. The nextcolumn reports the probability of the exchange rate return falling belowCRSP. This is 46 per cent for the BP. The next column, T < RF, estimates thefraction of time that the compounded return on the strategy was below thecompounded return on a risk-free bond. For the BP series this is 34 per cent.The ®nal column reports the average beta and the standard deviation of theestimated beta across the simulations. Beta is estimated weekly for eachsimulation. This again shows that there is very little correlation between thestrategy and the CRSP series. Results for the buy and hold strategy for the BPare shown in the next row. This gives a mean return of 11.0 with a standarddeviation of 15.7. Buy and hold falls below the dynamic strategy in meanreturn and it shows little improvement in riskiness.The distribution of these returns along with the CRSP distribution is shownin Figure 1.2. These are the one-year holding period simulated returns. This®gure clearly shows strong evidence that the BP series may ®rst-orderstochastically dominate its equivalent buy and hold position. The comparison
Technical trading rules and regime shifts in foreign exchange
31
Table 1.15 One-year horizonSeries
Return/year
Std
Prob(<RF
BP
19.5
13.7
BP BH
11.0
DM
5%)
Prob(<CRSP)
T < RF
0.15
0.46
0.34(0.31)
0.04(0.26)
15.7
0.42
0.59
0.50(0.39)
0.06(0.16)
14.3
18.7
0.32
0.50
0.46(0.34)
0.06(0.18)
DM BH
5.6
14.7
0.55
0.65
0.62(0.37)
0.07(0.19)
JY
22.2
24.0
0.19
0.37
0.39(0.35)
0.04(0.21)
JY BH
9.4
16.6
0.45
0.59
0.56(0.37)
0.03(0.17)
CRSP
16.2
18.0
0.29
CRSP  BP
17.9
11.2
0.14
0.46
0.36(0.32)
0.47(0.13)
3FX  CRSP 17.4
12.8
0.18
0.38
0.37(0.32)
0.47(0.10)
0.41(0.35)
Simulation results for one year trading business. Results of 1000 simulations of randomly selected2 year intervals during the sample. Prob(<RF 5%) is the probability of underperforming thedomestic risk-free rate by more than 5%. Prob(<CRSP) is the probability that the strategyunderperforms the CRSP index. T < RF is the fraction of time that the cumulative return on thestrategy spends below the cumulative return on the risk free asset. is again the beta estimatedagainst the CRSP index. CRSP  BP is a portfolio which is started with portfolio weights of 1/2and 1/2 on CRSP and the trading strategy respectively. 3FX  CRSP is a portfolio which isstarted with weights of 1/2 on a buy and hold stock position, and 1/2 on an equally weightedposition in the three foreign exchange strategies.
with CRSP is more dicult, but the graph suggests that the pound strategymay second-order stochastic dominate CRSP. Both these comparisons awaitmore detailed statistical testing. 12Results for the DM series are given in the next two rows of Table 1.15. Thisseries gives a mean return less than CRSP with similar risk characteristics. Itsreturns are again much larger than the equivalent buy and hold strategy.Figure 1.3 plots the distribution for the DM strategies. There is a clearindication that the strategy ®rst-order stochastically dominates the buy andhold strategy. No simple comparisons can be made between the DM strategyand CRSP.
32
Advanced Trading Rules
Figure 1.2 Simulated return distribution ± one-year period
Figure 1.3 Simulated return distribution ± one-year period
Results for the JY are given in the next two rows. The JY outperforms CRSPby 6 per cent and its buy and hold by 10 per cent. It has a larger standarddeviation, but its other risk measures are equivalent to CRSP. Figure 1.4shows the distributions. Once again, it appears that the strategy ®rst-orderstochastically dominates buy and hold. The strategy appears close to ®rst-
Technical trading rules and regime shifts in foreign exchange
33
Figure 1.4 Simulated return distribution ± one-year period
order stochastically dominating CRSP except for a small section. However, itshows strong evidence for second-order stochastic dominance.For all three currencies the betas are very low. This suggests the possibilityfor diversi®cation. The next row in Table 1.15, labelled `CRSP  BE' presentsresults for a portfolio formed by starting out invested half in stocks and halfin the BP dynamic strategy. The portfolio increases returns and reducesstandard deviation over the original CRSP portfolio. It is easy to select anoptimal portfolio using currencies determined by looking at the results expost. The next row tests a strategy that might have been followed had theinvestor not known the relatively poor performance of the DM. In thisstrategy, wealth is split equally between a buy and hold CRSP portfolio andthe dynamic portfolios. The half in the dynamic foreign exchange portfolios issplit 1/3 to each currency. This strategy performs similarly to the BP  CRSPportfolio, showing that there is probably little diversi®cation gain acrossforeign exchange strategies themselves.Results for all these strategies and CRSP are plotted in Figure 1.5. The twodynamic strategies are close to each other and appear close to second-orderdominating the CRSP returns alone. This is consistent with the properties ofthe dynamic foreign exchange strategies which suggested that they were zerobeta securities exhibiting similar risk±return characteristics to the stock portfolio.These results are further tested in Table 1.16. This table compares theprevious distributions using a myopic one-year investor with constant relative
34
Advanced Trading Rules
Figure 1.5 Simulated return distribution ± one-year period
Table 1.16 Utility distribution comparisons: one-year horizonSeries
BH
CRSP
3FX  CRSP
BPDMJYBP  CRSP3FX  CRSP
0.920.930.91
0.951.010.960.960.96
0.991.051.000.99
Utility comparisons of myopic 1 year crra investors. Fractionof wealth that would make an investor indierent betweenthe row rule used on W and the column rule on W. Degreeof relative risk aversion is ®xed at 4.
risk aversion utility. The coecient of relative risk aversion is set to four. Thetable ®nds that sets:Eu
W R~1   Eu W R~2 ;
u x 
11

x1
where R1 is the return given by the labels on the left side of the rows, and R2 isthe return given in the columns. For each currency, it is clear that thisconsumer would be willing to give up close to 8 per cent of the invested wealth
Technical trading rules and regime shifts in foreign exchange
35
to shift to the dynamic strategy from BH. Comparisons with CRSP, suggestthe consumer would be willing to give up 4±5 per cent of wealth for eachstrategy except for the DM where CRSP is preferred. This improvement holdsfor the three exchange rate CRSP portfolio. The last column compares thediversi®ed portfolio with each of the strategies. Interestingly, the portfolioshows little improvement over the BP and JY strategies separately.Finding an optimal portfolio ex post is not a con®rmation of an inecientmarket. It should always be easy to ®nd portfolios which dominate themarket portfolio in an ex post data search. The evidence shows someperformance improvements for currency and currency-stock portfolioswhen compared to the stock portfolio. This should be viewed with somecaution as it awaits further statistical testing. All the trading rules do oersimilar performance characteristics to the market portfolio with no beta risk.To the stock market investor wondering whether to speculate in the foreignexchange market, the evidence at this point appears somewhat uncertain.However, for any economic agent whose job requires some amount ofliquidity in various foreign exchange markets, the recommendation is clear.These agents are comparing the risk-free rates of return in all markets andwill have to maintain some exposure to foreign exchange risk. There are verydramatic improvements in moving from buy and hold strategies to thetrading rules for these agents. 13There are several problems that could move these conclusions in eitherdirection. First, the data used may not represent interest rates that traderscould actually use. Also, there may be some timing problems in terms ofsettlements. For example, the rules as implemented, assume that traders canget the closing price on the day of the signal. This may not always be the case.Also, settlement procedures are not considered here. 14 Finally, measurementof risk with respect to a US stock portfolio probably misses much of theexposure to international portfolio risk to which the exchange rate portfoliosare exposed. Estimating betas on a world portfolio or using a multifactormodel might be more appropriate here.There are some problems in the analysis which work in favour of thetrading rules. First, the rules used are very simple compared to what mosttraders use. Also, most traders would operate at the daily frequency orhigher. 15 Second, the comparison series, the CRSP index, may be dicult toobtain in practice. No attempt was made to adjust for transactions costs onthis series even though using the CRSP index implies that dividends are beingcontinuously reinvested. The ability of the average investor to track this indexshould be more carefully considered.There has been some recent evidence that the usefulness of technicaltrading strategies has diminished over time (Sweeney and Surajaras, 1989).
Advanced Trading Rules
Annual return
36
Figure 1.6 Rolling strategy returns ± two-year period
To check the possibility of a trend in trading rule pro®ts over time, a plot ismade of the trading-rule returns measured over two-year horizons for thethree currencies rolling the horizon forward one-quarter for each pointplotted. This is plotted in Figure 1.6. There is some evidence for a drop oin pro®ts in recent years. However, when analysing the entire series, it isunclear whether this period is at all unusual. There have been earlier periodswhen the rules did not perform very well. The time period around 1982±1983appears to have also been relatively poor. It is interesting that these might beperiods in which the two-year horizon is reaching into periods just after thePlaza Agreement when the dollar changed direction.1.6 CONCLUSIONSThis chapter has presented evidence supporting the premise that exchangerates do not follow a random walk. Moreover, these deviations are detectedby simple moving-average trading rules. These rules ®nd that, in general,returns during buy periods are higher than returns during sell periods.Volatility appears to be indistinguishable during these two periods. Also,skewness and kurtosis show no discernible patterns over buy and sell periods.These results are supportive of earlier work in Dooley and Shafer (1983),Schulmeister (1987), Sweeney (1986), Taylor (1980, 1986) and, more recently,Taylor (1992). These other authors perform extensive tests on the pro®tabilityof these tests and ®nd that, in general, the rules make money even when
Technical trading rules and regime shifts in foreign exchange
37
adjusted for transactions costs, interest rate dierentials and very simplemeasures of risk.In this chapter, the rules are ®rst used as speci®cation tests for severaldierent processes. The GARCH, regime shifting and interest rate adjustedmodels are unable to generate results consistent with the actual series. In eachcase, it is still possible that a modi®ed version of the model could be capableof generating results consistent with the actual data, but this awaits furtherexperimentation. The following two answers come from these results. First, itis possible that the series are non-stationary and are punctuated by strongchanges in regime that cannot be captured by these simple models. Second,none of the models considered here allow for any connection between trendand volatility changes. This possibility is considered in Taylor (1986) andresults in Bilson (1990), Kim (1989) and LeBaron (1992) suggest that theremay be some connection.The ®nal section of this chapter runs some experiments to test the economicsigni®cance of these results. The trading rules are implemented on the data asthey would be used in practice. Estimates for transaction costs and interestrate spreads are used to measure the realized returns from the strategies. Forthe three currencies tested, the trading-rule strategies generated returndistributions similar to those from the CRSP stock index with very lowcorrelation with the market. This suggests portfolios formed by combiningthe strategies with the CRSP index may dominate the stock index on its own.While these results are interesting, they should still be viewed with somecaution. There are still several interest rate and timing issues that are notworked out exactly. Also, the use of other risk factors than CAPM beta maybe important. As in any trading-rules test, there are further questions aboutthe parameters used and whether the prices used were actually tradeable.Given these issues and the lack of a statistical test on the distributioncomparisons, the results cannot be taken as clear evidence that everyeconomic agent is missing a big opportunity. However, for one group ofagents, the results are fairly strong. For people who are involved in foreignexchange markets, either in trading goods or securities, and who maintainpositions in foreign currencies, there appear to be major gains over buy andhold strategies. This is easily seen in Table 1.15 by comparing the buy andhold strategies with those for the rules. This may explain the large number oftechnical trading services available in the foreign exchange market. 16The results in this chapter may eventually lead to some better explanationsfor several eects in foreign exchange markets. Among these are the movements in forward and futures markets for foreign exchange. 17 Also, resultsfrom survey data found in Dominguez (1986) and Frankel and Froot (1990)may be relevant to some of the results found here. 18 Finally, foreign exchange
38
Advanced Trading Rules
markets dier from stock markets in that central banks play an importantrole. The behaviour of these large economic agents may dier greatly fromthat of ordinary traders. These agents may even be willing to lose money tosatisfy other objectives.This chapter has shown that technical trading rules may provide a usefulspeci®cation test for examining foreign exchange markets. This chapter usesthese rules to demonstrate some of the shortcomings of common parametricmodels for foreign exchange movements. Some evidence is given on theeconomic signi®cance of these results and shows that the strategies generatereturns similar to those from a domestic stock portfolio. Further tests will benecessary to completely answer the questions raised about the economicsigni®cance of these results.
REFERENCESBilson, J.F. (1990) `Technical' Currency Trading, The Currency-Hedging Debate(ed. L.R. Thomas), IFR Publishing, London.Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity,Journal of Econometrics, 21, 3, 307±328.Bollerslev, T. and Wooldridge, J.M. (1990) Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time-varying Covariances, MIT,Department of Economics.Bollerslev, T., Chou, R.Y., Jayaraman, N. and Kroner, K.F. (1990) ARCHModelling in Finance: A Review of the Theory and Empirical Evidence, Journalof Econometrics, 52, 1, 5±60.Brock, W.A., Lakonishok, J. and LeBaron, B. (1992) Simple Technical TradingRules and the Stochastic Properties of Stock Returns, Journal of Finance, 47,1731±64.Diebold, F.X. and Nason, J.M (1990) Nonparametric exchange rate prediction?,Journal of International Economics, 28, 315±32.Dominguez, K.M. (1986) Are foreign exchange forecasts rational? New evidencefrom survey data, Economics Letters, 21, 277±81.Domowitz, I. and Hakkio, C.S. (1985) Conditional variance and the risk premiumin the foreign exchange market, Journal of International Economics, 19, 47±66.Dooley, M.P. and Shafer, J. (1983) Analysis of short-run exchange rate behaviour: March 1973 to November 1981. In Exchange Rate and Trade Instability:Causes, Consequences and Remedies (eds D. Bigman and T. Taya) Cambridge,MA, Ballinger, pp. 43±72.Due, D. and Singleton, K. (1993) Simulated Moments Estimation of MarkovModels of Asset Prices, Econometrica, 61, 4, 929±52.Efron, B. (1979) Bootstrap methods: another look at the jackknife, The Annals ofStatistics, 7, 1, 1±26.Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates ofthe variance of United Kingdom in¯ation, Econometrica, 50, 987±1007.
Technical trading rules and regime shifts in foreign exchange
39
Engle, R.F. and Bollerslev, T. (1986) Modelling the persistence of conditionalvariances, Econometric Reviews, 5, 1, 1±50.Engel, C. and Hamilton, J.D. (1990) Long swings in the dollar: are they in thedata and do markets know it?, American Economic Review, 8, 4, 689±713.Frankel, J.A. and Froot, K.A. (1990) Chartists, fundamentalists and trading inthe foreign exchange market, American Economic Review, 80, 2, 181±5.Goodhart, C. (1988) The foreign exchange market: a random walk with adragging anchor, Economica, 55, 437±60.Hodrick, R.J. (1987) The Empirical Evidence on the Eciency of Forward andFutures Foreign Exchange Markets, New York, Harwood Academic Publishers.Hsieh, D. (1988) The statistical properties of daily foreign exchange rates: 1974±1983, Journal of International Economics, 24, 129±45.Hsieh, D. (1989) Testing for nonlinear dependence in daily foreign exchange rates,Journal of Business, 62, 3, 339±68.Kim, C.M. (1989) Volatility Eect on Time Series Behaviour of Exchange RateChanges, Korea Institute for International Economic Policy.Kunsch, H.R. (1989) The jacknife and the bootstrap for general stationaryobservations, Annals of Statistics, 17, 3, 1217±41.LeBaron, B. (1992) Forecast Improvements Using a Volatility Index, Journal ofApplied Econometrics, 7, S137±50.Lee, B-S. and Ingram, B.F. (1991) Simulation estimation of time-series models,Journal of Econometrics, 47, 197±205.Levich, R.M. and Thomas, L.R. (1991) The Signi®cance of Technical TradingRule Pro®ts in the Foreign Exchange Market: A Bootstrap Approach, New YorkUniversity, Graduate School of Business, New York.McFadden, D. (1989) A method of simulated moments for estimation of discreteresponse models without numerical integration, Econometrica, 57, 5, 995±1026.Meese, R. and Rogo, K. (1983) Empirical exchange rate models of the seventies:do they ®t out of sample?, Journal of International Economics, 14, 3±24.Meese, R.A. and Rose, K.A. (1990) Nonlinear, nonparametric, nonessentialexchange rate estimation, American Economic Review, 8, 2, 192±6.Newey, W. and West, K.D. (1987) A simple, positive de®nite, heteroskedasticityand autocorrelation consistent covariance matrix, Econometrica, 55, 703±8.Pakes, A. and Pollard, D. (1989) Simulation and the Asymptotics of OptimizationEstimators, Econometrica, 57, 5, 1027±58.Rothschild, M. and Stiglitz, J. (1970) Increasing risk I: a de®nition, Journal ofEconomic Theory, 2, 225±43.Schulmeister, S. (1987) An Essay on Exchange Rate Dynamics. Wissenschaftszentrum Berlin fuÈr Sozialforschung, Berlin, IIM/LMP, 87±8.Schwarz, G. (1978) Estimating the dimension of a model, Annals of Statistics, 6,461±4.Sweeney, R.J. (1986) Beating the foreign exchange market, Journal of Finance, 41,1, 163±82.Sweeney, R.J. and Surajaras, P. (1989) The stability of speculative pro®ts in theforeign exchanges. In A Reappraisal of the Eciency of Financial Markets (edsR. Guimaraes, B. Kingsman and S.J. Taylor), Heidelberg: Springer-Verlag.
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Advanced Trading Rules
Taylor, S.J. (1980) Conjectured models for trends in ®nancial prices, tests andforecasts, Journal of the Royal Statistical Society, 143, 3, 338±362.Taylor, S.J. (1986) Modelling Financial Time Series, Chichester: John Wiley &Sons.Taylor, S.J. (1992) Rewards Available to Currency Futures Speculators: Compensation For Risk or Evidence of Inecient Pricing, Economic Record, 68,105±16.
NOTES1. Recently, Levich and Thomas (1991) have obtained some related results forseveral foreign exchange futures series.2. There are many variations of this simple rule in use. One is to replace theprice series with another moving average. A second modi®cation is to onlygenerate signals when the price diers from the moving average by a certainpercentage. Many other modi®cations are discussed in Schulmeister (1987),Sweeney (1986) and Taylor (1992).3. For more extensive descriptions of these results on exchange rates, see Hsieh(1988, 1989) and other references contained in Bollerslev et al. (1990).4. Other speci®cations with changing conditional means related to volatility(GARCH-M) were also tried, but these turned out to be insigni®cant. Thisagrees with some of the results found in Domowitz and Hakkio (1985).5. The convergence of the bootstrap distribution has not been shown forGARCH models. Brock, Lakonishok and LeBaron (1992) use a similartechnique for stock returns. Their results are supported through largecomputer simulations.6. This is generally the case for all the exchange rate tests used here. It diersfrom some of the results in Brock et al. (1992), where stock returns werefound to be more volatile during sell periods than during buy periods.7. tanh x 
e x  exe x  ex
8. This condition brings in the problem of a free parameter, . This parameteris set to 1/10 the standard deviation of the price-moving-average ratio.Experiments with this parameter have shown the results to be insensitive tochanges in the parameter ranging from 1 to 1/100 standard deviations.9. This technique also allows the use of several trading-rule conditions simultaneously. Dependence across rules is captured in the variance±covariancematrix.10. This means that an investor with one dollar who receives a buy signal willborrow one dollar domestically and invest two dollars in the foreigncurrency. The reverse is followed for a sell.11. One drawback of this technique is that the ®rst and last part of the series willbe under-represented in simulations. One solution might be to think of theseries as rolling around back onto itself on a circle. However, this imposes asevere pasting together of disjointed parts of the series. Another solutionmight be to use the m-dependent bootstrap of Kunsch (1989). Both of these
Technical trading rules and regime shifts in foreign exchange
41
possibilities are left for the future. For the present, the reader should realizethat the simulation does not adequately sample some parts of the series.12. First-order stochastic dominance is obtained whenF x
G x  0
8
x
for the distribution functions F and G, where the inequality is strict over a setof positive measure. Any consumer preferring more to less will prefer thedistribution G. Second-order stochastic dominance is obtained whenx
F s1
G s ds  0
8
x
In this case, only risk-averse consumers will prefer G (Rothschild andStiglitz, 1970).13. This may explain the extensive use of technical trading advice by manymarket participants.14. An experiment was performed to test the robustness of the results to timing.The testing programs were modi®ed so that investors could not get theinterest rates at time t, but could get the rates given one day later. Results ofthis experiment are not presented since they are almost identical to thosefrom the original series.15. Most of the rules used here were repeated at daily frequency with littlechange in the results.16. See Frankel and Froot (1990) for some evidence on the number of chartists.17. See Hodrick (1987) for a survey of these results.18. These papers, using survey data, ®nd that short-range forecasts are moretrend-following while longer-range forecasts are more mean reverting.
ACKNOWLEDGMENTSThe author is grateful to Robert Hodrick, Simon Potter, seminarparticipants at the Mid-West International Economics Meetings, TheUniversity of Iowa and Carnegie-Mellon University for comments onan earlier draft. This research was partially supported by the EconomicsResearch Program at the Santa Fe Institute which is funded by grantsfrom Citicorp/Citibank, the Russell Sage Foundation, the Alex C.Walker Educational and Charitable Foundation ± Pittsburgh NationalBank, and by grants to SFI from the John D. and Catherine T.MacArthur Foundation, the National Science Foundation (PHY8714918) and the US Department of Energy (ER-FG05-88ER25054).The author also is grateful to the National Science Foundation (SES9109671) for support.
Chapter 2Foundations of technical analysis:computational algorithms, statisticalinference and empirical implementationANDREW W. LO, HARRY MAMAYSKY AND JIANG WANG
2.1 INTRODUCTIONOne of the greatest gulfs between academic ®nance and industry practice isthe separation that exists between technical analysts and their academiccritics. In contrast to fundamental analysis, which was quick to be adoptedby the scholars of modern quantitative ®nance, technical analysis has been anorphan from the very start. It has been argued that the dierence betweenfundamental analysis and technical analysis is not unlike the dierencebetween astronomy and astrology. Among some circles, technical analysisis known as `voodoo ®nance'. And in his in¯uential book A Random WalkDown Wall Street, Burton Malkiel (1996) concludes that `under scienti®cscrutiny, chart-reading must share a pedestal with alchemy'.However, several academic studies suggest that despite its jargon andmethods, technical analysis may well be an eective means for extractinguseful information from market prices. For example, in rejecting the RandomWalk Hypothesis for weekly US stock indexes, Lo and MacKinlay (1988,1999) have shown that past prices may be used to forecast future returns tosome degree, a fact that all technical analysts take for granted. Studies byTabell and Tabell (1964), Treynor and Ferguson (1985), Brown and Jennings(1989), Jegadeesh and Titman (1993), Blume, Easley, and O'Hara (1994),Chan, Jegadeesh, and Lakonishok (1996), Lo and MacKinlay (1997), Grundyand Martin (1998), and Rouwenhorst (1998) have also provided indirectsupport for technical analysis, and more direct support has been given byPruitt and White (1988), Neftci (1991), Brock, Lakonishok, and LeBaron(1992), Neely, Weller, and Dittmar (1997), Neely and Weller (1998),Chang and Osler (1994), Osler and Chang (1995), and Allen and Karjalainen(1999).
Foundations of technical analysis
43
One explanation for this state of controversy and confusion is the uniqueand sometimes impenetrable jargon used by technical analysts, some of whichhas developed into a standard lexicon that can be translated. But there aremany `homegrown' variations, each with its own patois, which can oftenfrustrate the uninitiated. Campbell, Lo, and MacKinlay (1997, pp. 43±44)provide a striking example of the linguistic barriers between technical analystsand academic ®nance by contrasting this statement:The presence of clearly identi®ed support and resistance levels, coupled with aone-third retracement parameter when prices lie between them, suggests thepresence of strong buying and selling opportunities in the near-term.
with this one:The magnitudes and decay pattern of the ®rst twelve autocorrelations and thestatistical signi®cance of the Box-Pierce Q-statistic suggest the presence of a highfrequency predictable component in stock returns.
Despite the fact that both statements have the same meaning ± that pastprices contain information for predicting future returns ± most readers ®ndone statement plausible and the other puzzling, or worse, oensive.These linguistic barriers underscore an important dierence betweentechnical analysis and quantitative ®nance: technical analysis is primarilyvisual, whereas quantitative ®nance is primarily algebraic and numerical.Therefore, technical analysis employs the tools of geometry and pattern