on whether positions should bestationary or not. We, therefore, ®t an autoregressive model on positions,adding price changes in the equation, as advocated by the neo-classicaltheory. We also introduce a pro®t indicator and test whether currentperformance, when poor, helps to explain current positions. Our empiricalstrategy does not restrict the value of the coecient on PO jt 1 so that wede®ne tests compatible with stationary as well as non-stationary positions.When PO jt is stationary, we run for each ®rm the regression:PO jt 
0
5Xk1
jik PO t k 
i
DPt 
j
PO jt 1 1fR j <c%g   jtt
11:8
where 1fR j <c%g is a dummy variable equal to 1 if R jt is lower than the bottomtc% of ®rm j 's realized pro®ts and 0 otherwise. The null hypothesis of no stoploss is H0 : j  0. The alternative hypothesis of stop-loss is H1 : j < 0.Under H1 , the position at t is smaller when realized pro®ts are very negative.In that case, the coecient on PO jt 1 is reduced by 1 . When the process ofPO jt contains a unit-root, Equation (11.8) is run in ®rst dierence:DPO jt 
0
jj1 DPO t 1 
j
DPt 
j
PO jt 1 1fR j <c%g   jtt
11:9
11.4 EMPIRICAL RESULTSAs justi®ed before, the two types of tests de®ned in the preceding section arenot run on the whole data set, but only on the sample of the accounts activemore than 95 per cent of the time (104 accounts). In particular, this sampledoes not include the error accounts. In this sample, each account will betreated as a trading unit.
Stop-loss rules as a monitoring device
325
11.4.1 Conditional distribution of z jt on pooled dataThe ®rst test is implemented by computing, for each account, the distributionof pro®ts and losses on the whole sample period. We de®ne classes of pro®tsand measure the probability of getting the dierent occurrences of z jt and z~ jt .Stop-loss orders are likely to occur, as shown in section 11.2, for extremelylow values of pro®ts. Hence, we condition z jt by dierent classes of realizedpro®ts. To avoid drawing conclusions from a very small subsample ofconditioned observations for each trading unit, we pool data across allaccounts. We compute pki , the proportion of realizations k  1; 0; 1 of z jtin each pro®t class i: 12 For each trading unit, 7 classes are de®ned: the 1 percent lowest pro®ts, the fractions 1±10 per cent, 10±25 per cent, 25±75 per cent,75±90 per cent, 90±99 per cent and the top 1 per cent pro®ts. For a giventrading unit and for each class of pro®t, the number of realizations of thedierent occurrences of z jt and z~ jt are computed. Then, aggregating thoseresults across all 104 trading units, class by class, one gets the conditionaldistribution of the proportion of the dierent occurrences of z jt and z~ jt for thewhole population.Note however, that in order to run the test, we remove from the sample allobservations corresponding to PO jt  PO jt 1 as well as the observationswhere PO jt 1  0. The aim of these exclusions is to avoid taking into accounthedging positions, in the ®rst case, as well as zero pro®ts, in the second case.As a consequence, z jt has two occurrences instead of three andPrfz jt > 0g  1
Prfz jt < 0g
The results for the conditional distribution of z jt are exhibited in Figure 11.2and Table 11.2 for pro®ts measured over one period. The bottom part of thegraph is Prfz jt < 0g. The dashed line is the unconditional proportion and thesolid line is the conditional distribution. Only the relevant window isdisplayed. The graph reveals general characteristics of trading as well asevidence in favour of stop-loss strategies.Concerning the general characteristics of trading, the ®rst remark that canbe made is that the unconditional distribution is slightly biased in favour ofz jt < 0 (54 per cent versus 46 per cent). Therefore, positions in our data setexhibit mean-reversion. One possible explanation is that ®rms active in theTreasury bond future market have to meet daily deposit requirements by theclearinghouse. Most ®rms have a target position, so that, when the target ismissed, the constraint of deposit requirement entails a liquidation of theposition, independently of the level of realized pro®ts. On the other hand, dueto aggregation problems positions may still be aected by noises and includesome components of `error accounts' ± for which the correlation betweenDPO jt and PO jt 1 is negative with probability 1 p, while DPO jt  PO jt  0
ClassFigure 11.2 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over 1 day)
Stop-loss rules as a monitoring device
327
Table 11.2 Conditional distribution of z jtPrfz jt < 0 j R j1t 2 ci gci are classes of one-period realized pro®tstu : Student t of dierence of proportion with uncond. proportiontc : Student t of dierence of proportion with 25±75% classt1 : Student t of dierence of proportion with 1% classt99 : Student t of dierence of proportion with 99% class104 accountsT  J  43 176 observationsci
p
p
tu
Unconditional proportion
54.06
0.002
±
R jt < 1%1%  R jt < 10%10%  R jt < 25%25%  R jt < 75%75%  R jt < 90%90%  R jt < 99%R jt  99%
63.2450.7651.5252.5558.0060.9765.92
0.0240.0080.0060.0030.0060.0080.021
3.634.124.094.446.368.635.28
tc
t1
t99
±
±
±
4.182.051.45±7.719.685.88
±4.704.49±±±±
±±±±3.442.13±
with probability p ± which induces a bias in the distribution of z jt in favour ofz jt  0 or z jt < 0 . As a consequence, our task is to prove that the global meanreversion behaviour of market participants is more pronounced for badrealizations of pro®ts.Figures 11.2 to 11.4 show that the conditional distribution of theoccurrence z jt < 0 is consistent with the existence of stop-loss strategies: theproportion of mean reverting changes in position is much higher in thebottom 1 per cent pro®t class (60.9 per cent) than in the unconditionalproportion (54 per cent), or the central 25±75 per cent pro®t class (52.6 percent) (the two-standard deviation band is drawn around the point estimates).It is also higher than in the adjacent classes (the Student t-statistic ofdierence of proportion is highly signi®cant, see Table 11.2).On the other hand the conditional distribution exhibits some of the basicfeatures of the standard neo-classical model of trading: except for the ®rst(bottom) class, the likelihood of the occurrence z jt < 0 is increasing withhigher classes of realized pro®ts. When pro®ts are very positive, investment®rms reduce their position in 66.1 per cent of the cases and realize theirpro®ts.
ClassFigure 11.3 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over 5 days)
ClassFigure 11.4 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over 15 days)
330
Advanced Trading Rules
As shown in Figures 11.3 and 11.4, when pro®ts are computed on a longerperiod (  5 and 15), stop-loss eects remain apparent, albeit in a weakerfashion. This result suggests that our ®ndings are independent of continuoushedging strategies, which would mainly appear at intradaily frequencies andwould be inexistent at other frequencies.Also notice the asymmetry between pro®ts and losses, with positionreversion concentrated in the lower bound of the distribution of pro®ts.Stop-loss orders are, therefore, triggered after signi®cant deviations in termsof pro®ts.To con®rm the robustness of the results and to avoid taking into accountin®nitesimal changes of positions that may blur the picture, the same test isrun for the indicatorz~ jt  z jt 1
j
fjDPO jt j> g4
This indicator excludes a high proportion of observations with very smallchanges (38 per cent of the changes in positions are smaller than  j =4). InFigures 11.5, 11.6 and 11.7, the two dashed lines are the unconditionalz jt > 0g (upper line). Theproportion of z~ jt < 0 (lower line) and 1 Prf~three solid lines correspond to the conditional distribution of the threeoccurrences z~ jt < 0, z~ jt  0 and z~ jt > 0. The occurrence z~ jt  0 correspondsto small changes in positions, either positive or negative.As in the case of z jt , the conditional distribution of z~ jt < 0 is V-shaped; itoers evidence in favour of pro®t-taking and stop-loss behaviours: in thebottom class of realized pro®ts, the proportion of z~ jt < 0 is 52 per cent versus34.6 per cent for the unconditional distribution and 31.2 per cent in thecentral class (see Table 11.3). The comparison of Figures 11.2 and 11.5 revealsthat, when pro®ts are very negative, the bulk of mean reverting positionchanges (z jt < 0) is largely supported by large position changes.The results of logistic regression models are exhibited in the appendix(Tables A.11.1 and A.11.2). For details see `Holding reversion behaviour'earlier. The pro®t dummy variables have a strong eect on z jt  1 for thebottom 1 per cent pro®ts and the top 25 per cent (i.e. above the 75th centile)in the binary response model. In the ordered response model they are the onlysigni®cant variables, with the largest eect for the ®rst and last classes.To conclude this section, we interpret the results presented so far asevidence in favour of stop-loss strategies. Although one may argue that thetests are simply tests of mean reversion, there is, to our knowledge, noalternative convincing explanation of the strong correlation between lowpro®ts over dierent horizons and reversion of positions.
ClassFigure 11.5 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over one day);Ð middle proportion is 0:25 std of position changes
104 accounts
ClassFigure 11.6 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over ®ve days);middle proportion is 0:25 std of position changes
104 accounts
ClassFigure 11.7 Indicator of sellos ± accounts active more than 95 per cent of the sample period without constant positions conditioned by pro®ts (over 15 days);middle proportion is 0:25 std of position changes
334
Advanced Trading RulesTable 11.3 Conditional distribution of z~ jtPrf~z jt < 0 j R j1t 2 ci gci are classes of one-period realized pro®tstu : Student t of dierence of proportion with uncond. proportiontc : Student t of dierence of proportion with 25±75% classt1 : Student t of dierence of proportion with 1% classt99 : Student t of dierence of proportion with 99% class104 accountsT  J  43 176 observations
ci
p
p
tu
Unconditional proportion:z~ jt  1z~ jt  1
34.6127.80
0.0020.002
±±
R jt < 1%1%  R jt < 10%10%  R jt < 25%25%  R jt < 75%75%  R jt < 90%90%  R jt < 99%R jt  99%
51.9335.9333.5331.2138.2943.7453.55
0.0250.0080.0060.0030.0060.0080.022
7.181.721.8210.516.2211.958.84
tc
t1
t99
±±
±±
±±
8.715.803.52±10.6415.2810.54
±6.217.41±±±±
±±±±6.684.12±
11.4.2 Causality tests on individual dataWe now go more deeply into our assessment of stop-loss strategies by:(1) controlling for price changes(2) providing estimates of the eect of losses on individual accounts.Using the framework of Equations (11.8) and (11.9), we carry out a t testseparately on each individual unit to determine whether the introduction ofrealized losses improves the forecasting power of the regression. 13We present the results for the sample of accounts active more than 95per cent of the time, distinguishing I 0 and I 1 accounts. In Table 11.4, wetally the results on each account to measure how widespread the causality is.We then discuss the robustness of the results, by introducing alternativespeci®cations.The RHS variables in the Equations (11.8) and (11.9) are the price changes,the lagged endogenous variable and the realized pro®t dummy variables.Price changes are computed by selecting the price of the most active contractat each date and removing the step that appears when the contact is changed
Stop-loss rules as a monitoring device
335
Table 11.4 OLS estimates on individual accounts: equations (11.8) and (11.9)`Tally' of accounts where realized pro®ts are signi®cantStudent t test signi®cantly dierent from 0 at 10% levelI 0 accounts ± total number: 73Number of accountsj
j
>0
<0
Def. of RHS (c%)
nb accounts
%
nb accounts
%
R jt < 1%R jt < 5%5%  R jt < 10%10%  R jt < 15%
5887
6.811119.6
161583
21.920.5114.1
85%  R jt < 90%90%  R jt < 95%R jt  95%R jt  99%
3379
4.14.19.612.2
661614
8.28.221.919.2
I 1 accounts ± total number: 31Number of accountsj
j
>0
<0
Def. of RHS (c%)
nb accounts
%
nb accounts
%
R jt < 1%R jt < 5%5%  R jt < 10%10%  R jt < 15%
2122
6.43.26.46.4
8822
25.825.86.46.4
85%  R jt < 90%90%  R jt < 95%R jt  95%R jt  99%
0000
±±±±
53103
16.19.732.29.7
(for details, see Bensaid and Boutillier, 1995). We use Akaike criterion todetermine the optimal number of lags in the AR part, with a maximal laglength of ®ve. Concerning the dummy variables, dierent indicators ofrealized pro®ts are introduced as shown in Table 11.4.For I 0 accounts, the top panel of Table 11.4 shows that, out of 73accounts, there is a signi®cant eect of pro®ts for 28.7 per cent of theaccounts. Among them, 16 (21.9 per cent) exhibit a signi®cant negative eectof low realized pro®ts (bottom 1 per cent and bottom 5 per cent). The eect of
336
Advanced Trading Rules
losses is statistically less signi®cant for upper classes of realized returns: only 3accounts react negatively to pro®ts in the 10±15 per cent class. This con®rmsthe idea that stop-loss occurs only for extremely low pro®ts. Evidencecontradicting such a strategy is scarce: very negative pro®ts have a positiveeect on positions for only 5 accounts (6.8 per cent). As predicted by the neoclassical theory, the highest realizations of returns also have a negative eecton positions: it is the case for 14 accounts (19.2 per cent).In the case of I(1) accounts, 8 out of 31 accounts (25.8 per cent) exhibit arestrictive eect of negative pro®ts on position changes. Evidence in favour ofstop-loss strategies is therefore stronger in the sample of I 1 accounts thanfor I 0 accounts. On the contrary, the impact of very positive pro®ts onsellos is somewhat weaker than in the I 0 case: the indicator of pro®ts inthe top 99 per cent class is signi®cant in 3 cases only (10 in the top 5 per centclass).To check the robustness of the results, we investigate more thoroughly theimpact of changes in prices, which are expected to be signi®cant according tothe standard neo-classical theory. First, the variable DPt =Pt 1 is substitutedto variable DPt , since Pt ¯uctuates in a narrow range, with very similar result.Second, we proceed to test how signi®cant the coecient of the pro®tindicator is. We use the subsample of accounts for which the neo-classicaltheory is not strongly rejected (i.e. where the coecient on prices is notpositive). Table A.11.3 in the Appendix shows that on 60 I 0 and 24 I 1accounts (columns t < 0 or n.s.), the pro®t indicator has a negative sign(t < 0) in one out of four cases.11.5 CONCLUSIONSAlthough we only investigate one (large) market, the French Treasury bondfutures market, the paper provides evidence in favour of stop-loss strategies,that we interpret as optimal responses to agency con¯icts inside theinvestment ®rms. Both types of tests presented here are consistent withmean-reversion of positions when performance is poor. Since sellos areassociated with very negative pro®ts, it appears that at least some ®rms usestop-loss strategies. Empirical evidence shows that around 20 per cent of the®rms follow that kind of strategy. Given the level of the threshold of pro®tstriggering stop-loss orders, the evidence favours the explanation of stop-lossorders by moral hazard rather than adverse selection. In the latter case,selection of traders is likely to induce smaller loss limits than the one we haveuncovered. Unfortunately, we cannot distinguish between the implications ofthe optimal level of liquidation under moral hazard and psychologicalmotives.
Stop-loss rules as a monitoring device
337
Such ®ndings may have systemic consequences, since sellos induced bystop-loss orders in a few ®rms may rapidly spread to a signi®cant proportionof other ®rms, triggering large adjustments in the market. However, otherstudies have shown that the market was ecient during the period underreview (Bensaid±Boutillier, 1995). We can, therefore, conclude that themarket was resilient enough to absorb the potentially destabilizing eect ofstop-loss strategies.11.6 STATISTICAL APPENDIXTable A.11.1 Logistic regression: binary response modelz jt  sgn DPO jt PO jt 1   1; 1logit Prf y jt  1g  X jtwhere X jt  X1 ; . . . ; X6  are pro®t dummy variablesNumber of observations: 43 176 2 6  497:310 ( p-value  0:0001)Variable
Estimate
Wald statistic
X1: R j1t  1%
p-value
0.5425
26.61
0.0001
X2: 1%  R j1t  25%
0.0495
6.33
0.0118
0.1030
57.07
0.0001
X4: 75%  R j1t  90%X5: 90%  R j1t  99%
0.32360.4460
164.75183.82
0.00010.0001
0.6599
48.22
0.0001
X3: 25%  R j1t  75%
X6: R j1t  99%
Table A.11.2 Logistic regression: ordered response modelz~ jt  sgn DPO jt PO jt 1 1jDPO j j j =4  1; 0; 1tlogit Prf~z jt  1g  X jtwhere X jt  X1 ; . . . ; X4  are pro®t dummy variablesNumber of observations: 43 176 2 4  254:32 ( p-value  0:0001)VariableFirst interceptSecond interceptX1: R j1t  1%
Estimate0.71400.8856
Wald statistic
p-Value
2836.155599.51
0.00010.0001
0.5650
34.85
0.0001
X2: 75%  R j1t  90%
0.2144
72.38
0.0001
0.3667
135.07
0.0001
X4: R j1t  99%
0.6860
63.79
0.0001
X3: 90%  R j1t  99%
338
Advanced Trading RulesTable A.11.3 OLS estimates on individual accounts: prices and pro®ts variablesComparison of signi®cance of j andt : Student t for DPtt : Student t for 1fR j <c%g
j
1;t
I 0 accountst <0
t n.s.
t >0
Total
nb
% column
nb
% column
nb
% column
nb
% column
t >0t <0
13
4.112.5
412
11.133.3
01
±7.6
516
6.821.9
Total (% line)
24
100 (32.8)
36
100 (49.3)
13
100 (17.8)
73
100 (100)
I 1 accountst <0
t n.s.
t >0
Total
nb
% column
nb
% column
nb
% column
nb
% column
t >0t <0
13
7.121.4
13
10.030.0
02
±28.5
28
6.425.0
Total (% line)
14
100 (19.1)
10
100 (32.2)
7
100 (22.5)
31
100 (100)
Table A.11.4 Whiteness of the residuals in equations (11.8)±(11.9)
I 0 : PO jt 
j0 
I 1 : DPO jt 
5Xk1
j0 
jj1;k PO t k 5Xk1
j
DPt 
jj1;k DPO t k 
j
j
PO jt 1 1R j <1%  " jt1t
PO jt 1 1R j <1%  " jt1t
Test of whiteness of " jt% of accounts where p-value is greater than 1% or 5%Final unit root diagnostic:I 0
I 1
Degrees of freedom
p > 1%
p > 5%
p > 1%
p > 5%
612182430
94%96%94%92%92%
83%93%93%92%90%
100%96%93%93%93%
90%86%93%93%93%
Table A.11.5 Whiteness of the residuals in Dickey±Fuller unit root tests
Eq.: DPO jt 
j

j
t  PO jt 1 
4Xk1
jjjk DPO t k  " t
Test of whiteness of " jt% of accounts where p-value greater than 1% or 5%Final unit root diagnostic:I 0 with trend and intercept
I 0 with intercept
I 0 without intercept
I 1 without trend, or intercept
Degrees of freedom
p > 1%
p > 5%
p > 1%
p > 5%
p > 1%
p > 5%
p > 1%
p > 5%
612182430
100%100%93%93%87%
100%93%93%80%87%
100%100%92%100%92%
100%100%92%92%92%
100%98%96%96%96%
100%98%93%76%74%
100%97%97%93%97%
100%97%93%93%93%
340
Advanced Trading Rules
11.7 MATHEMATICAL APPENDIXProof of Lemma1Let us denote  a 0  the multiplier associated with the incentive compatibility0condition IC aa ,  the multiplier associated with the trader's individualrationality constraint IR and  x f xja the multiplier associated with the^ wtrader limited liability condition LL. Then the Lagrangian of program Pis given by:L
x
w x
b x f xja dx
X
 a0
u w x
X
 a 0  c a

a 0 6a
0
a 6a
d xs x f xja
f xja 0  dx
c a 0    u w x d xs x f xja dx c a
  x w x w f xja dxA necessary condition for a; w x; s x to solve the principal agent problemis that:@L 0 and@w x
@L 0 if 0 < s x < 1;@s x 0 if s x  1;
 0 if s x  0
Let us compute@L@w x
@L@Land. One has:@w x@s xXf xja  a 0 u 0 w xa 0 6a
 f xja
and@L@s x
f xja 0   u 0 w x f xja   x f xja
d x @Lu 0 w x @w x
f xja b x  1
d x x 0u w x

@Lis thenThus, if w x > w, the exclusion condition gives  x  0 and@s xequal to:d xf xja b x  0<0u w xwhich in turn implies that s x  0. Moreover, when s x  1, one hasnecessarily  x > 0 and thus w x  w. Finally, since w x is determined
Stop-loss rules as a monitoring device
341
@L@L 0,is either strictly positive or@w x@s xstrictly negative and, therefore, s x can only take the values 0 or 1. Q.E.D.by the necessary condition
Proof of result 