University Press,Oxford.Press, W., Flannery, B., Teukolsky, S. and Vetterling, W. (1986) Numerical Recipes:The Art of Scienti®c Computing Cambridge University Press, Cambridge.Pruitt, S. and White, R. (1988) The CRISMA trading system: Who says technicalanalysis can't beat the market?, Journal of Portfolio Management, 14, 55±58.Riesenhuber, M. and Poggio, T. (1997) Common computational strategies in machineand biological vision, in Proceedings of International Symposium on System Life,Tokyo, Japan, 67±75.Rouwenhorst, G. (1998) International momentum strategies, Journal of Finance, 53,267±284.
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Simono, J. (1996) Smoothing Methods in Statistics, Springer-Verlag, New York.Smirnov, N. (1939a) Sur les eÂcarts de la courbe de distribution empirique, Rec. Math.(Mat. Sborn.), 6, 3±26.Smirnov, N. (1939b) On the estimation of the discrepancy between empirical curves ofdistribution for two independent samples, Bulletin. Math. Univ. Moscow, 2, 3±14.Tabell, A. and Tabell, E. (1964) The case for technical analysis, Financial AnalystJournal, 20, 67±76.Treynor, J. and Ferguson, R. (1985) In defense of technical analysis, Journal ofFinance, 40, 757±773.
NOTES1. A similar approach has been proposed by Chang and Osler (1994) and Osler andChang (1995) for the case of foreign-currency trading rules based on a head-andshoulders pattern. They develop an algorithm for automatically detecting geometric patterns in price or exchange data by looking at properly de®ned localextrema.2. See, for example, Beymer and Poggio (1996), Poggio and Beymer (1996), andRiesenhuber and Poggio (1997).3. Despite the fact that K x is a probability density function, it plays noprobabilistic role in the subsequent analysis ± it is merely a convenient methodfor computing a weighted average, and does not imply, for example, that X isdistributed according to K x (which would be a parametric assumption).4. However, there are other bandwidth-selection methods that yield the sameasymptotic optimality properties but which have dierent implications for the®nite-sample properties of kernel estimators. See HaÈrdle (1990) for furtherdiscussion.5. Speci®cally, we produced ®tted curves for various bandwidths and comparedtheir extrema to the original price series visually to see if we were ®tting more`noise' than `signal', and asked several professional technical analysts to do thesame. Through this informal process, we settled on the bandwidth of 0:3  h and used it for the remainder of our analysis. This procedure was followed beforewe performed the statistical analysis of section 2.4, and we made no revision tothe choice of bandwidth afterward.6. See Simono (1996) for a discussion of the problems with kernel estimators andalternatives such as local polynomial regression.7. After all, for two consecutive maxima to be local maxima, there must be a localminimum in between, and vice versa for two consecutive minima.8. If we are willing to place additional restrictions on m , e.g., linearity, we canobtain considerably more accurate inferences even for partially completedpatterns in any ®xed window.9. For example, Chang and Osler (1994) and Osler and Chang (1995) propose analgorithm for automatically detecting head-and-shoulders patterns in foreignexchange data by looking at properly de®ned local extrema. To assess the ecacyof a head-and-shoulders trading rule, they take a stand on a class of tradingstrategies and compute the pro®tability of these across a sample of exchange
Foundations of technical analysis
111
rates against the US dollar. The null return distribution is computed by abootstrap that samples returns randomly from the original data so as to inducetemporal independence in the bootstrapped time series. By comparing the actualreturns from trading strategies to the bootstrapped distribution, the authors ®ndthat for two of the six currencies in their sample (the yen and the Deutschemark), trading strategies based on a head-and-shoulders pattern can lead tostatistically signi®cant pro®ts. See also Neftci and Policano (1984), Pruitt andWhite (1988), and Brock et al. (1992).10. If the ®rst price observation of a stock is missing, we set it equal to the ®rst nonmissing price in the series. If the tth price observation is missing, we set it equal tothe ®rst non-missing price prior to t.11. In particular, let the price process satisfydP t  P t dt  P t dW twhere W t is a standard Brownian motion. To generate simulated prices for asingle security in a given period, we estimate the security's drift and diusioncoecients by maximum likelihood and then simulate prices using the estimatedparameter values. An independent price series is simulated for each of the 350securities in both the NYSE/AMEX and the Nasdaq samples. Finally, we use ourpattern-recognition algorithm to detect the occurrence of each of the ten patternsin the simulated price series.
ACKNOWLEDGMENTSThis research was partially supported by the MIT Laboratory forFinancial Engineering, Merrill Lynch, and the National Science Foundation (Grant SBR±9709976). We thank Ralph Acampora, FranklinAllen, Susan Berger, Mike Epstein, Narasimhan Jegadeesh, Ed Kao,Doug Sanzone, Je Simono, Tom Stoker, and seminar participants atthe Federal Reserve Bank of New York, NYU, and conferenceparticipants at the Columbia-JAFEE conference, the 1999 JointStatistical Meetings, RISK 99, the 1999 Annual Meeting of the Societyfor Computational Economics, and the 2000 Annual Meeting of theAmerican Finance Association for valuable comments and discussion.This chapter was published in The Journal of Finance, 55(4), pp. 1705±1765, August 2000 # Blackwell Publishers.
Chapter 3Mean-variance analysis, trading rulesand emerging marketsDAAN MATHEUSSEN AND STEPHEN SATCHELL
3.1 INTRODUCTIONIt has only been in the last few years that serious study on the statisticalproperties of emerging markets returns has commenced. Papers in this newliterature include, inter alia, Frankel (1995), Behaert and Harvey (1995), Erb,Harvey and Viskanter (1995). The purpose of this chapter is to examine thepossibility of using trading rules based on mean-variance analysis onemerging markets. The question, among many others, has been discussed inan important study by Harvey (1995). He ®nds on the basis of extensivetesting that standard global asset pricing models fail to explain expectedreturns in emerging markets due to large idiosyncratic (country-speci®c)informational eects. However, Eekhari and Satchell (1995) examine thevalidity of the capital asset pricing model (CAPM) using monthly emergingmarkets data and ®nd that, although the data is highly non-normal, theycannot conclude that the mean-variance CAPM gives dierent betas than themean semi-variance CAPM. This result is compatible with the data comingfrom an elliptical distribution and such a result implies the validity of meandispersion analysis where dispersion is a measure proportional to variance,thus the analysis is equivalent to mean-variance analysis. Considering twentyemerging market indices, we ®nd skewness present in ten cases whilst kurtosisis present in ®fteen. Kurtosis is compatible with elliptical distributions,whereas skewness is not.This discussion suggests that it would be an interesting, and possibly valid,exercise to construct one-period mean-variance optimized portfolios and basetrading rules on them. We do this as part of a broader aim which is toinvestigate the empirical performance of various dynamic trading rules andstrategies to our emerging market database. These new markets are characterized by high costs and optimized strategies may not perform as well as
Mean-variance analysis, trading rules and emerging markets 113simple rules such as rebalancing every six months, say, which will reduce`churning' the portfolios and keep costs down. There is a long history ofempirical work on strategies and trading rules in ®nance and we make noattempt to survey this vast literature. Some references on the empiricalperformance measurement of strategies include Johnston and Shannon(1975) and Bloom®eld, Left and Long (1977) to name but two. We presenta set of trading rules in section 3.2, our results in section 3.3 and ourconclusions in section 3.4.3.2 DATA AND PORTFOLIO CONSTRUCTIONThe data we use is the International Finance Corporation (IFC) monthly onecountry returns. Diagnostics of this data are presented in Harvey (1995) andEekhari and Satchell (1995). Essentially, it is highly non-normal, but oftenelliptical, possesses some but not excessive autocorrelation and in some butnot all cases has time-varying heteroskedasticity.The strategy of our investment is myopic, that is we choose our portfolioacross countries based on next period's return distribution. In certain casesthis can be shown to be optimal for N periods expected utility maximizations,see Stapleton and Subrahmanyan (1978), for example. The advantage of themyopic strategy is that it only considers one-period returns and, as referred toin section 3.1, we have some evidence that the returns are approximatelyelliptical. It is perfectly possible for a series that is elliptical through timerather than cross-sectionally to exhibit GARCH behaviour, see Satchell(1995) for results on this phenomenon. All our data is converted to USdollars so that our analysis is from the perspective of a US investor, ourreturns are arithmetic.The ®rst portfolio we consider was suggested to us by Kleinwort BensonInvestment Management (KBIM). This is the averaged portfolio. We notethat this assigns weights equal to 1=N where N is the number of countries atthe time of investment. This procedure is independent of any predictions ofmoments. There is an issue of timing; we consider two versions of the equalweight (EW) portfolio, one in which we hold 1= N  1 from the month anew market emerges. We call this EWII. The other we hold 1= N  1 10months after the new market emerges. We call this EWI. The reason for thisdistinction is to keep EWI compatible with the other portfolios, as describedlater in the text. Another simple strategy used by KBIM, but not included, isa two-tier weight system rather than the equally weighted examples considered in this study.The next portfolio to consider is the global minimum variance portfolio. Tosimplify this we assume that the cross-correlations are zero; this seems
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Advanced Trading Rules
reasonable for most of the period and agrees with the low empirical crosscorrelation in the data. Under this assumption it is a simple exercise to showthat the optimal weights Wi;t are given by:1 2i;tWi;t  NX 1i1
 2i;t
with i  1; . . . ; N and  2i;t is the variance of returns of asset i at time t. We notethat the weights are independent of the means, they depend only on thevariances, the larger the variance the less of the asset we hold.The third portfolio is the so-called Markowitz optimal portfolio. In thiscase, the weights Wi;t are given by:Wi;t 
Mi;tNXMi;ti1
with i  1; . . . ; Ni;t rtMi;t  2i;tand i;t is the expected return of Index i at time t, rt is the US dollar monthlymid-rate.In this case the Markowitz portfolio depends on both means and variances.Correlations are assumed to be zero as before. This portfolio corresponds tothe tangency portfolio of the familiar mean-variance diagram. It re¯ects theoptimal portfolio of an investor whose net position in bonds is zero andwhose total investment in the risky assets is held in proportion to the market.Because it is likely that short positions are not feasible, or, at least, veryexpensive, in emerging markets, we adjust our weights in the following way. IfMi;t < 0, then Mi;t  0, i.e. all short positions are set to zero and our weightsare rescaled to add up to one.The portfolio returns are presented in Table 3.2 for the cases where we haveused the past ten, twenty or all past observations for parameter estimation.We wait ten months before including new assets in our portfolio. Thelikelihood that the process governing returns are non-stationary throughtime means that we need to consider rolling windows of data for estimatingMi;t and  2i;t . We also present some results on rebalancing. This popular andsimple technique consists of ®xing a set of portfolio weights at some level andrebalancing the portfolio back to these benchmark weights after some ®xedtime period. We consider the time periods monthly, three monthly, six
Mean-variance analysis, trading rules and emerging markets 115monthly and annually. Our procedure is complicated by the introduction ofnew markets. Here we consider an equally weighted portfolio where we waitten periods until we invest in a market, i.e. we start the funds in period eleven.However, if a market emerges after period eleven and we rebalance before (orafter) ten months subsequent to the period of emergence, we invest in that market.We consider a snapshot of our portfolios. Thus, for June 1992, whenvolatility seemed rather high, we compute the weights of the two non-EWportfolios which we report in Table 3.1. This is simply to give an example ofthe sort of weight patterns that might emerge and to familiarize the readerwith the data and models.3.3 RESULTSWe ®rst discuss our snapshot values. In Table 3.1 we list the Markowitzoptimal (MO) weights, second column, the minimum variance weights in theTable 3.1 Weights of MO and MV in June 1992
ArgentinaBrazilChileColumbiaMexicoPeruVenezuelaChinaKoreaPhilippinesTaiwanIndiaIndonesiaMalaysiaPakistanSri LankaThailandGreeceHungaryJordanNigeriaPolandPortugalTurkeyZimbabweTotal
Log returnin June 1992
MOweights (%)
MVweights (%)
MOreturn
MVreturn
0.256 0690.036 660.020 2530.139 9190.152 879Not indexed0.007 495Not indexed0.004 5170.137 3730.00450.028 7650.031 2790.015 8680.059 784Not indexed0.093 7570.106 283Not indexed0.023 234 30.055 112Not indexed0.000 1450.245 490.118 589
1.580.1120.3611.984.95
20.413.351.692.036.74
0.004 0510.000 4110.004 1230.016 7640.007 56
0.052 250.048 960.000 3430.002 8410.010 30
3.56
5.46
0.000 267
0.000 41
5.6310.042.854.990.001.8211.89
1.983.226.642.692.901.780.95
0.000 2540.013 7920.000 1280.001 43400.000 2890.007 11
8.95 50.004 4220.000 2990.000 7750.000 9080.000 2820.000 567
7.053.13
2.164.48
0.006 610.003 33
0.002 0240.004 758
0.000.00
0.724.52
00
0.000 1670.002 492
4.210.565.3
5.1011.022.16
6.10 60.001 370.006 28
7.41 60.02700.0026
100.0
100.0
0.0361
0.0677
116
Advanced Trading Rules
Table 3.2 Sharpe ratio of the considered strategies over entire period taken with respectto averagesReturn (net cum) (%)
Mean
Standard deviation
Sharpe ratio
874865920797
0.023 7790.022 1390.022 7530.024 077
0.055 6050.043 6830.045 7520.088 297
0.333 9730.387 660.383 5450.213 703
645628631
0.021 8730.021 6570.024 556
0.048 9410.043 5910.094 221
0.337 0980.373 5140.203 575
1648868660
0.028 4820.022 2440.024 491
0.056 7690.043 4940.108 757
0.407 0280.387 8510.175 766
All observationsMOEWIEWIIMV20 ObservationsMOEWIMV10 ObservationsMOEWIMV
third column. Note that, although we do not record it, the weights for theequal weight (EW) portfolio would be 5 per cent. The date chosen, 30 June,1992, was purely for the purpose of providing a snapshot and has no specialsigni®cance other than showing large dierences in monthly returns betweenthe two portfolios.We concentrate on the dierences. The mean-variance portfolio is long inArgentina, Brazil and Turkey, whereas the MO portfolio is long in Chile,Columbia, The Philippines and Pakistan. In the fourth and ®fth columns wereport the portfolio weights multiplied by the asset return for that month. Wesee that the MO portfolio gained 4.1 per cent in the month whereas the meanvariance portfolio lost 7 per cent in the month. Overall, these portfoliosappear to select quite dierent assets, no clear picture emerges, the meanvariance portfolio has made substantial pro®ts in Turkey and large losses inBrazil and Argentina.We also compute the time-series means and variances of the dierentportfolios. These are presented in Table 3.2.We ®rst report results for the long period. Using all data MV has thehighest return but also the highest risk. On a risk-adjusted basis the KBIMIfund out-performs. Using 20 observations MV has the highest expectedreturn. Again the KBIMI fund out-performs. Using 10 observations theMV fund out-performs for both risk-adjusted and actual returns.
Mean-variance analysis, trading rules and emerging markets 117Table 3.3 Sharpe ratio of the considered strategies over short period taken with respect to averagesReturn (net cum) (%)
Mean
Standard deviation
Sharpe ratio
46.22443832
0.015 2310.012 990.011 4980.013 645
0.046 2710.045 1940.047 7720.059 122
0.267 230.224 010.180 700.182 33
314439
0.011 3720.012 990.014 416
0.046 6180.045 1940.052 549
0.181 470.222 990.218 91
494446
0.015 5940.012 990.016 299
0.044 3590.045 1940.055 647
0.285 90.222 990.240 56
All observationsMOEWIEWIIMV20 ObservationsMOEWIMV10 ObservationsMOEWIMV
Turning to the short period we see that for all data MO has the best riskadjusted and return characteristic; for 20 observations MV has the highestreturn whilst KBIM has the best risk-adjusted return, and for 10 observations, MV has the best return whilst MO has the best risk-adjustedperformance.Summarizing, MV performs well but typically has large volatility, MOperforms well based on a short window and has the best expected return (inthe long period). The KBIMI fund performs well and is in no sensedominated by the others.We repeated the exercise assuming transaction costs of 2 per cent for a oneway transaction. The results, see Table 3.4, are qualitatively the same as ourprevious results, this was done using all previous data in our estimations.We omit results for transaction costs for ten and twenty due to lack ofspace.Finally, we discuss our rebalancing results, see Table 3.5. Although theSharpe ratios are lower than those of the MO portfolio, the results areinteresting in terms of the optimal rebalancing time. It seems clear that forone-way costs of 2 per cent, a 6-monthly rebalance outperforms otherrebalancing periods. If costs are higher, and it may be reasonable to thinkthat they are, then a longer period is likely to be optimal. Such a procedureis a low-cost activity that lies somewhere between active and passivequantitive management, it is easy to understand and seems to producegood results.
118
Advanced Trading RulesTable 3.4 Transaction costs and their eect on the Sharperatio over the entire periodMO
EWI
MV
0.023 7790.055 6050.333 973
0.022 1390.043 6830.387 66
0.024 0770.088 2970.213 703
0.023 3320.055 540.326 33
0.022 110.043 6880.386 822
0.023 9030.088 3540.211 592
0.022 8860.055 4790.318 641
0.022 070.043 6940.385 979
0.023 7290.088 4130.209 481
0.022 4390.055 4220.310 915
0.022 0360.043 7010.385 132
0.023 5550.088 4730.207 37
0.021 9930.055 3680.303 152
0.022 0010.043 7080.384 281
0.023 380.088 5330.205 261
0.0% CommissionMeanStandard deviationSharpe0.5% CommissionMeanStandard deviationSharpe1% CommissionMeanStandard deviationSharpe1.5% CommissionMeanStandard deviationSharpe2% CommissionMeanStandard deviationSharpe
3.4 CONCLUSIONSIn conclusion, simple one-period models that are based on mean-varianceanalysis perform adequately in emerging markets con®rming a possibleimplication of Eekhari and Satchell (1995). The results are robust totransaction costs and suggest that these simple strategies could be pro®table.The diculty is the extent to which the increased participation by worldwideinvestors has changed the stochastic structure of these markets and eliminatedthe investment opportunity present in the historical data.To expand on the previous remark, it seems to be the case that the portfoliorisk structure has changed. Although we do not report the results, we foundthat the average correlation between emerging markets has increased in recentyears suggesting that common factors matter more than previously. The
Mean-variance analysis, trading rules and emerging markets 119Table 3.5 Sharpe ratio for rebalancing over entire period with dierent levels of transaction costs
Monthly
3 Monthly
6 Monthly
Annually
Notrebalanced
865.260.022 1390.043 6830.387 207
875.190.022 2790.044 5820.382 529
966.490.023 2130.046 5350.386 548
1038.530.024 0560.051 1650.368 047
881.080.022 7230.052 7410.331 782
800.950.021 4840.043 5740.373 132
840.40.021 9350.044 5750.374 879
957.080.023 1280.046 5050.384 965
1020.540.023 9110.051 2770.364 419
8680.022 6010.052 8190.328 985
740.870.020 8290.043 4690.358 959
806.810.021 5910.044 5740.367 172
947.730.023 0430.046 4820.383 329
1002.80.023 7670.051 3960.360 756
855.940.022 4800.052 9020.326 157
684.760.020 1730.043 3670.344 691
774.380.021 2480.044 580.359 41
938.450.022 9580.046 4660.381 642
985.290.023 6220.051 5230.357 059
842.250.022 3580.052 9930.323 30
632.340.019 5180.043 2680.330 329
743.080.020 9040.044 5940.351 597
929.240.022 8740.046 4560.379 905
968.020.023 4770.051 6580.353 331
829.580.022 2360.053 0890.320 417
0.0% CommissionCumulative returnMeanStandard deviationSharpe0.5% CommissionCumulative returnMeanStandard deviationSharpe1% CommissionCumulative returnMeanStandard deviationSharpe1.5% CommissionCumulative returnMeanStandard deviationSharpe2% CommissionCumulative returnMeanStandard deviationSharpe
double impact of international investment and ®nancial integration has led tojoint market features that represent, perhaps, a less interesting investmentenvironment. New constituent emerging markets may not have the samestatistical characteristics as the original ones did at the same stage of theirdevelopment. Poland, Hungary and China may not be statistically equivalentto Taiwan, Thailand and Columbia at the point at which they join theinvestment opportunity set.Why should trading rules based on our optimized portfolios lead to goodrisk-adjusted returns? One reason is that the MO portfolio is constructed to
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Advanced Trading Rules
have the maximum Sharpe ratio in a mean-variance world. Although ourreturns may appear highly non-normal, conditions for the validity of meanvariance analysis are known to be more general than just multivariatenormality. As proved in Chamberlain (1983), a necessary and sucientcondition for the validity of MV analysis is that the random returns are alinear transformation of a spherically distributed random vector, see Dempster (1969) for a de®nition of a spherical distribution. Furthermore, in theabsence of a riskless asset, mean-variance analysis is valid for a vector of Nreturns if and only the last (N 1) components are spherically