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Measures of Correlation

Pete Smith, PRM, FSA, MAAA

Correlation has many uses and definitions. As Carol Alexander, 2001 observes,orrelation may only be meanin!fully om"uted for stationary "roesses. Covariane

stationarity for a time series, yt, is defined as#

Constant, finite mean Constant, finite variane Covariane$yt, yt%s& de"ends only on the la! s

For finanial data, this im"lies that orrelation is only meanin!ful for variates suh as

rates of return or normally transformed variates, ', suh that#

' ( $x % &)

*here x is non%stationary and is the mean of x and the standard deviation. For non%stationary variates li+e "ries, orrelation is not usually meanin!ful.

A more oherent measure of relatedness is ointe!ration. Cointe!ration uses a to%ste"

"roess#

-on!%term euilibrium relationshi"s are established A dynami orrelation of returns is estimated

Cointe!ration ill not be disussed in these /RM sessions, hoever, it is very im"ortantin develo"in! dynami hed!es that see+ to +ee" stationary tra+in! error ithin "reset

bounds. ed!in! usin! orrelation measures ty"ially is not able to ahieve suh ontrol.

oever, instantaneous and terminal measures of orrelation are used in variousa""liations suh as develo"in! stohasti interest rate !enerators.

Definitions of Correlation

Pearsons correlation formula

-inear relationshi"s beteen variables an be uantified usin! the Pearson Produt%Moment Correlation Coeffiient, or

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he value of this statisti is alays beteen %1 and 1, and if and are unrelated itill eual 'ero.

$soure# htt"#))mai!ret."sy.ohio%

state.edu)trish)eahin!)3ntro4Stats)-eture45otes)ha"ter6)node6.html&

Spearman's Correlation Method

A non"arametri $distribution%free& ran+ statisti "ro"osed by S"earman in 1708 as ameasure of the stren!th of the assoiations beteen to variables $-ehmann and

9:Abrera 177;&. he S"earman ran+ orrelation oeffiient an be used to !ive anR%

estimate, and is a measure of monotone assoiation that is used hen the distribution ofthe data ma+e Pearson:sorrelation oeffiientundesirable or misleadin!.

he S"earman ran+ orrelation oeffiient is defined by

$1&

here dis the differene in statistial ran+of orres"ondin! variables, and is an

a""roximation to the exat orrelation oeffiient

$2&

om"uted from the ori!inal data.

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$soure# htt"#))mathorld.olfram.om)S"earmanRan+CorrelationCoeffiient.html&

The Simple Formula for rs, for Rankings without Ties

Here is the same table you saw above, except now we also take the

difference between each pair of ranks (D=XY), and then the suareof each difference! "ll that is reuired for the calculation of the

#pearman coefficient are the values of $ and% D&, accordin' to theformula

rs = D&

$($&)

$soure# htt"#))faulty.vassar.edu)lory)h=b.html&

here is no !enerally ae"ted method for om"utin! the standard error for smallsam"les.

Kendall's Tau Coefficient

S"earman>s r treats ran+s as sores and om"utes the orrelation beteen to sets ofran+s. ?endall>s tau is based on the number of inversions in ran+in!s.

Althou!h there is evidene that ?endall:s au holds u" better than Pearson:s r to extreme

nonnormality in the data, that seems to be true only at uite extreme levels.

wine X Y D D&

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-et inv #( number of inversions, i.e. reversals of "air%ise ran+ orders beteen n "airs.

/ual ran+in!s need an ad@ustment.

( 1 2B inv)$number of "airs of ob@ets&

( 1 % 2 B inv) $nB$n%1&)2& ( 1 8B inv)$nB$n%1&&

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BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

he "reedin! orrelation definitions ei!ht all observations eually. ene, lusterin!

of data assoiated ith "eriods of hi!h and lo volatility is not refleted. i!h volatility

"eriods often dis"lay hi!her levels orrelation ris+. For exam"le, hen the Russian

Rouble beame distressed and hi!hly volatile in 177;, the third orld debt mar+etex"eriened onta!ion.

he S"earman and ?endall au measures are more a""ro"riate for non%normal

distributions.

he above definitions of orrelation are absolute and ill only aidentally re"liate

im"lied mar+et orrelations.

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here# vi( 0i$u&2du

erminal orrelation is usually the more !ermane measure of orrelation for "riin! and

hed!in!.

$see Rebonato,Modern Pricing of Interest-Rate Derivatives, 2002&

Implied Correlation

he im"lied volatility on the ross rate on a forei!n exhan!e "roess may be ex"ressedas#

2x%y( 2xE 2y%2xy

hih may be solved o"tions are available on the f)x rates and ross, i.e.#

( $2xE 2y% 2x%y&)$2xy&

3n the above exam"le, is the im"lied orrelation.

$see Alexander,Market Models, 2001&

Confidence Intervals for Correlation Point Estimates: Fisher!

transformation

-et be a "oint estimate of the orrelation r, define the transformation '#

' ( ln$1Er&)$1%r&D

*hen the sam"le is lar!e $26 or more is a useful rule of thumb&, the distribution ' isa""roximately normal ith an a""roximate mean and variane#

/$'& ( ln$1E&)$1%&D and $'& ( 1)$$n%=&&

he standardi'ed statisti, $' /$'&&)$'&&, is a""roximately a standard normal variable.herefore, a""roximate 1 % onfidene limits for ' are# /$'& $1 % )2&$'&. he 1 % limits for are then obtained by transformin! the limits on ' by means of#

( $ex"$2'& 1&)$ex"$2'& E 1&

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"elationship #et$een Correlation and %olatilit&

3n Volatility and Correlation in Option Pricing,1,in the ontext of to im"erfetlyorrelated variables, Riardo Rebonato states,

GHnder these assum"tions e an no run to simulations, one ith a onstantIidential volatility for both variables and ith im"erfet orrelation, and the

other ith different instantaneous im"erfet orrelation, and the other ith

instantaneous volatilities Ibut "erfet orrelation. Jne an then evaluateorrelation, alulated alon! the "ath, beteen the han!es in the lo! of the to

variables in the to ases. IAs is a""arent from the to fi!ures, the same

sam"le orrelation an be obtained des"ite the fat that the to de%orrelation%!eneratin! mehanisms are very different.

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3nMarket Models, 2001,Carol Alexander observes,

GHnli+e "ries, volatility and orrelation are not diretly observable in the mar+et.

hey an only be estimated in the ontext of a model. 3t is im"ortant to

understand that im"lied and statistial volatility models "rovide estimates orforeasts of the same thin!Kthat is, the volatility "arameter in some assumed

underlyin! "rie "roess.

he folloin! em"irially derived !ra"hs sho instantaneous orrelations as a funtion

of o"tion value. hese relationshi"s allo foreast and simulation "roesses to vary

orrelation onditional on the level of the o"tion "rie.

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Correlation Methods

("C) Models: enerali!ed (utore*ressive Conditional )eteroscedastic

$Referene for LARC Models# Carol Alexander, Market Models, 2001&

)eteroscedasticmodels have varyin! volatility.LARC models are Conditionaltime series models here the urrent variable volatility

de"ends on "rior values.

(uto"e*ressivemodels have the mean reversion "ro"erty.he enerali!edform of the model results in instanes that onstitute a family of models

ith varyin! "arameters.

LARC models are im"ortant in finane beause they enable modelin! of volatilitylusterin!. olatility lusterin! ours beause "eriods of hi!h volatility in tradin!

finanial assets are inters"ersed ith "eriods of lo volatility. LARC models enable

lusterin! to be a"tured ithout im"lementin! full haos models. Full haos finanemodels often develo" inom"lete mar+ets, i.e. re"liatin! strate!ies ill not exist.

("C)+p,

he ARC$"& "roess a"tures the onditional heterosedatiity as a ei!hted avera!e

of "ast suared unex"eted returns#

2t( 0E 12t%1E I E "2t%"0N 0, 1, I, "0 t3t5$0, 2t&

ARC models are rarely used in finane as sim"le LARC models "erform so muhbetter.

("C)+p-.,: S&mmetric ("C)

2t( 0E 12t%1E I E "2t%"E 1t%1E I E 2t%0N 0, 1, I , ",1,I, 0

3t is rarely neessary to use more than a LARC$1,1& model for finanial a""liations.

("C)+/-/,

2t( E 2t%1E 2t%1

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N 0, , 0

he LARC$1,1& model is euivalent to an infinite ARC model ith ex"onentially

delinin! ei!hts.

I("C)

E O 1 if the returns "roess is to be stationary. *hih is neessary for meanrevertin! "roesses.

3f E ( 1, the return "roess is a random al+, and the LARC$1,1& "roess may beex"ressed as#

2t( E $1 % &2t%1E 2t%1

( , 1 0

Suh models are alled integratedLARC or 3%LARC models. 3%LARC models are

often a""ro"riate in forei!n exhan!e mar+ets.

*hen ( 0, the 3%LARC model beomes an /*MA $ex"onentially ei!hted movin!avera!e& model.

(("C): (s&mmetric ("C)

/uity mar+ets are ty"ially more volatile in fallin! mar+ets than risin! mar+ets.A%LARC models ere desi!ned to fit suh mar+ets.

E("C): E0ponential ("C)

he first A%LARC model as the /%LARC or /x"onential LARC model, of the

folloin! form#

ln 2t( E !$'t%1& E ln2t%1

here !$B& is an asymmetri res"onse funtion defined by#

!$'t& ( 'tE $ 't % $2&&

'tis the standard normal unex"eted return t)t

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1("C): nonlinear ("C)

rt( r % .62tE tt

2t( E 2t%1$t%1% % &2E t%1

t( tE 5%LARC models are ty"ially not ris+%neutral, althou!h loal ris+%neutrality may exist.

Models that are not ris+%neutral are inom"lete and !enerally do not have re"liatin!"ortfolios everyhere available. ene, hed!in! may not be "ossible. y"ially

5%LARC models are more om"lex than the "reedin! LARC models in this

"resentation. For these reasons 5%LARC models are not urrently used as extensivelyin finane as other LARC models.

)i*hFre.uenc& ("C) Models

Model freueny is a funtion of the time interval assoiated ith return observations.

So hourly observation is more freuent than daily and daily is more freuent than ee+ly,

and so on.

Most LARC models in "ratie assume that unex"eted "ortion of the atual return is

onditionally normally distributed. 5on%normal models are harateri'ed by fat taileddistributions for the unex"eted "ortion of the return. t("C) models develo" the

unex"eted "ortion of the return aordin! to a student%t distribution. 5on%normal

models may be develo"ed by modifyin! the li+elihood funtion used to estimate the

model "arameters.

i!h freueny models, es"eially intra%day models, are muh more li+ely to be more

volatile and fat%tailed than loer freueny models.

"elationship of Correlation and %a"

Covariane aR han!es linearly as the underlyin! model or fator variable

instantaneous orrelations han!e. -i+eise non%linear aR ill han!e non%linearly as

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the instantaneous orrelations han!e. 9ynami hed!in! osts should "ro"erly reflet the

aR ost of a"ital.

#i2lio*raph& and Future "eadin*s

arious readin!s have been referened throu!h the "ro!ress of this "resentation. As the

"resentation is intended to "rovide a summary of orrelation one"ts in "re"aration forthe subseuent sessions, the one"ts herein are not ori!inal to the "resenter. A !uide to

ori!inal or+ is "rovided so that attendees may develo" a dee"er understandin! of the

sub@et.

Carol (le0ander-Market Models, Wiley, 2001

3f you read one boo+ on these sub@ets, Ms. Alexander>sMarket Modelsis hi!hlyreommended. he boo+ ontains luid and intuitive summaries of hi!hly om"lex

ideas. Market Models is a "leasure to read.

"icardo "e2onato- Volatility and Correlation in Option Pricing, 3ile&- /444

Volatility and Correlationdisusses advaned to"is beyondMarket Models. -i+e all of

Rebonato>s boo+s, it is hi!hly luid and em"irially foused.

5ther "ecommended "eadin*s

/n!le, 200! Risk "#$"resentation

Mante!na and Stanley,$n Introd%ction to &conop'ysics ( Correlation and Co)ple*ity in+inance, Cambrid!e, 2000