Long/Short Investing Introduction U.S. institutions have invested using long/short strategies since at least the late 1980s. These strategies have generated controversy, and, over time, increasing acceptance as a worthwhile innovation. Long/short strategies offer a distinct advantage over long-only strategies: the potential for more efficient use of information, particularly but not exclusively downside information. This chapter will analyze several important aspects of long/short strategies and, by implication, some important and poorly understood aspects of long-only strategies. We will define long/short strategies and briefly introduce their advantages and the controversies surrounding these advantages. We will then analyze in detail the increased efficiency offered by long/short strategies and the subtle but pervasive effects of the long-only constraint. This analysis is therefore important to all managers—not just those offering long/short strategies. We will later discuss the appeal of long/short strategies and some empirical observations. The chapter ends with the usual notes, references, and technical appendix. The main results of this chapter include the following: • The benefits of long/short investing arise from the loosening of the (surprisingly important) longonly constraint. • Long/short implementations offer the most improvement over long-only implementations when the universe of assets is large, the assets' volatility is low, and the strategy has high active risk.
Page 420 • The long-only constraint tends to induce biases, particularly toward small stocks. Surprisingly, it can limit the ability to completely act on upside information, by not allowing short positions that could finance long positions. In this chapter, we will define long/short strategies specifically as equity market–neutral strategies. These strategies have betas of 0 and equal long and short positions. Some databases group these strategies in the more general category of ''hedge fund." However the hedge fund category can include almost any strategy that allows short positions. We will focus much more specifically on equity strategies managed according to the principles of this book, and with zero beta and zero net investment. Long/short investing refers to a method for implementing active management ideas. We can implement any strategy as long/short or long-only. Long/short investing is general. It does not refer to a particular source of information. Now, every long-only portfolio has an associated active portfolio with zero net investment and often zero beta. Therefore, every long-only portfolio has an associated long/short portfolio. But the longonly constraint has a significant impact on this associated long/short portfolio. Long/short strategies provide for more opportunities—particularly in the size of short positions in smaller stocks (assuming a capitalization-weighted benchmark). Long/short strategies are becoming increasingly popular. According to Pensions and Investments (May 18, 1998), 30 investment management firms offer market-neutral strategies, up from the 21 investment management firms one year earlier. Market-neutral strategies are something of a "phantom" strategy. The Pensions and Investments list does not include many large investment management firms that offer market-neutral strategies. It also appears to underreport the assets invested for some of the firms listed. Market-neutral strategies short stocks, a strategy frowned upon by some owners of funds. This apparently leads to enhanced discretion by the managers. But this is only part of the controversy.
Page 421 The Controversy Proponents of long/short investing offer several arguments in its favor. One simple argument depends on diversification. A long/short implementation includes effectively a long portfolio and a short portfolio. If each of these portfolios separately has an information ratio of IR, and the two portfolios are uncorrelated, then the combined strategy, just through diversification, should exhibit an information ratio of IR · √2. The problem with this argument is that it applies just as well to the active portfolio associated with any long-only portfolio. So this argument can't be the justification for long/short investing. A second argument for long/short investing claims that the complete dominance of long-only investing has preserved short-side inefficiencies, and hence the short side may offer higher alphas than the long side. The third and most important argument for long/short investing is the enhanced efficiency that results from the loosening of the long-only constraint. The critical issue for long/short investing isn't diversification, but rather constraints. These arguments in favor of long/short investing have generated considerable controversy. The first argument, based on diversification, is misleading if not simply incorrect. Not surprisingly, it has attracted considerable attack. The second argument is difficult to prove and brings up the issue of the high implementation costs associated with shorting stocks. The third argument is the critical issue, with implications for both long/short and long-only investors. The Surprising Impact of the Long-Only Constraint We are interested in the costs imposed by the most widespread institutional constraint—the restriction on short sales—or, equivalently, the benefits of easing that constraint. We will ignore transactions costs and all other constraints, and focus our attention on how this constraint affects the active frontier: the trade-off between exceptional return α and risk ω. Let's start with a simple market that has N assets and an equal-weighted benchmark. We presume in addition that all assets have
Page 422 identical residual risk ω and that residual returns are uncorrelated. This model opens a small window and allows us to view the workings of the long-only constraint. Let αn be the expected residual return on asset n and λR the residual risk aversion. In this setup, assuming that we want zero active beta, the active position for asset n is The overall residual (and active) risk ψP is We know from Chap. 10 that alphas have the form , where zn is a score with mean 0 and standard deviation 1, and we have invoked the fundamental law of active management to write the information coefficient in terms of the information ratio and the number of assets. Hence, the active positions and portfolio active risk become We can use Eqs. (15.3) and (15.4) to link the active position with the desired level of active risk ψP, the stock's residual risk ω, and the square root of the number of assets The limitation on short sales becomes binding when the active position plus the benchmark holding is negative. For an equal-weighted benchmark, this occurs when Figure 15.1 shows this information boundary as a function of the number of stocks for different levels of active risk.
Page 423 Figure 15.1 Sensitivity to portfolio active risk.
Page 424 Information is wasted if the z score falls below the minimum level. The higher the minimum level, the more information we are likely to leave on the table. As an example, if we consider a strategy with 500 stocks, active risk of 5 percent, and typical residual risk of 25 percent, we will waste information whenever our score falls below –0.22. This will happen 41 percent of the time, assuming normally distributed scores. This rough analysis indicates that an aggressive strategy involving a large number of lowervolatility assets should reap the largest benefits from easing the restriction on short sales. The more aggressive the strategy, the more likely it is to hit bounds. The lower the asset volatility, the larger the active positions we take. The more assets in the benchmark, the lower the average benchmark holding, and the more likely it is that we will hit the boundary. Indirect Effects In a long-only optimization, the restriction against short selling has both a direct and an indirect effect. The direct effect, studied above, is to preclude exploiting the most negative alphas. The indirect effect grows out of the desire to stay fully invested. In that case, we must finance positive active positions with negative active positions. Hence, a scarcity of negative active positions can affect the long side as well: Overweights require underweights. Put another way, without the long-only constraint we could take larger underweights relative to our benchmark. But since underweights and overweights balance, without the long-only constraint we will take larger overweights as well. We can illustrate this "knock-on" effect with a simple case. We start with an equal-weighted benchmark and generate random alphas for each of the 1000 assets. Then we construct optimal portfolios in the long-only and long/short cases. Figure 15.2 displays the active positions in the long/short and long-only cases, with assets ordered by their alphas from highest to lowest. In the long/short case, there is a rough symmetry between the positive and negative active positions. The long-only case essentially assigns all assets after the first 300 the same negative alpha. We expected that the long-only portfolio would less efficiently handle neg-
Page 425 Figure 15.2 Long/short and long-only active positions: equal weighted benchmark.
Page 426 ative alphas than the long/short portfolio. More surprisingly, Fig. 15.2 shows that it also less efficiently handles the positive alphas. The Importance of the Benchmark Distribution This knock-on effect is more dramatic if the benchmark is not equal-weighted. We can illustrate this with an extreme case where the benchmark consists of 101 stocks. Stock 1 is 99 percent of the benchmark, and stocks 2 through 101 are each 0.01 percent: Gulliver and 100 Lilliputians. To make this even simpler, suppose that 50 of the Lilliputians have positive alphas of 3.73 percent and 50 have negative alphas of the same magnitude. We consider two cases: Gulliver has a positive alpha, again 3.73 percent, and Gulliver has a negative alpha, –3.73 percent. In the long/short world, the capitalization of the stocks is irrelevant. Table 15.1 displays the active positions of the stocks, along with some portfolio characteristics: Gulliver does not merit special consideration in the long/short world. Gulliver, Ltd., receives the same active position as a Lilliputian company with a similar alpha. Note that since all of the active positions are smaller than , the no-short-sale restriction would not be binding if the benchmark assets were equal-weighted. We encounter significant difficulty with the highly imbalanced benchmark when we disallow short sales. In that case, it makes a TABLE 15.1 Long/Short Results Characteristic α (Gulliver) Positive Negative Gulliver active position 0.79% –0.79% Positive stock active position 0.79% 0.80% Negative stock active position –0.80% –0.79% Portfolio alpha 3.00% 3.00% Portfolio active risk 2.00% 2.00%
Page 427 TABLE 15.2 Long-Only Results Characteristic α (Gulliver) Positive Negative Gulliver active position 0.01% –1.55% Positive stock active position 0.01% 0.04% Negative stock active position –0.01% –0.01% Portfolio alpha 0.04% 0.15% Portfolio active risk 0.02% 0.39% great deal of difference whether Gulliver's alpha is positive or negative. With a negative alpha, we assume a very large negative active position (–1.55 percent) that allows us to finance the overweightings of the Lilliputians with positive alphas.1 But when Gulliver has a positive alpha, we can achieve only tiny active positions, both long and short. Table 15.2 displays the results. The Gulliver example illustrates another problem: the potential for a significant size imbalance. The shortage of negative active positions causes relatively larger underweighting decisions on the higher-capitalization stocks. If the alpha on Gulliver had a 50/50 chance of being a positive or negative 3.73 percent, the average active holding of Gulliver would be –0.77 percent. Capitalization-Weighted Benchmarks The Gulliver example shows that the distribution of capitalization in the benchmark is an important determinant of the potential for 1There is an alternative approach: • Relax the condition that the net overweight must equal the net underweight. • Use a short cash position (leverage!) to finance the overweights. • Sell the benchmark forward to cover the added benchmark exposure. In the Gulliver model, this procedure allows us to achieve an alpha of 1.53 percent with an active risk of 1.42 percent. The negative cash position is 39 percent of the portfolio's unlevered value.
Page 428 adding value in a long-only strategy. To calculate the benefits of long/short investing in realistic environments, we will need a model of the capitalization distribution. This requires a short detour. We will use Lorenz curves to measure distributions of capitalization. To construct them, we must • Calculate benchmark weight as a fraction of total capitalization. • Order the assets from highest to lowest weight. • Calculate the cumulative weight of the first n assets, as n moves from largest to smallest. The Lorenz curve plots the series of cumulative weights. It starts at 0 and increases in a concave fashion until it reaches 1. If all assets have the same capitalization, it is a straight line. Figure 15.3 shows Lorenz curves for the Frank Russell 1000 index, an equal-weighted portfolio, and a model portfolio designed (as we will describe below) to resemble the Frank Russell 1000 index. One summary statistic for the Lorenz curve is the Gini coefficient, which is twice the area under the curve less the area under the equal-weighted curve. Gini coefficients must range between 0 (for equal-weighted benchmarks) and 1 (for single-asset benchmarks). So we can draw Lorenz curves for benchmarks with any arbitrary distribution of capitalization, and summarize any distribution with a Gini coefficient. To progress further, we must assume a specific form for the distribution of capitalization. A Capitalization Model We will assume that the distribution of capitalization is log-normal. Here is a one-parameter model that will produce such a distribution. First, we order the N assets by capitalization, from largest (n = 1) to smallest (n = N). Define These values look like probabilities. They start close to 1 and move toward 0 as the capitalization decreases. Next, we calculate a nor-
Page 429 Figure 15.3 Lorenz curves: 1000 assets.
Page 430 TABLE 15.3 Country Index Assets Gini Constant c U.S. Frank Russell 1000 1000 0.71 1.55 U.S. MSCI 381 0.66 1.38 U.K. MSCI 135 0.63 1.30 Japan MSCI 308 0.65 1.35 The Netherlands MSCI 23 0.64 1.38 Freedonia Equal weight 101 0.00 0.00 Freedonia Cap weight 101 0.98 11.15 mally distributed quantity yn such that the probability of observing yn is pn: where Φ{ } is the cumulative normal distribution. So far, we have converted linear ranks to normally distributed quantities yn. To generate capitalizations, we use We can choose the constant c to match the desired Gini coefficient or to match the Lorenz curve of the market.2 We used this model to match the Frank Russell 1000 Index in Fig. 15.3. Table 15.3 contains similar results for several markets covered by Morgan Stanley Capital International (MSCI) Indexes as of September 1998. The equal-weighted and Gulliver examples reside in the hypothetical land of Freedonia.3 The constant c ranges from 1.30 to 1.60 in a large number of countries. To analyze the loss in efficiency due to the long-only 2As an alternative, set the constant c to the standard deviation of the log of the capitalization of all the stocks. The two criteria mentioned in the text place greater emphasis on fitting the larger-capitalization stocks. 3Freedonia appeared in the 1933 Marx Brothers movie Duck Soup. During a 1994 Balkan eruption, when asked if the United States should intervene in Freedonia, several U.S. congressmen laughed, several stated that it would require further study, and several more were in favor of intervention if Freedonia continued its policy of ethnic cleansing.
Page 431 constraint, we will use the value 1.55. This stems from a feeling that the MSCI indices necessarily trim out a great many of the smaller stocks in a market. The Freedonia rows show an equal and a very unequal benchmark for comparison purposes. Armed with this one-parameter model of the distribution of capitalization, we are ready to derive our rough estimates of the potential benefits of long/short investing. An Estimate of the Benefits of Long/Short Investing We cannot derive any analytical expression for the loss in efficiency due to the long-only constraint, since the problem contains an inequality constraint. But we can obtain a rough estimate of the magnitude of the impact with a computer simulation. As our previous simple analysis showed, the important variables in the simulation include the number of assets and the desired level of active risk. We considered 50, 100, 250, 500, and 1000 assets, with desired risk levels* from 1 to 8 percent by 1 percent increments, and from there to 20 percent by 2 percent increments. For each of the 5 levels of assets and the 14 desired risk levels, we solved 900 randomly generated long-only optimizations. For each case, we assumed uncorrelated residual returns, identical residual risks of 25 percent, a full investment constraint, and an information ratio of 1.5. We ignored transactions costs and all other constraints. We generated alphas using Figure 15.4 shows the active efficient frontier: the alpha per unit of active risk. We can roughly estimate the efficient frontiers in Fig. 15.4 as 4We used Eq. (15.4) to convert desired risk levels to risk aversions. We required extremely high levels of desired risk, since the long-only constraint severely hampers our ability to take risks.
Page 432 Figure 15.4
Page 433 and, as elsewhere in the book, we measure α and ω in percent. As anticipated, with the information ratio held constant, long-only implementations become less and less effective, the greater the number of assets. We can also see that higher desired active risk lowers efficiency. In fact, we can define an information ratio (and information coefficient) shrinkage factor as Figure 15.5 illustrates the dependence of shrinkage on risk and the number of assets. For typical U.S. equity strategies—500 assets and 4.5 percent risk—the shrinkage is 49 percent according to Eq. (15.13), Figure 15.5
Page 434 which agrees with Fig. 15.5. The long-only constraint has enormous impact: It cuts information ratios for typical strategies in half! Equation (15.13) also allows us to quantify the appeal of enhanced indexing strategies, i.e., lowactive-risk strategies. The shrinkage factor is 71 percent for a long-only strategy following 500 assets with only 2 percent active risk. At this lower level of risk, we lose only 29 percent of our original information ratio. At high levels of active risk, long/short implementations can have a significant advantage over longonly implementations. At low levels of active risk, this advantage disappears. And, given the higher implementation costs of long/short strategies (e.g. the uptick rule, costs of borrowing), at very low levels of active risk, long-only implementations may offer an advantage. With a large number of assets and the long-only constraint, it is difficult to achieve higher levels of active risk. Using Eq. (15.11), we can derive an empirical analog of Eq. (15.4): See the technical appendix for details. Figures 15.4 and 15.5 illustrate efficient frontiers under several assumptions: an inherent information ratio of 1.5, a log-normal size distribution constant c = 1.55, and identical and uncorrelated residual risks of 25 percent. We have analyzed the sensitivities of the empirical results to these assumptions. Changing the inherent information ratio does not affect our conclusions at all. As Eq. (15.11) implies, the efficient frontier simply scales with the information ratio. Changing the log-normal size distribution constant through the range from 1.2 to 1.6, a wider range than we observed in examining several markets, has a very minor impact. Lower coefficients are closer to equal weighting, so the long-only constraint is less restrictive. At 4.5 percent active risk and 500 assets, though, as we vary this coefficient, the shrinkage factor ranges only from 0.49 to 0.51. Figure 15.6 shows how our results change with asset residual risk. Our base-case assumption of 25 percent is very close to the median U.S. equity residual risk. But we may be focusing on a more narrow universe. As asset residual risk increases, we can achieve more risk with smaller active positions, making the long-
Page 435 Figure 15.6 Sensitivity to asset residual risk. only constraint less binding. At the extremely low level of 15 percent, the long-only constraint has very high impact. In the more reasonable range of 20 to 35 percent, the shrinkage factor at 4.5 percent risk and 250 assets ranges from 65 to 54 percent. We can also analyze the assumption that every asset has equal residual risk. Given an average residual risk of 25 percent, and assuming 500 assets, we analyzed possible correlations between size (as measured by the log of capitalization) and the log of residual risk. We expect a negative correlation: Larger stocks tend to exhibit lower residual risk. Looking at large U.S. equities (the BARRA HICAP universe of roughly the largest 1200 stocks), this correlation has varied from roughly –0.51 to –0.57 over the past 25 years. Figure 15.7 shows the frontier as we vary that correlation from 0 to –0.6. With a correlation of 0, we found a shrinkage factor of 49 percent at 4.5 percent active risk. With a correlation of –0.6, the situation improves, to a shrinkage factor of 0.63. Finally, Fig. 15.8 displays the size bias that we anticipated. Figure 15.8 shows the result for various correlations between size and the log of residual risk, though the correlation does not signifi-
Page 436 Figure 15.7 Sensitivity to size/volatility correlations.
Page 437 Figure 15.8 Size bias sensitivity to size/volatility correlations.
Page 438 cantly change the result. We have measured size as log of capitalization, standardized to mean 0 and standard deviation 1. So an active size exposure of –0.3 means that the active portfolio has an average size exposure 0.3 standard deviation below the benchmark. These size biases are significant. Figure 15.8 implies that a typical manager following 500 stocks and targeting 4.5 percent risk will exhibit a size exposure of –0.65. In the United States, from October 1997 through September 1998, the size factor in the BARRA U.S. equity model exhibited a return of 1.5 percent: Large stocks outperformed small stocks. This would have generated a 98basis-point loss, just due to this incidental size bet. From September 1988 through September 1998, the same size factor experienced a cumulative gain of 3.61 percent, generating a loss of 2.35 percent over that 10-year period. The Appeal of Long/Short Investing Who should offer long/short strategies? Who should invest in them? Clearly, long/short strategies are a ''pure" active management bet. The consensus expected return to long/short strategies is zero, since they have betas of zero. Put another way, the consensus investor does not invest in long/short strategies. Therefore, the most skillful active managers should offer long/short strategies. It allows them the freedom to implement their superior information most efficiently. Long/short strategies offer no way to hide behind a benchmark. A long-only manager delivering 15 percent while the benchmark delivers 20 percent is arguably in a better position than a long/short manager losing 5 percent. While this isn't an intrinsic benefit of long-only strategies, it can be a practical benefit for investment managers. Long/short strategies also offer investment managers the freedom to trade only on their superior information. They can build a long/short market-neutral portfolio using only utility stocks, if that is where they add value. They have no reason to buy stocks just because they are benchmark members. Both the long and the short sides of the portfolio may have large active risk relative to the S&P 500—just not to each other.
Page 439 For investors, long/short strategies offer the most benefit to those investors who are best able to identify skillful managers. Given that, long/short strategies are quite appealing because of the (engineered) low correlation with equity market benchmarks. Long/short strategies can, in this way, successfully compete against bonds. Long/short investing also offers the appeal of easy alpha portability. Futures contracts can move active return from one benchmark to another. If we start with a strategy managed relative to the S&P 500, and sell S&P 500 futures and buy FTSE 100 futures, we will transfer the alpha to the FTSE 100. In a conventional long-only strategy, this transfer requires an extra effort. It is not the natural thing to do. A long/short strategy places the notion of a portable alpha on center stage. With a long/short strategy, we start with the pure active return, and must chose a benchmark. The potential for transfer is thrust upon us. Finally, long/short investing offers the possibility of more targeted active management fees. Longonly portfolios contain, to a large extent, the benchmark stocks. Long-only investors pay active fees for that passive core.5 Long/short investors pay explicitly for the active holdings. Empirical Observations Here we wish to present preliminary observations on long/short strategies. These strategies do not have a sufficiently long track record to definitively compare their performance to that of long-only strategies. But we can begin to understand especially their risk profile, and observe at least initially their performance record. These results are based on the performance of 14 U.S. long/short strategies with histories of varying lengths, but all in the 1990s, ending in March 1998.6 These 14 strategies are those of large, sophisticated quantitative managers. Most of these managers are 5See Freeman (1997). 6See Kahn and Rudd (1998).
Page 440 TABLE 15.4 Percentile History (Months) Volatility Beta S&P 500 Correlation IR 90 96 10.90% 0.10 0.23 1.45 75 86 6.22% 0.06 0.15 1.23 50 72 5.50% 0.02 0.04 1.00 25 50 4.12% –0.03 –0.07 0.69 10 28 3.62% –0.16 –0.20 0.44 BARRA clients. Table 15.4 shows the relevant observations. It is important to keep in mind the small and potentially nonrepresentative sample behind these data. First, note that the risk levels here do not substantially differ from the typical active risk levels displayed in Table 5.8.7 So, at least based on these 14 sophisticated implementations, long/short strategies do not exhibit substantially higher levels of risk than long-only strategies. Second, according to Table 15.4, these strategies achieved market neutrality. They exhibit realized betas and market correlations very close to zero. In fact, the highest observed correlations corresponded to managers with the shortest track records. There is no statistical evidence that any of these strategies had true betas different from zero, and the realized numbers are all quite small. This (admittedly limited) sample refutes the argument that achieving market neutrality is difficult. Third, at least in this historical period, these long/short strategies as a group exhibited remarkable performance. The performance results of 14 strategies over a particular market period do not prove that long/short implementations will boost information ratios, but they do help explain the increasing popularity of these strategies. 7The standard error of the mean risk level for these long/short strategies is 0.64 percent. So while the medians displayed here exceed those in Table 5.8, the difference is not significant at the 95 percent confidence level.
Page 441 Summary Long/short investing is an increasingly popular approach to implementing active strategies. Long/short strategies offer the potential to more efficiently implement superior information. Because the long-only constraint is an inequality constraint, and because its impact depends on the distribution of benchmark holdings, we cannot derive many detailed analytical results on exact differences in efficiency. However, both simple models and detailed simulations show that the benefits of long/short investing can be significant, particularly when the universe of assets is large, the assets' volatility is low, and the strategy has high active risk. From the opposite perspective, long-only managers should understand the surprising and significant impact of the long-only constraint on their portfolios. Among the surprises: This constraint induces a significant negative size bias, and it constrains long positions. Empirical observations on long/short investing are preliminary, but should certainly inspire further interest and investigation. Notes The debate over long/short investing has been contentious, and even acrimonious. In part, the controversy arose because the initial arguments in favor of long/short investing relied on diversification, as we discussed first. That simple argument is misleading in several ways. The first serious criticism of long/short investing was by Michaud (1993). From there the debate moved to Arnott and Leinweber (1994), Michaud (1994), Jacobs and Levy (1995), and a conference of the Institute for Quantitative Research in Finance (the "Q Group"), "Long/Short Strategies in Equity and Fixed Income," at La Quinta, California, in October 1995. Jacobs and Levy (1996), Freeman (1997), Jacobs, Levy, and Starer (1998), and Levin (1998) have subsequently published further detailed analyses of aspects of long/short investing. More recent work has looked at how long/short strategies fit into overall pension plans [Brush (1997)] and the performance of long/short managers [Kahn and Rudd (1998)].
Page 442 Problems 1. Jill manages a long-only technology sector fund. Joe manages a risk-controlled, broadly diversified core equity fund. Both have information ratios of 0.5. Which would experience a larger boost in information ratio by implementing his or her strategy as a long/short portfolio? Under what circumstances would Jill come out ahead? What about Joe? 2. You have a strategy with an information ratio of 0.5, following 250 stocks. You invest long-only, with active risk of 4 percent. Approximately what alpha should you expect? Convert this to the shrinkage in skill (measured by the information coefficient) induced by the long-only constraint. 3. How could you mitigate the negative size bias induced by the long-only constraint? References Arnott, Robert D., and David J. Leinweber. "Long-Short Strategies Reassessed." Financial Analysts Journal, vol. 50, no. 5, 1994, pp. 76–78. Brush, John S. "Comparisons and Combinations of Long and Long/Short Strategies." Financial Analysts Journal, vol. 53, no. 3, 1997, pp. 81–89. Dadachanji, Naozer. "Market Neutral Long/Short Strategies: The Perception versus the Reality." Presentation at the Q-Group Conference, La Quinta, Calif, October 1995. Freeman, John D. "Investment Deadweight and the Advantages of Long/Short Portfolio Management." VBA Journal, September 1997, pp. 11–14. Jacobs, Bruce I. "The Long and Short on Long/Short." Journal of Investing, vol. 6, no. 1, Spring 1997, Presentation at the Q-Group Conference, La Quinta, Calif, October 1995. Jacobs, Bruce I., and Kenneth N. Levy. "More on Long-Short Strategies." Financial Analysts Journal, vol. 51, no. 2, 1995, pp. 88–90. ———. "20 Myths about Long-Short." Financial Analysts Journal, vol. 52, no. 5, 1996, pp. 81–85. Jacobs, Bruce I., Kenneth N. Levy, and David Starer. "On the Optimality of Long-Short Strategies." Financial Analysts Journal, vol. 54, no. 2, 1998, pp. 40–51. Kahn, Ronald N., and Andrew Rudd. "What's the Market for Market Neutral?" BARRA Preprint, June 1998. Levin, Asriel. "Long/Short Investing—Who, Why, and How." In Enhanced Index Strategies for the Multi-Manager Portfolio, edited by Brian Bruce (New York: Institutional Investor, Inc., 1998).
Page 443 Michaud, Richard O. "Are Long-Short Equity Strategies Superior?" Financial Analysts Journal, vol. 49, no. 6, 1993, pp.44–49. Presentation at the Q-Group Conference, La Quinta, Calif, October 1995. Michaud, Richard O. "Reply to Arnott and Leinweber." Financial Analysts Journal, vol. 50, no. 5, 1994, pp. 78–80. Pensions and Investments, May 18, 1998, and May 12, 1997, articles on market-neutral strategies. Technical Appendix We will present the derivation of Eq. (15.14), the risk aversion required to achieve a given level of risk. We express utility in terms of risk as Using Eq. (15.11), this becomes We solve for the optimal level of risk by taking the derivative of U with respect to ω and setting the result equal to zero. We find This leads directly to Eq. (15.14).
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