Modelling of Dependent Credit Rating Transitions

Transcription

1 ling of (Joint work with Uwe Schmock) Financial and Actuarial Mathematics Vienna University of Technology Wien,

2 Introduction Motivation: Volcano on Iceland erupted and caused that most of the flights in Europe had to be cancelled for a few days. That caused simultaneous losses of the airlines. Credit quality of the airlines is simultaneously affected. Previous literature: Dependence introduced by interacting intensities No simultaneous credit rating transitions possible! Main modeling assumption: Firms may simultaneously change their credit rating in continuous time. 2/23

3 Outline 1 General framework General model Examples 2 3 Maximum MLE for the extended strongly coupled random walk Asymptotic properties of the estimator 3/23

4 The Marked Point Process Definition (Marked point process) (τ i ) i N: random time with values in (0, ], and τ i < τ i+1 on {τ i < } and τ i = τ i+1 = on {τ i = } (ρ i ) i N: random mark with ρ i E on {τ i < } and ρ i := ρ on {τ i = }, where ρ external point of E. We call ( (τ i, ρ i ) ) i N a marked point process. General framework General model Examples Mark space E: E = { r : S I S r is ( P(S) I ) -P(S) measurable } S = {1,..., K}: credit rating classes, where K means firm is in default and 1 is best rating class Measurable space (I, I): state space of idiosyncratic component 4/23

5 The General Framework F = {1,..., n}: set of firms, n N is number of firms X = ( (X t (1),..., X t (n)) ) : credit rating process t 0 ( (τi, ρ i ) ) i N : marked point process U i (j): I -valued random variable for i N and j F. Definition (General framework) We say that the process X = (X t ) t 0 with state space S n follows the general framework, if 1 X t = X 0 for t [0, τ 1 ), and General framework General model Examples 2 for each i N and firm j F X t (j) = ρ i ( Xτi (j), U i (j) ) for t [τ i, τ i+1 ). Remark: Process is in general not Markovian. 5/23

6 Markov Process in the General Framework Additional assumptions to obtain a Markov process: 1 Random times (τ i ) i N: jump times of a Poisson process with intensity λ > 0 2 Random marks (ρ i ) i N: i. i. d. sequence General framework General model Examples 3 Idiosyncratic components {U i (j) : i N, j F }: i. i. d. collection 4 (ρ i ) i N, {U i (j) : i N, j F }, X 0 and the Poisson process are pairwise independent. In the following: We assume that these additional assumptions are satisfied. 6/23

7 Dynamics of the General Assumption for the general model: All firms with the same rating may simultaneously change only to the same rating class or remain in their rating class. Dynamics of the general model: Possible rating transitions are given by a map s : S S: Each firm with rating 1 either remains in this class or changes its rating to s(1), each firm with rating 2 remains in 2 or changes to s(2), and so on... The probability that a firm actually changes is given by p x, where x S is the current rating of the firm. General framework General model Examples 7/23

8 Definition of the General Definition (General model) We say that the Markov jump process X = (X t ) t 0 follows the general model with parameters (λ, P, p), if it follows the general framework with the additional assumptions: P probability distribution on S S and p [0, 1] S Each ρ i takes a. s. only values in {r s : s S S } E where General framework General model Examples r s (x, u) = P[ρ i = r s ] = P(s) for each s S { s(x), if u [0, px ], x, if u [p x, 1]. U i (j): uniformly distributed on I = [0, 1] for i N, j F 8/23

9 Example 1: The Strongly Coupled Random Walk Dynamics: Only firms in one rating class may simultaneously change to the same rating class or remain in their rating class. Parameters: Independent Poisson processes with intensity λ x > 0 for each rating class x S Stochastic transition function P c : S S [0, 1]: probability for transitions from x to y given Poisson process of x jumps p x [0, 1]: probability that a firm with rating x actually changes the rating General framework General model Examples 9/23

10 Embedding in the General Define λ = x S λ x and the distribution P on S S by λ x λ Pc (x, y), if there exist x, y S with x y, s(x) = y, s(u) = u for all u S \ {x} P(s) = λ x λ Pc (x, x), if s(x) = x for all x S, x S 0, otherwise. General framework General model Examples Definition (Strongly coupled random walk) We say that the Markov jump process X is a strongly coupled random walk process with parameters ( (λ x ) x S, P c, p ), if X follows the general model with parameters (λ, P, p). 10/23

11 Example 2: The Scheme Rating classes For each x S the interval [0, 1] is divided into K subintervals with Probabilities length p xy for the y-th subinterval. The subinterval containing V represents the rating class s(x). General framework General model Examples (p xy ) x,y S [0, 1] S S : stochastic transition function V : random variable, uniformly distributed on [0, 1] S S -valued random function s: { y 1 } s(x) = max y S : p xk V, for x S. k=1 11/23

12 Definition of the Scheme The distribution of s is given by P s (s) = max { min x S Definition (Scheme model) s(x) k=1 } s(x) 1 p xk max p xk, 0. x S k=1 (p xy ) x,y S [0, 1] S S : stochastic transition function P s : probability distribution of s λ > 0 and p = (p x ) x S is a vector in [0, 1] S We say that the Markov jump process X follows the scheme model with parameters ( λ, (p xy ) x,y S, p ), if X follows the general model with parameters (λ, P s, p). General framework General model Examples 12/23

13 Embedding of a with Fewer Firms Theorem (Embedding property) X rating process in general framework with n firms Y rating process in general framework with m < n firms Distribution of rating transitions of first m firms of X = Distribution of rating transitions of Y General framework General model Examples Q-matrix µ R K K of the transitions of the individual firms is the same for all firms. Correspondence of parameters: (µ, p) (λ x, P c, p) or (λ, P s, p) Extended strongly coupled random walk: p x = 0: independent rating transitions of firms in class x 13/23

14 Loss of a Credit Portfolio Credit portfolio: n = 100 credits with amount C = 1 and maturity T = 15. Obligors change credit rating according to process X. K = 8 rating classes, default K is an absorbing state. Recovery rate: δ = 0.4 Default-free interest rate is zero. Loss of the credit portfolio: n L(t) = C (1 δ)1 {Xt T (i)=k}, for t 0. i=1 14/23

15 Empirical Excess Loss Distribution X 0 : 16 firms in rating class 1, 14 firms each in 2 to 7 p x = p for all x S Intensity µ of individual credit rating transitions is based on data of Standard & Poor s. Empirical excess loss distribution (5 000 simulations): P(L(5) > x) p=0 p=0.3 p=0.7 p=1 P(L(5) > x) p=0.1 p=0.3 p=0.7 p= x strongly coupled random walk x scheme model 15/23

16 Histogram of Simulated Losses entire histogram coupled scheme detail of the histogram coupled scheme Number of Losses Number of Losses Loss Loss Figure: Histogram of the simulated losses for the strongly coupled random walk (coupled) and the scheme model (scheme) where p = 0.5 and t = 5, based on simulations. 16/23

17 for Strongly Coupled Random Walk Notation: Set of parameters: Θ = ( [0, ) K 1 [0, 1] ) K θ Θ: Set θ = (θ x ) x S with θ x = (µ x,1,..., µ x,x 1, µ x,x+1,..., µ x,k, p x ) X follows extended strongly coupled random walk process with true parameter θ 0 and n firms. MLE Asymptotic Properties Parameter estimation: Given observations of sample paths of X, which parameter ˆθ is likeliest to be the true parameter θ 0? 17/23

18 Function function: ( ) n ( L(θ) = µ xy px b 1 (1 p x ) a b) N x,y,a,b x,y S x y a,b=1 a b ( k P [ ] ) X 0 = z j exp j=1 { x S µ x n a=1 T x,a a 1 } (1 p x ) j j=0 MLE Asymptotic Properties N x,y,a,b N 0 : total number of simultaneous rating changes of b firms from x to y x, a firms originally with rating x in the observed k paths T x,a [0, T ]: total time that exactly a firms have rating x z j S n : initial rating in the j-th observed path 18/23

19 Maximum Estimator Theorem (Maximum Estimator) The parameters in ˆΘ Θ are exactly the MLE, where for ˆθ ˆΘ holds: 1 ˆp x for x S is either 0, 1, the unique root in (0, 1) of polynomial P x, or arbitrary depending on Ñ x,a,b and T x,a. 2 For x S with T x,a > 0 for a 1: MLE Asymptotic Properties ˆµ xy = n a N x,y,a,b a=1 b=1 n a 1 T x,a a=1 j=0 (1 ˆp x ) j, for y S with x y 3 ˆθx arbitrary for x S with T x,a = 0 for all a N. Ñ x,a,b N 0 : total number of rating changes of b firms with rating x, where a firms originally in class x 19/23

20 Maximum Estimator for p up=true Ñ x,a,b = 0 for all a b 2 Ñ x,a,1 = 0 for all a 2 Ñ x,1,1 > 0 T x,a > 0 for a 2 c l(x) < 0 3 ˆp x = 1 ˆp x [0, 1] ˆp x = 0 ˆp x unique root in (0,1) of P x MLE Asymptotic Properties down=false Ñ x,a,b = 0 for all a > b 3 ˆp x = 1 ˆp x unique root in (0,1) of P x 20/23

21 Definition of Polynomial P x Definition of polynomial P x : P x (p) = c 0 + c 1 p + c 2 p c n p n has coefficients c 0 = n (b 1)Ñx,a,b a,b=1 a b c j = ( 1) j n a,b=1 a b n i T x,i i=1 n Ñ x,a,b i=j ( ) i ( i j ) j j + 1 b + a i T x,i for j {1,..., n} MLE Asymptotic Properties Definition of c l(x) : l(x) {0,..., n} is the maximal index such that c j = 0 for all j {0,..., l(x) 1}. 21/23

22 Consistency of the Maximum Estimator Theorem (Consistency) Θ θ 0 = (µ x,1,..., µ x,x 1, µ x,x+1,..., µ x,k, p x ) x S ˆθ k for k N: MLE for the observed first k paths MLE Asymptotic Properties Assume µ x > 0 for all x S and the expected time is positive, that more than one firm has rating x in the path of X. Then the maximum likelihood estimator of θ 0 is strongly asymptotically consistent, i. e. ˆθ k θ 0, a. s. for k. 22/23

23 Asymptotic Normality of the MLE Theorem (Asymptotic normality) Θ θ 0 = (µ x,1,..., µ x,x 1, µ x,x+1,..., µ x,k, p x ) x S ˆθ k for k N: MLE for the observed first k paths MLE Asymptotic Properties Assume µ xy (0, ) and p x (0, 1) for x, y S with x y. Then the maximum likelihood estimator is asymptotically normal, i.e. k(ˆθ k θ 0 ) converges to a normal distribution for k with mean zero and the inverse Fisher matrix as covariance matrix. 23/23