3.2.2 Rating migrations and default events
Moody's simulates rating migrations and default events within
a multi-factor gaussian copula framework applied to a markovian multi-period
rating transition model. The first input of this model is a square rating
transition matrix over a given time horizon T, noted MT E Mp(R),
where p denotes the number of potential rating categories of the obligors,
including one default category. Moody's assumes there are 18 of them, the
mapping of which can be derived from figure (3.1) with categories Caa - C
merged and an extra default category D with rating 18.
The rating path of the nth obligor, j E {1, .., N},
until time horizon T is given by the random process (Rn(t))t?[0,T]:
Rn : Ù × [0,T] -? {1,..,18} (ù,t)
-? Rn(t)(w)
We then recall the definitions of a generator matrix and of a
time-homogeneous Markov process
Definition 5. Generator Matrix
Assuming A E Mp(R) with general term
(ëij)(i,j)?{1,..,p}2. Then A is called a generator matrix
if:
i) Vi E {1, ..,p}, >ip j=1 ëij = 0
ii) V(i,j) E {1,..,p}2, i =6 j ëij = 0
Definition 6. Time-homogeneous Markov process
X is a time-homogeneous Markov process with generator Ë
if:
?t = 0, ?Ät > 0, ?(i, j) ? {1, .., p}2,
P(X(t + Ät) = j | X(t) = i) = (eËÄt)ij
We now assume that (Rn(t))tE[0,T] is a Markov
time-homogeneous process with generator matrix Ë. We introduce the
transition matrix MÄt over time period Ät through its general term
(pij)(i,j)<p:
?t ? [0, T, ]?(i, j) ? {1, .., p}2, pij := P
(Rn(t + Ät) = j | Rn(t) = i)
It is worth noting that pij does not depend on t because of
the time-homogeneous property of (Rn(t))tE[0,T]. As a direct
consequence of Rn's definition, we have the following property:
Proposition 5. Composition of transition matrices
Assume Ät is such that TÄt ? N*.
Then:
T } k = MTTÄt
?k ? {1, ..
Ät
We shall now describe briefly Moody's multi-factor gaussian
copula model: similarly to the one factor gaussian copula model, the idea is to
draw a random vector X = (X1, .., XN) from a gaussian law with a given
correlation matrix Ó, where the latter depends on several factors. Let
us define (ZG, ZI, ZI,R) as three independent
standard gaussian factors that account for respectively the global state of the
economy, the state of any specific industrial sector and for a combination of
both industrial and regional factors. For any given obligor n ? {1, .., N}, let
us define en as an idiosyncratic factor that follows a standard
gaussian law and that is independent from the common factors (ZG,
ZI, ZI,R) and from all other idiosyncratic factors. Then,
for all n ? {1, .., N}, one can affect the state variable Xn to
nth obligor:
qXn ñG ZG
ñInZI VñIn ,R ZI,R \
+ 1 - ñG -ñIn -ñn I,Ren
The random vector X is a gaussian vector with zero mean and a
correlation matrix Ó given below. The correlation parametres
(ñGn , ñIn, ñn
I,R) are specific to each obligor and depend on some characteristics
of their businesses in terms of industry and operations' scale. They are picked
up from a subset of values subject to Ó remaining positive definite.
?
? ? ? ? ?
Ó=
?
? ? ? ? ?
1 ñ12 . . . ñ1p
.
.
ñ21 1 .
. .
.
...
...
. ..ñp-1,p
ñp1 . . . ñp,p-1 1
with:
q? (i, i) ? { 1, .., p}2 ñij ñG
Vñi ,Rñj,R
Applying well known results on the generalized invert of the
distribution function of (Rn(t + Ät)|Rn(t)), we can
write the following proposition:
Proposition 6. Rating transition simulation
Let us assume that each obligor's rating is likely to be
confirmed or revised only on
the following dates {Ät,..,kÄt,..,mÄt}, where m :=
T/Ät ? N*. Let F -1
k,Ät, k ?
{1, ..,p} denote the generalized invert of the cumulative
distribution function of rating transitions over time period Ät starting
from initial rating category k. We recall that Ö is the
cumulative distribution function of the standard gaussian law and that
(X1,..,XN) is the gaussian vector with correlation matrix Ó
describing the state of our N obligors. We finally assume that initial
ratings (R1(0),..,RN(0)) are known. Then the rating of each obligor on
discrete dates {Ät,..,kÄt,..,mÄt} can be expressed through the
recursive formula:
?k ? {1, .., m}, ?n ? {1, .., N}, Rn(kÄt)
= FR,1:((k-1)
Ät),Ät (Ö(Xn))
Moody's uses its historical database of rating transitions and
defaults over time horizon T to build the marginal cumulative distribution
functions (Fk,T)k?{1,..,p} and MT through some cohort method. Assuming
(Rn(t))t?[0,T] is a Markov time-homogeneous process, one can infer
MÄt thanks to proposition (5) and use proposition (6) to simulate N
correlated rating paths. The rescaling of matrix MT comes at a cost however:
given the choice of the gaussian copula, one can show that when Ät --? 0,
joint default times (ô(Ät)
1 , ..,ô(Ät) N) become independent: a
way to address this issue is to stress Ó as Ät gets smaller so that
the correlation structure is somehow preserved. In the case of the DPPI, T is
equal to 10 years and Ät to 6 months.
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