itive.''10 In fact, the conception of a trading system usually impliesseveral rules and indicators used in combination, and a systematicanalysis of the behavior of the system in order to re®ne it continuously.Therefore, not only would traders use several rules and/or indicatorsat a given time, including some highly proprietary, they would alsoprobably not use the same ones over a long period. An indication that asingle rule would yield 17.2 percent over a 100-year period thus cannotbe considered entirely re¯ective of an intrinsic pro®tability of technicaltrading in general.On the other end, it illustrates very clearly an earlier point:Computational and brain power dedicated to understanding and investigating market ef®ciency and trading opportunities are more and moresigni®cant. The study covered a 100-year period and almost 7,500 different trading con®gurations. Only 10 years ago such an undertakingwould have been impossible, whereas it is extremely reasonable today.Indeed, many trading ®rms and hedge funds have developed tradingsystems that monitor virtually thousands of stocks and have the abilityto run back-testing simulations for decades or more.If we use these results as a starting point, we can explore anotheravenue to get a sense of how pro®table pair and technical trading canbe. Again hedge funds provide the easiest access to this information bynature of their business. Table 10-3 provides year-to-year performanceof four categories of hedge funds. These ®gures give a sense of thevariability of the returns and their average magnitude. As a ®rstapproximation, we can place the typical pro®tability between 10 and20 percent, in some cases as high as 25 to 30 percent, or as low as 5 to 10percent. Again these ®gures have to be adjusted for leverage, which wewill take between 1 and 2, lower than before to re¯ect the fact that thelevel of risk is higher than risk arbitrage. All in all, we could estimatean average pro®tability around 10 percent with a reasonable rangebeing 5 to about 15 percent.11Like risk arbitrage, the purpose of the exercise is not to obtain aprecise ®gure; this would naturally be impossible. Rather, we want todemonstrate how public data can help us estimate an order of magnitude with minimal knowledge of what the reality is because we do not10
By ``primitive'' we mean to emphasize the fact that these rules have been around forsuch a long time that they are probably fully ``priced'' in the marketplace by now.11This result is consistent with quantitative returns found in other studies. See, forexample, Evan Gates, William Goetzmann, and K. Gast Rowenhorst, ``PairsTrading: Performance of a Relative Value Arbitrage Rule,'' National Bureau ofEconomic Research, Working Paper 7032, March 1999, Cambridge.
468
COMMON ARBITRAGE SITUATIONS
TABLE 10-3Year
MarketNeutral
RiskArbitrage
StatisticalArbitrage
Other RelativeValue
1975197619771978197919801981198219831984198519861987198819891990199119921993199419951996199719981999
37.11%23.93%7.15%6.56%18.61%32.50%4.91%21.55%22.55%6.27%31.75%18.68%5.26%21.84%26.80%8.33%23.37%12.61%22.68%4.48%18.11%19.84%17.64%10.53%12.65%
12.45%5.18%19.66%7.15%23.48%12.53%17.37%9.18%12.84%
31.18%37.23%16.86%9.27%20.61%12.17%15.32%12.20%
4.83%13.47%16.06%4.68%20.34%
CAGR
16.0%
13.2%
19.0%
8.2%
(Source: HedgeFund.net, November 1999, reprinted with permission.)
have access to detailed information. The legitimate conclusion is thattechnical trading and more generally speculative arbitrage is likely toyield signi®cantly more than risk arbitrageÐand naturally muchhigher than index arbitrage. Again these ®gures are nominal returnsand therefore are not risk adjusted.
RISKSConvergence
There are not many ways to manage a convergence risk in a technicaltrading position. Indeed, there is only one, and it is called discipline. In
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469
virtually every book published by traders or experienced investorstrying to explain the basic principles of investing, discipline systematically comes back as a critical requirement, and for good reason.Experience and history show invariably that successful traders areextremely rigorous about the management of their risksÐalthoughrigor is certainly not enough to succeed.Discipline and rigor in that context simply point toward a totalelimination of the ``human factor'' from the trading decision. We mentioned earlier that risk and the perspective of a loss are likely to affectsigni®cantly human behavior, all the more if it is one's own moneyinvolved. Therefore, being disciplined simply means applying systematicrules, no matter what. And with respect to risk management and speculative trading, these rules are extremely simple: They are called stoppro®t and stop-loss rules.Taking a position can then be described as an exercise in applying Boolean logic. A particular position is initiated when a rule istriggeredÐfor example, if two moving averages cross over. Atinitiation of the positionÐand not afterÐa set of risk parametersis ®xed, speci®cally stop pro®t and stop loss. They are de®nedaccording to the expectations associated with the strategy itselfand probably to re¯ect also current market conditions. For example,if we are looking at a moving average crossover trying to capturelong-term trends, it does not make any sense to take a pro®t at 5percent and cut a loss at 3 percent. A long-term trend by de®nition is more likely to create moves of wider amplitudes, so thresholds, for example, at 20 percent and 15 percent would probablymake more sense. In contrast, a day trader would probably be wiseto consider 5 and 3 percent because the nature of his or herbusiness is such that these amplitudes are more reasonable. Inany case, both types of investor, will also look, for example, atthe overall volatility of the market. In a period of high volatility,it is probably safe to reduce the amplitude between stop-pro®t andstop-loss triggers. Eventually whenever any of these thresholds isreached or when enough time has lapsed to turn the position intoa probable loser, it should be cut immediately, without any considerationÐor remorseÐwhatsoever for its immediate pro®tability.The thresholds can be variable in many ways, and they do nothave to be constant from one position or one day to the next.They could be based, for example, on the success rate of a givenstrategy in the six months prior to the current opportunity. In anycase, they should be rigorously set when the position is effectivelytaken, and more importantly, respected.
470
COMMON ARBITRAGE SITUATIONS
Naturally this is a schematic presentation and it should not betaken as a rule that is rigid and absolute. There may be many validreasons that a particular situation would require a trader to revisit theoriginal assumptions and thereby the original thresholds. This isexactly what judgment and experience are trying to capture: the abilityto anticipate when to stick by the rules in a systematic way and whento reevaluate an opportunity. The important point is to realize that themental process here is fundamentally different from what it wasbefore. In a situation of index arbitrage, the primary concern is tocarry the position with the minimum peripheral losses in order tokeep the largest part of the original pro®t, assuming there was anoriginal pro®t. In risk arbitrage, the original pro®t is less certain, sothe appropriate question is whether a position should be kept at allbased on a case-by-case analysis of facts that are absolutely not systematic or statistical in nature but situation speci®c. In speculativearbitrage the appropriate question is not whether one particular position should be examined more carefully and possibly reevaluated orunwound but rather what are the appropriate risk and reward thresholds that will be applied systematically and will eventually guarantee apositive return on average.Naturally the appropriateness of a particular stop-pro®t or stoploss threshold depends on a number of factors and is not a numbercreated out of thin air. Typically a given strategy can be evaluated for ascenario of maximum loss based on historical simulations, and thisanalysis would naturally be highly instrumental in determining boththe expected and maximum admissible loss from the strategy. This isessentially what we did for risk arbitrage, looking at BT/MCI andSPLS/ODP and extracting an estimate of the impact of a negativeevent on the spread.In order to illustrate the basic principles behind the process, let'scall p the average success rate of the strategy sL and sP the stop loss andstop pro®t in percentage terms, and P the expected yield from thestrategy over a reasonably long period. In other words, out of 100positions, a number p are pro®table and yield a pro®t sP percent onaverage, the remaining losing sL percent on average. P is then equal to:P  p  sP
1
p  sL
This equation implies that we can stop every position at its preset stop,which is unlikely in practice.12 Experience shows that it is much more12
The equation also implies that all positions have a similar size, an assumption weconsider for the sake of simplicity.
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dif®cult to stop losses than pro®ts because in general, negativeresponses from the market are associated to higher volatility comparedto positive events. Therefore, P is probably an overestimate of theoverall pro®t.Consider that historical simulations suggest that 10 percent is areasonable estimate of the loss associated to a failed opportunity and60 percent an acceptable average success rate. Stated otherwise, whenwe lose 10 percent, it is very unlikely that the position will turn overand become pro®table, so we might as well stop it. P is then equal to:P  60 percent  sP
40 percent  10 percent
If we want a yield of 10 percent on average over a reasonably longperiod of time, we have to choose sP such that:60 percent  sP 40 percent  10 percent 10 percent ) sP  23:3 percentThis result gives a numerical value to the stop-pro®t threshold weshould use if we want to be consistent with the expectations underlying the model. More importantly, it shows that stop loss, stop pro®t,success rate, and overall pro®tability are related, and thus one shouldnot be guessed independently from the others but they should beconsidered as a coherent set of parameters.The apparent simplicity behind these results should not occult thefact that there is little in practice directly under a trader's control.Beyond the entry signalÐarguably a critical partÐand speci®c valuesfor sP and sL , a trader has little in¯uenceÐwhich is exactly why stopsshould be managed very carefully. It is certainly possible to have areasonably sound strategy give disappointing results because it is notstopped adequately, on either side, pro®t or loss.At the same time the above equation should not be considered formore than what it really is: a reality check and one more tool to makebetter decisions. For example, the following relationship can be used asa back-of-the-envelope proportion:  sP1 pP0)psLStated clearly, there is a direct dependence between the risk-rewardratio and the success rate of a particular model. The higher the successrate, the lower the required return in terms of expected pro®t compared to expected loss. For example, a strategy for which p  60 percent could function with sP /sL = 66 percent at a minimum. This resultdoes not give an indication of the absolute value of sP and sL , but if we
472
COMMON ARBITRAGE SITUATIONS
can extract sL from a rigorous risk analysis, sP is automatically ®xed. Ifwe realize that the value of sP we obtain is unrealistic, it might verywell indicate that we need to re®ne the strategy considerably. Forexample, consider that sL = 10 percent, then sP should be equal to 6.6percent. If it turns out that historical pro®ts are below 6.6 percent onaverage, even with a success rate as high as 60 percent, it is fair toconclude that we are investigating a strategy that is simply not pro®table enough for the level of risk it creates. On the other hand, if p  40percent, we know that we should consider having sP /sL above 1.5Ðthat is, sP  15 percent if sL = 10 percent. Again looking at past performance, it is possible to get a sense of whether sP  15 percent is realisticand make a reasonably con®dent decision about the viability of thesystem.13In conclusion, it is important to realize how different the thoughtprocess can be for a speculative arbitrage compared to situations inwhich the convergence is more certain, even though the same framework can be used in practice.
Other Risks: Execution, Financing, Dividends, andShort Squeezes
Even more than any other component of a strategy, execution andmore generally risks other than convergence are dif®cult to encapsulatewhen we move away from absolute convergence. The obvious reasonfor that is the diversity of opportunities pursued in the marketplace. Interms of execution, for example, a situation of pair trading with a cycleof convergence of a few days will probably present no particular dif®culty because like risk arbitrage, the spread is likely to exist for morethan a few minutes at a time. At the other extreme, certain forms ofstatistical arbitrage are trying to capture tiny movements in prices onvery frequent trades and therefore are more dependent upon the abilityto execute instantaneously at the right price.Along the same lines, ®nancing and dividends may vary widelyfrom one strategy to another. If, for example, one leg of a strategy is aderivativeÐfutures or optionsÐ®nancing should be monitored withspecial care because of the leverage on the derivative position andassociated margin calls. A futures contract, as we have seen, does13
Incidentally, the entire discussion also applies to other forms of arbitrage. The maindifference is the magnitude of p. In index arbitrage, p is virtually equal to 100 percentin the sense that a position executed adequately will eventually turn a sure pro®t. Inrisk arbitrage, p is high but certainly not close to 100 percent anymore.
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not have a major impact on treasury because the ®nancing of margincalls can be hedged. It does however create an exposure to interestrates because of its pricing model. In contrast, a stock position createsan immediate need for cash because it has to be paid for when it isacquired, but it does not create any exposure to interest rates per se.Dividends are another illustration of the same phenomenon. A spreadposition like risk arbitrage or pair trading creates an exposure to thedividend differential because this differential enters into the P&L formula. Alternatively, a position of statistical arbitrage is mostly immuneto dividends, simply because the average holding period of a givenstock is in general short enough that dividends are irrelevant.In the same direction, short squeezes are a threat only when theposition is highly dependent on maintaining a short in one particularsecurity for a reasonable period of time and for a decent size. Pairtrading, for example, may fall in that category, like risk arbitrage.Statistical arbitrage, on the other hand, considers many stocks aspurely random variables, which means that if one is unavailable orexpensive to short sell, it can most probably be replaced by anotherwithout any dif®culty.Again, it is dif®cult to offer generic answers here because so manyquestions can be asked. There are, however, some recurrent themes:. Derivatives versus nonderivatives:
The inclusion of a derivative in thestrategy may have a signi®cant impact on ®nancing and interestrate exposure, as well as on dividends, for example.. Cash ¯ows: A careful examination of cash ¯ows has allowed us toprice futures with the cash-and-carry model and to price a riskarbitrage opportunity. It should be clear by now that a careful¯ow diagram is a powerful tool to understand the different aspectsof a particular implementation, notably in terms of treasury, butnot only. This is not an almighty tool, however, because in somecases, in particular when delta hedging and/or options areinvolved, ¯ow diagrams are almost useless.. Time horizons: A good estimation of what the typical horizon is fora given strategy is also a safe principle. For example, a strategywith a pro®t cycle of a few days will require an analysis differentfrom a model based on patterns expressed over several months.Stops are naturally different, but execution should also be considered from two different angles, as should probably be risks associated with interest rates, dividends, and so on.. Convergences: If the convergence risk is managed appropriately, thestrategy is likely to to be pro®table overall. It is, for example, very
474
COMMON ARBITRAGE SITUATIONS
unlikely to see a position of pair trading or statistical arbitragesigni®cantly affected by a change in interest rates or dividends,although they may have a second-order effect.. Hedging: Naturally hedging the position appropriately remains avery important part of risk management. Whether the hedge is®xed quantity or nominal, it should be monitored carefully.. Short versus long positions: A short position is always a potentialcandidate for a short squeeze. If it is kept for a brief period of time,the risk is mitigated, but as a general rule, traders should neverexclude the possibility of a short squeeze.In practice, managing a position requires having every single aspect incheck without taking anything for granted. Some risks are more pervasive than others, but the risk analysis engine should be ¯exibleenough to describe and evaluate all of them.
ILLUSTRATIONRoyal Dutch and Shell
The arbitrage of Royal Dutch (RD) and Shell (SC) is very popular onWall Street because it is a pair trading model with characteristic closeto those of an absolute convergence. RD and SC are essentially twosides of the same coin, because the two companies are two ``shells''deriving their revenues through a ®xed partition of the income generated by the Royal Dutch±Shell Group. RD and SC are linked by acorporate charter stipulating that 60 percent of the income receivedby the Group will be allocated to RD, while the 40 percent remainingis allocated to SC. Therefore, there is a very clear relationship betweenRD and SC with respect to their respective incomes, and as a result,there should also be a relatively simple relationship between their stockprices.14
14
For more information, see Kenneth Froot and AndreÂ Perold, Global Equity Markets: TheCase of Royal Dutch and Shell, Harvard Business School, Case 9-296-077, July 1997.In order to have a rigorous determination of this ratio, let us call nRD and nSC thenumber of shares outstanding for RD and SC respectively. For every dollar of incomegenerated by the Group, RD shareholders receive $0.60Ðthat is, $0.60/nRD per sharewhereas SC shareholders receive $0.40Ðthat is, $0.40/nSC per share. Therefore, the``exact'' ratio between RD and SC should be:
CHAPTER 10
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For the purpose of this discussion, let's imagine that we knownothing of the relationship and that we want to trade the pair becausethere is historically a very high correlation between the two stocks. Thiscorrelation manifests itself notably through a relatively stable ratiobetween the prices of the two companies, a property that we willexploit to generate trading signals. Figure 10-1, for example, showsthe ratio between the closing prices of SC and RD over the two-yearperiod from June 1998 to June 2000. The dotted line is a 30-day movingaverage of the ratio, and it is designed to capture its ``stable value.'' In
Figure 10-1 SC/RD(Source: Compiled by author from Datastream data.)
(Footnote continued)
0
10:4=0:4:nRD 2 nRDnSC A :rSC=RD  @0:6=0:6:nSC 3 nSCnRD
In addition, RD and SC are listed on the AMEX as ADRs, with one SC ADR beingequivalent to six common shares of SC. Therefore, the ratio between the prices of theADRs is:!nRD2nrSC=RD   4 RD3 nSC=6nSCTraditionally, this ratio has been around 0:65  0:85.
476
COMMON ARBITRAGE SITUATIONS
other words, we make the assumption that the observed value of theratio is the superposition of a fundamental function and market noise,and we chose the moving average as an estimator of the fundamentalfunction.15 Thirty days is short enough to be reactive to the market andlong enough to appear reasonably ef®cient in stripping noise out. Wecould have chosen other values, and in reality, we would certainly putother values to the test to determine if 30 is really the best we can ®nd.Once we have an estimate of this ratio, we can compare the marketspread between RD and SC with what we think this spread should be.For example, on July 8, 1998, SC closed at $411/4, RD at $5315/16, and the30-day moving average of their ratio was 0.7725. Based on this ratio,SC ``should be worth'' $41.66 (= $5315/16  0.7725). Because SC closed at$411/4, it is undervalued by $0.4187 (= $41.25 $41.66). We could at thispoint use the value of the difference as a trading indicator. For example, if it is below $1, we could buy SC and sell RD because SC isundervalued, and vice versa. This is not the best way to proceed,however, because an absolute value of $1 does not have any particularmeaning for the stocks we are considering. It is smarter to introduceone more step, in which we normalize the value of the difference.16 Ifwe denote  as this difference, we want to compute norm such that:norm 

MA30 30 
where MA30() is the 30-day moving average of , and 30() its 30day standard deviation. The value of norm then represents the ``mispricing'' of the spread expressed in terms of standard deviations. Theimplicit approximation in the process is that  is a random variabledistributed normally, so that norm is normally distributed with mean0 and standard deviation 1. The advantage in using norm over  isthat we know its distribution, and choosing a value for norm hastherefore a lot of meaning compared to choosing a value for . Forexample, there is a 15 percent chance that norm will be above 1 or
15
Note that what we call a stable value is obtained from the market with a movingaverage and has nothing to do with the rigorous calculation of the ratio rSC=RD weperformed earlier by looking at fundamentals of the companies. Along the same lines,we are not trying to explain or understand why the market is pricing the ratio withsuch variability; we are only taking a very super®cial approach by considering purelymarket prices.16This is exactly what we did already for the mean-reversal strategy in the ®rst part.This procedure is called a z transform.
CHAPTER 10
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below 1, and a 2.3 percent chance that it will be above 2 or below 2.For our original purpose of creating trading signals, we would thenknow that trading when norm is above 1 means that we are considering market events that are relatively rare, or even rarer with normabove 2. This is indeed exactly what we want to do because wewant to ®nd an indicator that will detect uncommon events, our expectations being that these events will disappear very fast, allowing us torealize a pro®t.As an illustration, Figure 10-2 shows the value of norm over theperiod from June 1998 to June 2000. As we might expect, norm is anoscillator whose values change quite often. The chart con®rms thatnorm is only infrequently above +1 or below 1, and even more rarelyabove +2 or below 2. To transform this chart into a trading strategy,we need to ®x a threshold and implement trading signals whenevernorm exits this preset range. We choose here 1.5 standard deviationsas the thresholdÐthat is, an occurrence of 6.7 percent if the distributionof norm was perfectly normal. Every time norm is above 1.5, we sellSC and buy RD, and we enter the opposite position if norm is below1:5. Table 10-4 shows the trading signals over a short period fromJuly 17, 1998, to July 24, 1998.
Figure 10-2 SC/RD Norm(Source: Compiled by author from Datastream data.)
478
COMMON ARBITRAGE SITUATIONS
TABLE 10-4DateJuly 17, 1998July 20, 1998July 21, 1998July 22, 1998July 23, 1998July 24, 1998
SC
RD
MA30()

norm
Signal
$417/ 16$411/ 16$403/ 8$411/ 16$401/ 8$401/ 8
$535/ 8$535/16$521/ 4$539/ 16$517/ 8$523/ 4
0.76980.76950.76920.76870.76850.7681
0.15940.03690.18660:11300.25990:3903
1.79091.28891.75310.75061.84240:2013
111
P&L$0.13$0:33$0.68
On July 17, 1998, for example, norm is 1.7909, indicating that SC istoo expensive compared to our model because it is above 1.5 standarddeviations; therefore, we sell it and buy RD at the closing price. Tokeep the position hedged, we choose a ®xed-quantity hedge with aratio equal to MA30() because this is our estimator of what the stablevalue of the ratio should be. In other words, we enter the followingdeals:17. Sell 1 share of SC at $417/16.
. Buy 0.7698 share of RD at $535/8.
For the sake of simplicity, we ®x a stop in absolute terms on the holding periodÐthat is, we close the position regardless of its pro®tabilitythe next day at the close. Therefore, on July 20, 1998, we effect thefollowing transactions:. Buy 1 share of SC at $411/16.
. Sell 0.7698 share of RD at $535/16
Naturally the P&L for the position is easy to compute:P&L  1  $417/16
$411/16) + 0:7698  $535/16
$535/8)  $0:1344
If we apply the same principles over a two-year period from June 1998to June 2000, we have a total of 118 positions, and the daily andaccumulated P&L look like the one shown in Figure 10-3. The totalP&L is $20.57, and the strategy has the following characteristics:. Maximum pro®t: 2.6 percent, maximum loss
2 percent
. P&L on long trades: $8.74, on short trades $11.83. Success rate: 69.5 percent
17
Note that the position is not hedged in nominal because the residual exposure is0.7698  $535/ 8 1  $417/ 16  $0.157. We are still appropriately hedged accordingto our model.
CHAPTER 10
Figure 10-3
PAIR TRADING AND TECHNICAL TRADING
479
SC/RD Arbitrage
Indeed, this is a very successful strategy, for a number of reasons:. Both long and short trades are equally pro®table.
. The overall success rate is better than two in three.
. The accumulated pro®t is never negative and only rarely decreas-
ing. Even when trades are losing, they are not losing much, whichis also visible in the fact that the maximum pro®t is higher than themaximum loss in percentage terms.
We know in this particular case that the spread is much more predictable than a common pair trading relationship because of the corporatelineage between RD and SC. Our approach also generates losing tradesin some cases, however, which is not entirely surprising because weintentionally restrict the information we use.It is interesting at this point to consider a sensitivity analysis todetermine if we were simply lucky in coming up with this model. If,for example, we vary the threshold between 0.25 and 3 standarddeviations, Figure 10-4 shows the corresponding accumulated P&Land total success rate.18 This picture con®rms what was already18
Note that in reality, we would carry a sensitivity analysis on many more parameterssuch as the periodicity for the moving averages for MA30  and 30 , the holdingperiods, the implementation of distinct percentage stops for long and short, or theimplementation of distinct thresholds for long and short.
480
COMMON ARBITRAGE SITUATIONS
Figure 10-4 SC/RD Sensitivity
apparent: The strategy is extraordinarily successful because it is alwayspro®table regardless of the value of the threshold, and its success rateincreases steadily to 100 percent. The decreasing P&L might be relatively worrying, but it is simply a result of the lesser number ofpositions taken because the trading signals are getting sparser andsparser. Indeed, if we ever were to encounter such a diagram foranother strategy, it would probably mean that we are in front of arelatively solid convergence without knowing it.This strategy is relatively peculiar, because we traded pretendingwe did not know anything about it beforehand. Nevertheless, thebasic principles still apply, and in this particular case our primaryriskÐconvergenceÐwas handled by a very strict holding periodlimited to one trading day. Naturally this procedure does not ensurethat we will never incur