky securtes s zero, meanng that h =1 H= = 0 Ths condton s ndependent of h R, the fracton of wealth held n the rsky portfolo. Applyng the condton gven n Equaton (5) to the optmal weghts from Equaton (4), together wth a smplfyng assumpton regardng the covarance matrx, t can be shown that the dollar-neutral portfolo s equal to the mnmally constraned optmal portfolo when: 8 r H ( ξ ξ) = 0 σ = 1 where σ s the standard devaton of the return of stock, ξ = 1/σ s a measure of the stablty of the return of stock, and ξ s the average return stablty of all stocks n the nvestor s unverse. The term r /σ s a rskadjusted return, and the term ξ ξ can be regarded as an excess stablty, or a stablty weghtng. Hghly volatle stocks wll have low stabltes, so ther excess stabltes wll be negatve. Conversely, low-volatlty (5) (6) 8 LOG-SHORT PORTFOLIO MAAGEMET: A ITEGRATED APPROACH WITER 1999

7 stocks wll have hgh stabltes, so ther excess stabltes wll be postve. The condton shown n Equaton (6) states that the optmal net holdng of rsky securtes s proportonal to the unverse s net stablty-weghted rskadjusted expected return. If ths quantty s postve, the net holdng should be long; f t s negatve, the net holdng should be short. The optmal rsky portfolo wll be dollar-neutral only under the relatvely unlkely condton that ths quantty s zero. Beta-eutral Portfolos. We next consder the condtons under whch a beta-neutral portfolo maxmzes the mnmally constraned utlty functon. Once the nvestor has chosen a benchmark, each securty can be modeled n terms of ts expected excess return α and ts beta β wth respect to that benchmark. Specfcally, f r B s the expected return of the benchmark, then the expected return of the th securty s r = α + β r B (7) The expected return of the portfolo can be modeled n terms of ts expected excess return α P and beta β P wth respect to the benchmark r P = α P + β P r B (8) where the beta of the portfolo s expressed as a lnear combnaton of the betas of the ndvdual securtes, as follows: β P = h β =1 From Equaton (8), t s clear that any portfolo that s nsenstve to changes n the expected benchmark return must satsfy the condton (9) β P = 0 (10) Applyng the condton gven n Equaton (10) to the optmal weghts from Equaton (4), together wth the model gven n Equaton (7), t can be shown that the beta-neutral portfolo s equal to the optmal mnmally constraned portfolo when: β r = 0 = 1ω (11) where s the varance of the excess return of securty. Equaton (11) descrbes the condton that a unverse of securtes must satsfy n order for an optmal portfolo constructed from that unverse to be unaffected by the return of the chosen benchmark. The summaton n Equaton (11) can be regarded as the portfolo s net beta-weghted rsk-adjusted expected return. Only under the relatvely unlkely condton that ths quantty s zero wll the optmal portfolo be beta-neutral. ω Optmal Equtzaton Usng varous benchmark return vectors, one can construct an orthogonal bass for a portfolo s returns. 9 The portfolo can then be characterzed as a sum of components along (or exposures to) the orthogonal bass vectors. Consder a two-dmensonal decomposton. The expected return of the chosen benchmark can be used as the frst bass vector and an orthogonalzed cash return as the second. The expected return of a beta-neutral portfolo s ndependent of the returns of the chosen benchmark. That s, ts returns are orthogonal to the returns of the benchmark, and can therefore be treated as beng equvalent to an orthogonalzed cash component. In ths sense, the beta-neutral portfolo appears to belong to a completely dfferent asset class from the benchmark. It can be transported to the benchmark asset class by usng a dervatves overlay, however. A long-short portfolo can be constructed to be close to orthogonal to a benchmark from any asset class, and can be transported to any other asset class by use of approprate dervatves overlays. But because long-short portfolos comprse exstng underlyng securtes, they nhabt the same vector space as exstng asset classes; they do not consttute a separate asset class n the sense of addng a new dmenson to the exstng asset class vector space. Some practtoners nevertheless treat long-short portfolos as though they represent a separate asset class. They do ths, for example, when they combne an optmal neutral long-short portfolo wth a separately optmzed long-only portfolo so as to optmze return and rsk relatve to a chosen benchmark. The long-only portfolo s n effect used as a surrogate benchmark to transport the neutral long-short portfolo toward the desred rsk and return profle. Although unlkely, t s possble that the resultng WITER 1999 THE JOURAL OF PORTFOLIO MAAGEMET 9

8 combned portfolo can optmze the nvestor s orgnal utlty functon. It can do so, however, only f the portfolo h that maxmzes that utlty can be constructed from a lnear combnaton of the long-only portfolo and the neutral long-short portfolo. Specfcally, f h LO represents the holdngs of the long-only portfolo and h LS those of the neutral longshort portfolo, the combned portfolo can be optmal f h belongs to the range of the transformaton nduced by vectors h LO and h LS ; that s, f h R[h LO h LS ] (1) In general, however, there s nothng forcng the three portfolos to satsfy such a condton. How, then, should one combne ndvdual securtes and a benchmark securty to arrve at an optmal portfolo? The answer s straghtforward: One ncludes the benchmark securty explctly n the formulaton of the nvestor s utlty functon and performs a sngle ntegrated optmzaton to obtan the optmal ndvdual securty and benchmark securty holdngs smultaneously. Consder the problem of maxmzng a longshort portfolo s return wth respect to a benchmark whle smultaneously controllng for resdual rsk. The varables that can be controlled n ths problem are h and the benchmark holdng denoted by h B. We make the smplfyng assumpton that benchmark holdngs consume no captal. Ths s approxmately true for benchmark dervatves, such as futures and swaps. The portfolo s expected excess return s thus E F =1 B B B r = r + hr + h r r (13) portfolo to the varance of that return. Clearly, as the expected excess return to the MRR portfolo ncreases, or the varance of that return decreases, the rato m ncreases, and a larger proporton of the rsky portfolo should be assgned to the MRR portfolo. Conversely, as m decreases, more of the rsky portfolo should be assgned to the standard portfolo φ. As the nvestor s rsk tolerance ncreases, the amount of wealth assgned to the rsky portfolo ncreases. The exposure to the benchmark that maxmzes the nvestor s utlty s h B = 1 mτ Ths exposure decreases as the MRR portfolo becomes more attractve and as the nvestor s rsk tolerance ncreases. The exposure may be negatve, under whch condton the nvestor sells the benchmark securty short. Conversely, as the nvestor s rsk tolerance or the attractveness of the MRR portfolo decreases, the benchmark exposure should ncrease. In the lmt, as ether m or τ tends toward zero, the optmal benchmark exposure reaches 100% of the nvested wealth. An optmally equtzed portfolo, however, wll generally not nclude a full exposure to the benchmark securty. In the lmt, as m approaches zero (and h B approaches one), the rsky portfolo h becomes proportonal to the standard portfolo; for ths rsky portfolo to be optmally beta- or dollar-neutral, the same condtons must be satsfed as those gven n Equatons (6) and (11) for the unequtzed portfolo defned by Equaton (4). The rsky part of the equtzed portfolo s optmally dollar-neutral when It can be shown (see Jacobs, Levy, and Starer [1998]) that the optmal rsky portfolo h n ths case s: r + mq ( ξ ξ) σ = 1 = 0 (14) h =(φ + mψ)τ where φ = Q 1 r s the standard portfolo that would be chosen by an dealzed nvestor wth unt rsk tolerance who optmzes Equaton (1) wthout any constrants; ψ = Q 1 q s a mnmum-resdual rsk (MRR) portfolo; q = cov(r, r B ) s a vector of covarances between the rsky securtes returns and the benchmark return; and m s the rato of the expected excess return of the MRR The term on the left-hand sde of Equaton (14) can be nterpreted as a net stablty-weghted rsk-adjusted expected return. The rsky part of the optmally equtzed portfolo should be net long f ths quantty s postve and net short f t s negatve. Ths s analogous to the condton gven n Equaton (6) for an unequtzed long-short portfolo. The equtzed case ncludes an addtonal term, mq, that captures the attractveness of the MRR portfolo and the correla- 30 LOG-SHORT PORTFOLIO MAAGEMET: A ITEGRATED APPROACH WITER 1999

9 tons between the rsky securtes and the benchmark s returns. Smlarly, the rsky part of the optmally equtzed portfolo s beta-neutral when β = 1ω Ths s analogous to the condton gven n Equaton (11) for an unequtzed portfolo. Agan, the condton for the equtzed portfolo to be beta-neutral ncludes the addtonal term mq. COCLUSIO ( r + mq ) = 0 The freedom to sell stocks short allows the nvestor to beneft from stocks wth negatve expected returns as well as from those wth postve expected returns. The advantages of combnng long and short portfolo postons, however, depend crtcally on the way the portfolo s constructed. Tradtonally, longshort portfolos have been run as two-portfolo strateges, where a short-only portfolo s added to a longonly portfolo. Ths s suboptmal compared wth an ntegrated, sngle-portfolo approach that consders the expected returns, rsks, and correlatons of all securtes smultaneously. Such an approach maxmzes the nvestor s ablty to trade off rsk and return for the best possble performance. Also generally suboptmal are constructon approaches that constran the short and long postons of the portfolo to be dollar- or beta-neutral. Only under very lmted condtons wll such a constraned portfolo provde the same utlty as an unconstraned portfolo. In general, rather than usng long-short balance to acheve a desred exposure (ncludng no exposure at all) to a partcular benchmark, nvestors wll be better off consderng benchmark exposure as an explct element of ther utlty functons. Long-short management s often perceved as substantally rsker or costler than long-only management. Much of any ncremental cost or rsk, however, reflects ether the long-short portfolo s degree of leverage or ts degree of actveness ; both of these parameters are under the explct control of the nvestor. Addtonally, proper optmzaton ensures that expected returns compensate the nvestor for rsks ncurred. Gven the added flexblty that a long-short portfolo affords the nvestor, t can be expected to perform better than a long-only portfolo based on the same set of nsghts. EDOTES The authors thank Clarence C.Y. Kwan for helpful comments, and Judth Kmball for edtoral assstance. 1 As the medan-captalzaton stock n the Russell 3000 ndex has a weghtng of 0.01%. The ablty to short wll be partcularly valuable n a market n whch short-sellng s restrcted and nvestment opnon dverse. When nvestors hold dverse opnons, some wll be more pessmstc than others. Wth short-sellng restrcted, however, ths pessmsm wll not be fully reflected n securty prces. In such a world, there are lkely to be more proftable opportuntes for sellng overprced stocks short than there are proftable opportuntes for purchasng underprced stock. See Mller [1977]. 3 Ths assumes symmetry of neffcences across attractve and unattractve stocks. It also assumes that portfolo constructon proceeds dentcally and separately for the long and short sdes as t does n long-only portfolo constructon. Although these assumptons may appear unduly restrctve, they have often been nvoked. See Jacobs, Levy, and Starer [1998] for a dscusson of ths lterature and our counterponts. 4 In dervng the formula, t s assumed that the beta of the short sde equals the beta of the long sde. 5 Furthermore, under Regulaton T, a long-only portfolo can engage n leverage to the same extent as a long-short portfolo. Long-short has an advantage here, however, because purchasng stock on margn can gve rse to a tax lablty for taxexempt nvestors. Accordng to Internal Revenue Servce Rulng 95-8, borrowng shares to ntate short sales does not consttute debt fnancng, so any profts realzed when short postons are closed out do not gve rse to unrelated busness taxable ncome. 6 For analytcal tractablty and expostonal smplcty, we use the tradtonal mean-varance utlty functon, although t s only a sngle-perod formulaton and s not senstve to nvestor wealth. Also, behavoral research may queston the use of an analytc utlty functon n the presence of apparently rratonal nvestor behavor. evertheless, we beleve our conclusons hold for more elaborate descrptons of nvestor behavor. 7 A dollar balance may appear to provde tangble proof of the market neutralty of the portfolo. But unless a dollar-balanced portfolo s also beta-balanced, t s not market-neutral. 8 The smplfyng assumpton appled s the constant-correlaton model of Elton, Gruber, and Padberg [1976]. 9 One could, for example, use the Gram-Schmdt procedure (see Strang [1988]). REFERECES Elton, Edwn J., Martn J. Gruber, and Manfred W. Padberg. Smple Crtera for Optmal Portfolo Selecton. Journal of Fnance, December 1976, pp WITER 1999 THE JOURAL OF PORTFOLIO MAAGEMET 31

10 Jacobs, Bruce I. Controlled Rsk Strateges. In ICFA Contnung Educaton: Alternatve Assets. Charlottesvlle, VA: Assocaton for Investment Management and Research, Jacobs, Bruce I., Kenneth. Levy, and Davd Starer. On the Optmalty of Long-Short Strateges. Fnancal Analysts Journal, March/Aprl Markowtz, Harry. Portfolo Selecton. Journal of Fnance, March 195, pp Mller, Edward M. Rsk, Uncertanty, and Dvergence of Opnon. Journal of Fnance, September 1977, pp Sharpe, Wllam F. Captal Asset Prces wth and wthout egatve Holdngs. Journal of Fnance, June 1991, pp Strang, Glbert. Lnear Algebra and Its Applcatons, 3rd ed. ew York: Harcourt Brace Jovanovch, To order reprnts of ths artcle, please contact Ajan Malk at or Reprnted wth permsson from the Wnter 1999 of The Journal of Portfolo Management. Copyrght 1999 by Insttutonal Investor Journals, Inc. All rghts reserved. For more nformaton call (1) Vst our webste at 3 LOG-SHORT PORTFOLIO MAAGEMET: A ITEGRATED APPROACH WITER 1999