ling of (Joint work with Uwe Schmock) Financial and Actuarial Mathematics Vienna University of Technology Wien, 15.07.2010
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
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
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
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
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
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
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
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
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
Example 2: The Scheme Rating classes 1 2 3 4 1 2 3 4 For each x S the interval [0, 1] is divided into K subintervals with 0 0.2 0.4 0.6 0.8 1 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
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
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
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
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) 1 0.8 0.6 0.4 p=0 p=0.3 p=0.7 p=1 P(L(5) > x) 1 0.8 0.6 0.4 p=0.1 p=0.3 p=0.7 p=1 0.2 0.2 0 0 5 10 15 20 25 x strongly coupled random walk 0 0 5 10 15 20 25 x scheme model 15/23
Histogram of Simulated Losses 1500 1250 entire histogram coupled scheme 150 125 detail of the histogram coupled scheme Number of Losses 1000 750 500 250 Number of Losses 100 75 50 25 0 0 5 10 15 20 25 Loss 0 18 20 22 24 26 28 30 32 34 36 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 10 000 simulations. 16/23
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
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
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
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
Definition of Polynomial P x Definition of polynomial P x : P x (p) = c 0 + c 1 p + c 2 p 2 +... + 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
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
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