endogenous trading rules, we use the Equation (10.5)as de®ned in Acar and Lequeux (1995):Ft 
mX1
m
j1
jXt j1  Ft  Pt
mX1
Pt jmj0
10:5
In the description of system A (endogenous) we de®ne the moving average oforder m by the function:9m 1X>>>Ft m jXt j1>>=j110:6>>Xt  Ln St =St 1 >>>;St  spot price of the currency pairIn the description of system B (exogenous) the exogenous trading rules aresimilar to the moving-average endogenous trading rule but for the source ofthe signal:Ft 
mX1
m
10:7
jYt j1
j1
With Yt  It for the exogenous trading rules using only one of the twointerest rate series to generate their signal and Yt  Idt for the ones using theinterest rates dierential. It and Idt are de®ned respectively in Equations(10.2) and (10.3). The return Rt generated by the trading rules at time t iscalled `rule return' and its value expressed as in Equation (10.8).Rt  Xt if Ft 1 < 010:8Rt  Xt if Ft 1 > 0where Xt is the price change during a given period t
1; t.
10.3.2 The time horizon of the trading rulesTo chose the various order of the trading rules that we would test, we ®rstneed to recognize that not all investors work their strategies over the sametime horizons. For example, a day trader can trade anonymously with a
300
Advanced Trading RulesTable 10.9 Trading rules correlation

S 2
S 2
1
S 3S 5
S 3
S 5
S 9
S 17
S 32
S 61
S 117
0.705
0.521
0.378
0.272
0.196
0.142
0.102
1
0.71
0.512
0.366
0.264
0.19
0.137
1
0.705
0.501
0.361
0.26
0.187
1
0.699
0.501
0.359
0.258
1
0.707
0.504
0.361
1
0.705
0.502
1
0.704
S 9S 17S 32S 61S 117
1
pension fund; the former trades frequently for short-terms gains, the lattertrades infrequently for long-term ®nancial security. Each participates simultaneously and each diversi®es the other. To adequately represent all timehorizons in our study, we use the statistical properties of technical indicators(Acar and Lequeux, 1996). Assuming that an underlying time series, Xtfollows a centred normal law, the returns R1;t and R2;t generated by movingaverages of order m1 and m2 exhibit linear correlation coecient R given by:10Min m;m212XBm1 i 1 m2 i 1 CCBCB2i0B10:9R m1 ; m2   ArcsinBvvCCuumCBumX2 21 2Xu@tm2 i 1 2Am1 i 1 2 ti0
i0
For example it is possible to evaluate the level of correlation between the cash¯ows generated by two moving averages of respective order ®ve and nine byusing Equation (10.9), S 5; S 9 would be 0.705 independently of theunderlying series. Table 10.9 shows the coecient of correlation between therule returns triggered by two dierent moving averages applied to the sameunderlying market.The orders of the rules have been chosen such that they are almostequicorrelated (around 0.70) under the normal random walk assumption.As such, we can demonstrate that the simple moving averages of order {2, 3,5, 9, 17, 32, 61, 117} are representative of holding periods from 2 to 120 days.It is possible to demonstrate that the reduction in risk of such an equallydistributed portfolio is very close to that which would be obtained by using
Informative spillovers in the currency markets 301quadratic minimization on the whole universe of moving averages of order 2±120. Therefore, it can be said that our elementary basket of trading rulesachieves satisfactory diversi®cation when compared to an optimal portfolioof moving averages. In addition, from a practitioner point of view, an equallyweighted portfolio of trading rules is much easier to implement than aminimum variance portfolio involving heavily unbalanced and non-tradableweights.10.3.3 Transaction costsBecause we wanted to look at the performance of the forecaster and not theoverall return of a trading strategy, the results expressed in the followingtables do not take into account the cost of transactions that would begenerated by using such trading rules. This cost can be easily evaluated byusing the statistical properties of technical indicators (Acar and Lequeux,1995). The expected number of round turns generated by a moving-averagemethod of order m supposing that a position is opened at the beginning of theperiod and that the last position is closed at the end day of the period, is:1 1E N  1  T 2arcsin F 10:102 whereF 
mX2
m
i
1 m
i
2
Xm 2
m
i
1 2
i0
i0
if m  2. Then the expected value of a transaction costs is given by:E TC 
cE N where c is the trading cost per round turn
10:11
Table 10.10 indicates the expected number of round turns in a year of 250days under the random walk assumption. The last column gives the resultingyearly cost in percentage terms for a cost equal to c  0.03 per cent per roundturn.10.3.4 Performance evaluationTo evaluate the robustness of our results, we use the Student t-statistic whichis generally accepted by academics and practitioners to test the hypothesisthat the returns generated by technical analysis are zero. Its popularity is duein part to its simplicity as well as its intuitive appeal. It can be de®ned as:T
p RNR
10:12
302
Advanced Trading Rules
Table 10.10 Expected number of round turns and cost under the random walk assumptionMoving average E N  Expected number of round turns Expected yearly cost %c  0:03%S 2S 3S 5S 9S 17S 32S 61S 117
125.0092.5267.4048.6234.9225.4618.6213.68
3.752.772.021.461.050.760.560.41
Assumes a year of 250 trading days.
with N number of (daily) observations, R the average (daily) rule returns andR the standard deviation of (daily) rule returns.If T is above 1.645, the returns are said to be signi®cantly positive at thecritical threshold of 5 per cent. If T is above 0.841, the returns are said to besigni®cantly positive at the critical threshold of 20 per cent (that is, 5 per centand 20 per cent probability, respectively, that this conclusion is incorrect).It is interesting to note that the t-statistic is similar to a ratio widely used bythe managed funds industry, the Sharpe ratio ± Equation (10.13). It is generalpractice to report performance in absolute terms as well as in a risk adjustedform (De Rosa, 1996; Murphy, 1990). It has been shown that since investorsare, generally speaking, risk averse, a good measure of performance would bethe ratio of reward per unit of risk taken (Murphy, 1986). Sharpe de®ned thedierence between the return on a risky asset and the risk-free return onanother secure asset as a good measure of the reward and the variance of thereturn on the asset as being an appropriate measure of risk. The formula ofthe Sharpe ratio is:Sharpe ratio 
R
RfR
10:13
with R the annualized return of the trading rule, Rf , the annualized risk freereturns of the asset under management, and R annualized standard deviationof (daily) rule returns.One of the drawbacks of the Sharpe ratio compared with the t-statistic isthat it is not weighted by the number of observations. Hence, it does notproperly re¯ect the impact of time and does not reward long-term performance. The Sharpe ratio will mechanically decrease over time. Neither ratiocan distinguish between intermittent and consecutive losses. This is becausethe measure of risk (standard deviation) that they both use is independent onthe order of the data. Various attempts have been made to design a modi®ed
Informative spillovers in the currency markets 303measure to overcome this shortcoming, but as to date such proposals havebeen unable to retain the simplicity of the t-statistic and the Sharpe ratio,which has impeded their acceptance and implementation.
10.3.5 Performance of the trading rulesFor each currency we look at the gross returns generated by the exogenoustrading rule from the point of view of each interest rate, as well as from theirdierential we discussed previously in the data section (Tables 10.11, 10.13and 10.15). The period where the absolute dierential between the interestrates of the currencies involved is below 2 per cent has been highlighted ingrey on the Figures 10.10 to 10.12. We also show the correlation between theendogenous and exogenous trading rules of same order (Tables 10.12, 10.14and 10.16).Overall, it would seem that the exogenous trading rules performed betterthan the endogenous ones over the period studied. We note that there weremany more positive and signi®cant t-statistics for the exogenous trading rulesthan for the endogenous ones. The USD interest rate would seem to explainmost of the pro®tability of the endogenous systems for the USD-DEM andUSD-GBP parities. Nevertheless, the best results are obtained when using theinterest rate dierential. The volatility of the cash ¯ows is similar forendogenous and exogenous trading strategies. One of the most interestingfeatures for a system trader is the level of correlation between endogenousand exogenous trading rules, which would provide a high level of expecteddiversi®cation. The performance outside the 2 per cent interest rates dierential bands appears to be superior, but further study should be made toverify this point. Most of the short-term rules would be eliminated because ofthe high level of transactions and hence costs that they generate.
10.4 CONCLUSIONSThe use of interest rates in the design of exogenous trading rules can representa simple and ecient tool for systems traders. Not only can it provide a newsource of information for forecasting, but also provide an improved level ofdiversi®cation within a multisystem 4 environment. The study of the relationship between markets aords new areas of research for a technical trader,whether the focus is on the issues of forecasting, diversi®cation, assetallocation or risk management.
Table 10.11 Results of the trading rules for USD-CHFEndogenous rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
2.55%13.13%0.41
5.88%13.13%0.94
9.00%13.12%1.44
9.30%13.12%1.48
5.28%13.13%0.84
3.23%13.13%0.52
1.77%13.13%0.28
1.29%13.13%0.20
3.21%9.08%0.74
Euro USD rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.49%13.13%0.71
8.15%13.12%1.30
5.97%13.13%0.95
9.98%13.12%1.59
15.84%13.09%2.53
12.89%13.11%2.06
6.62%13.13%1.05
3.86%13.13%0.62
8.47%9.80%1.81
Euro CHF rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.15%13.13%0.66
1.95%13.13%0.31
2.68%13.13%0.43
10.71%13.12%1.71
7.39%13.12%1.18
11.28%13.11%1.80
16.92%13.09%2.70
11.81%13.11%1.88
8.36%10.07%1.74
Di. rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
1.96%13.13%0.31
8.30%13.12%1.32
6.86%13.13%1.09
11.74%13.11%1.87
15.29%13.10%2.44
12.29%13.11%1.96
6.56%13.13%1.05
2.66%13.13%0.42
8.21%9.70%1.77
Table 10.12 Correlation of endogenous versus exogenous trading rules, USD-CHFCorrelation
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Euro USDEuro CHFDi.
0.030.050.14
0.060.010.10
0.100.000.10
0.110.030.13
0.100.060.11
0.100.090.10
0.150.140.15
0.110.250.12
0.090.060.10
Table 10.13 Results of the trading rules for USD-DEMEndogenous rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.32%11.91%0.76
7.93%11.90%1.39
5.03%11.91%0.88
5.23%11.91%0.92
3.68%11.91%0.65
3.71%11.91%0.65
1.21%11.91%0.21
1.27%11.91%0.22
2.80%8.22%0.71
Euro USD rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.52%11.91%0.79
6.11%11.90%1.07
6.19%11.90%1.09
9.88%11.89%1.74
13.93%11.88%2.45
9.94%11.89%1.75
5.49%11.91%0.97
0.85%11.91%0.15
7.12%9.02%1.65
Euro DEM rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
1.10%11.91%0.19
4.41%11.91%0.77
1.46%11.91%0.26
5.02%11.91%0.88
0.63%11.91%0.11
1.92%11.91%0.34
1.63%11.91%0.29
0.87%11.91%0.15
1.72%8.28%0.44
Di. rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.94%11.91%0.87
8.75%11.90%1.54
1.50%11.91%0.26
8.03%11.90%1.41
3.60%11.91%0.63
0.53%11.91%0.09
2.80%11.91%0.49
5.87%11.90%1.03
4.50%8.58%1.10
Table 10.14 Correlation of endogenous versus exogenous trading rules, USD-DEMCorrelation
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Euro USDEuro DEMDi.
0.030.100.16
0.030.080.24
0.110.100.17
0.080.130.11
0.160.110.12
0.200.100.12
0.320.050.20
0.320.100.05
0.190.080.19
Figure 10.10 Performance of the basket of moving averages on USD-CHF (exogenous and endogenous)
Figure 10.11 Performance of the basket of moving averages on USD-DEM (exogenous and endogenous)
Figure 10.12 Performance of the basket of moving averages on GBP-USD (exogenous and endogenous)
Table 10.15 Results of the trading rules for GBP-USDEndogenous rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
0.85%11.18%0.16
2.23%11.18%0.42
3.65%11.18%0.68
4.25%11.18%0.79
4.65%11.18%0.87
2.58%11.18%0.48
2.80%11.18%0.52
5.25%11.18%0.98
3.28%8.02%0.86
Euro USD rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
1.64%11.18%0.31
1.43%11.18%0.27
2.09%11.18%0.39
0.40%11.18%0.08
5.67%11.17%1.06
1.40%11.18%0.26
0.08%11.18%0.01
4.98%11.18%0.93
0.44%8.45%0.11
Euro GBP rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
4.18%11.18%0.78
5.92%11.17%1.11
4.25%11.18%0.80
2.27%11.18%0.42
0.36%11.18%0.07
4.88%11.18%0.91
0.82%11.18%0.15
0.54%11.18%0.10
0.78%7.93%0.20
Di. rule
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Yearly returnYearly volatilityT-statistic
3.35%11.18%0.63
7.59%11.17%1.42
4.03%11.18%0.75
8.24%11.17%1.54
6.64%11.17%1.24
9.40%11.16%1.76
8.72%11.17%1.63
6.07%11.17%1.14
6.76%7.70%1.83
Table 10.16 Correlation of endogenous versus exogenous trading rules GBP-USDCorrelation
S 2
S 3
S 5
S 9
S 17
S 32
S 61
S 117
Average
Euro USDEuro GBPDi.
0.090.140.10
0.010.210.09
0.040.110.13
0.080.120.20
0.050.260.21
0.040.200.19
0.050.180.37
0.040.270.20
0.020.210.21
310
Advanced Trading Rules
REFERENCESAcar, E. and Lequeux, P. (1995) Trading rules pro®ts and the underlying timeseries properties, First International Conference on High Frequency Data inFinance, Zurich, Switzerland.Acar, E. and Lequeux, P. (1996) Dynamic strategies: a correlation study. InForecasting Financial Markets (ed. C. Dunis), Wiley, pp. 93±123.Arnott, R.D. and Pham, T.K. (1993) Tactical currency allocation, FinancialAnalysts Journal, September, 47±52.Bilson, O. (1993) Value, yield and trend: A composite forecasting approach toforeign exchange trading. In Strategic Currency Investing (ed. A. Gitlin),Probus, Chicago, pp. 366±403.DeRosa, D.F. (1996) Managing Foreign Exchange Risk, Irwin, Chicago.Dunis, C. (1989) Computerized technical systems and exchange rate movements.In Exchange Rates Forecasting (eds C. Dunis and M. Feeny), WoodheadFaulkner, Cambridge, pp. 165±205.Johnson, T. ( 1994) Interest rate comparison reveal currency trends, FuturesMagazine, February, pp. 44±6.Kaufman, P.J. (1987) The New Commodity Trading Systems and Methods, JohnWiley & Sons, New York.Kritzman, M. (1989) Serial dependence in currency returns: investment implications, Journal of Portfolio Management, Fall, 96±102.LeBaron, B. (1992) Persistence of the Dow Jones index on rising volume,Working Paper 9201, University of Wisconsin, Social Science Research.Levich, R.M. (1989) Forward rates as the optimal future spot rate forecast. InExchange Rates Forecasting (eds C. Dunis and M. Feeny), Woodhead-Faulkner, Cambridge, pp. 75±98.Murphy, J.A. (1986) Futures fund performance: a test of the eectiveness oftechnical analysis, Journal of Futures Markets, 6, 2, 175±85.Murphy, J.E. (1990) The Random Character of Interest Rates, McGraw-Hill,London.Olsen, A. (1995) High Frequency Data in Financial Markets: Issues and Applications, First International Conference on High Frequency Data in Finance,Zurich, Switzerland.Schwager, J. (1984) A Complete Guide to the Futures Markets, John Wiley & Sons,New York.Sease, D.R. (1992) Pound crisis shakes world currency markets, Wall StreetJournal, 28 September 1992.Silber, L.W. (1994) Technical trading: when it works and when it doesn't, TheJournal of Derivatives, Spring, 39±44.Taylor, S.J. (1986) Modelling Financial Time Series, John Wiley & Sons, Chichester, England.
NOTES1. Endogenous trading rule: trading rule that uses the price of one market toforecast this same market (e.g. forecasting USD-DEM with USD-DEM).
Informative spillovers in the currency markets 3112. Exogenous trading rule: trading rule that uses the price of one market toforecast another market (e.g. forecasting USD-DEM with USD and DEMinterest rates).3. Leptokurtic: characteristic of a distribution exhibiting an abnormally highnumber of unexpected events under the assumption of a normal distribution.The well-known bell-shaped curve of the distribution will then have fattertails and a higher peak due to the number of `abnormal' events.4. Multisystem: trading strategy that involves more than one trading ruleapplied to the same underlying (e.g. trading USD-DEM with a movingaverage of order twenty and a moving average of order ®ve).
Chapter 11Stop-loss rules as a monitoring device:theory and evidence from the bondfutures marketBERNARD BENSAID AND OLIVIER DE BANDT
11.1 INTRODUCTIONSince Schiller's (1981) seminal paper, the economic profession has faced along-standing debate about market volatility, opposing two con¯icting viewson the functioning of ®nancial markets. On the one hand, advocates of therational expectation hypothesis have stressed the forward-looking propertiesof ®nancial markets regarding changes in business conditions. On the otherhand, models have been developed highlighting various `anomalies' in®nancial markets, like the existence of bubbles or `fad' eects, among others.Actually the opposition between these two views is not as irreductible as itmight seem at ®rst glance and large movements of prices can be explained bythe optimizing behaviour of rational agents. We argue in the chapter that it isthe case, in particular, of stop-loss strategies which can trigger dramatic pricechanges in equity, exchange rates and/or future markets. Those tradingstrategies leading investors to liquidate massively portfolios exhibitingheavy losses seem to have played a major role during the 1929 and 1987crashes (Schiller, 1989), as well as during other periods of speculative attacksin foreign exchange rate markets (see Krugman and Miller, 