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CORRELATION BETWEEN TRADING MODELS King's College London Tuesday 2 December 2008 This presentation is for informational/academic purposes only. Emmanuel Acar emmanuel.acar@directionaltrading.com

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Trading models What are we talking about ? 1988 - 12 technical trading systems [7], [8] 2008 - 7,846 technical trading rules [10] broken into 5 families* Is the distinction arbitrary ? What about econometrics models ? What about fundamental strategies ? Similarities and Differences Can it be quantified ? Does it requires back-testing ? What can be assessed ex-ante ? What needs to be estimated ? * Filter, Moving Average, Support and Resistance, Channel Break-outs, and On-balance Volume Motivations & uses

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Academics and Researchers - Difficult to test Random Walk Hypothesis using technical indicators [12] - Avoid pitfalls and duplication - Facilitate research Investment Managers - Strengthen portfolio construction - Allow quantification of revisions for single strategy that needs refinements Motivations & uses

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1) A few theoretical results 2) Application to the FX markets 3) Portfolio implications 4) Challenges ahead

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Simplified Notations Two underlying assets whose passive Buy & Hold returns are denoted and correlated (Gbp/Usd & Eur/Usd), (Ftse & Dax) or (Usd/Jpy and Ftse) To generate his position in each markets, the trader uses a forecasting technique respectively denoted and correlated Momentum of length 5 days on Gbp/Usd, Simple Moving average of length 20 days on Eur/Usd

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Units in quantity are held when the forecast is positive (negative) with i=1,2. The returns generated by the forecasting rules are denoted. That is:

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7 Momentum of length 5 days applied to Gbp[/Usd] Sep Contract

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8 Simple Moving Average of length 20 days applied to Eur[/Usd] Sep Contract

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1) A few theoretical results

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10 Problem: Distribution of H or at least moments E(H),Stdev(H).. Assumption: With Introduction: Modelling Single Trading Rule on Single Market See [1] for further results

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11 Assumptions (See [2], with # notations )

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12 Correlation between rule returns Under assumptions specified in [2]:

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13 What an improvement !!! Even I assume the position sizes to be given to estimate correlation between rule returns I need to: (1) estimate correlation between markets (2) correlation between forecasting strategies (3) use a complicated mathematical formula ! Not always so…………

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14 Symmetrical strategies Position sizes and

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15 Different rules applied to the same underlying process Gbp[/Usd]

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16 Different rules applied to the same underlying process Example Trading Gbp/Usd only Simple moving average of length 5 days Simple moving average of length 20 days In that particular case when using past prices only in the forecasting strategies is a known number that does not have to be estimated

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17 Correlation coefficient between Simple MA* [3] shows that the returns generated by moving averages of order m1 and m2 exhibit linear correlation coefficient given by: * most popular trading rule ?? [9],[11] For mathematical proofs, see: Acar, E. and Lequeux, P. (1996), " Dynamic Strategies: A Correlation Study ", in C.Dunis (ed), Forecasting Financial Markets, Wiley, London, pp 93-123

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18 Moving averages Equivalence

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19 Theoretical Correlation between rule returns Equi-correlation between simple MA achieved for 2, 3, 5, 9, 17, 32, 61, 117, 225 Close to Fibonacci numbers ! 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

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20 Does not have to be estimated The same relationship between technical indicators across markets Even when analytical formulae do not exist proceed to Monte-carlo simulations Relationship between two technical indicators applied to the same market should be more or less independent on the market itself when measured by the correlation coefficient

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21 Same [technical] rules applied to different underlying process Gbp[/Usd]

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22 Eur[/Usd]

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23 Same [technical] rules applied to different underlying process Example Trading Gbp/Usd and Eur/Usd using Simple moving average of length 20 days Only correlation between markets has to be estimated Then results independent on the rule itself

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2) Application to the FX markets

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26 Chf[/Usd] Futures markets Daily data from 1978 to 2008 Establishing returns generated by simple moving averages S(5), S(9) and S(225) Calculating correlation coefficients between S(5) and S(9) S(5) and S(225) Different [technical] rules applied to the same underlying process

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29 Testing equality of correlation R empirical correlation coefficient calculated over N observations R0 theoretical value Testing R=R0 requires Transformation 5% confidence interval set to detect statistically different coefficients

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30 Transformed Correl(S(5),S(9))

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31 Transformed Correl(S(5),S(225))

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32 Theoretical Correlation Deviations with empirical values over a year and/or on specific markets. Yet overall adequation over the long- term and across currency pairs Does not require any estimation in the case of different rules applied to the same market - Analytical formula or - Monte-carlo simulations once and for all because - independent on the market itself Sure ??? Monthly data across 45 ccy pairs

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33 FX markets Monthly Spot, Forwards and interest rates data from end of Aug 1982 to end of Aug 2008 45 currency pairs crossing USD, EUR (DEM), JPY, CHF, GBP, AUD, CAD, NZD, SEK, NOK Different rules applied to the same underlying process

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34 Correlation between SMA(2) and SMA(3)

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35 Monthly strategies Buying or Selling One month Forward - Momentum If previous B&H return positive (negative), buy (sell) - Carry If positive (negative) interest rate differential, buy (sell) Under the RW hypothesis, Correlation between Forecasts = 0 Therefore Correlation between returns = 0 See [5] for an application to emerging markets

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36 Correlation Momentum, Carry

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37 Same rules applied to different underlying process Example Trading Usd/Chf and Eur/Jpy using the same momentum strategy or the same carry methodology Same strategy applied to 45 Markets => 990 correlations PS Analytical formula on valid for momentum rule. Strictly speaking not applicable to carry strategy

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3) Portfolio Implications

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40 Chf[/Usd] Futures markets Daily data from 1978 to 2008 Establishing returns generated by simple moving averages S(32), S(61) and S(117) Calculating rolling annualised volatility over the past 250 days for - Buy and Hold, - Individual moving averages 32, 61, 117 - Equally weight portfolio of moving averages Different [technical] rules applied to the same underlying process

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41 Theory tells us Vol of any (+1,-1) strategy = Vol (B&H) Vol (portfolio) = K * Vol (B&H) where K= Function (correlation coefficients) For portfolio of moving averages 32,61,117 K=0.871 See [6] Only one estimate required: markets volatility Irrespective of the strategy

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4) Challenges ahead

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44 Reformulating existing strategies as sum of elementary components (1,-1) Higher moments (skewness and kurtosis) generated by portfolio of strategies have been quantified ([4]) How to incorporate these results in portfolio construction ? Isnt the goal to maximize risk-adjusted returns ? Return expectations will always be subjective. Yet no universal definition of risk. (Stdev, VaR,…)

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45 On the risk side (higher moments), Using theoretical results allow to build an unified framework across strategies and shift the focus on measuring/predicting: - Market volatility - Market correlations Enough uncertainties not to use analytical results when available. Freeing time to investigate the primary question: Which strategy makes money and when….?

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References 1)Acar, E (2004), Modelling directional hedge funds-mean, variance and correlation with tracker funds, in Satchell and Scowcroft eds, Advances in Portfolio Construction and Implementation, Elsevier pp 193-214 2)Acar, E and Middleton, A (2004), "Active Correlations: New Findings and More Challenges", presented at the September EIR Conference in London 3)Acar, E. and Lequeux, P. (1996), " Dynamic Strategies: A Correlation Study ", in C.Dunis (ed), Forecasting Financial Markets, Wiley, London, pp 93-123 4)Acar, E. and S.E. Satchell (2002), The portfolio distribution of directional strategies, in Acar and Satchell eds, Advanced Trading Rules, 2d edition, Butterworth-Heinemann, Oxford, pp 174-182 5)De Zwart, G., Markwat, T., Swinkels, L. and D. van Dijk (2007), The economic value of fundamental and technical information in emerging currency markets, ERIM Report Series 2007-096-F&A. 6)Lequeux, P. and E. Acar (1998), A Dynamic Index for Managed Currencies Funds Using CME Currency Contracts, European Journal of Finance, 4(4), 311-330

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References 7)Lukac, L.P., Brorsen B.W. and S.H. Irwin (1988), Similarity of computer guided technical trading systems, Journal of Futures Markets, Vol 8(1), pp 1 – 13 8)Lukac, L.P., Brorsen B.W. and S.H. Irwin (1988), A test of futures market disequilibrium using twelve different technical trading systems, Applied Economics, Vol 20(5), pp 623 – 639 9)Maditinos, D.I., Z. Sevic, N.G. Theriou, (2006), Users' Perceptions and the Use of Fundamental and Technical Analyses in the Athens Stock Exchange: A Second View, AFFI 2006 International Congress, Finance d'entreprise et finance de marche: quelles complementarites, Poitiers, France, 26-27 June 2006 10)Marshall, B.R, Cahan, R.H. & J.M. Cahan, (2008), Can Commodity Futures Be Profitably Traded with Quantitative Market Timing Strategies?, Journal of Banking and Finance, 32, pp. 1810-1819 11)Menkhoff, L and U. Schmidt, (2005), The use of trading strategies by fund managers: some first survey evidence, Applied Economics, Vol 37(15), pp. 1719-1730 12)Shintani, M., Yabu, T., and D. Nagakura (2008), Spurious Regressions in Technical Trading: Momentum or Contrarian?, IMES Discussion Paper Series 2008-E-9

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