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Modelling of Dependent Credit Rating Transitions Verena Goldammer - - PowerPoint PPT Presentation
Modelling of Dependent Credit Rating Transitions Verena Goldammer - - PowerPoint PPT Presentation
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
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Outline
1
Model General framework General model Examples
2
Simulation
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Maximum Likelihood Estimation MLE for the extended strongly coupled random walk Asymptotic properties of the estimator
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The Marked Point Process
Definition (Marked point process)
(τi)i∈◆: random time with values in (0, ∞], and τi < τi+1 on {τi < ∞} and τi = τi+1 = ∞ on {τi = ∞} (ρi)i∈◆: random mark with ρi ∈ E on {τi < ∞} and ρi := ρ∞ on {τi = ∞}, where ρ∞ external point of E. We call
- (τi, ρi)
- i∈◆ a marked point process.
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
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The General Framework
F = {1, . . . , n}: set of firms, n ∈ ◆ is number of firms X =
- (Xt(1), . . . , Xt(n))
- t≥0: credit rating process
- (τi, ρi)
- i∈◆: marked point process
Ui(j): I-valued random variable for i ∈ ◆ and j ∈ F.
Definition (General framework)
We say that the process X = (Xt)t≥0 with state space Sn follows the general framework, if
1 Xt = X0 for t ∈ [0, τ1), and 2 for each i ∈ ◆ and firm j ∈ F
Xt(j) = ρi
- Xτi−(j), Ui(j)
- for t ∈ [τi, τi+1).
Remark: Process is in general not Markovian.
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Markov Process in the General Framework
Additional assumptions to obtain a Markov process:
1 Random times (τi)i∈◆:
jump times of a Poisson process with intensity λ > 0
2 Random marks (ρi)i∈◆: i. i. d. sequence 3 Idiosyncratic components {Ui(j) : i ∈ ◆, j ∈ F}:
- i. i. d. collection
4 (ρi)i∈◆, {Ui(j) : i ∈ ◆, j ∈ F}, X0 and the Poisson
process are pairwise independent. In the following: We assume that these additional assumptions are satisfied.
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Dynamics of the General Model
Assumption for the general model: All firms with the same rating may simultaneously change
- nly 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 px, where x ∈ S is the current rating of the firm.
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Definition of the General Model
Definition (General model)
We say that the Markov jump process X = (Xt)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 SS and p ∈ [0, 1]S Each ρi takes a. s. only values in {rs : s ∈ SS} ⊂ E where rs(x, u) = s(x), if u ∈ [0, px], x, if u ∈ [px, 1]. P[ρi = rs] = P(s) for each s ∈ S Ui(j): uniformly distributed on I = [0, 1] for i ∈ ◆, j ∈ F
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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 λx > 0 for each rating class x ∈ S Stochastic transition function Pc : S × S → [0, 1]: probability for transitions from x to y given Poisson process of x jumps px ∈ [0, 1]: probability that a firm with rating x actually changes the rating
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Embedding in the General Model
Define λ =
x∈S λx and the distribution P on SS by
P(s) =
λx λ Pc(x, y),
if there exist x, y ∈ S with x = y, s(x) = y, s(u) = u for all u ∈ S \ {x}
- x∈S
λx λ Pc(x, x),
if s(x) = x for all x ∈ S, 0,
- therwise.
Definition (Strongly coupled random walk)
We say that the Markov jump process X is a strongly coupled random walk process with parameters
- (λx)x∈S, Pc, p
- ,
if X follows the general model with parameters (λ, P, p).
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Example 2: The Scheme Model
0.2 0.4 0.6 0.8 1 4 3 2 1 Probabilities Rating classes 1 2 3 4
For each x ∈ S the interval [0, 1] is divided into K subintervals with length pxy for the y-th subinterval. The subinterval contain- ing V represents the rating class ˜ s(x). (pxy)x,y∈S ∈ [0, 1]S×S: stochastic transition function V : random variable, uniformly distributed on [0, 1] SS-valued random function ˜ s: ˜ s(x) = max
- y ∈ S :
y−1
- k=1
pxk ≤ V
- ,
for x ∈ S.
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Definition of the Scheme Model
The distribution of ˜ s is given by Ps(s) = max
- min
x∈S s(x)
- k=1
pxk − max
x∈S s(x)−1
- k=1
pxk, 0
- .
Definition (Scheme model)
(pxy)x,y∈S ∈ [0, 1]S×S: stochastic transition function Ps: probability distribution of ˜ s λ > 0 and p = (px)x∈S is a vector in [0, 1]S We say that the Markov jump process X follows the scheme model with parameters
- λ, (pxy)x,y∈S, p
- , if X follows the
general model with parameters (λ, Ps, p).
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Embedding of a Model 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 Q-matrix µ ∈ ❘K×K of the transitions of the individual firms is the same for all firms. Correspondence of parameters: (µ, p) ⇒ (λx, Pc, p)
- r
(λ, Ps, p) Extended strongly coupled random walk: px = 0: independent rating transitions of firms in class x
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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: L(t) =
n
- i=1
C (1 − δ)✶{Xt∧T (i)=K}, for t ≥ 0.
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Empirical Excess Loss Distribution
X0: 16 firms in rating class 1, 14 firms each in 2 to 7 px = p for all x ∈ S Intensity µ of individual credit rating transitions is based
- n data of Standard & Poor’s.
Empirical excess loss distribution (5 000 simulations):
5 10 15 20 25 0.2 0.4 0.6 0.8 1 P(L(5) > x) x p=0 p=0.3 p=0.7 p=1 5 10 15 20 25 0.2 0.4 0.6 0.8 1 P(L(5) > x) x p=0.1 p=0.3 p=0.7 p=1
strongly coupled random walk scheme model
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Histogram of Simulated Losses
entire histogram detail of the histogram
5 10 15 20 25 250 500 750 1000 1250 1500 Number of Losses Loss coupled scheme 18 20 22 24 26 28 30 32 34 36 25 50 75 100 125 150 Number of Losses Loss coupled scheme
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.
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Estimation for Strongly Coupled Random Walk
Notation: Set of parameters: Θ =
- [0, ∞)K−1 × [0, 1]
K θ ∈ Θ: Set θ = (θx)x∈S with θx = (µx,1, . . . , µx,x−1, µx,x+1, . . . , µx,K, px) X follows extended strongly coupled random walk process with true parameter θ0 and n firms. Parameter estimation: Given observations of sample paths of X, which parameter ˆ θ is likeliest to be the true parameter θ0?
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Likelihood Function
Likelihood function: L(θ) =
x,y∈S x=y n
- a,b=1
a≥b
- µxypb−1
x
(1 − px)a−bNx,y,a,b
- ×
- k
- j=1
P
- X0 = zj
- exp
- −
- x∈S
µx
n
- a=1
Tx,a
a−1
- j=0
(1 − px)j
- Nx,y,a,b ∈ ◆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 Tx,a ∈ [0, T]: total time that exactly a firms have rating x zj ∈ Sn: initial rating in the j-th observed path
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Maximum Likelihood Estimator
Theorem (Maximum Likelihood Estimator)
The parameters in ˆ Θ ⊂ Θ are exactly the MLE, where for ˆ θ ∈ ˆ Θ holds:
1
ˆ px for x ∈ S is either 0, 1, the unique root in (0, 1) of polynomial Px, or arbitrary depending on ˜ Nx,a,b and Tx,a.
2 For x ∈ S with Tx,a > 0 for a ≥ 1:
ˆ µxy =
n
- a=1
a
- b=1
Nx,y,a,b
n
- a=1
Tx,a
a−1
- j=0
(1 − ˆ px)j , for y ∈ S with x = y
3
ˆ θx arbitrary for x ∈ S with Tx,a = 0 for all a ∈ ◆. ˜ Nx,a,b ∈ ◆0: total number of rating changes of b firms with rating x, where a firms originally in class x
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Maximum Likelihood Estimator for p
˜ Nx,a,b = 0 for all a ≥ b ≥ 2 up=true down=false
✑✑ ✑ ✸ ◗◗ ◗ s
˜ Nx,a,1 = 0 for all a ≥ 2 ˜ Nx,a,b = 0 for all a > b
✏✏✏ ✶ PPP q ✘ ✘ ✿ ❳ ❳ ③
˜ Nx,1,1 > 0 Tx,a > 0 for a ≥ 2 cl(x) < 0 ˆ px = 1 ˆ px unique root in (0,1) of Px
✏ ✏ ✶ P P q ✘ ✘ ✿ ❳ ❳ ③
ˆ px = 1 ˆ px ∈ [0, 1] ˆ px = 0 ˆ px unique root in (0,1) of Px
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Definition of Polynomial Px
Definition of polynomial Px: Px(p) = c0 + c1p + c2p2 + . . . + cnpn has coefficients c0 =
n
- a,b=1
a≥b
(b − 1) Nx,a,b
n
- i=1
i Tx,i cj = (−1)j
n
- a,b=1
a≥b
- Nx,a,b
n
- i=j
i j i − j j + 1 b + a − i
- Tx,i
for j ∈ {1, . . . , n} Definition of cl(x): l(x) ∈ {0, . . . , n} is the maximal index such that cj = 0 for all j ∈ {0, . . . , l(x) − 1}.
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Consistency of the Maximum Likelihood Estimator
Theorem (Consistency)
Θ ∋ θ0 = (µx,1, . . . , µx,x−1, µx,x+1, . . . , µx,K, px)x∈S ˆ θk for k ∈ ◆: MLE for the observed first k paths Assume µx > 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 θ0 is strongly asymptotically consistent, i. e. ˆ θk → θ0,
- a. s. for k → ∞.
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