hullrmfi3rdedch11

7/29/2019 HullRMFI3rdEdCh11

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Correlations andCopulas

Chapter 11

Risk Management and Financial Institutions 3e, Chapter 11, Copyright John C. Hull 2012 1


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Correlation and Covariance

The coefficient of correlation between two

variables V1 and V2 is defined as

The covariance isE(V1V2)E(V1 )E(V2)


)()(

)()()(

21

2121

VSDVSD

VEVEVVE


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Independence

V1 and V2 are independent if the

knowledge of one does not affect the

probability distribution for the other

wheref(.) denotes the probability densityfunction


)()( 212 VfxVVf


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Independence is Not the Same as

Zero Correlation

Suppose V1 =1, 0, or +1 (equally

likely)

IfV1 = -1 orV1 = +1 then V2= 1 IfV1= 0 then V2 = 0

V2 is clearly dependent on V1 (and vice

versa) but the coefficient of correlationis zero



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Types of Dependence (Figure 11.1, page 235)


E(Y)

X

E(Y)

E(Y)

X

(a) (b)

(c)

X


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Monitoring Correlation Between

Two Variables Xand Y

Definexi=(XiXi-1)/Xi-1 andyi=(YiYi-1)/Yi-1

Also

varx,n: daily variance ofXcalculated on day n-1vary,n: daily variance ofYcalculated on day n-1

covn: covariance calculated on day n-1

The correlation is

nynx

n

,, varvar

cov



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Covariance

The covariance on day n is

E(xnyn)E(xn)E(yn)

It is usually approximated asE(xnyn)



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Monitoring Correlation continued

EWMA:

GARCH(1,1)

111 )1(covcov nnnn yx

111 covcov nnnn yx



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Positive Finite Definite Condition

A variance-covariance matrix, W, isinternally consistent if the positive semi-definite condition

wTWw 0

holds for all vectors w



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Example

The variance covariance matrix

is not internally consistent


1 0 0 9

0 1 0 9

0 9 0 9 1

.

.

. .


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V1 and V2 Bivariate Normal

Conditional on the value ofV1, V2 is normal with

mean

and standard deviation where m1,, m2, s1,

and s2 are the unconditional means and SDs ofV1 and V2 and r is the coefficient of correlation

between V1 and V2


1

1122

s

mrsm V

2

2 1 rs


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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 239-240)

=NORMSINV(RAND()) gives a random

sample from a normal distribution in

Excel

For a multivariate normal distribution a

method known as Choleskys

decomposition can be used to generaterandom samples



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Factor Models (page 240)

When there areNvariables, Vi (i = 1,2,..N), in a multivariate normal distributionthere areN(N1)/2 correlations

We can reduce the number of correlationparameters that have to be estimated witha factor model



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One-Factor Model continued


IfUi have standard normal distributions

we can set

where the common factorFand the

idiosyncratic componentZihave

independent standard normal

distributions Correlation between Uiand Ujis ai aj

iiii ZaFaU21


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Gaussian Copula Models:Creating a correlation structure for variables that are not

normally distributed

Suppose we wish to define a correlation structure between

two variable V1 and V2 that do not have normal distributions

We transform the variable V1 to a new variable U1 that has astandard normal distribution on a percentile-to-percentile

basis.

We transform the variable V2 to a new variable U2 that has a

standard normal distribution on a percentile-to-percentilebasis.

U1 and U2 are assumed to have a bivariate normal

distribution

Risk Management and Financial Institutions 3e, Chapter 11,

Copyright John C. Hull 201216


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The Correlation Structure Between the Vs is

Defined by that Between the Us


-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

V1V2

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

U1U2

One-to-one

mappings

Correlation

Assumption

V1V2

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

U1U2

One-to-one

mappings

Correlation

Assumption


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Example (page 241)


V1 V2


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V1 Mapping to U1


V1 Percentile U10.2 20 -0.84

0.4 55 0.13

0.6 80 0.84

0.8 95 1.64


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V2 Mapping to U2


V2 Percentile U20.2 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 V1 and V2 are both less

than 0.2 is the probability that U1 < 0.84

and U2 < 1.41 When copula correlation is 0.5 this is

M( 0.84, 1.41, 0.5) = 0.043

whereMis the cumulative distributionfunction for the bivariate normal

distribution



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Other Copulas

Instead of a bivariate normal distribution

forU1 and U2 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 V1, V2,Vn

We transform each variable Vito a newvariable Ui that has a standard normal

distribution on a percentile-to-percentile

basis.

TheUs 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 Us 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, Ti, to a

new variable Ui and assume

WhereFand theZi have independent standard

normal distributions

The copula correlation is r=a2

ii ZaaFU21



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Analysis

To analyze the model we

Calculate the probability that, conditional on the value

ofF, Ui is less than some value U

This is the same as the probability that Ti is less that Twhere Tand Uare the same percentiles of their

distributions

This leads to

where PD is the probability of default in time T


r

r

1

PD)(Prob

1 FNNFTTi


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The Model continued

TheX% worst case value ofFisN-1(X)

The worst case default rate during time Twith a

confidence level ofXis therefore

The VaR for this time horizon and confidence limit

is

whereL is loan principal and LGD is loss given default


rr

1

)()]([WCDR

11 XNTQNN(T,X)

),(WCDRLGD),(VaR XTLXT


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Gordys Result

In a large portfolio ofMloans where each

loan is small in relation to the size of the

portfolio it is approximately true that


M

i

iii XTLXT1

),(WCDRLGD),(VaR


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Estimating PD and r We can use data on default rates in

conjunction with maximum likelihood

methods

The probability density function for the

default rate is

Risk Management and Financial Institutions 3e Chapter 11 Copyright John C Hull 2012 32

rr

rr

2

1121 )PD()DR(1))DR((21exp1)DR( NNNg

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