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

2. Correlation and Covariance The coefficient of correlation between two variables V 1 and V 2 is defined as The covariance is E ( V 1 V 2 )− E ( V 1 ) E ( V 2 ) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 2

3. 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 density function Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 3

4. 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 correlation is zero Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 4

5. Types of Dependence (Figure 10.1, page 204) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 5 E(Y)E(Y) X E(Y)E(Y) E(Y)E(Y) X (a) (b) (c) X

6. Monitoring Correlation Between Two Variables X and Y Define x i =(X i −X i-1 )/X i-1 and y i =(Y i −Y i-1 )/Y i-1 Also var x,n : daily variance of X calculated on day n -1 var y,n : daily variance of Y calculated on day n -1 cov n : covariance calculated on day n -1 The correlation is Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 6

7. Covariance The covariance on day n is E ( x n y n )− E ( x n ) E ( y n ) It is usually approximated as E ( x n y n ) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 7

8. Monitoring Correlation continued EWMA: GARCH(1,1) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 8

9. Positive Finite Definite Condition A variance-covariance matrix,  is internally consistent if the positive semi- definite condition holds for all vectors w Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 9

10. Example The variance covariance matrix is not internally consistent Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 10

11. V 1 and V 2 Bivariate Normal Conditional on the value of V 1, V 2 is normal with mean and standard deviation where  1,,  2,  1, and  2 are the unconditional means and SDs of V 1 and V 2 and  is the coefficient of correlation between V 1 and V 2 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 11

12. Multivariate Normal Distribution Fairly easy to handle A variance-covariance matrix defines the variances of and correlations between variables To be internally consistent a variance- covariance matrix must be positive semidefinite Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 12

13. 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 generate random samples Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 13

14. Factor Models (page 209) When there are N variables, V i ( i = 1, 2,.. N ), in a multivariate normal distribution there are N ( N −1)/2 correlations We can reduce the number of correlation parameters that have to be estimated with a factor model Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 14

15. One-Factor Model continued Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 15 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 a i a j

16. 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 V 1 and V 2 that do not have normal distributions We transform the variable V 1 to a new variable U 1 that has a standard normal distribution on a “percentile-to-percentile” basis. We transform the variable V 2 to a new variable U 2 that has a standard normal distribution on a “percentile-to-percentile” basis. U 1 and U 2 are assumed to have a bivariate normal distribution Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 16

17. The Correlation Structure Between the V’s is Defined by that Between the U’s Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 17 -0.200.20.40.60.811.2 -0.200.20.40.60.811.2 V 1 V 2 -6-4-20246246 -6-4-20246 U 1 U 2 One-to-one mappings Correlation Assumption

18. Example (page 211) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 18 V1V1 V2V2

19. V 1 Mapping to U 1 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 19 V1V1 Percentile U1U1 0.220-0.84 0.4550.13 0.6800.84 0.8951.64

20. V 2 Mapping to U 2 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 20 V2V2 Percentile U2U2 0.28−1.41 0.432−0.47 0.6680.47 0.8921.41

21. 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 distribution function for the bivariate normal distribution Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 21

22. 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 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 22

23. 5000 Random Samples from the Bivariate Normal Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 23

24. 5000 Random Samples from the Bivariate Student t Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 24

25. 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 new variable 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 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 25

26. Factor Copula Model In a factor copula model the correlation structure between the U ’s is generated by assuming one or more factors. Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 26

27. 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 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 27

28. 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 Q i as the cumulative probability distribution of T i Prob(U i <U) = Prob(T i <T) when N(U) = Q i (T) Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 28

29. The Model continued Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 29

30. 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 Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 30