Econophysics V: Credit Risk

Transcription

1 Fakultät für Physik Econophysics V: Credit Risk Thomas Guhr XXVIII Heidelberg Physics Graduate Days, Heidelberg 2012

2 Outline Introduction What is credit risk? Structural model and loss distribution Numerical simulations Random matrix approach Conclusions general, present credit crisis

3 Introduction

4 Diversification in a Stock Portfolio No Correlations empirical distribution of normalized returns (400 stocks)

5 Diversification in a Stock Portfolio No Correlations empirical distribution of normalized returns (400 stocks) portfolio: superposition of stocks

6 Diversification in a Stock Portfolio No Correlations empirical distribution of normalized returns (400 stocks) portfolio: superposition of stocks risk reduction by diversification (no correlations yet!): returns are more normally distributed, market risk reduced by approx. 50 percent

7 Correlations stocks highly correlated to overall market risk reduction by diversification (with correlations): unsystematic risk can be removed, systematic risk (overall market) remains

8 Credits and Stability of the Economy credit crisis shakes economy dramatic instability physics: model building based on empirical information econophysics: treat economy as complex system

9 Credits and Stability of the Economy credit crisis shakes economy dramatic instability physics: model building based on empirical information econophysics: treat economy as complex system risk reduction by diversification?

10 Zero Coupon Bond t = 0 Creditor Principal Obligor t = T Creditor Face value Obligor principal: borrowed amount face value F : borrowed amount + interest + risk compensation credit contract with simplest cash-flow credit portfolio comprises many such contracts

11 Defaults and Losses default occurs if obligor fails to repay loss between 0 and face value F possible losses have to be priced into credit contract correlations are important to evaluate risk of credit portfolio statistical model yields loss distribution

12 Modeling Credit Risk

13 Structural Models of Merton Type V k (t) V k (0) F T t microscopic approach dynamical description of risk elements V k (t), k = 1,..., K default occurs if asset value V k (T ) falls below face value F k then the (normalized) loss is L k = F k V k (T ) F k

14 Structural Models of Merton Type V k (t) V k (0) F T t microscopic approach dynamical description of risk elements V k (t), k = 1,..., K default occurs if asset value V k (T ) falls below face value F k then the (normalized) loss is L k = F k V k (T ) F k e.g. credits with stock portfolio or houses as securities

15 Geometric Brownian Motion with Jumps choose the stock prices as risk elements V k (t), k = 1,..., K we include jumps! dv k (t) V k (t) = µ k dt + σ k ε k (t) dt + dj k (t) deterministic term µ k dt diffusion term σ k ε k (t) dt jump term dj k (t), governed by a Poisson process parameters can be tuned to describe the empirical price and return distributions

16 Jump Process and Price or Return Distributions Jump with a magnitude of: 65% No Jump V(t) Jump t jumps reproduce empirically found heavy tails

17 Financial Correlations stock prices V k (t), k = 1,..., K measured at t = 1,..., T returns R k (t) = dv k(t) V k (t) S(t) / S(Jan 2008) Jan 2008 Feb 2008 Mar 2008 Apr 2008 May 2008 Jun 2008 Jul 2008 Aug 2008 Sep 2008 Oct 2008 Nov 2008 Dec 2008 IBM MSFT normalization M k (t) = correlation C kl = M k (t)m l (t), R k(t) R k (t) R 2 k (t) R k(t) 2 u(t) = 1 T T u(t) t=1 K T data matrix M such that C = 1 T MM

18 Inclusion of Correlations in Risk Elements ε i (t), i = 1,..., I set of random variables K I structure matrix A correlated diffusion, uncorrelated drift, uncorrelated jumps dv k (t) V k (t) = µ k dt + σ k I A ki ε i (t) dt + dj k (t) i=1 for T correlation matrix is C = AA covariance matrix is Σ = σcσ with σ = diag (σ 1,..., σ K )

19 Loss Distribution

20 Individual Losses V k(t) normalized loss at maturity t = T V k(0) F T t L k = F k V k (T ) F k Θ(F k V k (T )) if default occurs

21 Portfolio Loss Distribution homogeneous portfolio K portfolio loss L = 1 K k=1 stock prices at maturity V = (V 1 (T ),..., V K (T )) distribution p (mv) (V, Σ) with Σ = σcσ L k want to calculate p(l) = d[v ]p (mv) (V, Σ) δ ( L 1 K ) K L k k=1

22 Large Portfolios Real portfolios comprise several hundred or more individual contracts K is large. Central Limit Theorem: For very large K, portfolio loss distribution p(l) must become Gaussian. Question: how large is very large?

23 Typical Portfolio Loss Distributions Frequency Unexpected loss Expected loss α-quantile Loss in % Economic capital of exposure highly asymetric, heavy tails, rare but drastic events mean of loss distribution is called expected loss (EL) standard deviation is called unexpected loss (UL) kurtosis excess (KE) to measure heavy tails: γ 2 = µ 4 /µ 2 2 3

24 Simplified Model No Jumps, No Correlations analytical, good approximations slow convergence to Gaussian for large portfolio kurtosis excess of uncorrelated portfolios scales as 1/K

25 Simplified Model No Jumps, No Correlations analytical, good approximations slow convergence to Gaussian for large portfolio kurtosis excess of uncorrelated portfolios scales as 1/K diversification works slowly, but it works!

26 Numerical Simulations

27 Numerical Simulations: Influence of Correlations, No Jumps fixed correlation C kl = c, k l, and C kk = 1 c = 0.2 c = 0.5

28 Tail as Function of Fixed Correlation kurtosis excess standard deviation (UL) limiting tail behavior quickly reached diversification does not work, it does not reduce risk! standard deviation decreases, bad measure for credit risk

29 Value at Risk versus Fixed Correlation Value at Risk VaR 0 p(l)dl = α here α = 0.99 K = 10, 100, % quantile, portfolio losses are with probability 0.99 smaller than VaR, and with probability 0.01 larger than VaR

30 Numerical Simulations: Correlations and Jumps correlated jump diffusion fixed correlation c = 0.5 jumps change picture only slightly tail behavior stays similar with increasing K

31 Numerical Simulations: Correlations and Jumps correlated jump diffusion fixed correlation c = 0.5 jumps change picture only slightly tail behavior stays similar with increasing K diversification does not work

32 Random Matrix Approach

33 Search for Generic Features large portfolio large K correlation matrix C is K K second ergodicity : spectral average = ensemble average set C = WW and choose W as random matrix

34 Search for Generic Features large portfolio large K correlation matrix C is K K second ergodicity : spectral average = ensemble average set C = WW and choose W as random matrix additional motivation: correlations vary over time

35 Price Distribution at Maturity Brownian motion, V = (V 1 (T ),..., V K (T )), price distribution ( p (mv) 1 1 (V, Σ) = exp 1 ) K 2πT det Σ 2T (V µt ) Σ 1 (V µt ) C = WW with W rectangular real K N, N free parameter, such that Σ = σww σ assume Gaussian distribution for W with variance 1/N p (corr) (W ) = KN ( N exp N ) 2π 2 tr W W average correlation is zero, that is WW = 1 K

36 Average Price Distribution p (mv) (ρ) = N = 2πT with hyperradius ρ = K d[w ]p (corr) (W )p (mv) (V, σww σ) K 2 1 N 2 Γ(N/2) ρ N+K 1 2 k=1 V 2 k (T ) σ 2 k similar to statistics of extreme events N K ( N 2 K N K ρ T 2 ) N T easily transferred to geometric Brownian motion

37 Heavy Tailed Average Price Distribution p Ρ 0.3 p Ρ Ρ K = 50 and N = K, 2K, 5K, 30K N smaller stronger correlated heavier tails

38 Average Loss Distribution p(l) = d[v ] p (mv) (ρ) δ ( L 1 K ) K L k k=1 100 C kl = 0, k l p L 10 N = 5 std (C kl ) = K = 10, 100, 1000, best case scenario, heavy tails remain, little diversification benefit

39 General Conclusions uncorrelated portfolios: diversification works (slowly) correlations lead to extremely fat tailed distribution fixed correlations: diversification does not work ensemble average reveals generic features of loss distributions average correlation zero, but still: heavy tails remain, little diversification benefit

40 Conclusions in View of the Present Credit Crisis credit contracts with high default probability, e.g. houses as securities credit institutes resold the risk of credit portfolios, grouped by credit rating lower ratings higher risk and higher potential return problems: rating agencies rated way too high effect of correlations underestimated benefit of diversification overestimated

41 R. Schäfer, M. Sjölin, A. Sundin, M. Wolanski and T. Guhr, Credit Risk - A Structural Model with Jumps and Correlations, Physica A383 (2007) 533 M.C. Münnix, R. Schäfer and T. Guhr, A Random Matrix Approach to Credit Risk, arxiv: both ranked for several months among the top ten new credit risk papers on

42