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expected returns may be lowest when economic Asset Mispricing, risks are perceived to be low, at or near a business- cycle peak. Thus, the simple random-walk model , and of returns may be false, but a relevant notion of market efficiency survives because high returns William R. Emmons and Frank A. Schmid are earned only by taking large amounts of risk. A different type of explanation of return predictability fter nearly four decades, academic econ- rejects market efficiency and focuses on market omists continue to debate financial-market imperfections of various sorts, such as incomplete Aefficiency as vigorously as ever.1 The original stock-market participation by households, significant theoretical arguments put forward in favor of effi- transactions costs, changes in sentiment, cient markets were based on the notion of stabiliz- or limited wealth and liquidity resources to conduct 2 ing in the form of arbitrage (Friedman, arbitrage (as in the current article). 1953). Simply put, arbitrage is “the simultaneous Whatever its economic explanation, mounting purchase and sale of the same, or essentially similar, evidence of return predictability leads Campbell, Lo, in two different markets for advantageously and MacKinlay (1997, p. 24) to suggest that it is time different prices” (Sharpe and Alexander, 1990). In for financial economists to focus their attention theory, a perfectly hedged trading of this on the “relative efficiency” of a market instead of sort could be executed at no cost (as the -sale continuing the all-or-nothing battle of attrition that proceeds are used to finance the position). is characteristic of much of the earlier market effi- Vigilant traders on the look-out for just such arbi- ciency literature. trage opportunities would ensure that no one could As we now understand more clearly, the original consistently “beat the market”—the hallmark of case for efficient markets probably leaned too heav- efficient markets theory. ily on the notion of risk-free, cost-free arbitrage to The academics’ logical case for efficient markets eliminate all profitable trading strategies immedi- boils down to a pair of simple rhetorical questions: ately. In real markets, arbitrage is neither as easy Why would utility-maximizing traders leave unex- nor as effective as economists once had assumed. ploited any profitable opportunities (after adjusting For one thing, financial markets are not complete properly for risk)? And if no risk-adjusted “free and frictionless, so arbitrage in general is risky and lunches” exist, how could market prices be predict- costly. In addition, it is not realistic to assume that able enough to make money? For several decades, the number of informed arbitrageurs or the supply empirical evidence piled up both for and against of financial resources they have to invest in arbitrage market efficiency. As of the early 1990s, neither strategies is limitless. side could claim total vindication. As the 1990s This article builds on an important and insight- progressed, however, the weight of the evidence ful recent model of arbitrage by professional traders seemed to tip toward those who claimed asset prices who need—but lack—wealth of their own to trade were, at least to some extent, predictable (Campbell, (Shleifer and Vishny, 1997). Professional arbitrageurs Lo, and MacKinlay, 1997, Chaps. 2 and 7). must convince wealthy but uninformed The academic asset-pricing literature today is to entrust them with investment capital in order to dominated by attempts to explain why and to what exploit mispricing and push the market back toward extent the price movements of financial assets are the ideal of efficiency. Unfortunately, arbitrageurs predictable. One potential explanation of stock- cannot prove that they recognize the intrinsic (or return predictability is that markets are efficient “fundamental”) values of the assets they claim are (“no free lunch”) but expected returns are time- mispriced. Even worse, it is possible the assets will varying, perhaps being linked to the business cycle. For example, expected returns may be highest when 1 For early statements of the theory of efficient markets and the unpre- economic risks are perceived to be high, such as at dictability of asset-price movements, see Fama (1965), Muth (1960), or near the bottom of a business cycle. Conversely, or Samuelson (1965). For a recent summary of the evidence for return predictability and its implications for efficient-markets theory, see Campbell, Lo, and MacKinlay (1997, Chap. 2). William R. Emmons is an economist and Frank A. Schmid is a senior 2 economist at the Federal Reserve Bank of St. Louis. William V. Bock Ironically, Keynes (1936, Chap. 12) clearly foreshadowed the recent provided research assistance. interest in investor sentiment and liquidity for understanding behavior, but was forgotten for decades as the efficient-markets © 2002, The Federal Reserve Bank of St. Louis. hypothesis dominated the academic discussion.

NOVEMBER/DECEMBER 2002 19 Emmons and Schmid R EVIEW become even more mispriced before reverting first period, at which time the investors may “roll eventually to their intrinsic values. Having incurred over” their funds with the arbitrageur or demand losses, the outside investors may demand their their money back if they have lost confidence in his money back at this point even though the expected ability. The asset will assume its intrinsic value at the profit of staying invested actually has increased. end of the second period with certainty, although Thus, market efficiency may depend ultimately only the arbitrageur knows in advance what that on the successful resolution of a principal-agent value is. Consequently, the two-period return on the problem that exists between informed but wealth- arbitrageur’s private information would be both posi- constrained arbitrageurs and uninformed wealthy tive and risk-free if he could be assured of financing. investors. The resulting degree of market efficiency Our set-up highlights the fact that a two-period may change over time and differ across markets, risk-free arbitrage nevertheless can be risky over a and it could depend importantly on factors such as one-period horizon in the presence of noise traders the outside investors’ use of performance-based and financial constraints on the arbitrageur. The (“feedback”) strategies when deciding on the possi- risk arises because the arbitrageur needs outside ble termination of ongoing investment mandates. investors, and these outside investors might revise After developing a simple model of wealth- their beliefs about the arbitrageur’s talent at the constrained professional arbitrage that departs in interim date, based on the return the arbitrageur several important aspects from the canonical Shleifer and Vishny (1997) model, we calibrate our model achieved in the first period. If the investors down- to illustrate its qualitative features. We show that the wardly revise their beliefs about the arbitrageur’s existence of professional arbitrageurs mitigates— abilities because the fund lost money due to a deep- but cannot eliminate—mispricing in the market ening of the mispricing, they might withdraw their relative to intrinsic values, regardless of how sensi- money precisely when the expected return on the tive the outside investors are to arbitrageurs’ past arbitrage is at its maximum. One implication is that performance in deciding whether to remain invested the arbitrageur will invest “strategically”—that is, with them. We also show that arbitrage dampens he will not invest as much initially as he would in a the unconditional volatility of asset returns, which world without wealth constraints—in order to we measure as the expected value of squared returns. against the possibility of being unable to exploit Most importantly, the presence of arbitrageurs limits even greater mispricing should it occur one period both the degree of increased mispricing and level ahead. Of course, this is not a new finding; for papers of volatility during a financial crisis, which we define with similar results, see Grossman and Vila (1992), as a period of heightened volatility and acute short- Shleifer and Vishny (1997), or Gromb and Vayanos age of liquidity.3 This result points out that profes- (2001).4 Our paper’s contributions in this respect sional arbitrageurs tend to stabilize markets even when they are wealth-constrained. Other papers 3 Myron Scholes (2000) suggests that the global financial crisis of 1997- show that investors who use “positive feedback” 98 was characterized by an increase in volatility, especially in equity markets, and a flight to liquidity (that is, a preference by many investors trading strategies—such as portfolio insurers—tend for assets whose liquidity was expected to be good). The crisis was to destabilize markets (Grossman and Zhou, 1996). accentuated by the “negotiated bankruptcy” of Long-Term Capital We analyze a three-date (two-period) model of Management (LTCM), a in which Scholes himself was a partner. According to Scholes, prior to the crisis, LTCM “was in the an aspiring professional arbitrageur (or “convergence business of supplying liquidity” and therefore its demise worsened ” in the language of Kyle and Xiong, 2001, the crisis by eliminating the liquidity it had been supplying. A theoretical and Xiong, 2001) who must obtain financing from analysis relevant to this episode is Xiong (2001). investors less informed than he is about the intrin- 4 The first rigorous investigations of the multi-period investment problem sic value of a financial asset—that is, its liquidation were Merton (1971, 1973) and Breedon (1979). Merton concluded that a trader should keep a constant fraction of his wealth invested in the value at the end of the second period. In addition to risky asset at all times. The fraction depends on the asset’s expected these two types of individuals, there are noise traders return and risk and the investor’s degree of risk aversion. Grossman who have wealth to invest but who misperceive the and Vila (1992) added leverage and solvency constraints to the dynamic trader’s problem. Their trader optimally commits more wealth to the asset’s intrinsic value. It is the noise traders who risky asset the shorter is the investment horizon and the further from drive the asset’s price away from the intrinsic value. the leverage constraint (not just today but prospectively in the future) The investors provide the arbitrageur with funds the trader finds himself. Campbell and Viceira (1999) is a recent exami- nation of the problem under the assumption that the investor is aware to invest in an underpriced asset at the outset of the that the probability distributions from which asset returns are drawn model. The price is observed again at the end of the change over time.

20 NOVEMBER/DECEMBER 2002 FEDERAL RESERVE BANK OF ST.LOUIS Emmons and Schmid

Figure 1 Timeline of the Model

t1 t2 t3

Noise traders’ Noise traders’ Noise traders’

misperception, S1, misperception misperception in place. might deepen to S2. corrects (S3 = 0). Arbitrageur allocates If asset price reverts Hedge fund financial resources to intrinsic value, V, winds down. to asset and cash. arbitrageur liquidates. are a more realistic objective function for the arbi- process of acquiring a long-short portfolio and hold- trageur and a set-up in which the arbitrageur’s trad- ing it until its price returns to the portfolio’s intrinsic ing significantly affects the asset’s price. Our model value. The long-short portfolio that any arbitrageur generates interior solutions and we provide cali- might hold defines a market segment of a larger brated illustrations of the model’s results. While arbitrage industry. We assume that arbitrageurs are Shleifer and Vishny (1997) assume that the arbi- highly skilled people who pursue proprietary trading trageur maximizes , we strategies and therefore enjoy a monopoly in their assume that he maximizes his income. The arbi- segment. For simplicity only, we make the assump- trageur’s income is determined by an incentive tion that the operating costs in the arbitrage industry scheme that resembles real-world contracts of hedge are zero. fund managers. The risk-free , and therefore the opportunity cost of capital, is zero. For simplicity, THE MODEL we assume that risky assets trading at fair value— There are three types of agents in the model. including the —also have an Noise traders have wealth but misperceive the intrin- expected return of zero. This implies that there are sic value of a financial asset. Professional arbitrageurs no priced systematic risk factors in the economy, have no wealth or borrowing capacity but know the that is, there is no equity risk premium. intrinsic value of the financial asset. Investors have The asset trades at three moments in time, t wealth but no insight into the financial asset’s intrin- (t=1,2,3). We capture the influence of the noise sic value. Unlike noise traders, investors know that traders’ misperceptions of the intrinsic value of the they cannot recognize the asset’s intrinsic value. asset at times t1 and t2 with the parameters S1 and All parties are risk-neutral. S2, respectively. There is no fundamental risk in the The investors may provide the arbitrageur with model because the price of the asset will revert to funds to invest in an underpriced asset at the outset the intrinsic value at a known date (t3) with certainty of the model (see Figure 1). We refer to this arrange- (so S3=0). ment as a hedge fund. Noise traders misperceive The supply of the financial asset is unity. Noise the intrinsic value of at least one financial asset in traders’ demand for the financial asset at time t the economy, which generates arbitrage opportuni- (t=1,2,3) is expressed as ties that so-called “long-short” investment strategies VS− (1)QN = t , 0 ≤ S

NOVEMBER/DECEMBER 2002 21 Emmons and Schmid R EVIEW financial asset. Without misperception (S=0), the quently the sole source of additional equity capital noise traders would be willing to absorb the unit in the second period. supply of the asset or, in other words, the asset would At time t2, the price of the asset either reverts trade at the intrinsic value ( pt=V). to V or it does not. If the asset price is V at t2, the The arbitrageur is compensated in two ways in arbitrageur liquidates the fund and holds cash until accord with actual practice—via an up-front “man- t3. If the asset price does not equal V at t2, the arbi- agement fee” and an after-the-fact performance- trageur invests aggressively—albeit not all of the based “incentive fee.”5 At the beginning of each fund’s cash—in the underpriced asset. This portfolio period, he receives a fraction (α ) of the assets under then generates a risk-free return because the asset management, and at the end of the period he receives price rises to V at t3 with certainty. a fraction (β ) of any positive return on the portfolio. The arbitrageur’s (that is, the hedge fund’s) This corresponds to compensation structures in demand for the asset at the interim date, t2, is given real-world hedge funds, where managers typically by α α collect =1 percent or =2 percent of the equity =≤≤D2 β (2) QA2 , 0DF22 , capital, plus =20 percent of any positive return p2 on the fund’s equity. We assume that the arbitrageur where D is the amount of the hedge fund’s demand invests his entire fee income in the fund. This is 2 in dollars. The amount F – D ≥ 0 is held in cash. because he recognizes the profitability of the fund’s 2 2 Because total demand aggregated across noise activities. traders and the arbitrageur must equal the asset The variable F denotes the total financial t supply of one unit (QN +QA =1), the price of the resources available to the arbitrageur at time t 2 2 financial asset at t is determined by combining (1) (t=1,2,3). The value of F is exogenous, while the 2 1 and (2): quantities F2 and F3 are determined in the model. The startup capital, F , is provided solely by the =− + ≤ < 1 (3) pVSD22222, 0 DS . investors, while the arbitrageur acquires the share α The condition D2

22 NOVEMBER/DECEMBER 2002 FEDERAL RESERVE BANK OF ST.LOUIS Emmons and Schmid

=− + < The arbitrageur knows that—despite a temporary (5) pVSDDS11111, . deepening of the mispricing—the price of the asset The condition D1

NOVEMBER/DECEMBER 2002 23 Emmons and Schmid R EVIEW

SS= V SS= misperception deepen in the first period, while 2 =⋅+−2 S =0 (9) F3 SS= DF22() D 2 MF 2 is the fee income if the asset price reverts p 2 2 2 to intrinsic value. The arbitrageur also captures capi- V = γ tal gains on the equity he builds from the reinvested =⋅+⋅+−SS2 SS= DF21(1 R 2 ) D 2 , management fees. The expected value of the capital p 2 2 gains, CG, equals where (14) SS= p 2 ==== −+⋅2 − =⋅SS2222 ⋅ ⋅+ SS + SS ⋅ SS ()FD11 D 1 F1 CG q([] R21332 MF1 R R MF ) = p SS2 = 1 = R2 . +−⋅S2 0 ⋅ (1qR ) 21 MF. F1 ≤ The arbitrageur’s choice variables are D1( F1) ≤ S2=S THE ARBITRAGEUR’S OPTIMIZATION and D2( F2 ), which are the amounts the arbi- PROGRAM trageur invests in the asset at t1 and t2, respectively. Unless the asset reverts to intrinsic value at t2( p2=V ), The arbitrageur’s total income consists of the t price of the asset given in equation (3) is a management fees and capital gains on reinvested 2 function of the t2 choice variable, D2. Similarly, the management fees. The expected value of the man- t price of the asset given in equation (5) is a function agement fees, MF, equals the sum of the expected 1 of the choice variable, D1. values of the management fees collected at t1(MF1), at t2(MF2), and at t3(MF3). The expected value of the SOLUTION TO THE MAXIMIZATION capital gains is CG. The arbitrageur’s maximization PROBLEM problem therefore is +++ We solve the maximization problem numeri- (10) Max {, MF1 MF23 MF CG} DD, cally. We hold constant all of the following: V=1; 12 α F1=S1=0.2; S2=0.4; q=1– q=0.5; =0.02; and where the management fees are β =0.2. Note that F1=S1=0.2 means that the arbi- =⋅α trageur has sufficient buying power to eliminate the (11) MF11 F , t1 mispricing entirely if so desired. Also, note that == 0.4=S2>S1=0.2 means that noise trader mispercep- (12)MF=+ MFSS22 MF S0, and 22 2 tion may deepen between t1 to t2—that is, the asset may become even more mispriced. For the values = =⋅⋅β SS2 MF33 q R chosen for S1, S2, and q, noise trader misperception, (13) == = S, is as likely to double as it is to vanish. Thus, the ⋅−+⋅−((1)FSS22 R SS MF MF SS 2 ) 2212 expected value of noise trader misperception in the and where second period, q·S2, equals the noise trader misper- == = ception observed in the first period, S1. SS22=⋅⋅αα SS −⋅+ SS 2 ⋅ MF22 q ([] F1 R 21 F We vary γ, the responsiveness to past perfor- = γ −⋅βαSS2 ⋅ − ⋅ mance of fund withdrawals, from =1 (no responsive- max{0,RF21 } (1 ) ) = ness by the investors to past investment performance, +⋅⋅βαSS2 ⋅ − ⋅ γ qRmax{0,21 } (1 ) F , that is, no withdrawals) to =20 (extreme respon- siveness) with a step length of unity. We use a grid === SSS2220 =⋅−⋅αα0 −⋅+0 ⋅ MF2 (1 q ) ( F2 [1 R2 ] F1 search method to solve the maximization problem. S =0 This involves varying D and D independently in −⋅βαRF2 ⋅(1 − ) ⋅ ) 1 2 2 1 very small increments within their bounds, 0 ≤ D ≤ = i +⋅βα − ⋅S2 0 ⋅ − ⋅ (1qR )2 (1 ) F1 , and Fi (i=1,2), to find the maximum of the objective function. == =V = SS22−+⋅− SS SS 2 SS2 The findings of the grid search are displayed ()FD22 D 2SS= F2 = p 2 in Figures 2 through 5. The first important point to SS2 = 2 R3 = . SS2 make concerns the extent to which the presence F2 of the hedge fund affects asset mispricing. Figure 2 S2=S The quantity MF2 represents the income the shows that the mispricing is less pronounced in each arbitrageur collects at t2 should the noise traders’ period than it would be without the hedge fund.

24 NOVEMBER/DECEMBER 2002 FEDERAL RESERVE BANK OF ST.LOUIS Emmons and Schmid

Figure 2 Figure 3 Effect of Investor Responsiveness on Effect of Investor Responsiveness on Asset Prices Asset Price Volatility

0.90 0.090

0.88 0.085 ) 2 p 0.86 0.080 , E( 1 p 0.84 0.075

0.82 p1 0.070 Asset Prices Asset Prices

E(p2) Unconditional Volatility 0.80 0.065

0.78 0.060 0246810 12 14 16 18 20 0246810 12 14 16 18 20 Responsiveness of Cash Flows (γ ) Responsiveness of Cash Flows (γ )

Remember that, without arbitrage, the first-period Figure 3 shows the unconditional volatility of price, p1, and the expected value of the second-period the asset’s returns for various degrees of investor γ price, E[p2], both would equal 0.8 (shown as a dashed responsiveness, . The unconditional volatility is line). On the other hand, without noise traders, the calculated as the expected value of the squared asset would trade at unit value in both periods (not returns over the two periods. For low values of shown). The hedge fund almost halves the difference investor responsiveness, volatility increases as γ between the expected value of the second-period increases. For high values of responsiveness, a fur- γ price, E[p2] (shown as solid circles), and the asset’s ther increase in reduces volatility monotonically. intrinsic, unit value. In fact, the degree of investor As γ goes to infinity, volatility approaches a level γ responsiveness, , has little bearing on E[p2], which (as shown by the solid line) that is lower than the approaches the value of approximately 0.8873 volatility level at γ=1 (as indicated by the leftmost (shown as a solid horizontal line) as γ approaches symbol), which is the benchmark case of unwaver- infinity. By comparison, the degree of responsiveness ing investor confidence in the hedge fund manager. has a strong impact on the first-period price, p1 The reason for this “volatility hump” lies in the exis- (shown as open boxes). This is because the arbitra- tence of two opposite effects. All else equal, the geur treads even more cautiously when putting on higher γ is, the bigger is the drop in the asset’s price this trade in the first period when he knows that the from t1 to t2 should the noise traders’ misperception investors penalize negative returns with sizeable deepen. On the other hand, the higher γ is, the lower γ withdrawals. In fact, the higher is , the more cash is the price of the asset at t1 because the arbitrageur the arbitrageur holds in the first period, and therefore, puts less money to work. For low values of investor the lower is p1. As the degree of investor responsive- responsiveness, the volatility-increasing effect domi- ness, γ, goes to infinity, the amount the arbitrageur nates. For increasingly higher values of γ, this effect invests in the first period goes to zero and, conse- becomes progressively weaker until it vanishes for quently, the first-period price, p1, converges to 0.8— an infinitely large degree of investor responsiveness. the value the asset would adopt if there were no It is important to note that the hedge fund hedge fund in the market (shown as a dashed line). greatly reduces asset price volatility, regardless of Thus we conclude that the hedge fund pushes the the degree of investor responsiveness. The uncon- price of the asset (or its respective expected value) ditional volatility without the hedge fund runs at toward the intrinsic, unit value in both periods. This 0.5694 (not shown), which is a multiple of the volatil- is our first main finding. ity that we observe even at the degree of responsive-

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Figure 4 Figure 5 Effect of Investor Responsiveness on Effect of Investor Responsiveness on Asset Return When Misperception Arbitrageur’s Profit Deepens 0.0142 –0.025

0.0140 2 S

–0.050 0.0138

0.0136 s Expected Profit –0.075 ’

0.0134 Arbitrageur –0.100 0.0132 First-Period Asset Return for S= Asset Return First-Period

0.0130 –0.125 0246810 12 14 16 18 20 0246810 12 14 16 18 20 Responsiveness of Cash Flows (γ ) Responsiveness of Cash Flows (γ ) ness that generates the highest level of volatility. profit, that is, his incentive to set up a hedge fund Thus, we conclude that the hedge fund unambigu- and engage in arbitrage. Figure 5 shows the arbi- ously reduces unconditional volatility. This is our trageur’s profit as a function of γ. Not surprisingly, second main finding. the profit of the arbitrageur decreases monotonically Another way to look at the impact of arbitrage with increased investor responsiveness to past per- on volatility is to ask how the market behaves when formance. The monotonic decline in the profitability asset mispricing deepens. Such an event—if severe— of arbitrage with increasing investor responsiveness might cause, or occur alongside, a financial crisis. to past performance is a manifestation of the fact Figure 4 shows, for the case of a deepening noise that liquidating a hedge portfolio when the expected trader misperception of the asset’s intrinsic value, return from arbitrage is highest is counterproduc- the first-period asset return as a function of investor tive—that is, it runs against “the nature of the trade.” responsiveness. The absolute value of the percentage decline of the asset price increases with investor CONCLUSION responsiveness, γ. For an infinitely high value of γ, the arbitrageur holds cash in the first period and Even financially constrained professional arbitrageurs may be able to exploit asset mispricing then invests aggressively at t2, although he does not invest all the cash available. The horizontal line in if they can link up with rational but uninformed Figure 4 signifies the first-period return for this investors. To achieve this goal, the two parties must borderline case of an infinite degree of responsive- overcome—at least to a degree—the problem of ness. Note that, without a hedge fund, the first-period asymmetric information about the arbitrageur’s return would amount to a negative 25 percent (not talent. The result of such an endeavor is a hedge shown), which is more than twice as much (in abso- fund that goes long on (comparatively) underpriced lute value) as what is observed even with a degree assets and short on (comparatively) overpriced assets. of responsiveness of zero (that is, γ equal to one). As a byproduct, the impacts of noise trader misper- Hence, we conclude that the presence of a hedge ceptions on asset prices and volatility are reduced. fund dampens volatility in the event of a deepening This holds for any degree of responsiveness to past of noise trader misperception, as might occur in a performance (“feedback”) of the investors’ confi- financial panic. This is our third main finding. dence in the arbitrageur’s talent. Finally, we are interested in the question of how This article builds on the dynamic-investment investor responsiveness affects the arbitrageur’s literature that reaches back at least to Merton (1971).

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Shleifer and Vishny (1997) provided an insightful Working paper, London Business School and Massachusetts model of wealth-constrained arbitrageurs that can Institute of Technology, October 2001. be, and has been, extended in several directions. We add several realistic features to the professional Grossman, Sanford J. and Vila, Jean-Luc. “Optimal Dynamic arbitrageur’s problem in the canonical model, includ- Trading with Leverage Constraints.” Journal of Financial ing the ability to build an equity stake in his hedge and Quantitative Analysis, June 1992, 27(2), pp. 151-68. fund over time, and a potentially more realistic multi- plicative (rather than linear) investor feedback rule. ______and Zhou, Zhongquan. “Equilibrium Analysis Like Shleifer and Vishny, we assume that the hedge of Portfolio .” Journal of Finance, September fund can influence the market price. Hedge funds 1996, 51(4), pp. 1379-1403. do, in fact, sometimes move market prices because they operate in specialized market segments that Keynes, John Maynard. The General Theory of Employment, have limited liquidity. It is also true, however, that Interest, and Money. New York: Harcourt, Brace & World, hedge funds alone cannot prevent asset-price volatil- 1936. ity or occasional mispricing—which might deepen before it eventually corrects. Kyle, Albert S. and Xiong, Wei. “Contagion as a Wealth Effect.” Journal of Finance, August 2001, 56(4), pp. 1401-40. REFERENCES Merton, Robert C. “Optimal Consumption and Portfolio Amin, Guarav S. and Kat, Harry M. “Hedge Fund Performance Rules in a Continuous-Time Model.” Journal of Economic 1990-2000: Do the ‘Money Machines’ Really Add Value?” Theory, December 1971, 3(4), pp. 373-413. Working paper, University of Reading (UK), 15 May 2001. ______. “An Intertemporal Capital Asset Pricing Model.” Breedon, Douglas T. “An Intertemporal Asset Pricing Model Econometrica, September 1973, 41(5), pp. 323-61. with Stochastic Consumption and Investment Opportuni- ties.” Journal of Financial Economics, September 1979, Muth, J. “Optimal Properties of Exponentially Weighted 7(3), pp. 265-96. Forecasts.” Journal of the American Statistical Association, 1960, 55, pp. 299-306. Campbell, John Y.; Lo, Andrew and MacKinlay, A. Craig. The Econometrics of Financial Markets. Princeton, N.J.: Samuelson, Paul. “Proof That Properly Anticipated Prices Princeton University Press, 1997. Fluctuate Randomly.” Industrial Management Review, 1965, 6, pp. 41-49. ______and Viceira, Luis M. “Consumption and Portfolio Decisions When Expected Returns Are Time Scholes, Myron S. “Crisis and Risk Management.” American Varying.” Quarterly Journal of Economics, May 1999, 114(2), Economic Review Papers and Proceedings, May 2000, 90(2), pp. 433-95. pp. 17-21.

Fama, Eugene F. “The Behavior of Stock Market Prices.” Sharpe, William P. and Alexander, Gordon. Investments. Journal of Business, 1965, 38, pp. 34-105. 4th Edition. Englewood Cliffs, N.J.: Prentice Hall, 1990.

Friedman, Milton. “The Case for Flexible Exchange Rates,” Shleifer, Andrei and Vishny, Robert W. “The Limits of in Essays in Positive Economics. Chicago: University of Arbitrage.” Journal of Finance, March 1997, 52(1), pp. 35-55. Chicago Press, 1953. Xiong, Wei. “Convergence Trading with Wealth Effects: An Gromb, Denis and Vayanos, Dimitri. “Equilibrium and Welfare Amplification Mechanism in Financial Markets.” Journal in Markets with Financially Constrained Arbitrageurs.” of Financial Economics, November 2001, 62(2), pp. 247-92.

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