A mixed Weibull model for counterparty credit risk in reinsurance Jurgen Gaiser-Porter, Ian Cook ASTIN Colloquium 24 May 2013
Standard credit model Time 0 Prob default pd (1.2%) Expected loss el = pd x expo = 100,000 x 1.2% = 1,200 Exposure expo (100,000) Year 1
S&P default rates Rating 1 2 3 4 5 6 7 8 AAA 0.002% 0.008% 0.043% 0.076% 0.130% 0.228% 0.336% 0.520% AA 0.010% 0.030% 0.080% 0.160% 0.260% 0.400% 0.560% 0.710% A 0.050% 0.163% 0.325% 0.538% 0.825% 1.125% 1.450% 1.763% BBB 0.240% 0.712% 1.225% 1.961% 2.690% 3.434% 4.039% 4.634% BB 1.200% 3.713% 6.863% 9.853% 12.525% 15.300% 17.663% 19.791% B 6.040% 13.871% 20.879% 26.590% 30.849% 34.333% 37.469% 40.354% CCC 30.410% 39.810% 45.742% 50.433% 55.922% 58.461% 59.570% 60.256%
Loss given default In case of default, remaining assets are distributed to creditors Sometimes called recoveries (can cause confusion in reinsurance) Call it loss given default lgd, for example 40% Which makes the expected loss el el el = pd x expo x lgd = 1.2% x 100,000 x 40% = 480
Portfolio: Dependence between counterparties Exposure to several counterparties with Exposure amounts expo i Binary variables B i with probability of default pd i Loss given default lgd i Correlation between defaults Correlation matrix Explicitly set the correlation of default between c parties i and j i.e. ρ(b i, B j ) Copula?
Latent variable A hidden variable Z causes the dependence (Common shock model) Z often vaguely described as state of economy or health of reinsurance industry Could be major catastrophe events or systemic reserving strain But in practice rarely fitted to such data Large values of Z would increase the probability of default and joint probability of 2 counterparties
Solvency 2 Current spec (QIS 5): Peter ter Berg, Portfolio Modelling of Counterparty Reinsurance Default Risk 1-year default rates from S&P Latent variable Z driving correlation between counterparties
Multi-year Time 0 Year 1 Year 2 ok ok default ok default default
Multi-year (2) The longer the term the higher the probability of default Business motivation: long-term casualty A model as described in chart would become very complicated Chosen a simpler model
Time to default Random variable is time to default T i Connected by latent variable Z, survivor function: Conditional on Z, the time to default variables have independent (shifted) distributions
Gamma-Weibull mixture Conditional distribution Candidate Exponential distribution rejected Needed more flexibility to match S&P term structure Fit a Weibull distribution to default rates for each rating category Mixing distribution Gamma distribution with 1 parameter determines the whole dependence structure Unconditional distribution Multi-Burr distribution (when shifts are zero) Clayton copula (when shifts are zero)
Model properties Unconditional survivor function When shift = 0, the unconditional distribution is multi-burr B j counterparty j in default at end of year 1 :
Calibration Base rates of default c j (expert opinion?) Year one rates ξ j determine first Weibull parameter Second Weibull parameter ϕ j fitted to term structure Use the unconditional distribution notation ( Burr ) for calibration Use mixed distribution notation ( Weibull ) for simulation Fitting the gamma parameter ν implied correlation comparison with other models
Model comparison Portfolio 1: one counterparty in each rating category AAA-CCC Seven parties, each with notional of 1/7 Portfolio 2: two counterparties in each rating category AAA-CCC 14 exposures, each with notional of 1/14 Also compared mixed Weibull with McNeil et. al.
Summary Counterparty credit model suitable for cash flows over several years Multi-year to adequately measure risk of long lines Integrated into in-house software product Dependence structure between counterparties credible Further thoughts: Is the term structure of default rates matched well enough? Opportunity to calibrate latent variable (correlate with cat losses?) Dependence structure between counterparties fixed by latent variable parameters (do we need more flexibility?) Data for correlation (old problem)
A mixed Weibull model for counterparty credit risk in reinsurance Jurgen Gaiser-Porter, Ian Cook ASTIN Colloquium 24 May 2013
References P. ter Berg. Portfolio Modelling of Counterparty Reinsurance Default Risk. Life & Pensions Magazine S. P. Britt and Y. Krvavych. Reinsurance Credit Risk Modelling-DFA Approach A. McNeil, R. Frey, and P. Embrechts. Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press
Appendix: Distributions