Modelling of Dependent Credit Rating Transitions Verena Goldammer (Joint work with Uwe Schmock) Financial and Actuarial Mathematics Vienna University of Technology Wien, 15.07.2010
Introduction Dependent Credit Rating Transitions Verena Goldammer Motivation: Model Volcano on Iceland erupted and caused that most of the Simulation flights in Europe had to be cancelled for a few days. Likelihood Estimation 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 Dependent Credit Rating Transitions Verena Goldammer Model Model 1 Simulation General framework Likelihood Estimation General model Examples Simulation 2 Maximum Likelihood Estimation 3 MLE for the extended strongly coupled random walk Asymptotic properties of the estimator 3/23
The Marked Point Process Dependent Credit Rating Transitions Verena Goldammer Definition (Marked point process) Model ( τ i ) i ∈ ◆ : random time with values in (0 , ∞ ], and General framework τ i < τ i +1 on { τ i < ∞} and τ i = τ i +1 = ∞ on { τ i = ∞} General model Examples ( ρ i ) i ∈ ◆ : random mark with ρ i ∈ E on { τ i < ∞} and Simulation ρ i := ρ ∞ on { τ i = ∞} , where ρ ∞ external point of E . Likelihood Estimation � � We call ( τ i , ρ i ) i ∈ ◆ a marked point process . Mark space E : � � � � � � r is E = r : S × I → S 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 Dependent Credit Rating Transitions Verena Goldammer F = { 1 , . . . , n } : set of firms, n ∈ ◆ is number of firms � � Model X = ( X t (1) , . . . , X t ( n )) t ≥ 0 : credit rating process General framework � � General model ( τ i , ρ i ) i ∈ ◆ : marked point process Examples Simulation U i ( j ): I -valued random variable for i ∈ ◆ and j ∈ F . Likelihood Estimation 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 2 for each i ∈ ◆ and firm j ∈ F � � X t ( j ) = ρ i X τ i − ( 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 Dependent Credit Rating Transitions Verena Goldammer Model Additional assumptions to obtain a Markov process: General framework General model 1 Random times ( τ i ) i ∈ ◆ : Examples Simulation jump times of a Poisson process with intensity λ > 0 Likelihood 2 Random marks ( ρ i ) i ∈ ◆ : i. i. d. sequence Estimation 3 Idiosyncratic components { U i ( j ) : i ∈ ◆ , j ∈ F } : i. i. d. collection 4 ( ρ i ) i ∈ ◆ , { U i ( j ) : i ∈ ◆ , 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 Model Dependent Credit Rating Transitions Verena Goldammer Assumption for the general model: Model General framework All firms with the same rating may simultaneously change General model Examples only to the same rating class or remain in their rating class. Simulation Likelihood Estimation 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. 7/23
Definition of the General Model Dependent Credit Rating Transitions Verena Goldammer Model Definition (General model) General framework We say that the Markov jump process X = ( X t ) t ≥ 0 follows General model Examples the general model with parameters ( λ, P , p ), if it follows Simulation the general framework with the additional assumptions: Likelihood Estimation 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 � s ( x ) , if u ∈ [0 , p x ], r s ( x , u ) = x , if u ∈ [ p x , 1]. P [ ρ i = r s ] = P ( s ) for each s ∈ S U i ( j ): uniformly distributed on I = [0 , 1] for i ∈ ◆ , j ∈ F 8/23
Example 1: The Strongly Coupled Random Walk Dependent Credit Rating Transitions Verena Goldammer Dynamics: Model General Only firms in one rating class may simultaneously change to framework General model Examples the same rating class or remain in their rating class. Simulation Likelihood Estimation 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 9/23
Embedding in the General Model Dependent Credit Rating Transitions Verena Goldammer Define λ = � x ∈ S λ x and the distribution P on S S by Model General framework General model λ x λ P c ( x , y ) , if there exist x , y ∈ S with x � = y , Examples s ( x ) = y , s ( u ) = u for all u ∈ S \ { x } Simulation � P ( s ) = Likelihood λ x λ P c ( x , x ) , if s ( x ) = x for all x ∈ S , Estimation x ∈ S 0 , otherwise. 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 Model Dependent Credit Rating Transitions Verena Goldammer For each x ∈ S the 1 2 3 4 interval [0 , 1] is divided Model 1 General into K subintervals with framework General model Rating classes 2 Examples length p xy for the y -th Simulation subinterval. 3 Likelihood The subinterval contain- Estimation 4 ing V represents the rating class ˜ s ( x ). 0 0.2 0.4 0.6 0.8 1 Probabilities ( 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 Model Dependent Credit Rating Transitions Verena Goldammer The distribution of ˜ s is given by Model � � General s ( x ) s ( x ) − 1 framework � � General model P s ( s ) = max min p xk − max p xk , 0 . Examples x ∈ S x ∈ S Simulation k =1 k =1 Likelihood Estimation Definition (Scheme model) ( 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 ). 12/23
Embedding of a Model with Fewer Firms Dependent Credit Rating Transitions Verena Goldammer Theorem (Embedding property) Model X rating process in general framework with n firms General framework General model Y rating process in general framework with m < n firms Examples Simulation ⇒ Distribution of rating transitions of first m firms of X Likelihood Estimation = Distribution of rating transitions of Y Q -matrix µ ∈ ❘ 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 Dependent Credit Rating Transitions Verena Goldammer Credit portfolio: Model Simulation n = 100 credits with amount C = 1 and maturity T = 15. Likelihood Estimation 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 ) ✶ { X t ∧ T ( i )= K } , for t ≥ 0. i =1 14/23
Empirical Excess Loss Distribution Dependent Credit Rating Transitions Verena X 0 : 16 firms in rating class 1, 14 firms each in 2 to 7 Goldammer p x = p for all x ∈ S Model Intensity µ of individual credit rating transitions is based Simulation on data of Standard & Poor’s. Likelihood Estimation Empirical excess loss distribution (5 000 simulations): 1 1 p=0 p=0.1 p=0.3 p=0.3 0.8 0.8 p=0.7 p=0.7 p=1 p=1 P(L(5) > x) P(L(5) > x) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x x strongly coupled random walk scheme model 15/23
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