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