CORRELATION BETWEEN

TRADING MODELS

King's College LondonTuesday 2 December 2008

This presentation is for informational/academic purposes only.

Emmanuel Acar

emmanuel.acar@directionaltrading.com

• Trading modelsWhat 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 DifferencesCan 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

• 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

1) A few theoretical results

2) Application to the FX markets

3) Portfolio implications

4) Challenges ahead

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

1X 2X x

1F 2F f

• 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:

0FifXb

0FifXaH

iii

iiii

)b(a ii

iF

iH

7

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

F1=P1[t]-P1[t-5]

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

14/0

7/20

08

16/0

7/20

08

18/0

7/20

08

20/0

7/20

08

22/0

7/20

08

24/0

7/20

08

26/0

7/20

08

28/0

7/20

08

30/0

7/20

08

01/0

8/20

08

03/0

8/20

08

05/0

8/20

08

07/0

8/20

08

09/0

8/20

08

11/0

8/20

08

13/0

8/20

08

15/0

8/20

08

17/0

8/20

08

19/0

8/20

08

21/0

8/20

08

23/0

8/20

08

25/0

8/20

08

27/0

8/20

08

29/0

8/20

08

31/0

8/20

08

02/0

9/20

08

04/0

9/20

08

-8%

-4%

0%

4%

8%

12%

16%

20%

P1[t] P1[t-5]

X1=Ln(P1[t+1]/P1[t])

8

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

F2=P2[t]-SMA2(20)

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

14/0

7/20

08

16/0

7/20

08

18/0

7/20

08

20/0

7/20

08

22/0

7/20

08

24/0

7/20

08

26/0

7/20

08

28/0

7/20

08

30/0

7/20

08

01/0

8/20

08

03/0

8/20

08

05/0

8/20

08

07/0

8/20

08

09/0

8/20

08

11/0

8/20

08

13/0

8/20

08

15/0

8/20

08

17/0

8/20

08

19/0

8/20

08

21/0

8/20

08

23/0

8/20

08

25/0

8/20

08

27/0

8/20

08

29/0

8/20

08

31/0

8/20

08

02/0

9/20

08

04/0

9/20

08

-8%

-4%

0%

4%

8%

12%

16%

20%

P2[t] SMA2(20)

X2=Ln(P2[t+1]/P2[t])

1) A few theoretical results

10

•

Problem: Distribution of H or at least moments E(H),Stdev(H)..Assumption:

With

Introduction: Modelling Single Trading Rule on Single Market

0FifXb

0FifXaH

0Fifb

0FifaB

),(N~F

X2ffxxf

fxxf2x

f

x

xx* /)X(X ff

* /)F(F

*xx

*xx XBB)X(BXBH )XB(E)B(E)H(E *

xx

)XbXa()](b)(a[)H(E*X *F

*

*X *F

*x

f

f

f

fx

f

f

f

f

)5.0exp(2

)ba()](b)(a[)H(E

2f

2f

xfxf

f

f

fx

See [1] for further results

11

Assumptions (See [2], with # notations)

)

00

00

00

00

,

0

0

0

0

(~

F

F

X

X

22f2f1ff

2f1ff21f

22x2x1xx

2x1xx2

1x

2

1

2

1

12

Correlation between rule returns

• Under assumptions specified in [2]:

))sin(Arc

2

1

4

1(aa[

baba5.0)H,H(Corr f212

222

21

21

x21h

))]sin(Arc2

1

4

1(bb))sin(Arc

2

1

4

1(ab))sin(Arc

2

1

4

1(ba f21f21f21

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…………

)b(a ii

h

fx

14

Symmetrical strategies

• Position sizes and 1aa 21 1bb 21

)sin(Arc2

fxh

15

Different rules applied to the same underlying process

F1=P[t]-SMA(5)

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

14/0

7/20

08

16/0

7/20

08

18/0

7/20

08

20/0

7/20

08

22/0

7/20

08

24/0

7/20

08

26/0

7/20

08

28/0

7/20

08

30/0

7/20

08

01/0

8/20

08

03/0

8/20

08

05/0

8/20

08

07/0

8/20

08

09/0

8/20

08

11/0

8/20

08

13/0

8/20

08

15/0

8/20

08

17/0

8/20

08

19/0

8/20

08

21/0

8/20

08

23/0

8/20

08

25/0

8/20

08

27/0

8/20

08

29/0

8/20

08

31/0

8/20

08

02/0

9/20

08

04/0

9/20

08

-12%

-8%

-4%

0%

4%

8%

12%

16%

20%

P[t] SMA(5) SMA(20)

X=Ln(P[t+1]/P[t])

F2=P[t]-SMA(20)

Gbp[/Usd]

16

Different rules applied to the same underlying process

• Example Trading Gbp/Usd onlySimple moving average of length 5 daysSimple 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

)sin(Arc2

fh

f

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

)

)1im()1im(

)1im()1im(sin(Arc

22m

0i

22

2m

0i

21

2)m,m(Min

0i21

h21

21

18

Moving averages EquivalenceP r o p o s i t i o n

A n y m e c h a n i c a l s y s t e m t r i g g e r i n g a s e l l s i g n a l f r o m a f i n i t e l i n e a r c o m b i n a t i o n o f p a s t p r i c e s o f t h e f o r m :

s e l l B a Pt j t jj

m

1 00

1

w h e r e m b e i n g a n i n t e g e r l a r g e r t h a n o n e , a n d a j c o n s t a n t s ,

a d m i t s a n ( a l m o s t ) e q u i v a l e n t l i n e a r r e t u r n f o r m u l a t i o n o f t h e f o r m :

s e l l ~B d Xt j t j

j

m

1 00

2

w h e r e X t = L n ( P t / P t - 1 ) , = a jj

m

0

1

, d j =

a ii j

m 2

T h e o n l y r e q u i r e d a s s u m p t i o n i s t h a t r a t e s o f r e t u r n s c a n b e a p p r o x i m a t e d b y t h e i r l o g a r i t h m i c v e r s i o n s .

T h a t i s f o r j = 1 , m - 1 .

)P/P(Ln~P/P1 jtttjt

19

Theoretical Correlation between rule returns

• Equi-correlation between simple MA achieved for2 , 3, 5, 9, 17, 32, 61, 117, 225Close to Fibonacci numbers !2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Length 2 3 5 9 17 32 61 117 2252 1 0.70 0.52 0.38 0.27 0.20 0.14 0.10 0.073 1 0.71 0.51 0.37 0.26 0.19 0.14 0.105 1 0.71 0.50 0.36 0.26 0.19 0.139 1 0.70 0.50 0.36 0.26 0.19

17 1 0.71 0.50 0.36 0.2632 1 0.70 0.50 0.3661 1 0.70 0.50117 1 0.70225 1

20

• Does not have to be estimatedThe 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

21

Same [technical] rules applied to different underlying process

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

14/0

7/20

08

16/0

7/20

08

18/0

7/20

08

20/0

7/20

08

22/0

7/20

08

24/0

7/20

08

26/0

7/20

08

28/0

7/20

08

30/0

7/20

08

01/0

8/20

08

03/0

8/20

08

05/0

8/20

08

07/0

8/20

08

09/0

8/20

08

11/0

8/20

08

13/0

8/20

08

15/0

8/20

08

17/0

8/20

08

19/0

8/20

08

21/0

8/20

08

23/0

8/20

08

25/0

8/20

08

27/0

8/20

08

29/0

8/20

08

31/0

8/20

08

02/0

9/20

08

04/0

9/20

08

-12%

-8%

-4%

0%

4%

8%

12%

16%

20%

P1[t] SMA(20)

X1=Ln(P1[t+1]/P1[t])

F1=P1[t]-SMA(20)

Gbp[/Usd]

22

F2=P2[t]-SMA2(20)

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

14/0

7/20

08

16/0

7/20

08

18/0

7/20

08

20/0

7/20

08

22/0

7/20

08

24/0

7/20

08

26/0

7/20

08

28/0

7/20

08

30/0

7/20

08

01/0

8/20

08

03/0

8/20

08

05/0

8/20

08

07/0

8/20

08

09/0

8/20

08

11/0

8/20

08

13/0

8/20

08

15/0

8/20

08

17/0

8/20

08

19/0

8/20

08

21/0

8/20

08

23/0

8/20

08

25/0

8/20

08

27/0

8/20

08

29/0

8/20

08

31/0

8/20

08

02/0

9/20

08

04/0

9/20

08

-8%

-4%

0%

4%

8%

12%

16%

20%

P2[t] SMA2(20)

X2=Ln(P2[t+1]/P2[t])

Eur[/Usd]

23

Same [technical] rules applied to different underlying process

• Example Trading Gbp/Usd and Eur/Usdusing Simple moving average of length 20 days

Only correlation between markets has to be estimatedThen results independent on the rule itself

)sin(Arc2

xxh

x

24

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Correlation Coefficient between Different Markets

Co

rre

lati

on

Co

eff

icie

nt

be

twe

en

th

e S

am

e S

tra

teg

y A

pp

lie

d t

o D

iffe

ren

t M

ark

ets

Symmetrical Strategy B&H

2) Application to the FX markets

26

• Chf[/Usd] Futures marketsDaily data from 1978 to 2008

• Establishing returns generated by simple moving averages S(5), S(9) and S(225)

• Calculating correlation coefficients betweenS(5) and S(9)S(5) and S(225)

Different [technical] rules applied to the same underlying

process

27

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00M

ay-7

9

May

-80

May

-81

May

-82

May

-83

May

-84

May

-85

May

-86

May

-87

May

-88

May

-89

May

-90

May

-91

May

-92

May

-93

May

-94

May

-95

May

-96

May

-97

May

-98

May

-99

May

-00

May

-01

May

-02

May

-03

May

-04

May

-05

May

-06

May

-07

May

-08

Co

rrel

atio

n C

oef

fici

ent,

On

e Y

ear

(250

day

s) R

oll

ing

Chf/Usd Theoretical

Chf/Usd TheoryCorrel(S(5),S(9)) 0.692 0.705

28

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40M

ay-7

9

May

-80

May

-81

May

-82

May

-83

May

-84

May

-85

May

-86

May

-87

May

-88

May

-89

May

-90

May

-91

May

-92

May

-93

May

-94

May

-95

May

-96

May

-97

May

-98

May

-99

May

-00

May

-01

May

-02

May

-03

May

-04

May

-05

May

-06

May

-07

May

-08

Co

rrel

atio

n C

oef

fici

ent,

On

e Y

ear

(250

day

s) R

oll

ing

Chf/Usd Theoretical

Chf/Usd TheoryCorrel(S(5),S(225)) 0.094 0.134

29

Testing equality of correlation

• R empirical correlation coefficient calculated over N observationsR0 theoretical value

• Testing R=R0 requiresTransformation

5% confidence interval set to detect statistically different coefficients

r

rLogz

1

1

2

1

)3N(12z

30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8M

ay-7

9

May

-80

May

-81

May

-82

May

-83

May

-84

May

-85

May

-86

May

-87

May

-88

May

-89

May

-90

May

-91

May

-92

May

-93

May

-94

May

-95

May

-96

May

-97

May

-98

May

-99

May

-00

May

-01

May

-02

May

-03

May

-04

May

-05

May

-06

May

-07

May

-08

Tran

sfo

rmed

Co

rrel

atio

n

Chf/Usd Theoretical Percentile(97.5%) Percentile(2.5%)

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

31

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5M

ay-7

9

May

-80

May

-81

May

-82

May

-83

May

-84

May

-85

May

-86

May

-87

May

-88

May

-89

May

-90

May

-91

May

-92

May

-93

May

-94

May

-95

May

-96

May

-97

May

-98

May

-99

May

-00

May

-01

May

-02

May

-03

May

-04

May

-05

May

-06

May

-07

May

-08

Tran

sorm

ed C

orr

elat

ion

Chf/Usd Theoretical Percentile(97.5%) Percentile(2.5%)

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

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

33

• FX marketsMonthly Spot, Forwards and interest rates data from end of Aug 1982 to end of Aug 2008

• 45 currency pairscrossing USD, EUR (DEM), JPY, CHF, GBP, AUD, CAD, NZD, SEK, NOK

Different rules applied to the same underlying process

34

Correlation between SMA(2) and SMA(3)

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

Usd/E

ur

Usd/J

py

Usd/C

hf

Usd/G

bp

Usd/A

ud

Usd/C

ad

Usd/N

zd

Usd/S

ek

Usd/N

ok

Eur/J

py

Eur/C

hf

Eur/G

bp

Eur/A

ud

Eur/C

ad

Eur/N

zd

Eur/S

ek

Eur/N

ok

Jpy/

Chf

Jpy/

Gbp

Jpy/

Aud

Jpy/

Cad

Jpy/

Nzd

Jpy/

Sek

Jpy/

Nok

Chf/G

bp

Chf/A

ud

Chf/C

ad

Chf/N

zd

Chf/S

ek

Chf/N

ok

Gbp

/Aud

Gbp

/Cad

Gbp

/Nzd

Gbp

/Sek

Gbp

/Nok

Aud/C

ad

Aud/N

zd

Aud/S

ek

Aud/N

ok

Cad/N

zd

Cad/S

ek

Cad/N

ok

Nzd/S

ek

Nzd/N

ok

Sek/N

ok

SMA(2,3) Theory

35

• Monthly strategiesBuying or Selling One month Forward- MomentumIf previous B&H return positive (negative), buy (sell)- CarryIf positive (negative) interest rate differential, buy (sell)

• Under the RW hypothesis, Correlation between Forecasts = 0Therefore Correlation between returns = 0

• See [5] for an application to emerging markets

36

Correlation Momentum, Carry

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Usd/E

ur

Usd/Jp

y

Usd/C

hf

Usd/G

bp

Usd/A

ud

Usd/C

ad

Usd/N

zd

Usd/S

ek

Usd/N

ok

Eur/Jp

y

Eur/C

hf

Eur/G

bp

Eur/A

ud

Eur/C

ad

Eur/N

zd

Eur/S

ek

Eur/N

ok

Jpy/C

hf

Jpy/G

bp

Jpy/A

ud

Jpy/C

ad

Jpy/N

zd

Jpy/S

ek

Jpy/N

ok

Chf/G

bp

Chf/A

ud

Chf/C

ad

Chf/N

zd

Chf/S

ek

Chf/N

ok

Gbp/A

ud

Gbp/C

ad

Gbp/N

zd

Gbp/S

ek

Gbp/N

ok

Aud/C

ad

Aud/N

zd

Aud/S

ek

Aud/N

ok

Cad/N

zd

Cad/S

ek

Cad/N

ok

Nzd/S

ek

Nzd/N

ok

Sek/N

ok

-0.8

-0.4

0.0

0.4

Aug-06 Aug-08

2 ye

ars

Co

rrel

atio

n

Usd/Gbp

37

Same rules applied to different underlying process

• Example Trading Usd/Chf and Eur/Jpyusing 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

38

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Correlation Coefficient between Different Markets

Co

rrel

atio

n C

oef

fici

ent

bet

wee

n t

he

Sam

e S

trat

egy

Ap

pli

ed t

o D

iffe

ren

t M

arke

ts

Carry Momentum Theoretical B&H

3) Portfolio Implications

40

• Chf[/Usd] Futures marketsDaily 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

41

• Theory tells usVol of any (+1,-1) strategy = Vol (B&H)Vol (portfolio) = K * Vol (B&H)where K= Function (correlation coefficients)

For portfolio of moving averages 32,61,117K=0.871 See [6]

• Only one estimate required: market’s volatilityIrrespective of the strategy

42

5%

7%

9%

11%

13%

15%

17%

19%

21%M

ay-7

9

May

-80

May

-81

May

-82

May

-83

May

-84

May

-85

May

-86

May

-87

May

-88

May

-89

May

-90

May

-91

May

-92

May

-93

May

-94

May

-95

May

-96

May

-97

May

-98

May

-99

May

-00

May

-01

May

-02

May

-03

May

-04

May

-05

May

-06

May

-07

May

-08

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0.000

0.125

0.250

0.375

0.500

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1.000

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(K)

B&H 32 61 117 Portfolio (32,61,117) K Coefficient Theory

4) Challenges ahead

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 ?

• Isn’t the goal to maximize risk-adjusted returns ?Return expectations will always be subjective.Yet no universal definition of risk. (Stdev, VaR,…)

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….?

References1) 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

References7) 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