Copulas and credit risk models: some potential developments

Transcription

1 Copulas and credit risk models: some potential developments Fernando Moreira CRC Credit Risk Models 1-Day Conference 15 December 2014

2 Objectives of this presentation To point out some limitations in some credit risk models (special focus on calculations involving probability of default) To give an initial intuition about how copula functions can be used to overcome some of these limitations To show that you can improve the models you use (assuming you have a basic knowledge of statistics) 2

3 Probability of default (PD) Based on Merton (1974): it is the probability that a latent variable associated to borrowers (e.g. log-return of assets) falls bellow a cut-off (outstanding debt) Let Y denote the latent variable and y c denote the cut-off: Asset return (Y) Expected Y y t Probability of all possible future values y c Probability of default = PD = Prob[Y<y c ] 0 t Time Liability value 3

4 Probability of default (PD) A closer look at the asset return distribution (without considering the stochastic movement) PD = Pr[Y < y c ] The original model assumes that Y is normally distributed y c Y 4

5 A Single Factor Model The root of our problems: Idiosyncratic ( specific ) risk Y i X 1 ij i ij Borrowers (i and j) asset return (not observable) Systematic ( market ) risk (single factor) Correlation between asset returns (Y i and Y j ) Y j X 1 ij j ij Idiosyncratic ( specific ) risk Y, X, and ε ~ N(0,1); ε i X; ε i ε j (for i j); 0 1 5

6 A statistical convenience A linear relationship: Y (Very) Special coefficients: 1 2 i X i Y i X 1 ij i ij where YX ij Y j X 1 ij j ij For more details, see: Moreira, Fernando (2011b). 6

7 A by-product Extreme PD Expected Probability of Default = F(y c ) Unfavourable economic status ( quantile ) K V 1 ( PD) 1 1 (0.999) Correlation (dependence) across latent variables regarding borrowers This formula is used, for instance, in Basel Accords to calculate the capital necessary to cover extreme (unexpected) credit losses 7

8 Copulas Functions that link joint distributions to marginal distributions Probability Joint cumulative distribution functions Copula Individual cumulative distribution functions Pr[X x,y y] = F XY (x,y) = C(F X (x), F Y (y)) In other words, copulas use individual cdfs as inputs and return joint probabilities Formal definition: 1959 First (published) application in risk management:

9 Copulas Pr[X x,y y] = F XY (x,y) = C(F X (x), F Y (y)) Important advantage: compatible with any distribution F(.). Therefore we can relax the assumption of specific distributions and capture potential asymmetric dependence (e.g. high losses more associated than low losses) 9

10 Copula families The most popular copulas used in finance/risk management: Copula Dependence Tail dependence Gaussian (Normal) Symmetric No Student t Symmetric Both (left and right) Clayton Asymmetric Left Gumbel Asymmetric Right Note: tail dependence (= extreme values are more associated than values around the mean); left tail dependence refers to small values and right tail dependence refers to large values 10

11 x2 y2 Visual example of tail dependence Without tail dependence With tail dependence Joint extreme (tail) cases 1 Two normal variables (rho = 0.5) with Gaussian dependence 1 Two normal variables (rho = 0.5) with Gumbel dependence x y1 PS: these pairs of data have the same (linear) correlation! 11

12 An empirical application in credit risk Crook and Moreira (2011a) see reference ahead Credit card portfolio (UK bank; April/07 Mar/09) Estimated probability of joint high default rates in different segments (credit quality) Compared (traditional) model based on the assumption of normality with copulas In general, non-normal copulas outperformed normality-based model since they (copulas) yielded estimates (of joint extreme events) closer to the default rates observed in the portfolio 12

13 Copulas and conditional distributions F( y x) Pr( Y y X x) C Y X ( F Y ( y) F X ( x)) C YX ( FY ( y), F F ( x) X X ( x)) The conditional distribution and the conditional C Y X follow the same property of the copula C YX in terms of tail dependence 13

14 An implicit application in credit risk The formula used in Basel Accords to estimate unexpected credit losses corresponds to the first derivative of a copula family (Gaussian) that does not capture tail dependence (i.e. does not capture higher dependence across default rates in unfavourable scenarios) For more details, see: Moreira, Fernando (2015). 14

15 An implicit application in credit risk C Y X ( F Y ( y) F X ( x)) C YX ( FY ( y), F F ( x) X X ( x)) First derivative of the Gaussian Copula: F( y x) 1 ( F Y ( y)) 1 1 ( F X ( x)) Basel formula: K V 1 ( PD) Probability of default 1 1 (0.999) Economic status ( quantile ) For more details, see: Moreira, Fernando (2015). Correlation (dependence) across latent variables regarding obligors 15

16 Default rates (in %) Empirical tests Evolution of default rates in all American commercial banks credit cards mortgages corporate /1 1989/4 1994/4 1999/4 2004/4 2009/4 Year/Quarter Source: Federal Reserve Economic Data (FRED) Federal Reserve Bank of St. Louis For more details, see: Moreira, Fernando (2015). 16

17 Default rates (in %) Question Evolution of default rates in all American commercial banks credit cards mortgages corporate /1 1989/4 1994/4 1999/4 2004/4 2009/4 Year/Quarter 2008/4 17

18 Results credit cards Year/ Quarter Observed default rates Basel 99.9% Basel 99.99% Clayton Copula Student t Copula 2009Q Q Q Q Q Q The best estimates are highlighted in boldface 18

19 Results corporate (business) Year/ Quarter Observed default rates Basel 99.9% Basel 99.99% Clayton Copula Student t Copula 2009Q Q Q Q Q Q The best estimates are highlighted in boldface 19

20 Results mortgages Year/ Quarter Observed default rates Basel 99.9% Basel 99.99% Clayton Copula Student t Copula 2009Q Q Q Q Q Q The best estimates are highlighted in boldface 20

21 Possibly yes, if: Do you need copulas in your work? you are working with dependence you are working with association among two or more variables (including regressions) Especially easy to apply if the values you are using can be interpreted as the area below a particular (even if unknown) point (i.e. your data can be seen as F(x) ). Examples in credit risk: PD and LGD. Just plug your data into copulas without assuming any distribution 21

22 How about data? (Example: PD) We are measuring dependence between Y i and Y j (that drive PD) but they are unobservable We can use historical data on PD to estimate the copula between Y i and Y j Convenience: the copula of the unobserved variables in structural models (Y) is the same as the copula of PD (= F(y)) because... PD is a monotonic transformation of Y, i.e. the smallest y results in the lowest PD, the second smallest y results in the second lowest PD, and so on 22

23 Other potential applications (beyond credit risk ) Joint (stochastic) modelling of assets or economic variables considering that their dependence may be asymmetric (i.e. different intensity in different scenarios). For example, creating scenarios in which losses (or gains) are proportionally higher when interest rates reach extreme levels Identification of more profitable portfolios: cases where high returns tend to cluster (increasing profit) and low returns tend to be less connected (reducing losses) Insurance companies: broken-heart syndrome 23

24 Final thoughts Copulas don t solve all the problems in credit risk models Copulas also have some limitations (in particular, when we work in higher dimensions) But copulas help us improve the models we have (use) nowadays until we find another approach that yields even better results 24

25 Some references General introduction to copulas Genest, C., A. Favre (2007). Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask. Journal of Hydrologic Engineering, Vol. 12, No. 4 (PDF available online) Nelsen, Roger (2006). An Introduction to Copulas. New York: Springer, 2 nd edition Copulas in finance/risk management Cherubini, U., E. Luciano, W. Vecchiato (2004). Copula methods in finance. Chichester: John Wiley & Sons. McNeil, A., R. Frey, P. Embrechts (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton: Princeton University Press. 25

26 Some references Copulas in credit risk Crook, J., F. Moreira (2011a). Checking for asymmetric default dependence in a credit card portfolio: A copula approach. Journal of Empirical Finance, Vol. 18, Issue 4, pp Das, S., G. Geng (2006). Correlated Default Processes: A Criterion-Based Copula Approach. In: Fong, H. (ed.) The Credit Market Handbook. Advanced Modeling Issues. Hoboken, New Jersey: John Wiley & Sons, pp Frey, R., A. McNeil, M. Nyfeler (2001). Copulas and Credit Models. Risk, October, pp Moreira, Fernando F. (2011b). Inaccurate dependence measures in credit models for non-normal variables. Banking and Finance Review, Vol. 3, No. 2, pp Moreira, Fernando (2015). Estimating Portfolio Credit Losses in Downturns. Financial Markets, Institutions and Instruments (forthcoming). 26

27 Contact information Fernando Moreira Lecturer in Business Economics University of Edinburgh Business School/ Credit Research Centre 29 Buccleuch Place Edinburgh EH8 9JS