Vasicek Single Factor Model

Transcription

1 Alexandra Kochendörfer 7. Februar / 33

2 Problem Setting Consider portfolio with N different credits of equal size 1. Each obligor has an individual default probability. In case of default of the n th obligor we lose the whole n th position in portfolio. What can we say about the loss distribution? 2 / 33

3 Contents Default Correlation Definition Why is default correlation important Independent/perfectly dependent defaults Modelling Default Correlation Data sources Default triggered by firm s value Loss distribution in finite portfolio Large Homogeneous Portfolio Approximation Conclusion 3 / 33

4 Default Correlation Definition Definition Default correlation is the phenomenon that the likelihood of one obligor defaulting on its debt is affected by whether or not another obligor has defaulted on its debts. Positive correlation: one firm is the creditor of another Negative correlation: the firms are competitors Drivers of Default Correlation State of the general economy Industry-specific factors Oil industry: 22 companies defaulted over Thrifts: 19 defaults over Casinos/hotel chains: 10 defaults over / 33

5 Default Correlation Definition U.S. Corporate Default Rates Since / 33

6 Default Correlation Why is default correlation important Why is default correlation important? Consider, for two default events A and B default probabilities p A and p B joint default probability p AB conditional default probability p A B correlation ρ AB between default events These quantities are connected by p A B = p AB p B 6 / 33

7 Default Correlation Why is default correlation important ρ AB = Cov(A, B) Var(A) Var(B) = p AB p A p B pa (1 p A )p B (1 p B ) The default probabilities are usually very small p A = p B = p 1 p AB = p A p B + ρ AB pa (1 p A )p B (1 p B ) p 2 + ρ AB p ρ AB p p A B = p A + ρ AB p B pa (1 p A )p B (1 p B ) ρ AB The joint default probability and conditional default probability are dominated by the correlation coefficient. 7 / 33

8 Default Correlation Independent/perfectly dependent defaults Independent Defaults Consider N independent default events D 1,..., D N with p D1 = = p DN = p Number of defaults B(p, N).For N = 100, p = 0.05 p (%) (%) VaR / 33

9 Default Correlation Independent/perfectly dependent defaults Perfectly dependent defaults Consinder default correlation ρ ij = 1 for all pairs ij. 1 = p D1 D 2 p D1 p D2 pd1 (1 p D1 )p D2 (1 p D2 ) = p D 1 D 2 p 2 p(1 p) p D1 D 2 = p D1 = p D2 = p i.e. D 1 D 2 = D 1 = D 2 a.s. p D1 D 2 D 3 = p D2 D 3 = p D 1 D 2 D 3 = D 1 a.s.... D 1 D N = D 1 a.s. p D1...D N = p All loans in the portfolio defaults with probability p, none with probability 1 p. 9 / 33

10 Default Correlation Independent/perfectly dependent defaults Perfectly dependent defaults 10 / 33

11 Modelling Default Correlation Data sources Data Sources Actual Rating and Default Events. + Objective and direct. Joint defaults are rare events, sparse data sets. Credit Spread. + Incorporate information on markets, observable. No theoretical link between credit spread correlation and default correlation. Equity correlation. + Data easily available, good quality. Connection to credit risk not obvious, needs a lot of assumptions. 11 / 33

12 Modelling Default Correlation Default triggered by firm s value Default triggered by Firm s Value The firm value (A n,t ) 0 t 1 of each obligor n {1,..., N} is modelled as in Black-Scholes model, hence at terminal time t = 1 with A n,1 = A n we have {( ) } A n = A n,0 exp µ n σ2 n + σ n B n 2 with some standard normal variable B n. The r.v. (B 1,..., B N ) are jointly normally distributed with covariance matrix Σ = (ρ ij ) ij, where ρ ij denotes the asset correlation between assets i and j. 12 / 33

13 Modelling Default Correlation Default triggered by firm s value The obligor n defaults if the asset value falls below a perspecified barrier C n (debts) D n = 1 {An<Cn} The default probability of the n s debtor is p Dn = P(D n = 1) = P(A n < C n ) = P(B n < c n ) = Φ(c n ) with default barrier c n = log Cn A n,0 µ n σ n We can assume the individual default probabilities p Dn and compute c n = Φ 1 (p Dn ) and vice versa. as given 13 / 33

14 Modelling Default Correlation Default triggered by firm s value The joint distribution of B i determines the dependency structure of default variables uniquely P(D 1 = 1,..., D N = 1) = P(B 1 < c 1,..., B N < c N ) = Φ N (Φ 1 (p D1 ),..., Φ 1 (p Dn ); Σ) In case with two assets with correlation ρ 1,2 = ρ 2,1, the default correlation can be computed via ρ = P(D 1 = 1, D 2 = 1) p D1 p D2 pd1 (1 p D1 )p D2 (1 p D2 ) = Φ 2(Φ 1 (p D1 ), Φ 1 (p D2 ); ρ 1,2 ) p D1 p D2 pd1 (1 p D1 )p D2 (1 p D2 ) 14 / 33

15 Modelling Default Correlation Default triggered by firm s value We need N(N 1)/2 asset correlations of Σ N individual default probabilities Additional assumptions on the structure of B i to reduce the number of parameters. 15 / 33

16 Assume, that the logarithmic return B n can be written as B n = ρ Y + 1 ρ ɛ n with some constant ρ [0, 1] and N + 1 independent standard normally distributed r.v. Y, ɛ 1,..., ɛ N. Interpretation Y is a common systematic risk factor affecting all firms (state of economy) ɛ n are idiosyncratic factors independent across firms (management, innovations) Corr(B i, B j ) = ρ controls the proportions between systematic and idiosyncratic factors, empirically around 10%. 16 / 33

17 Conditional on the realisation of the systematic factor Y the logarithmic returns B n are independent ( for a constant y variables ρ y + 1 ρ ɛ n are independent) default variables D n = 1 {Bn<c n} are independent as function of B n The only effect of Y is to move B n closer or further away from barrier c n. 17 / 33

18 Loss distribution in finite portfolio Theorem For ρ (0, 1) and same default probabilities p = p D1 = = p DN the conditional default probability is given by ( Φ 1 (p) ) ρ y p(y) := P[B n < c Y = y] = Φ. 1 ρ The number of defaults L = N i=1 D i has the following distribution m ( ) N P(L m) = p(y) k (1 p(y)) N k φ(y)dy k k=0 18 / 33

19 Loss distribution in finite portfolio Proof The probability of k defaults is P(L = k) = E(P({L = k} Y )) = P(L = k Y = y)φ(y)dy, where φ is density of Y. The defaults D n are independent conditional on Y, hence ( ) N P(L = k Y = y) = p(y) k (1 p(y)) N k k Thus, for m {1,..., N} we have P(L m) = m k=0 ( ) N p(y) k (1 p(y)) N k φ(y)dy k 19 / 33

20 Loss distribution in finite portfolio Loss Distibutions for different ρ 20 / 33

21 Loss distribution in finite portfolio VaR Levels for different ρ with N = 100 and p = 5% Independent defaults ρ(%) 99.9(%)VaR 99.(%)VaR p (%) (%) VaR / 33

22 Large Homogeneous Portfolio Approximation Large Homogeneous Portfolio Approximation Definition (Large Homogeneous Portfolio LHP) p D1 = = p DN = p portfolio is weighted with ω (N) 1,..., ω (N) N, N such that N lim (ω n (N) ) 2 = 0 N n=1 n=1 ω(n) n = 1, The portfolio is not dominated by few loans much larger then the rest. 22 / 33

23 Large Homogeneous Portfolio Approximation Definition (Loss Rate) The portfolio loss rate is defined by L (N) = N n=1 ω (N) n D n [0, 1] Lemma Following holds for the LHP ( Φ E(L (N) 1 (p) ) ρ Y Y ) = p(y ) = Φ 1 ρ Var(L (N) Y ) = N n=1 (ω (N) n ) 2 p(y ) (1 p(y )) 23 / 33

24 Large Homogeneous Portfolio Approximation Proof Linearity of conditional expectation yields E(L (N) Y ) = = N n=1 N n=1 ω (N) n E(D n Y ) ω (N) n P(D n Y ) = p(y ) D n are independent conditional on Y, thus N n=1 ω (N) n = p(y ) Var(L (N) Y ) = = N n=1 N n=1 (ω (N) n ) 2 Var(D n Y ) (ω (N) n ) 2 p(y ) (1 p(y )) 24 / 33

25 Large Homogeneous Portfolio Approximation Theorem The portfolio loss rate in LHP converges in probability for N. ( L (N) P Φ 1 (p) ) ρ Y p(y ) = Φ 1 ρ Proof For the large portfolio the variation of loss rate given Y tends to 0 Var(L (N) Y ) = 1 4 N n=1 (ω (N) N n=1 n ) 2 p(y ) (1 p(y )) (ω (N) n ) 2 N 0 25 / 33

26 Large Homogeneous Portfolio Approximation This provides convergence in L 2 : E((L (N) p(y )) 2 ) = E((L (N) E(L (N) Y )) 2 ) = E(E((L (N) E(L (N) Y )) 2 Y )) = E(Var(L (N) Y )) N 0 Convergence in L 2 implies convergence in probability i.e. for all ɛ > 0: ( ) L lim P (N) p(y ) > ɛ = 0 N The law of L (N) converges weakly to the law of p(y ), i.e. P(L (N) x) N P(p(Y ) x) for all x, where the distribution function of p(y ) is continuous. 26 / 33

27 Large Homogeneous Portfolio Approximation Theorem (Approximative Distribution of Loss Rate in LHP) ( 1 ρ Φ 1 (x) Φ 1 ) (p) P(p(Y ) x) = Φ, x [0, 1] ρ Proof ( ( Φ 1 (p) ) ) ρ Y P(p(Y ) x) = P Φ x 1 ρ ( 1 ρ Φ 1 (x) Φ 1 ) (p) = P Y ρ ( 1 ρ Φ 1 (x) Φ 1 ) (p) = Φ ρ 27 / 33

28 Large Homogeneous Portfolio Approximation Approximative density of loss rate with p = 2%, ρ = 10% 28 / 33

29 Large Homogeneous Portfolio Approximation Properties of Loss Rate Distribution E(p(Y )) = lim N E(L(N) ) = lim N N n=1 ω n (N) p = p Because of convergence we can easily compute α-quantiles of loss rate distribution for large N ( 1 ρ Φ P(L (N) 1 (α) Φ 1 ) (p) α) Φ ρ 29 / 33

30 Large Homogeneous Portfolio Approximation When ρ 1 P(L ( ) α) = 1 p = P(L ( ) = 0) for all α (0, 1) P(L ( ) = 1) = p All loans default with prob. p, none with 1 p. When ρ 0 P(L ( ) α) = 0 for α < p P(L ( ) α) = 1 for α p P(L ( ) = p) = 1 With the Law of Large Numbers the loss in Binomial model tends almost surly to 1 N D i p N i=1 30 / 33

31 Large Homogeneous Portfolio Approximation Simulated Loss Distibution 31 / 33

32 Conclusion Conclusion The provides a closed form Loss Rate Distribution ( 1 ρ Φ 1 (x) Φ 1 ) (p) lim N P(L(N) x) = Φ ρ for a Large Homogeneous Portfolio, which depends only on two parameters p and ρ and gives a good fit to market data. 32 / 33

33 Conclusion Bibliography Vasicek : The Distribution of Loan Portfolio Value, Risk (2002). Martin, Reitz, Wehn : Kredit und Kreditrisikomkodelle, Vieweg, (2006). Schönbucher : Faktor Models: Portfolio Credit Risks When Defaults are Correlated, Journal of Risk Finance (2001). Elizalde : Credit Risk Models IV: Understanding and pricing CDOs, discussion paper (2005). 33 / 33