chapter 6 correlations and copulas

8/18/2019 Chapter 6 Correlations and Copulas

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Correlations andCopulas

Chapter 10

Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 1


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Correlation and Covariance

The coefficient of correlation between two

variables V 1 and V 2 is defined as

The covariance is E (V 1V 2)− E (V 1 ) E (V 2)


)()(

)()()(

21

2121

V SDV SD

V E V E V V E −


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Independence

V 1 and V 2 are independent if the

knowledge of one does not affect the

probability distribution for the other

where f (.) denotes the probability densityfunction


)()( 212 V f xV V f ==


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Independence is Not the Same as

Zero Correlation

Suppose V 1 = –1, 0, or +1 (equally

likely)

If V 1 = -1 or V 1 = +1 then V 2 = 1 If V 1 = 0 then V 2 = 0

V 2 is clearly dependent on V 1 (and vice

versa) but the coefficient of correlationis zero



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Types of Dependence (Figure 10.1, page 204)


E (Y )

X

E (Y )

E (Y )

X

(a) (b)

(c)

X


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Monitoring Correlation Between

Two Variables X and Y

Define xi=( X i− X i-1) /X i-1 and yi=(Y i−Y i-1) /Y i-1

Also

var x,n: daily variance of X calculated on day n-1var y,n: daily variance of Y calculated on day n-1

covn: covariance calculated on day n-1

The correlation is

n yn x

n

,, var var

cov



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Covariance

The covariance on day n is

E ( xn yn)− E ( xn) E ( yn)

It is usually approximated as E ( xn yn)



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Monitoring Correlation continued

EWMA:

GARCH(1,1)

111 )1(covcov −−− λ−+λ= nnnn y x

111 covcov −−− β+α+ω= nnnn y x



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Positive Finite Definite Condition

A variance-covariance matrix, , is

internally consistent if the positive semi-definite condition

holds for all vectors w


0≥Ωww

T


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Example

The variance covariance matrix

is not internally consistent


1 0 0 9

0 1 0 90 9 0 9 1

.

.. .


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V 1 and V 2 Bivariate Normal

Conditional on the value of V 1, V 2 is normal with

mean

and standard deviation where µ1,, µ2, σ1,

and σ2 are the unconditional means and SDs ofV 1 and V 2 and ρ is the coefficient of correlation

between V 1 and V 2


1

1122

σµ−ρσ+µ V

2

2 1 ρ−σ


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Multivariate Normal Distribution

Fairly easy to handle

A variance-covariance matrix defines

the variances of and correlationsbetween variables

To be internally consistent a variance-

covariance matrix must be positivesemidefinite



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Generating Random Samples for

Monte Carlo Simulation (pages 207-208)

=NORMSINV(RAND()) gives a random

sample from a normal distribution in

Excel

For a multivariate normal distribution a

method known as Cholesky’s

decomposition can be used to generaterandom samples



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One-Factor Model continued


If U i have standard normal distributions

we can set

where the common factor F and the

idiosyncratic component Z i have

independent standard normal

distributions Correlation between U i and U j is ai a j

iiii Z aF aU 21−+=


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The Correlation Structure Between the V’s is

Defined by that Between the U’s


-0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

V 1V 2

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U 1U 2

One-to-one

mappings

Correlation

Assumption

V 1V 2

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U 1U 2

One-to-one

mappings

Correlation

Assumption


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Example (page 211)


V 1 V 2


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V 1 Mapping to U 1


V 1 Percentile U 1

0.2 20 -0.84

0.4 55 0.13

0.6 80 0.84

0.8 95 1.64


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V 2 Mapping to U 2


V 2 Percentile U 2

0.2 8 −1.41

0.4 32 −0.47

0.6 68 0.47

0.8 92 1.41


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Example of Calculation of Joint

Cumulative Distribution

Probability that V 1 and V 2 are both less

than 0.2 is the probability that U 1 < −0.84

and U 2 < −1.41 When copula correlation is 0.5 this is

M ( −0.84, −1.41, 0.5) = 0.043

where M

is the cumulative distributionfunction for the bivariate normal

distribution



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Other Copulas

Instead of a bivariate normal distribution

for U 1 and U 2 we can assume any other

joint distribution

One possibility is the bivariate Student t

distribution



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5000 Random Samples from the

Bivariate Normal

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5



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5000 Random Samples from the

Bivariate Student t


-10

-5

0

5

10

-10 -5 0 5 10


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Multivariate Gaussian Copula

We can similarly define a correlation

structure between V 1, V 2,…V n

We transform each variable V i to a newvariable U i that has a standard normal

distribution on a “percentile-to-percentile”

basis.

The U’s are assumed to have a

multivariate normal distribution



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Factor Copula Model

In a factor copula model the correlation

structure between the U ’s is generated by

assuming one or more factors.



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Credit Default Correlation

The credit default correlation between two

companies is a measure of their tendency

to default at about the same time

Default correlation is important in risk

management when analyzing the benefits

of credit risk diversification

It is also important in the valuation of some

credit derivatives



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Model for Loan Portfolio

We map the time to default for company i, T i, to a

new variable U i and assume

where F and the Z i have independent standard

normal distributions

Define Qi as the cumulative probability distributionof T i

Prob(U i


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The Model continued

[ ]

[ ]

ncorrelatiocopulatheiswhere

Prob

companiesallfor sametheares'ands'the Assuming

Prob

Hence

Prob

ρ

ρ−

ρ−

=<

−

−=<

−

−=<

−

−

1

)(

)(

1

)()(

1)(

1

2

1

2

F T Q N

N F T T

aQ

a

F aT Q N N F T T

a

F aU N F U U

i

i

iii

i

ii



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The Model continued

The worst case default rate for portfolio for a

time horizon of T and a confidence limit of X is

The VaR for this time horizon and confidence

limit is

where L is loan principal and R is recovery rate


ρ−

ρ+

=

−−

1

)()]([ 11 X N T Q N

N WCDR(T,X)

),()1(),( X T WCDR R L X T VaR ×−×=

chapter 6 correlations and copulas

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