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Correlations and Copulas
Chapter 10
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
1
Correlation and Covariance
m2
rs2
V1 m1 s1
and standard deviation s2 1 r2 where m1,, m2, s1, and s2 are the unconditional means and SDs of V1 and V2 and r is the coefficient of correlation between V1 and V2
8
Positive Finite Definite Condition
A variance-covariance matrix, W, ithe positive semidefinite condition
w TWw 0
holds for all vectors w
is not internally consistent
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
10
V1 and V2 Bivariate Normal
Conditional on the value of V1, V2 is normal with mean
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
9
Example
The variance covariance matrix
1 0 0.9
0 1 0.9
0.9 0.9 1
4
Types of Dependence (Figure 10.1, page 204)
E(Y) X
E(Y) X
(a)
(b)
E(Y)
X
(c)
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
The coefficient of correlation between two variables V1 and V2 is defined as
E(V1V2 ) E(V1)E(V2 ) SD(V1)SD(V2 )
The covariance is E(V1V2)−E(V1 )E(V2)
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
2
Independence
V1 and V2 are independent if the knowledge of one does not affect the probability distribution for the other
EWMA:
covn covn1 (1 )xn1 yn1
GARCH(1,1)
covn xn1 yn1 covn1
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
V2 is clearly dependent on V1 (and vice versa) but the coefficient of correlation is zero
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
It is usually approximated as E(xnyn)
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
7
Monitoring Correlation continued
5
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
3
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
f (V2 V1 x) f (V2 )
where f(.) denotes the probability density function
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
c ovn varx,n vary,n
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009
6
Covariance
The covariance on day n is E(xnyn)−E(xn)E(yn)
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