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 . . . .