Asymptotic behaviour of multivariate default probabilities and default correlations under stress

Transcription

1 Asymptotic behaviour of multivariate default probabilities and default correlations under stress 7th General AMaMeF and Swissquote Conference EPFL, Lausanne Natalie Packham joint with Michael Kalkbrener and Ludger Overbeck 7 September 2015

2 Stress testing of bank portfolios Calculate risk measures (expected loss, value-at-risk, economic capital) and regulatory capital under adverse market conditions Stress tests are typically conducted within models Crucial inputs of any portfolio model: Distribution assumption on portfolio constituents, e.g. normally distributed asset returns fat-tailed asset returns Dependence assumption among portfolio constituents, e.g. correlation c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

3 Stress testing of bank portfolios Questions: What is the model behaviour under stress? What are model side effects when stress testing? In a series of papers we investigate these questions: M. Kalkbrener and N. Packham. Correlation under stress in normal variance mixture models. Mathematical Finance, 25:2 (2015), M. Kalkbrener and N. Packham. Stress testing of credit portfolios in light- and heavy-tailed models. J. Risk Management in Financial Institutions, 8:1 (2015), N. Packham, M. Kalkbrener, and L. Overbeck. Asymptotic behaviour of multivariate default probabilities and default correlations under stress. J. Applied Probability, 53:1 (2016). c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

4 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

5 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

6 Structural credit model Merton-model (Merton, 1974) links default of firm to relationship between assets and liabilities Counterparty i in default at T if asset value Zi below debt value Di Assets Assets Liabilities Debt Equity Default event: {Z i < D i } Portfolio loss: d L := l i 1 {Zi D i }, i=1 with l i loss-at-default of counterparty i c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

7 Credit portfolio risk Risk concentrations, correlations: Z i = R 2 i m j=1 w ij X j + 1 Ri 2ε i, i = 1,..., d, where X1,..., X m : systematic factors or risk factors, εi : firm-specific factor c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

8 Credit portfolio risk measures Expected loss: E(L) = d i=1 l i P(Z i D i ) Value-at-risk (at level β): β-quantile of L: VaR β (L) = inf{x R : P(L x) β} Default correlations as measure of dependence: Corr(1 {Zi D i }, 1 {Zj D j }) = P(Z i D i, Z j D j ) p i p j, pi (1 p i )p j (1 p j ) where p i := P(Z i D i ), i = 1,..., d c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

9 Stress testing Translate stress scenario into constraints on risk factors Truncate risk factor variable Z 0 (which is typically one of the systematic factors X i ): Z 0 C, C R stress level Portfolio risk is evaluated under P( Z 0 C) Consistent framework that associates severity of stress scenario with probability of stress scenario See e.g. Bonti et al. (2006); Duellmann and Erdelmeier (2009); Kalkbrener and Packham (2015a) c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

10 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

11 Elliptically distributed random variables Standard approach in credit risk portfolio modelling: risk factors and asset variables normally distributed Generalisations: normal variance mixture distribution elliptical distribution Cover variety of light-tailed to heavy-tailed distribution Example bivariate density: c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

12 Elliptically distributed random variables Let random vector Z = (Z 0,..., Z d ) T follow elliptical distribution with representation Z L = GAU, where G > 0 is a scalar random variable, the mixing variable, A is a deterministic (d + 1) (d + 1) matrix with AA T := Σ, which in turn is a (d + 1) (d + 1) nonnegative definite symmetric matrix of rank d + 1, U is a (d + 1)-dimensional random vector uniformly distributed on the unit sphere S d+1 := {z R d+1 : z T z = 1}, U is independent of G. c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

13 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

14 Asset correlations Asset correlations under stress (Kalkbrener and Packham, 2015a): multivariate normal or t-distribution: explicit formulas for asset correlations under stress Corr C (Z i, Z j ) Normal variance mixture distribution: ρ i ρ j + (ρ ij ρ i ρ j ) (α 1) lim C CorrC (Z i, Z j ) =, (ρ 2 i + (1 ρ 2 i ) (α 1)) (ρ2 j + (1 ρ 2 j ) (α 1)) with α > 2 the tail index of asset returns and risk factor and ρ i = ρ 0i c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

15 Asset correlations In a typical scenario, assets de-correlate with increasing stress and increasing tail index Helps explain why the relative impact of stress on EL of portfolio is stronger than on VaR normally dist. t dist. Ν 10 t dist. Ν 4 asset correlation stress probability Left: Conditional asset correlations; right: Asymptotic asset correlations c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

16 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

17 Preliminaries Fréchet max domain / regular variation: RVα : regularly varying functions with index α R G in Fréchet max domain iff P(G > ) RV α for some α > 0 Gumbel max domain / rapid variation: RV : rapidly varying functions If G in Gumbel max-domain, then P(G > ) RV c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

18 Preliminaries Fréchet max domain / regular variation: RVα : regularly varying functions with index α R G in Fréchet max domain iff P(G > ) RV α for some α > 0 Gumbel max domain / rapid variation: RV : rapidly varying functions If G in Gumbel max-domain, then P(G > ) RV Random vector Z = GAU standardised, so that Σ = AA T is the correlation matrix Correlations are positive, i.e., ρ ij > 0 A i : i-th row of A F U : uniform distribution on S d+1 c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

19 Default probabilities Theorem (Packham et al. (2016)) (i) If P(G > ) RV α, then = lim P(Z 1 D 1,..., Z d D d Z 0 C) C ( 1 (A 0 u) α df U (u) (A 0 u) α df U (u)). u S d+1,a 0 u>0 u S d+1,a 0 u>0,...,a d u>0 (ii) If P(G > ) RV, then lim P(Z 1 D 1,..., Z d D d Z 0 C) = 1. C Asymptotic default probabilities do not depend on default thresholds. c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

20 Default probabilities Theorem (Packham et al. (2016)) Let P(G > ) RV α. (i) For d = 1, (ii) For d = 2, lim C P(Z 1 D 1 Z 0 C) = t α+1 ( α + 1 ρ01 1 ρ 2 01 ) [1/2, 1). lim P(Z 1 D 1, Z 2 D 2 Z 0 C) C ( ) = 1 (α + 1) t 2 t α+1 + some long integrals. 1 t 2 c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

21 Default probabilities PD asympt Ρ bivariate PD asympt Ρ Α Α Asymptotic univariate PD s (left) and bivariate PD s (right) as a function of the tail index. Correlations: ρ 01 = ρ 02 = ρ and ρ 12 = ρ 2. c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

22 Default correlations Regularly varying case easily calculated from previous Theorem. Theorem Let P(G > ) RV. Then, lim C CorrC (1 {Z1 D 1}, 1 {Z2 D 2}) = 0, where Corr C denotes the correlation under P( Z 0 C). c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

23 Default probabilities vs. tail dependence Asymptotic default probability: lim C P(Z 1 D Z 0 C). Coefficient of (lower) tail dependence: λ l (Z 0, Z 1 ) := lim P(Z 1 C Z 0 C). C In the light-tailed case, tail dependence is 0, which is in contrast to the asymptotic default probability Tail dependence function that captures both: λ(z 0, Z 1, x) := lim P(Z 1 x C Z 0 C), x R. C c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

24 Relation to tail dependence Closed formula for tail dependence function λ(z 0, Z 1, x) in paper Special cases: stressed PD s: x = 0 tail dependence: x = 1 c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

25 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

26 Risk measures Risk measures for portfolio consisting of 60 homogeneous counterparties, each with a PD of 1%. Left: Value-at-risk at 99% confidence level Right: Expected loss c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

27 Risk measures Risk measures for portfolio consisting of 60 homogeneous counterparties, each with a PD of 1%. Economic Capital (VaR-EL) c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

28 Overview Structural credit portfolio models and stress testing Distribution of model variables Asset correlations under stress Default probabilities and default correlations under stress Risk measures Conclusion c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

29 Conclusion Stress tests are an integral part of risk management and banking supervision, and the analysis and understanding of risk model behaviour under stress has become ever more important. We analyse asset correlations, default probabilities and default correlations under stress in a generalised Merton-type credit portfolio setup covering light- and heavy-tailed distributions. It turns out that the model behaviour under stress depends on the heaviness of the tails of the risk factors. Contrary to popular belief, light-tailed models show a higher impact in extreme stress scenarios. We use our results to study the implications for credit reserves and capital requirements under stress. c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

30 References G. Bonti, M. Kalkbrener, C. Lotz, and G. Stahl. Credit risk concentrations under stress. Journal of Credit Risk, 2(3): , K. Duellmann and M. Erdelmeier. Crash testing German banks. International Journal of Central Banking, 5(3): , M. Kalkbrener and N. Packham. Correlation under stress in normal variance mixture models. Mathematical Finance, 25(2): , M. Kalkbrener and N. Packham. Stress testing of credit portfolios in light- and heavy-tailed models. Journal of Risk Management in Financial Institutions, 8(1):34 44, R. C. Merton. On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance, 29(2): , May N. Packham, M. Kalkbrener, and L. Overbeck. Asymptotic behaviour of multivariate default probabilities and default correlations under stress. Journal of Applied Probability, 53(1), c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September

31 Thank you! Prof. Dr. Natalie Packham Assistant Professor / Juniorprofessorin of Quantitative Finance Department of Finance Frankfurt School of Finance & Management Sonnemannstr Frankfurt am Main n.packham@frankfurt-school.de c N. Packham, Frankfurt-School.de Asymptotic behaviour under stress, 7 September