Correlatons and Copulas Chapter 9 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.
Coeffcent of Correlaton The coeffcent of correlaton between two varables V and V 2 s defned as E( VV 2 ) E( V ) E( V 2 ) SD( V ) SD( V 2 ) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.2
Independence V and V 2 are ndependent f the knowledge of one does not affect the probablty dstrbuton for the other f ( V2 V x) f ( V2 ) where f(.) denotes the probablty densty functon Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.3
Independence s Not the Same as Zero Correlaton Suppose V =, 0, or + (equally lkely) If V = - or V = + then V 2 = If V = 0 then V 2 = 0 V 2 s clearly dependent on V (and vce versa) but the coeffcent of correlaton s zero Correlaton measures lnear dependence Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.4
Types of Dependence E(Y\X ) X E(Y\X ) X (a) E(Y\X ) (b) X (c) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.5
Montorng Correlaton Varance rate per day of a varable: varance of daly returns Covarance rate per day between two varables: covarance between the daly returns of the varables Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.6
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.7 ) ( ) ( ) ( ) ( cov rate on day n : Covarance,, are : on day the returns, day at theend of and two varables of values and n n n n n n n y x E y E x E y x E Y Y Y y X X X x Y X Y X
Montorng Correlaton Between Two Varables X and Y Defne var x,n : daly varance rate of X estmated on day n- var y,n : daly varance rate of Y estmated on day n- cov n : covarance rate estmated on day n- The correlaton s var cov n x, n vary, n Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.8
Montorng Correlaton contnued EWMA: cov n cov n ( ) x n y n GARCH(,) cov n x nyn cov n Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.9
Postve Fnte Defnte Condton A varance-covarance matrx, W, s nternally consstent f the postve semdefnte condton w T Ww 0 holds for all Nх vectors w, W postvesemdefnte Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.0
Example The varance covarance matrx 0 0. 9 0 0. 9 0. 9 0. 9 s not nternally consstent Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.
V and V 2 Bvarate Normal Condtonal on the value of V, V 2 s normal wth mean V m E( V2 V ) m2 rs 2 s 2 and standard devaton s where m,, m 2, 2 r s, and s 2 are the uncondtonal means and SDs of V and V 2 and r s the coeffcent of correlaton between V and V 2 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.2
Multvarate Normal Many varables can be handled A varance-covarance matrx defnes the varances of and correlatons between varables To be nternally consstent a varancecovarance matrx must be postve semdefnte Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.3
Generatng Random Samples for Monte Carlo Smulaton Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.4
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.5
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.6
Factor Models When there are N varables, V (=,2,..N), n a multvarate normal dstrbuton there are N(N-)/2 correlatons We can reduce the number of correlaton parameters that have to be estmated wth a factor model Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.7
Factor Models contnued If U have standard normal dstrbutons we can set 2 U a F a Z F and Z have ndependent standard normal dstrbutons, a s a constant between - and +, Z uncorrelated wth each other and F All the correlaton between U and U j arses from F Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.8
Factor Models contnued Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.9
M-factor Model Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.20 M m jm m j M M M M a a s F Z F F Z a a a F a F a F a U 2 2 2 2 2 2 ' uncorrelated wth each other and varables, standardzed uncorrelated normal,...,... r
Gaussan Copula Models: Creatng a correlaton structure for varables that are not normally dstrbuted Suppose we wsh to defne a correlaton structure between two varable V and V 2 that do not have normal dstrbutons We transform the varable V to a new varable U that has a standard normal dstrbuton on a percentle-topercentle bass. We transform the varable V 2 to a new varable U 2 that has a standard normal dstrbuton on a percentle-topercentle bass. U and U 2 are assumed to have a bvarate normal dstrbuton Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.2
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.22
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.23
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.24
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.25
The Correlaton structure between the V s s defne by that between the U s -0.2 0 0.2 0.4 0.6 0.8.2-0.2 0 0.2 0.4 0.6 0.8.2 V V 2 One-to-one mappngs -6-4 -2 0 2 4 6-6 -4-2 0 2 4 6 U U 2 Correlaton Assumpton Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.26
Other Copulas Instead of a bvarate normal dstrbuton for U and U 2 we can assume any other jont dstrbuton One possblty s the bvarate Student t dstrbuton Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.27
5000 Random Samples from the Bvarate Normal 5 4 3 2 0-5 -4-3 -2-0 2 3 4 5 - -2-3 -4-5 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.28
5000 Random Samples from the Bvarate Student t 0 5 0-0 -5 0 5 0-5 -0 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.29
Multvarate Gaussan Copula We can smlarly defne a correlaton structure between V, V 2, V n We transform each varable V to a new varable U that has a standard normal dstrbuton on a percentle-to-percentle bass. The U s are assumed to have a multvarate normal dstrbuton Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.30
Factor Copula Model In a factor copula model the correlaton structure between the U s s generated by assumng one or more factors. Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.3
Credt Default Correlaton The credt default correlaton between two companes s a measure of ther tendency to default at about the same tme Default correlaton s mportant n rsk management when analyzng the benefts of credt rsk dversfcaton It s also mportant n the valuaton of some credt dervatves Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.32
The Model We map the tme to default for company, T, to a new varable U and assume U a F where F and the Z have ndependent standard normal dstrbutons Defne Q as the cumulatve probablty dstrbuton of T Prob(U <U) = Prob(T <T) when N(U) = Q (T) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.33 a 2 Z
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.34 The Model contnued (*) ) ( ) Prob( companes thesame for all 's are 's and Assumng the ) ( ) Prob( Hence ) Prob( 2 2 r r F Q T N N F T T a Q a F a T Q N N F T T a F a U N U F U
The Model contnued For a large portfolo of loans, equaton (*) provdes a goodestmate of the prercentage of loans defaultng by tme T condtonal on F, the default rate. P(F N N ( Y [ Q( T )) Y, so there s a probablty of Y that the default rate wll be greater than N )] r N - r ( Y ) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.35
The Model contnued We call the default rate that weare X% certan wll not be exceeded n tme T, the"worst - case default rate".substtute Y - X nto the prevous expresson, then WCDR( T, N X ) N [ Q( T )] r r N ( X ) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.36
Example Suppose that a bank has a total of $00 mllon of retal exposures. The one-year probablty of default averages 2% and the recovery rate averages 60%. The copula correlaton parameter s estmated as 0.. WCDR(,0.999) Losses n ths N N 0.28 case are 000.28( 0.6) (0.02) $5.3 mllon 0.N 0. (0.999) Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.37
Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6.38