1993).Obviously, given the mean-reverting property of ®nancial prices, suchstrategies cannot be justi®ed by classical arguments. As indicated by Friedman (1953), speculators tend to buy when prices are low and to sell whenprices are high, otherwise they lose money. In addition, information theory inthe line of Grossman and Stiglitz (1978), Kyle (1985), Grossman (1988) andGenotte and Leland (1991) does not provide satisfactory explanations ofstop-loss strategies. According to these studies, in a context of asymmetricinformation, uninformed traders infer from falling prices that informedtraders have received bad news about the future return of the stock. As a
Stop-loss rules as a monitoring device
313
consequence, uninformed investors, independently of their past performance,adjust their expectations downward as prices fall.Stop-loss orders have to be distinguished from other strategies that mayalso entail selling o a risky asset when its price falls, but in less dramaticproportions: for example, in the case of portfolio insurance, it is used toreplicate a put option designed to limit potential losses. Hence, positions areadjusted quasi-continuously. Similarly, with `drawdown' rules (gradual sellos), traders are forced to pull out of risky assets, as the net worth of theirassigned portfolio declines.Regarding the motives of stop-loss strategies, there is no shortage ofexplanations. First of all, one can mention psychological motives. A stoploss order allows investors to calm down after heavy losses and to make afresh start. Carrying over and over a losing position may prevent traders fromtaking the right decisions. In the line of Kahneman and Tversky (1979)`prospect theory', traders may also be psychologically more reluctant torealize losses than gains (see also Shefrin and Statman, 1985), so that it issuggested to impose stop-loss orders on them.In this chapter, we focus on motives related to the internal organization ofthe investment ®rm. In such organizations, relationships between employeesand the ®rm are usually plagued with adverse selection and/or moral hazardproblems. With adverse selection, the ®rm, searching for a trader in a pool oftraders of unknown quality, uses the whole series of observations on thetrader's past performance to revise its beliefs on the trader's actual quality.Following bad performances the ®rm decides to ®re the trader and toliquidate its position. However, this explanation of stop-loss strategies isonly relevant for young traders and cannot explain the bulk of stop-lossstrategies.To get a more convincing explanation of stop-loss orders, the chaptersuggests a model with moral hazard. The model is based on a con¯ict ofinterest between an investment ®rm and its traders. The investment ®rm relieson the technical expertise of traders who do not naturally maximize the ®rm'spro®ts. In this context, we show that the optimal behaviour of the ®rm is toenforce a stop-loss rule, i.e. to constrain the trader to liquidate the asset whenhis or her performance is poor. There are many possible causes of divergencebetween the investment ®rm and the traders. First of all a trader may deriveprestige from trading and will prefer a high to a low level of activity, hence asuboptimal point on the mean-variance frontier. A trader may also beconcerned by his or her own value on the job market and try to accumulatehuman capital at the expenses of the ®rm, by investing in exotic assets or newproducts, or by keeping, at the individual level, minimal turnover in somesegments of the market that would not be justi®ed at the level of the
314
Advanced Trading Rules
investment ®rm. Finally, traders may not be able to resist the temptation tobuy and sell shares on the ®rm's behalf to generate commissions paid to theirprivate account (bribery). As in the principal±agent model, the trader canchoose among a set of ®nancial strategies, i.e. among dierent distributionfunctions of the return on the asset. If the trader derives more utility fromselecting another strategy than the one which is the most preferred by the®rm, a variable wage is designed to make the choice of the latter strategyincentive compatible. Under reasonable conditions on the distribution ofreturns, the wage schedule is monotonic. When the trader has limited liability,we show that the ®rm may actually increase its pro®t by committing toliquidate the asset when the return is too low.Section 11.2 of this chapter develops a theoretical model which gives arationale for stop-loss rules. Section 11.3 presents the methodology we use inorder to ®nd out empirical evidence of stop-loss strategies in the FrenchTreasury bond future market. Section 11.4 gives the results of the tests.11.2 THE MODELIn this section, we consider a situation in which a fund provider hires a fundmanager in order to enhance his capital returns in ®nancial markets. The fundmanager is hired because of his or her skills and expertise in highly leveraged,volatile and sophisticated markets or in exotic markets. Accordingly, his orher performance determines his or her appointments. The fund provider canbe any ®nancial institution with market oriented activities in which case, thefund manager is said to be a proprietary trader. Like in the literature ofcorporate governance, we assume some con¯ict of interests between the fundprovider and the fund manager. The fund manager does not naturallymaximize the provider's gains because he can derive prestige and/or bribefrom his position by following suboptimal strategies. Because the ®nancialstrategies carried out by the trader are dynamic and complex, they are notobservable or they are observable at a very high cost. No complete contractsbased on such ®nancial strategies can be written. Moral hazard problem willtherefore spoil the relationships between the fund provider and the fundmanager.We use the principal±agent framework to model this con¯ict of interest. Inthe standard principal±agent model, the principal (the fund provider)observes perfectly the realized gains of the agent (the fund manager) andpays him or her a bonus. Here, we make the principal±agent model morerealistic by constraining the bonus to be lower bounded. 1 Moreover, weassume that the principal can enforce some kind of stop-loss rules when theagent's results are strongly de®cient. Ex post, this rule is always inecient,
Stop-loss rules as a monitoring device
315
since it denies the trader expertise. The aim of this section is to show that,because of the incentive compatibility conditions, the commitment to followsuch a rule can however become optimal under certain conditions.The assumptions and notations of the model are the following. The realizedgains are denoted x and the bonus w x. The fund provider is risk-neutral andthe trader's von Neumann±Morgenstern utility function is u . He or she canchoose a ®nancial strategy a in a ®nite set A and his or her preferences overthe dierent ®nancial strategies are represented by a desutility function c .Given the bonus scheme w and the strategy a, his or her utility is separableand writes u w c a.The usual assumptions are made concerning u  and c . Function u isincreasing and concave and function c is increasing and convex. Despite thetrader's expertise, the gains of the fund provider are unknown. These gainsdepend on the ®nancial strategy carried out; the trader can only control thedistribution of the gains. We denote f xja the density function of these gainsf xja > 0, and F xja the associated distribution function. These functionsre¯ect the ®nancial expertise of the trader. Then, the standard principal'sproblem, denoted by P is to:maximizea 2 A;w 
x
w x f xja dx
subject to the incentive compatibility condition:a 2 arg maxa 0 2 A
u w x f xja 0  dx
c a0
and to the agent's individual rationality condition:u w x f xja dx
c a  U
Under perfect observability and veri®ability of the strategy a (no moralhazard problem), the optimal contract is characterized by full insurance of thetrader (the funds provider is risk-neutral while the trader is risk-adverse) i.e.by a ®xed wage w  (no bonus). The ®xed wage w  is de®ned by:u w  U  c a
and the ®rst best strategy a  maximizes:x f xja dx
u 1 U  c a
In general, the ®rst best contract a  ; w   is dierent from the solutiona ; w   x of P because of the incentive compatibility condition. The usualprincipal agent model is commonly solved under the Mirlees's conditions(Mirlees and Milgrom, 1975). These conditions are that the density function
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Advanced Trading Rules
with the agent's strategy as parameter has a monotone likelihood ratio(MLRC) and, in addition, that the distribution function of gains is convexfor each level of gain (CDFC). Formally:if c a > c a 0 ; x 7!
f xja 0 is decreasing in xf xja
(MLRC)
and,if 0 
 1; for all x;
F xj a  1
0
a   F xja  1
(CDFC)0
F xja 
MLRC has the natural interpretation that a less preferred strategy for thetrader implies higher gains for the principal. This condition also ensures thatthe incentive scheme is increasing with the realized gains. CDFC has the morecontroversial interpretation (see Jewitt, 1988) that the trader's ®nancialexpertise exhibits decreasing returns with respect to less preferred strategy.In this chapter, we depart from the standard principal±agent model as justdescribed. First, we require the incentive scheme w to be lower bounded, i.e.:w x  w
(LL)
This very natural constraint can be interpreted as the trader's limited liability.It could also be some institutional constraint imposed on the incentivescheme, for example a minimal wage requirement. 2In addition to the standard principal±agent framework, we also assumethat the principal can enforce a more or less stringent liquidation rule. Moreprecisely, the principal can commit to liquidate the portfolio into a risk-freeasset according to the observed trader's results, whether results are potentialor realized. The enforcement of the rule is assumed to be costly to both theprincipal and the agent. The cost for the principal can be interpreted as thecost associated with the abrupt liquidation of the strategy. The portfolio issold in very bad conditions of liquidity and fundamentals. This cost can alsobe interpreted as the fund provider's opportunity cost of a risk-free portfolio.The cost for the trader can be interpreted as the cost due to the loss of prestigeand the bribe option implicit to the trader's activity. 3 Actually, all these costsdepend both on the trader's performance and on the intensity of theliquidation rule.We denote by s x 2 0; 1 the liquidation rule chosen by the fund provider.Given the stylized approach of the modelling, there is no loss of generality toassume that the liquidation cost incurred by the fund provider can be simplywritten b xs x. Similarly, we write d xs x the liquidation cost incurred bythe trader with b x; d x  0. Then, the fund provider's utility becomesx w b xs x and the trader's utility writes u w c a d xs x.Under these two extensions, the principal's problem becomes:
Stop-loss rules as a monitoring devicemaximizea 2 A;w ;s  UP a; w; s 
x
w x
317
b xs x f xja dx
subject to the incentive compatibility conditions:for all a 0 6 a; UA a; w; s 
u w x
d xs x f xja dx
 UA a 0 ; w; s;
c a0
IC aa 
the trader's individual rationality condition:u w x
d xs x f xja dx
c a  U
(IR)
and the trader's limited liability condition:w x  w
(LL)
^ w this problem and a^; w^ ; s^  a solution. 4 In a model ofWe denote Pperfect information, there is no rationale for the fund provider to enforce expost an inecient rule. No stop-loss rule (s  0) should be observed. But ifthere is some imperfect observability of the trader's strategy, the principal canbe induced to enforce s 6 0 in order to soften the incentive compatibilitycondition. This intuition is developed in the following.We ®rst prove that the optimal liquidation rule is necessarily binary: eithers  0 (no liquidation) or s  1 (full liquidation of the portfolio).Lemma 1^ w, then ^s x can take only two^ ; s^  is a solution of program PIf a^; w^ x > w, one has s^ x  0. Moreover, if w  1values: 0 or 1. When w^a   ; w   x; 0 is a solution of program P.Proof (see Appendix)This lemma says that without the limited liability condition, committing totake an ex post inecient decision does not actually enlarge the set ofadmissible incentive compatible contracts. Conversely, committing to expost ineciencies can be valuable only if the limited liability conditionholds. Thus, a stop-loss rule can only be optimal when the fund providerfaces some kind of limited liability of the fund manager.We need an additional assumption to insure stop-loss to be triggered only inbad performance states. This assumption is called decreasing ineciency (DI).Assumption DIFunctions d x and d x=b x are non-increasing.The fact that function d x is decreasing is quite natural. It says thatineciency of the stop-loss rule is decreasing with a higher performance of the
318
Advanced Trading Rules
portfolio. Liquidating a portfolio which generates positive pro®ts is simplyrealizing gains and should not generate costs for the fund manager. Thesecond part of assumption DI must be interpreted as the fact that the cost ofabrupt liquidation of the portfolio decreases more for the manager than forthe provider when the performance improves. Note that assumption DI issatis®ed when functions b and d are constants.
Result^ ; ^s  ofUnder assumptions MLRC, CDFC and DI, a solution a^; w^ necessarily satis®es that:program P^  is non-decreasing with respect to x(1) w(2) s^  is non-increasing with respect to x.ProofSee Appendix.Result 1 is depicted in Figure 11.1. This result has a straightforwardinterpretation in terms of stop-loss strategies. It concludes that the fundprovider, by committing to ex post ineciencies, 5 under the form of stop-lossrules, will be able to increase his pro®t when the limited liability constraint ofthe fund manager is binding. This rule ensures that stop-loss orders will occurin the lower range of the support of returns. As a consequence, the providerwill only use stop-loss orders when the trader's results are poor. Anotherinterpretation of the model is that the fund provider follows a `stick andcarrot' policy. To save on the carrot (the bonus), the fund provider can usesticks of dierent sizes and shapes. One big stick would be to ®re the trader.However, this is very costly for both the trader and the fund provider sincethe trader's human capital and knowledge are lost and, after the ®ring, theposition cannot remain unchanged. Liquidation of the position in extremesituations, without ®ring the trader is therefore less costly and still allows himor her to save on the bonus.s x  1w x
w x  w
s x  0
Figure 11.1 The optimal liquidation rule s x and the optimal reward w x
Stop-loss rules as a monitoring device
319
11.3 A TEST OF THE EXISTENCE OF STOP-LOSS STRATEGIESWe now provide empirical evidence in favour of the existence of stop-lossstrategies, using data on the French Treasury bond futures market, one of themost active contracts in the world. At this point, the focus is on stop-lossstrategies at the individual level of investment ®rms; the eect of stop-lossstrategies on the aggregate market equilibrium is reserved for future work.The empirical strategy has to face an identi®cation problem. It is necessary tode®ne a test of the existence of stop-loss strategies, as opposed to othertrading strategies that may have similar eects. In our view, a usefuldistinction is between strategies exclusively determined by prices, and thosewhich also depend on other factors. Hedging, portfolio rebalancing andpositive-feedback strategies 6 fall into the ®rst category. This is also the case,as will be stressed later, of the `neo-classical' model of demand for riskyassets. On the contrary, the second category includes stop-loss strategies,where realized losses become a crucial determinant of the demand for assets.However, it seems dicult to distinguish between the dierent explanationsof stop-loss strategies. In particular, as mentioned above, psychologicalmotives, highlighted by `prospect theory', imply traders' reluctance to realizelosses so that principals are also induced to use stop-loss orders to avoidcarrying over losing positions. The following subsections include:(1) a de®nition of stop-loss strategies(2) the description of the dataset used in the analysis, namely the end-ofday positions of all direct participants in the French Treasury bondfuture market(3) summary statistics on the accounts available in the dataset(4) the derivation of the two tests suggested to measure stop-loss orders.
11.3.1 DefinitionLoosely speaking, a stop-loss order means the liquidation of the investor'sportfolio following heavy losses. A stop-loss rule is associated with athreshold of losses under which stop-loss orders are triggered. In thefollowing, we de®ne formally a stop-loss rule. First, we denote by PO jt theposition of ®rm j  1; . . . ; J at date t  1; . . . ; T. (This is the number ofcontracts the trader has bought or sold.) DPt is the ®rst dierence in the priceof the contract, and R jt is the corresponding pro®t of ®rm j between datest  and t. The latter variable is simply de®ned as:R jt 
 1Xk0
PO jt k 1 DPt k
11:1
320
Advanced Trading Rules
Then, a stop-loss rule is described by the commitment:R jt  R j < 0
)
PO jt  0
11:2
which can be interpreted as saying that, below a given threshold, badperformances trigger liquidation of the portfolio. It is important to notethat the standard neo-classical model of demand for risky assets is unable toaccount for stop-loss rules. In this model, investors have dierent horizonsand dierent expectations. They determine their portfolios according to thecurrent price as given by:PO jt  O j E jt Ptnj
Pt 
11:3
where O is the ratio of their absolute risk aversion and the conditionalvariance of the price. Without revision of expectations, since Pt entersnegatively in Equation (11.3), the main implications of the neo-classicalmodel are:(1) the negative response of the position to an increase in prices, akin to a`pro®t-taking' behaviour(2) the positive response to a decrease in prices, akin to a `cost-averaging'behaviour.11.3.2 Data sourcesThe data used are end-of-day positions of all direct participants (seat owners)in the French Treasury bond future market (future contracts). They areaccounting data compiled by MATIF SA, the market's clearinghouse. Foreach of the 140 seat owners, we have information on their `customer' account(132 accounts) as well as on their `own' account (94 accounts, including theaccount of 9 market makers for the corresponding option contract). 7 Eachseat owner may have customers which are indirect participants to the futuremarket. The `customer' account of a seat owner is an aggregate account of theend-of-day positions of all its customers. It is also important to note that the`own account' of a direct participant may also be an aggregation of dierentprivate accounts, including his or her error account, if any. All accounts areanonymous, only identi®ed by a code number and by the statute of the ownerof the seat (bank, broker, `local', Globex-associated dealer, interbank marketdealer). For each account, daily positions (short/long) in number of lots on allcontracts are available from 1 May, 1993, to 31 March, 1995 (565 tradingdays). Actually contracts on four dierent maturities are available at eachpoint in time (e.g. March, June, September, December), so that PO jt iscomputed as the combined position on the dierent contracts, namely:PO jt 
4X
m1
PO jt;m
11:4
Stop-loss rules as a monitoring device
321
Similarly, the pro®t is computed using Pt;m , the price of the contract ofmaturity m and Equation (11.1) becomes:R jt 
4 1XX
PO jt k 1;m DPt k;m
11:5
k0 m1
Such a comprehensive view of investors (for all maturities) oers a bettermeasurement of investors' eective exposure to price movements in the bondfuture market, and takes into account the common strategy of buying thedierent maturities of the contract if one anticipates non-parallel shifts in theterm structure of interest rates (in that case the notional position for eachmaturity has little meaning). Since positions are given in terms of futurecontracts, the total net aggregate position is necessarily null: the equalitiesPP jj8j PO t  0 andj R t  0 hold at each date t:11.3.3 Summary statisticsSeveral remarks can be made concerning the relative size of the dierentaccounts and their time-series properties.First, the market is highly concentrated. On the one hand, among non-zeroaccounts at each date (i.e. for which PO jt 6 0 at date t), the top quartile of®rms ranked by jPO jt j hold about 80 per cent of the daily open interest. 9 Onthe other hand, when the average position is computed from the ®rst to thelast day where a ®rm is active (i.e. PO jt 6 0), 7.2 per cent of the accounts,holding more than 6000 lots in average (0.6 billion dollars), share about 50per cent of the daily open interest, with a high variability (the minimum is 40per cent and the maximum is 70 per cent). Concerning the nature of theposition, the smallest accounts at each date are slightly more likely to have ashort than a long position. When accounts are ranked by jPO jt j, the accountsbelow the median are short with a probability of 40±45 per cent. However,according to the Student t-test, the proportion is not signi®cantly dierentfrom 50 per cent (t  1.49).Second, an important feature of the data is that a signi®cant number ofaccounts are either inactive for most of the time or active during a fraction ofthe sample period only (this includes the opening of new accounts, but alsothe closure of accounts, at the beginning/end of the calendar year). 10 Asindicated in Table 11.1, 122 out of 226 accounts (53.9 per cent) are in thatcase. This includes 27 accounts present in the market less than 30 days, as wellas 95 accounts which, from the ®rst to the last day of presence have a positiondierent from zero less than 95 per cent of the time. Among the latter 95accounts, some share a very similar pattern in the sense that they exhibit azero position most of the time and a few extremely short-lived peaks and
322
Advanced Trading RulesTable 11.1 Results from Dickey±Fuller unit root tests
I 0+intercept I 0
Present<30 days
30 days  active< 95%(error accounts) Total
Accounts
I 1
I 0trend
Customer accountsOwn accounts± Market makers± Non-marketmakers
19
10
6
30
17
50 (38)
132
111
14
±6
412
19
2 (1)45 (36)
985
31
15
12
46
27
95 (75)
226
Total
troughs. Two explanations of this behaviour are possible. Either theycorrespond to the positions of ®rms with a very short horizon (one day orless), but in that case stop-loss strategies cannot be tested; or they areeectively what is known as `error accounts'. Seventy-®ve accounts appearunder this heading in Table 11.1. Inspection of their graph reveals asystematic behaviour: those operators have a target position equal to zero,but may miss their target and adjust their position back to zero the followingday. Therefore, they use the following rule: PO jt 1  0 with probability p, andPO jt 1  "t with probability 1 p. These accounts are not very wellapproximated by a white noise process, since the realization PO jt  0 shouldnot obtain with such a high frequency. Among them, most of the customeraccounts are locals or Globex-associated dealers, whose horizon is extremelyshort.This justi®es why in the following only the 104 remaining accounts will beinvestigated. Using Dickey±Fuller unit root tests on individual positions, itturns out that 31 accounts are non-stationary, whereas 73 are I 0 asindicated in Table 11.1. 1111.3.4 Tests of stop-loss strategiesThe formal de®nition of a stop-loss rule provided earlier is of little help inuncovering stop-loss strategies when data are observed at an aggregated level.In most cases, it seems dicult to disentangle the dierent strategies (hedging,arbitrage, speculation) followed by investment ®rms on their own account. Inother cases, data can be only observed with aggregation of dierentinvestment ®rms on the same customer account. For these reasons, weprovide tests which can only validate less restrictive de®nitions of stop-lossrules. Loosely speaking, our intent is to prove that poor past performancesare signi®cant determinants of current positions. Two tests are presented. The
Stop-loss rules as a monitoring device
323
®rst one looks for evidence of holding reversion behaviours associated withvery negative pro®ts. The second is based on causality tests of lowperformance indicators in the contract holding equation.Holding reversion behaviourThe ®rst method measures the eect of large losses on the likelihood ofclosing down a position. Since investors are allowed to choose between a longor a short position, closing down a position means reverting to the zeroposition, so that the movement is in opposite direction for long and shortpositions. Hence, the relevant indicator is DPO jt =PO jt 1 . However, PO jt 1may be very close to zero during some periods. In addition, due toheterogeneity of investors, one faces aggregation problems with the indicatorsgn PO jt 1 DPO jt when data are pooled across time and investors. We ®nd it,therefore, more convenient to concentrate on another statistic, namely:z jt  sgn PO jt 1 DPO jt which can take the values: 1; 0; 1. Only the valuereversion to the zero position. We also de®nez jt  z jt 1
jfjDPO jt j>s g
11:61 can be assimilated to a11:7
where  j is the standard deviation of DPO jt , s is a constant that we choosejjarbitrarily, and 1j j is a dummy variable equal to 1 when jDPO t j > sfjDPO t j> s g
and 0 otherwise. This indicator focuses on signi®cant changes in positions.In accordance with our previous remarks, under the neo-classical modelPrfz jt < 0g and Prfz jt > 0g only depend on changes in prices. However, giventhe correlation between changes in prices and realized one-period pro®ts, anadditional implication of the neo-classical model is the negative correlationbetween z jt and realized one-period pro®ts. Therefore, according to thismodel, z jt should be positive (respectively negative) more frequently whenone-period pro®ts are negative (respectively positive). Conversely, if stop-lossrules are followed by investment ®rms, Prfz jt < 0g should be higher whenpro®ts are negative. Based, on the preceding remark, our test computes thedistribution of Prfz jt g conditional on dierent classes of realized pro®ts, usingpooled data. If stop-loss strategies are eectively used, the conditionaldistribution should give more weight to the event z jt < 0 for the lowest classesof pro®ts. The ®rm's investment horizon may well exceed one day, so that thisproperty is expected to remain true when pro®ts over several periods areconsidered.To con®rm the preceding analysis, we use logistic regressions measuring theeect of pro®t dummy variables X jt on the probability of reversing a position.The dependent variable, z jt or z~ jt is still de®ned as a discrete indicator, taking
324
Advanced Trading Rules
the value 1, 0 or 1. In the case of a binary response model, we assume that z jtjtakes the value 1 whenever a latent variable z jt is such that z t > r1 .jjOtherwise, z t is equal to 1. z t depends linearly on realized pro®ts:jjjjjz jt  X t b  u t , where u t is a measurement error on z t . u t is assumed to beidentically logistically distributed. Finally:Prfz jt 
1jX jt g  Prfu jt > r1
X jt bg  1
F r1
X jt b
Causality of poor performanceTo test the eect of pro®ts on positions, we now use a non-structuralapproach. We allow current holdings to be partially determined by pastholdings, but we do not take a stand