Correlations andCopulasChapter 11RiskManagementandFinanciallnstitutions3e,Chapter11,CopyrightJohnC.Hull2012K
Correlations and Copulas Chapter 11 Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 1
Correlation and CovarianceThe coefficient of correlation between twovariables V, and V, is defined asE(VV)- E(V)E(V2)SD(V)SD(V2)The covarianceisE(V,V2)-E(V )E(V2)2RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Correlation and Covariance ⚫ The coefficient of correlation between two variables V1 and V2 is defined as ⚫ The covariance is E(V1V2 )−E(V1 )E(V2 ) Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 2 ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 SD V SD V E VV − E V E V
Independence V, and V, are independent if theknowledge of one does not affect theprobability distribution for the otherf(V2|V = x) = f(V2)where f(.) denotes the probability densityfunction3RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Independence ⚫ V1 and V2 are independent if the knowledge of one does not affect the probability distribution for the other where f(.) denotes the probability density function Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 3 ( ) ( ) 2 1 V2 f V V = x = f
IndependenceisNotthe SameasZero CorrelationSuppose V, = -1, 0, or +1 (equally likely)If Vi = -1 or Vi = +1 then V2 = 1If V, = O then V, = 0V, is clearly dependent on V, (and viceversa) but the coefficient of correlationis zeroRiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull20124
Independence is Not the Same as Zero Correlation ⚫ Suppose V1 = –1, 0, or +1 (equally likely) ⚫ If V1 = -1 or V1 = +1 then V2 = 1 ⚫ If V1 = 0 then V2 = 0 V2 is clearly dependent on V1 (and vice versa) but the coefficient of correlation is zero Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 4
Types of Dependence (Figure11.1, page 235)E()E()XX(a)(b)E()X(c)5RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Types of Dependence (Figure 11.1, page 235) Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 5 E(Y) X E(Y) E(Y) X (a) (b) (c) X
Monitoring Correlation BetweenTwo Variables X and YDefine x;=(X,-X,-1)/X,-1 and y;=(Y;-Yi-1)/Yi-1Alsovarx.n: daily variance of X calculated on day n-1vary.n: daily variance of Y calculated on day n-1covn: covariance calculated on day n-1The correlation iscOV,varvarx.ny,n6RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Monitoring Correlation Between Two Variables X and Y Define xi =(Xi−Xi-1 )/Xi-1 and yi =(Yi−Yi-1 )/Yi-1 Also varx,n: daily variance of X calculated on day n-1 vary,n: daily variance of Y calculated on day n-1 covn : covariance calculated on day n-1 The correlation is x n y n n , , var var cov Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 6
CovarianceThe covariance on day n isE(xnyn)-E(xn)E(yn) It is usually approximated as E(xnyn)RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull20127
Covariance ⚫ The covariance on day n is E(xn yn )−E(xn )E(yn ) ⚫ It is usually approximated as E(xn yn ) Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 7
Monitoring Correlation continuedEWMA:coV, = >coV n-I +(1 -2)xn-IYn-1GARCH(1,1)cOVn =O +αxn-1n-I +βcOVn-1RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull20128
Monitoring Correlation continued EWMA: GARCH(1,1) 1 1 1 cov cov (1 ) n = n− + − n− n− x y 1 1 1 covn = + n− n− +covn− x y Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 8
Positive Finite Definite ConditionA variance-covariance matrix, Q, isinternally consistent if the positive semidefinite conditionWTQW ≥ 0holds for all vectors w9RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Positive Finite Definite Condition A variance-covariance matrix, W, is internally consistent if the positive semidefinite condition wTWw ≥ 0 holds for all vectors w Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 9
ExampleThe variance covariance matrix00.91010.90.9(0.91is not internally consistent10RiskManagementandFinancialInstitutions3e,Chapter11,CopyrightJohnC.Hull2012
Example The variance covariance matrix is not internally consistent Risk Management and Financial Institutions 3e, Chapter 11, Copyright © John C. Hull 2012 10 1 0 0 9 0 1 0 9 0 9 0 9 1 . . . .