Long-Short Portfolio Management: an Integrated Approach

Long-Short Portfolio Management: An Integrated Approach

The real benefits of long-short are released only by an integrated portfolio optimization.

Bruce I. Jacobs, Kenneth N. Levy, and David Starer

ost investors focus on the management of long portfolios and the selection of “winning” securities. Yet the identifica- tion of winning securities ignores by definitionM a whole class of “losing” securities. The ability to sell short frees the investor to take advantage of the full array of securities and the full complement of investment insights by holding expected winners long and selling expected losers short. A long-short portfolio, by expanding the scope of the investor’s sphere of activity, can be expected to result in improved performance from active security selection vis-à-vis a long-only portfolio. But the benefits of long-short are to a large extent dependent on proper portfolio construction. Only an integrated optimization of long and short positions has the potential to maximize the value of investors’ insights. The benefits that emerge from integrated optimization encompass not only freedom from the short-selling constraint but also freedom from the restrictions imposed by individual securities’ benchmark weights. Of course, these benefits do not come without some cost. Much of the incremental cost associated with a given long-short portfolio reflects the strategy’s BRUCE I. JACOBS and degree of leverage. Nevertheless, as we will see, long- KENNETH N. LEVY are princi- short is not necessarily much costlier or, indeed, much pals and DAVID STARER is a riskier than long-only. senior quantitative analyst at Jacobs Although most existing long-short portfolios are Levy Equity Management in constructed to be neutral to systematic risk, we will see Roseland (NJ 07068) that neutrality is neither necessary nor, in most cases,

WINTER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 23 optimal. Furthermore, we show that long-short portfo- of the long-plus-short portfolio will equal the alpha of lios do not constitute a separate asset class; they can, the short side, which will equal the alpha of the long- however, be constructed to include a desired exposure to only portfolio. Furthermore, the residual risk of the the return (and risk) of virtually any existing asset class. long side of the long-plus-short portfolio, ω, will equal the residual risk of the short side, which will equal the LONG-SHORT: BENEFITS AND COSTS residual risk of the long-only portfolio. These equivalencies reflect the fact that all the Consider a long-only investor who has an portfolios, the long-only portfolio and the long and extremely negative view about a typical stock. The short components of the long-plus-short portfolio, are investor’s ability to benefit from this insight is very lim- constructed relative to a benchmark index. Each port- ited. The most the investor can do is exclude the stock folio is active in pursuing excess return relative to the from the portfolio, in which case the portfolio will have underlying index only insofar as it holds securities in about a 0.01% underweight in the stock, relative to the weights that depart from their index weights. The abil- underlying market.1 Those who do not consider this to ity to pursue such excess returns may be limited by the be a material constraint should consider what its effect need to control the portfolio’s residual risk by main- would be on the investor’s ability to overweight a typi- taining portfolio weights that are close to index cal stock. It would mean the investor could hold no weights. Portfolio construction is index-constrained. more than a 0.02% long position in the stock — a Consider, for example, an investor who does not 0.01% overweight — no matter how attractive its have the ability to discriminate between good and bad expected return. oil stocks, or who believes that no oil stock will signif- The ability to short, by increasing the investor’s icantly outperform or underperform the underlying leeway to act on insights, has the potential to enhance benchmark in the near future. In long-plus-short, this returns from active security selection.2 The scope of the investor may have to hold some oil stocks in the long improvement, however, will depend critically on the portfolio and short some oil stocks in the short portfo- way the long-short portfolio is constructed. In particu- lio, if only to control each portfolio’s residual risk. lar, an integrated optimization that considers both long The ratio of the performance of the long-plus- and short positions simultaneously not only frees the short portfolio to that of the long-only portfolio can be investor from the non-negativity constraint imposed on expressed as follows:4 long-only portfolios, but also frees the long-short portfolio from the restrictions imposed by securities’ IR 2 benchmark weights. To see this, it is useful to examine LS+ = one obvious (if suboptimal) way of constructing a long- IRLO1 + ρ L+ S short portfolio. Long-short portfolios are sometimes constructed where IR is the information ratio, or the ratio of excess by combining a long-only portfolio, perhaps a preexist- return to residual risk, α/ω, and ρL+S is the correlation ing one, with a short-only portfolio. This results in a between the alphas of the long and short sides of the long-plus-short portfolio, not a true long-short portfo- long-plus-short portfolio. lio. The long side of this portfolio is identical to a long- In long-plus-short, the advantage offered by the only portfolio; hence it offers no benefits in terms of flexibility to short is curtailed by the need to control incremental return or reduced risk. risk by holding or shorting securities in index-like In long-plus-short, the short side is statistically weights. A long-plus-short portfolio thus offers a ben- equivalent to the long side, hence to the long-only efit over a long-only portfolio only if there is a less- portfolio.3 In effect: than-one correlation between the alphas of its long and short sides. In that case, the long-plus-short portfolio

αL = αS = αLO will enjoy greater diversification and reduced risk relative to a long-only portfolio. A long-only portfolio can

ωL = ωS = ωLO derive a similar benefit by adding a less than fully cor- related asset with comparable risk and return, however, That is, the excess return or alpha, α, of the long side so this is not a benefit unique to long-short.

24 LONG-SHORT PORTFOLIO MANAGEMENT: AN INTEGRATED APPROACH WINTER 1999 The Real Benefits of Long-Short Costs Related to Shorting. To engage in short- selling, an investor must establish an account with a The real benefits of long-short emerge only prime broker. The broker clears all trades for the long- when the portfolio is conceived of and constructed as a short portfolio and arranges to borrow stock for short- single, integrated portfolio of long and short positions. ing. For some shares, especially those of the smallest- In this framework, long-short is not a two-portfolio capitalization companies, borrowability may be prob- strategy. It is a one-portfolio strategy in which the long lematic. Even when such shares are available for bor- and short positions are determined jointly within an rowing, they may pose a problem for the short-seller if optimization that takes into account the expected they are later called back by the stock lender. In that returns of the individual securities, the standard devia- case, the broker may not be able to find replacement tions of those returns, and the correlations between shares, and the long-short investor will be subject to a them, as well as the investor’s tolerance for risk. “buy-in” and have to cover the short positions. Within an integrated optimization, there is no The financial intermediation cost of borrow- need to converge to securities’ benchmark weights in ing, which includes the costs associated with securing order to control risk. Rather, offsetting long and and administering lendable stocks, averages 25 to 30 short positions can be used to control portfolio risk. basis points and may be higher for harder-to-borrow This allows the investor greater flexibility to take names. This cost is incurred as a “haircut” on the active positions. short rebate received from the interest earned on the Suppose, for example, that an investor’s short sale proceeds. strongest insights are about oil stocks, some of which Short-sellers may also incur trading opportuni- are expected to do especially well and some especially ty costs because exchange rules delay or prevent short poorly. The investor does not have to restrict the port- sales. Securities and Exchange Commission Rule 10a- folio’s weightings of oil stocks to index-like weights in 1, for example, states that exchange-traded shares can order to control the portfolio’s exposure to oil sector be shorted only at a price that is higher than the last risk. The investor can allocate much of the portfolio to trade price (an uptick) or the same as the last trade oil stocks, held long and sold short. The offsetting long price if that price was higher than the previous trade and short positions control the portfolio’s exposure to (zero-plus-tick). the oil factor. Such tick tests can be circumvented by the use Conversely, suppose the investor has no insights of “principal packages” (traded outside U.S. markets) into oil stock behavior. Unlike the long-only and long- or the sale of call options, but the costs involved may be plus-short investors discussed above, the long-short higher than the costs exacted by the rules themselves. investor can totally exclude oil stocks from the portfolio. For a long-short strategy that engages in patient trad- The exclusion of oil stocks does not increase portfolio ing, where the plan is to sell short only after a price rise, risk, because the long-short portfolio’s risk is indepen- the incremental impact of uptick rules will be minimal. dent of any security’s benchmark weight. The flexibility Trading Costs. Some other costs of long-short afforded by the absence of the restrictions imposed by may seem as though they should be high relative to securities’ benchmark weights enhances the long-short long-only and are often portrayed as such. For example, investor’s ability to implement investment insights. a long-short portfolio that takes full advantage of the Costs: Perception versus Reality leverage allowed by Federal Reserve Board Regulation T (two-to-one leverage) will engage in about twice as Long-short construction maximizes the benefit much trading activity as a comparable unlevered long- obtained from potentially valuable investment insights only strategy. The differential, however, is largely a func- by eliminating long-only’s constraint on short-selling tion of the portfolio’s leverage. Long-short management and the need to converge to securities’ index weights in does not require leverage. Given capital of $10 million, order to control portfolio risk. While long-short offers for example, the investor can choose to invest $5 million advantages over long-only, however, it also involves long and sell $5 million short; trading activity for the complications not encountered in long-only manage- resulting long-short portfolio will be roughly equivalent ment. Many of these complications are related to the to that for a $10 million long-only portfolio.5 use of short-selling. Aside from the trading related to the sheer size of

WINTER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 25 the investment in long-short versus long-only, the more risk than a long-only portfolio to the extent that mechanics of long-short management may require some it engages in leverage and/or takes more active posi- incremental trading not encountered in long-only. As tions. A long-short portfolio that takes full advantage of security prices change, for example, long and short posi- the leverage available to it will have at risk roughly dou- tions may have to be adjusted in order to maintain the ble the amount of assets invested in a comparable desired degree of portfolio leverage and to meet collater- unlevered long-only strategy. And, because it does not alization requirements (including margin requirements have to converge to securities’ benchmark weights in and marks to market on the shorts). When a long-short order to control risk, a long-short strategy may take portfolio is equitized by a position in stock index futures larger positions in securities with higher (and lower) contracts, the need for such trading is reduced because expected returns compared with an index-constrained price changes in the long futures positions will tend to long-only portfolio. offset marks to market on the short stock positions. (For But both the portfolio’s degree of leverage and some examples, see Jacobs [1998].) its “activeness” are within the explicit control of the Management Fees. Management fees for a long- investor. Furthermore, proper optimization should short portfolio may appear to be higher than those for ensure that incremental risks, and costs, are compensat- a comparable long-only portfolio. Again, the differen- ed by incremental returns. tial is largely a reflection of the degree to which leverage is used in the former and not in the latter. If one THE OPTIMAL PORTFOLIO considers management fees per dollar of securities positions, rather than per dollar of capital, there should not Here we consider what proper optimization be much difference between long-short and long-only. involves, and what the resulting long-short portfolio Furthermore, investors should consider the looks like. There are some surprises. In particular, a rig- amount of active management provided per dollar of orous look at long-short optimality calls into question fees. As noted, long-only portfolios must be managed the goals of dollar- and beta-neutrality — common with an eye to the underlying benchmark, as departures practices in traditional long-short management. from benchmark weights introduce residual risk. In We use the utility function:6 general, long-only portfolios have a sizable “hidden passive” component; only their overweights and under- 1 2 weights relative to the benchmark are truly active. By Ur=−PPστ/ (1) contrast, virtually the entire long-short portfolio is 2 active. In terms of management fees per active dollars, then, long-short may be substantially less costly than where rP is the expected return of the portfolio over the 2 long-only. Furthermore, long-short management is investor’s horizon, σP is the variance of the portfolio’s almost always offered on a performance-fee basis. return, and τ is the investor’s risk tolerance. This utili- Risk. Long-short is often portrayed as inherent- ty function, favored by Markowitz [1952] and Sharpe ly riskier than long-only. In part, this view reflects a [1991], provides a good approximation of other, more concern for potentially unlimited losses on short posi- general, functions and has the agreeable characteristics tions. Although it is true that the risk of a short posi- of providing more utility as expected return increases tion is theoretically unlimited because there is no and less utility as risk increases. bound on a rise in the price of the shorted security, this Portfolio construction consists of two interrelat- source of risk is considerably mitigated in practice. It is ed tasks: 1) an asset allocation task for choosing how to unlikely, for example, that the prices of all the securi- allocate the investor’s wealth between a risk-free secu- ties sold short will rise dramatically at the same time, rity and a set of N risky securities, and 2) a risky port- with no offsetting increases in the prices of the securi- folio construction task for choosing how to distribute ties held long. And the investor can guard against pre- wealth among the N risky securities. cipitous rises in the prices of individual shorted stocks Let hR represent the fraction of wealth that the by holding small positions in a large number of stocks, investor specifically allocates to the risky portfolio, both long and short. and let hi represent the fraction of wealth invested in In general, a long-short portfolio will incur the ith risky security. There are three components of

26 LONG-SHORT PORTFOLIO MANAGEMENT: AN INTEGRATED APPROACH WINTER 1999 T capital that earn interest at the risk-free rate. The first rR = h r (2-B) is the wealth that the investor specifically allocates to T T the risk-free security, and this has a magnitude of 1 – where h = [h1, h2, ..., hN] and r = [r1, r2, ..., rN] . It hR. The second is the balance of the deposit made can be shown that the variance of the risky return com- 2 with the broker after paying for the purchase of shares ponent, σR , is long, and this has a magnitude of hR – Σi∈Lhi, where L is the set of securities held long. The third is the 2 T proceeds of the short sales, and this has a magnitude of σR = hQh (3)

Σi∈S|hi| = –Σi∈Shi, where S is the set of securities sold short. (For simplicity, we assume no “haircut” on the where Q is the covariance matrix of the risky securi- short rebate.) ties’ returns. The variance of the overall portfolio is 22 Summing these three components gives the σσPR= . total amount of capital hF that earns interest at the With these expressions, the utility function in risk-free rate as Equation (1) can be expressed in terms of controllable variables. We determine the optimal portfolio by max- N imization of the utility function through appropriate hhFi=−1 ∑ choice of these variables. This maximization is per- i=1 formed subject to the appropriate constraints. A minimal set of appropriate constraints consists of 1) the A number of interesting observations can be made Regulation T margin requirement, and 2) the require- about hF. First, note that it is independent of hR. ment that all the wealth allocated to the risky securities Second, observe that, in the case of short-only man- is fully utilized. The solution (providing Q is non-sin- N agement in which ∑i=1hi =−1 , the quantity hF is equal gular) gives the optimal risky portfolio as to two; that is, the investor earns the risk-free rate twice. Third, in the case of dollar-balanced long-short h = τQ–1r (4) N management in which ∑i=1hi = 0, the investor earns the risk-free rate only once. where Q–1 is the inverse of the covariance matrix. We Let rF represent the return on the risk-free secu- refer to the portfolio in Equation (4) as the minimally th rity, and let Ri represent the expected return on the i constrained portfolio. risky security. The expected return on the investor’s The optimal portfolio weights depend on pre- total portfolio is dicted statistical properties of the securities. Specifically, the expected returns and their covariances must be

N quantities that the investor expects to be realized over rhrhRPFFii=+∑ the portfolio’s holding period. As no investor knows i=1 the true statistical distribution of the returns, expected returns and covariances are likely to differ between Substituting the expression derived above for hF into investors. Optimal portfolio holdings will thus differ this equation gives the total portfolio return as the sum from investor to investor, even if all investors possess the of a risk-free return component and a risky return same utility function. component, expressed as rP = rF + rR. The optimal holdings given in Equation (4) The risky return component is have a number of important properties. First, they define a portfolio that permits short positions because N no non-negativity constraints are imposed during its rhrRii= ∑ (2-A) construction. Second, they define a single portfolio that i=1 exploits the characteristics of individual securities in a single integrated optimization. Even though the single th where ri = Ri – rF is the expected return on the i risky portfolio can be partitioned artificially into one sub- security in excess of the risk-free rate. The risky return portfolio of only stocks held long and another sub- component can also be expressed in matrix notation as portfolio of only stocks sold short, there is no benefit

WINTER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 27 to doing so. Third, the holdings need not satisfy any short balance; the derivatives manager can be instruct- arbitrary balance conditions; dollar- or beta-neutrality ed to augment or offset the long-short portfolio’s mar- is not required. ket exposure. Because optimal portfolio weights are deter- Imposing the condition that the portfolio be mined in a single integrated optimization, without insensitive to the equity market return (or to any other regard to any index or benchmark weights, the portfo- factor) constitutes an additional constraint on the port- lio has no inherent benchmark. This means that there folio. The optimal neutral portfolio is the one that exists no inherent measure of portfolio excess return or maximizes the investor’s utility subject to all constraints, residual risk; rather, the portfolio will exhibit an abso- including that of neutrality. lute return and an absolute variance of return. This This optimal neutral portfolio need not be, return can be calculated as the weighted spread and generally is not, the same as the portfolio given between the returns to the securities held long and the by Equation (4) that maximizes the minimally con- returns to the securities sold short. strained utility function. To the extent that the opti- Performance attribution cannot distinguish mal neutral portfolio differs from the minimally con- between the contributions of the securities held long strained optimal portfolio, it will involve a sacrifice in and those sold short; the contributions of the long investor utility. In fact, a neutral long-short portfolio and short positions are inextricably linked. Separate will maximize the investor’s minimally constrained long and short alphas (and their correlation) are utility function only under the very limited condi- meaningless. tions discussed below. Dollar-Neutral Portfolios. We consider first the Neutral Portfolios conditions under which a dollar-neutral portfolio max- The flexibility afforded by the ability to short imizes the minimally constrained utility function. By stocks allows investors to construct long-short portfo- definition, the risky portfolio is dollar-neutral if the net lios that are insensitive to chosen exogenous factors. holding H of risky securities is zero, meaning that In practice, for example, most long-short portfolios are designed to be insensitive to the return of the N equity market. This may be accomplished by con- Hh= ∑ i = 0 (5) structing the portfolio so that the beta of the short i=1 positions equals and offsets the beta of the long positions, or (more problematically) the dollar amount of This condition is independent of hR, the frac- securities sold short equals the dollar amount of secu- tion of wealth held in the risky portfolio. Applying the rities held long.7 condition given in Equation (5) to the optimal weights Market neutrality, whether achieved through a from Equation (4), together with a simplifying assump- balance of dollars or betas, may exact costs in terms of tion regarding the covariance matrix, it can be shown forgone utility. If more opportunities exist on the short that the dollar-neutral portfolio is equal to the mini- 8 than the long side of the market, for example, one mally constrained optimal portfolio when: might expect some return sacrifice from a portfolio N that is required to hold equal-dollar or equal-beta posi- ri tions long and short. Market neutrality could be H ∝−∑()ξξi = 0 (6) i=1 σ achieved by using the appropriate amount of stock i index futures, without requiring that long and short security positions be balanced. where σi is the standard deviation of the return of stock Investors may nevertheless prefer long-short bal- i, ξi = 1/σi is a measure of the stability of the return of ances for “mental accounting” reasons. That is, stock i, and ξ is the average return stability of all stocks investors may prefer to hold long-short portfolios that in the investor’s universe. The term ri/σi is a risk- have no systematic risk, without requiring seemingly adjusted return, and the term ξi – ξ can be regarded as separate management of derivatives overlays. Even if an excess stability, or a stability weighting. Highly separate managers are used for long-short and for volatile stocks will have low stabilities, so their excess derivatives, however, there is no necessity for long- stabilities will be negative. Conversely, low-volatility

28 LONG-SHORT PORTFOLIO MANAGEMENT: AN INTEGRATED APPROACH WINTER 1999 2 stocks will have high stabilities, so their excess stabilities where ωi is the variance of the excess return of will be positive. security i. The condition shown in Equation (6) states that Equation (11) describes the condition that a the optimal net holding of risky securities is propor- universe of securities must satisfy in order for an opti- tional to the universe’s net stability-weighted risk- mal portfolio constructed from that universe to be adjusted expected return. If this quantity is positive, the unaffected by the return of the chosen benchmark. net holding should be long; if it is negative, the net The summation in Equation (11) can be regarded as holding should be short. The optimal risky portfolio the portfolio’s net beta-weighted risk-adjusted expect- will be dollar-neutral only under the relatively unlikely ed return. Only under the relatively unlikely condition condition that this quantity is zero. that this quantity is zero will the optimal portfolio be Beta-Neutral Portfolios. We next consider the beta-neutral. conditions under which a beta-neutral portfolio maxi- Optimal Equitization mizes the minimally constrained utility function. Once the investor has chosen a benchmark, each security can Using various benchmark return vectors, one can construct an orthogonal basis for a portfolio’s be modeled in terms of its expected excess return αi returns.9 The portfolio can then be characterized as a and its beta βi with respect to that benchmark. sum of components along (or exposures to) the orthog- Specifically, if rB is the expected return of the benchmark, then the expected return of the ith security is onal basis vectors. Consider a two-dimensional decomposition. The expected return of the chosen benchmark can be used as ri = αi + βirB (7) the first basis vector and an orthogonalized cash return as The expected return of the portfolio can be the second. The expected return of a beta-neutral portfolio is independent of the returns of the chosen bench- modeled in terms of its expected excess return αP and mark. That is, its returns are orthogonal to the returns of beta βP with respect to the benchmark the benchmark, and can therefore be treated as being

rP = αP + βPrB (8) equivalent to an orthogonalized cash component. In this sense, the beta-neutral portfolio appears to belong to a where the beta of the portfolio is expressed as a linear completely different asset class from the benchmark. It combination of the betas of the individual securities, can be “transported” to the benchmark asset class by as follows: using a derivatives overlay, however. A long-short portfolio can be constructed to be N close to orthogonal to a benchmark from any asset class, ββPi= ∑h i (9) and can be transported to any other asset class by use of i=1 appropriate derivatives overlays. But because long-short portfolios comprise existing underlying securities, they From Equation (8), it is clear that any portfolio inhabit the same vector space as existing asset classes; that is insensitive to changes in the expected bench- they do not constitute a separate asset class in the sense mark return must satisfy the condition of adding a new dimension to the existing asset class vector space. β = 0 (10) P Some practitioners nevertheless treat long-short Applying the condition given in Equation (10) to portfolios as though they represent a separate asset class. the optimal weights from Equation (4), together with the They do this, for example, when they combine an opti- model given in Equation (7), it can be shown that the mal neutral long-short portfolio with a separately opti- beta-neutral portfolio is equal to the optimal minimally mized long-only portfolio so as to optimize return and constrained portfolio when: risk relative to a chosen benchmark. The long-only portfolio is in effect used as a surrogate benchmark to transport the neutral long-short portfolio toward the N β r ii = 0 (11) desired risk and return profile. ∑ 2 i=1 ωi Although unlikely, it is possible that the resulting

WINTER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 29 combined portfolio can optimize the investor’s original portfolio to the variance of that return. utility function. It can do so, however, only if the port- Clearly, as the expected excess return to the folio h that maximizes that utility can be constructed MRR portfolio increases, or the variance of that from a linear combination of the long-only portfolio return decreases, the ratio m increases, and a larger and the neutral long-short portfolio. proportion of the risky portfolio should be assigned

Specifically, if hLO represents the holdings of the to the MRR portfolio. Conversely, as m decreases, long-only portfolio and hNLS those of the neutral long- more of the risky portfolio should be assigned to the short portfolio, the combined portfolio can be optimal standard portfolio φ. As the investor’s risk tolerance if h belongs to the range of the transformation induced increases, the amount of wealth assigned to the risky by vectors hLO and hNLS; that is, if portfolio increases. The exposure to the benchmark that maximizes

h ∈ R[hLO hNLS] (12) the investor’s utility is

In general, however, there is nothing forcing the three hB = 1 – mτ portfolios to satisfy such a condition. How, then, should one combine individual This exposure decreases as the MRR portfolio securities and a benchmark security to arrive at an opti- becomes more attractive and as the investor’s risk toler- mal portfolio? The answer is straightforward: One ance increases. The exposure may be negative, under includes the benchmark security explicitly in the for- which condition the investor sells the benchmark secu- mulation of the investor’s utility function and performs rity short. Conversely, as the investor’s risk tolerance or a single integrated optimization to obtain the optimal the attractiveness of the MRR portfolio decreases, the individual security and benchmark security holdings benchmark exposure should increase. simultaneously. In the limit, as either m or τ tends toward zero, Consider the problem of maximizing a long- the optimal benchmark exposure reaches 100% of the short portfolio’s return with respect to a benchmark invested wealth. An optimally equitized portfolio, how- while simultaneously controlling for residual risk. The ever, will generally not include a full exposure to the variables that can be controlled in this problem are h benchmark security. In the limit, as m approaches zero and the benchmark holding denoted by hB. We make (and hB approaches one), the risky portfolio h becomes the simplifying assumption that benchmark holdings proportional to the standard portfolio; for this risky consume no capital. This is approximately true for portfolio to be optimally beta- or dollar-neutral, the benchmark derivatives, such as futures and swaps. The same conditions must be satisfied as those given in portfolio’s expected excess return is thus Equations (6) and (11) for the unequitized portfolio defined by Equation (4).

N The risky part of the equitized portfolio is opti- rrEF=+∑ hrhrr iiBBB + − (13) mally dollar-neutral when i=1 N rmq+ It can be shown (see Jacobs, Levy, and Starer ∑()ξξ− ii= 0 (14) i σ [1998]) that the optimal risky portfolio h in this case is: i=1 i

h =(φ + mψ)τ The term on the left-hand side of Equation (14) can be interpreted as a net stability-weighted risk-adjusted where φ = Q–1r is the standard portfolio that would be expected return. The risky part of the optimally equi- chosen by an idealized investor with unit risk tolerance tized portfolio should be net long if this quantity is who optimizes Equation (1) without any constraints; ψ positive and net short if it is negative. This is analo- = Q–1q is a minimum-residual risk (MRR) portfolio; q gous to the condition given in Equation (6) for an

= cov(r, rB) is a vector of covariances between the risky unequitized long-short portfolio. The equitized case securities’ returns and the benchmark return; and m is includes an additional term, mqi, that captures the the ratio of the expected excess return of the MRR attractiveness of the MRR portfolio and the correla-

30 LONG-SHORT PORTFOLIO MANAGEMENT: AN INTEGRATED APPROACH WINTER 1999 tions between the risky securities’ and the bench- portfolio affords the investor, it can be expected to per- mark’s returns. form better than a long-only portfolio based on the Similarly, the risky part of the optimally equi- same set of insights. tized portfolio is beta-neutral when ENDNOTES N β i (rmq+=) 0 The authors thank Clarence C.Y. Kwan for helpful com- ∑ 2 ii i=1ωi ments, and Judith Kimball for editorial assistance. 1As the median-capitalization stock in the Russell 3000 index has a weighting of 0.01%. This is analogous to the condition given in Equation 2The ability to short will be particularly valuable in a mar- (11) for an unequitized portfolio. Again, the condition ket in which short-selling is restricted and investment opinion for the equitized portfolio to be beta-neutral includes diverse. When investors hold diverse opinions, some will be more the additional term mq . pessimistic than others. With short-selling restricted, however, this i pessimism will not be fully reflected in security prices. In such a world, there are likely to be more profitable opportunities for sell- CONCLUSION ing overpriced stocks short than there are profitable opportunities for purchasing underpriced stock. See Miller [1977]. The freedom to sell stocks short allows the 3This assumes symmetry of inefficiencies across attractive investor to benefit from stocks with negative expected and unattractive stocks. It also assumes that portfolio construction returns as well as from those with positive expected proceeds identically and separately for the long and short sides as it does in long-only portfolio construction. Although these assump- returns. The advantages of combining long and short tions may appear unduly restrictive, they have often been invoked. portfolio positions, however, depend critically on the See Jacobs, Levy, and Starer [1998] for a discussion of this literature way the portfolio is constructed. Traditionally, long- and our counterpoints. short portfolios have been run as two-portfolio strate- 4In deriving the formula, it is assumed that the beta of the gies, where a short-only portfolio is added to a long- short side equals the beta of the long side. 5 only portfolio. This is suboptimal compared with an Furthermore, under Regulation T, a long-only portfolio can engage in leverage to the same extent as a long-short integrated, single-portfolio approach that considers the portfolio. Long-short has an advantage here, however, because expected returns, risks, and correlations of all securities purchasing stock on margin can give rise to a tax liability for tax- simultaneously. Such an approach maximizes the exempt investors. According to Internal Revenue Service investor’s ability to trade off risk and return for the best Ruling 95-8, borrowing shares to initiate short sales does not possible performance. constitute debt financing, so any profits realized when short positions are closed out do not give rise to unrelated business Also generally suboptimal are construction taxable income. approaches that constrain the short and long positions 6For analytical tractability and expositional simplicity, we of the portfolio to be dollar- or beta-neutral. Only use the traditional mean-variance utility function, although it is under very limited conditions will such a constrained only a single-period formulation and is not sensitive to investor portfolio provide the same utility as an unconstrained wealth. Also, behavioral research may question the use of an ana- portfolio. In general, rather than using long-short bal- lytic utility function in the presence of apparently irrational investor behavior. Nevertheless, we believe our conclusions hold for more ance to achieve a desired exposure (including no expo- elaborate descriptions of investor behavior. sure at all) to a particular benchmark, investors will be 7A dollar balance may appear to provide tangible proof of better off considering benchmark exposure as an the market neutrality of the portfolio. But unless a dollar-balanced explicit element of their utility functions. portfolio is also beta-balanced, it is not market-neutral. 8 Long-short management is often perceived as The simplifying assumption applied is the constant-correlation model of Elton, Gruber, and Padberg [1976]. substantially riskier or costlier than long-only manage- 9One could, for example, use the Gram-Schmidt proce- ment. Much of any incremental cost or risk, however, dure (see Strang [1988]). reflects either the long-short portfolio’s degree of leverage or its degree of “activeness”; both of these param- REFERENCES eters are under the explicit control of the investor. Additionally, proper optimization ensures that expected Elton, Edwin J., Martin J. Gruber, and Manfred W. Padberg. returns compensate the investor for risks incurred. “Simple Criteria for Optimal Portfolio Selection.” Journal of Finance, Given the added flexibility that a long-short December 1976, pp. 1341-1357.

WINTER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 31 Jacobs, Bruce I. “Controlled Risk Strategies.” In ICFA Continuing Miller, Edward M. “Risk, Uncertainty, and Divergence of Education: Alternative Assets. Charlottesville, VA: Association for Opinion.” Journal of Finance, September 1977, pp. 1151-1168. Investment Management and Research, 1998. Sharpe, William F. “Capital Asset Prices with and without Negative Jacobs, Bruce I., Kenneth N. Levy, and David Starer. “On the Holdings.” Journal of Finance, June 1991, pp. 489-509. Optimality of Long-Short Strategies.” Financial Analysts Journal, March/April 1998. Strang, Gilbert. Linear Algebra and Its Applications, 3rd ed. New York: Harcourt Brace Jovanovich, 1988. Markowitz, Harry. “Portfolio Selection.” Journal of Finance, March 1952, pp. 77-91. To order reprints of this article, please contact Ajani Malik at ama- [email protected] or 212-224-3205.

Reprinted with permission from the Winter 1999 of The Journal of Portfolio Management. Copyright 1999 by Institutional Investor Journals, Inc. All rights reserved. For more information call (212) 224-3066. Visit our website at www.iijournals.com.

32 LONG-SHORT PORTFOLIO MANAGEMENT: AN INTEGRATED APPROACH WINTER 1999