Download The Hazard Rate Matrix Approach to Credit Rating Transitions

QUANT PERSPECTIVES
The Hazard Rate Matrix
Approach to Credit
Rating Transitions
By J. M. Pimbley 1
D
ebt securities are financial instruments that obligate
the issuer of the debt to make principal and interest
payments at specified times to the investor. Failure of
the issuer to make such payments is “default.” One may characterize all outstanding debt instruments as being “in default”
or “not in default.”
The great majority of debt instruments — to which we now
refer as “debt” or “bonds” — will not default prior to maturity. Let’s consider the designations of “in default” or “not in
default” as a primitive rating system for bond performance.
Every outstanding bond falls into one of these two categories.
Credit rating agencies (CRAs) are incorporated financial
firms that analyze bonds and provide opinions on bond default risk in the form of “ratings” that extend the “not in default” category of the primitive rating system.2 These CRAs
generally give a symbol D to the “in default” category and
give symbols such as AAA, AA, A, BBB, BB, B, and CCC to
seven “not in default”categories that represent increasing risk
of future default. The precise symbols for these “letter grades”
vary from one CRA to another. Further, each CRA has an
expanded scale that boosts the number of “not in default” ratings from seven to roughly 20. In this article, we will generally
use these seven “major” rating levels merely to simplify the
discussion.
Meaning of the CRA Credit Ratings
The bond rating that carries the least risk is AAA. The “lower” rating levels of AA down to CCC denote progressively increasing default risk. The four largest CRAs have published
www.garp.org
“default tables” that show either targets or historical assessments of default probability over time for each rating level.3
These tables would seem to provide a definition, or at least a
benchmark, for the meaning of the CRA ratings. The CRAs,
however, have generally avoided making the explicit claim that
their default tables define their ratings.4
The CRAs also publish “rating transition matrices” that
show the frequency of rating changes from one level (such as
AA) to another (such as A) over periods of one year or longer.5
These transition matrices are consistent with and complementary to the default tables since “default” is synonymous with
rating transition to the default rating D. Thus, for example,
the default probabilities for each rating for a tenor of one year
form the (bottom) row of the one-year transition matrix for the
final rating of D.6
Time-Dependence of the Credit Rating
Contemplation of the CRA rating transition matrix (RTM)
leads us to consider the process for time evolution of credit
ratings. Previous work begins with the one-year transition matrix and makes the assumption that rating changes are Markovian.7 In this context, the description “Markovian” means
simply that the likelihood of a rating change depends only on
a bond’s current rating and the RTM.
In practice, rating changes are not Markovian because the
CRAs — inadvertently or otherwise — behave in a manner
that depresses rating volatility. For example, if a currently AArated bond deserves a downgrade to BB, CRAs will typically
downgrade the bond to A or BBB rather than impose the full
A U G U S T 2 0 1 2 RISK PROFESSIONAL
1
not in default and that the probability of the bond not being in default at time
is ( ). The initial condition is ( )
. We specify the probability of the bond
(which
is identical
the T
probability
Q U A defaulting
NT P
ER
S P Eto C
I V EofSthe bond being downgraded to
the “in default” rating level) during the time period (
) as
( ). Since a
rating transition to the “in default” rating is a rating transition out of the “not in
default” rating, we write
( )
. (1)
rating change in one rating action.8 Moody’s Investors
Service
Equation
(1) shows
familiar
that is
Equation
(1) shows
the familiar
hazardthe
rate
processhazard
that is rate
well process
known for
10
states: “Moody’s ratings management practice, by avoiding re- well known for
bond
defaults
and
other
applications.
The
bond defaults and other applications.9 The solution of equation (1) for the survival
versals, necessarily produces positive serial momentum in rat- solution of equation (1) for the survival time S(t) with a time) with
) isis
time to( be
a time-dependent
rate
ings changes. A downgrade is much more likely
followed
dependenthazard
hazard
rate((t)
by another downgrade than it is an upgrade.”9
(2)
( )
( )] . (2)
[ ∫
Before continuing, let us emphasize this point. Existing treatments of rating transitions employ a Markov paradigm. This
( )
We specify
for
to indicate
presence
of default
risk at of
all detreatment is most tractable and most sensible:
ratings should
Weallspecify (t)>0
for allthet>0
to indicate
the presence
be Markovian to maintain accuracy of default
risk
estimation.
fault
risk
at
all
times.
While
the
probability
S(t)
of
the
bond
times. While the probability ( ) of the bond being in the “not in default” rating
Yet CRA rating transitions are clearly not Markovian in prac- being in the “not in default” rating level declines from 1 at
level
declines
from 1 at
timet=0 to zero
to zero
t goes
infinity,the
theprobability
probabilityofof the
tice. Hence, we will not try to fit empirical
rating
transition
time
as as
t goes
to toinfinity,
data with our model results. Rather, we view
the
relevance
of
bond
being
in
default
1–S(t)moves
in
the
opposite
manner
( ) - moves in the opposite manner from zero
the bond being in default at
this study to be the development of idealized (Markov) rat- from zero at t=0 to 1 as t goes to infinity. The default state is
to 1 as t goes to infinity. The default state is “absorbing” in the sense that a
ing transition behavior to assist banks, portfolio managers
and “absorbing” in the sense that a bond remains in default once
bond remains
defaultit once
it reaches
default.
other practitioners in the development of accurate
defaultinratreaches
default.
ing scales.
Consequently, we develop in this article an
alternative of
Marof Rating
Transitions: Full Rating Scale
Mathematics
RatingMathematics
Transitions – Full
Rating Scale
kovian process for rating changes that we find to be more intui- The purpose of our explaining the well-known hazard rate
The purpose
our ofexplaining
well-known
rate model
tive than existing methods that begin with the one-year
RTM. of
model
equationsthe
(1) and
(2) for thehazard
bond default
processofis to
The new treatment, which ultimately merges
with
the
convenshow
motivation
for
a
similar
“hazard
rate
matrix”
equations (1) and (2) for the bond default process is to show motivation approach
for a
tional method, has the advantage of both an easier derivation for credit rating transitions. Instead of beginning with a bond
2012 similar “hazard rate matrix” approach for credit rating transitions.J. M.
Pimbley
Instead
ofthat
and better examination of the bond rating
behavior for large that is simply “in default” or “not in default,” we
specify
times. We find, for example, that a certain matrix
eigenvector
has a “in
designated
of beingwe
in one
of the
beginning
with a bondthe
thatbond
is simply
default” probability
or “not in default”,
specify
sum toallone
and rating
be written
as the
column
vector (AAA,
⃗. This
vector
has components
forms a “natural rating distribution” to which
initial
eight
rating
categories
AA,
A, BBB,
BB, B, CCC, or D).
that the bond has a designated probability of being in one of the eight rating
states evolve.
Let these
eightrating
probabilities
one and
be written
with
designating
the eight
levels. Ifsum
onetowishes
to specify,
for as the
→ CCC,
categories
(AAA, AA,column
A, BBB,vector
BB, B,
or D).has
Letcomponents
these eight probabilities
The nature of this finding — that we identify
an eigenvecr. This vector
ri with i=1,...,8
example,
current designating
bond rating the
is definitely
BBB,
thenIfone
would
setto specify, for
tor with a credit rating distribution — implies
thatthat
thethe
matheight rating
levels.
one
wishes
9
See, for example,
Löffler andexample,
Posch (2011).
ematical development and the finance implication
are
strongly
that
the
current
bond
rating
is
definitely
and all other to zero. More generally, let us say there are N rating categories BBB,
with then
linked. Much of the remaining exposition of this article is un- one would set r4=1
and
all
other
r
to
zero.
More
generally,
i
4
J. M.NPimbley
category
being “default”.
avoidably mathematical. The following sections (up to “Practi- let us say there are N rating categories with category N being
cal Result – Examples”) constitute this analysis, Analogous
and the ensu“default.”
Formatted: Font: 12 pt, Italic
to equation
(1), the rating probability vector ⃗( ) evolves in time
matics of Rating Transitions
Simple Case
ing sections–provide
the finance interpretation.
Analogous to equation (1), the rating probability vector r(t)
as
evolves in time as
To begin, let us consider
the
“primitive
rating
system”
in
which
a bond
is
Mathematics of Rating Transitions: Simple
Case
⃗
(3)
⃗ (3)
⃗( )
Toinbegin,
let usAt
consider
rating
“in default” or “not
default”.
time the “primitive
, we specify
that system”
the bondiniswhich
a bond is either “in default” or “not in default.” At time t=0,
default and that the probability of the bond not being in default at time( )
where
is the
time-dependent
matrix (HRM)hazard
for rating
we specify that the bond is not in default
and that the
probwhere Q(t)hazard
is therate
time-dependent
ratetransitions.
matrix (HRM)
). The initial condition
(
)
is
.
We
specify
the
probability
of
the
bond
ability of the bond not being in default atIntime
S(t). The , the
for rating
In the limit
the t>0
limitis as
HRM transitions.
has the meaning
that as ∆t→0,isthe
theHRM
rating Q has
initialtocondition
is S(0)=1.
Webond
specify
thedowngraded
probabilitytoof the the meaning that I–Q∆t is the rating transition matrix for the
ting (which is identical
the probability
of the
being
transition
matrix
timeperiod
period∆t where
where
is identity
the identity
matrix.
bond defaulting (which is identical to the
probability
of for
the thetime
I is the
matrix.
More More
specifically,
(
) as ( ). Since a
n default” rating level)
during
the
time
period
bond being downgraded to the “in default”
rating level)
is the probability
a bond
specifically,
the ijdurelementthe
ofij elementisoftheI–Q∆t
probability
that a bond that
in rating
levelinj rating
ingdefault”
the timerating
period
S(t)dt. Since
rating
level j will transition to rating level i in the time period ∆t. We
transition to the “in
is (t,t+dt)
a ratingastransition
out ofa the
“nottransition
in
will transition to rating level i in the time period . We specify a negative sign in
to
the
“in
default”
rating
is
a
rating
transition
out
of
the
“not
specify a negative sign in equation (3) to maintain consonance
t” rating, we write
equation (3) to maintain consonance
with(1).
equation
(1). As
a result,
the components
in default” rating, we write
with equation
As a result,
the
components
Qij of Q satisfy
the
following:
of satisfy
( )
. (1)
(1)
→
Equation (1) shows the familiar hazard rate process that is well known for
defaults and other applications.9 The solution of equation (1) for the survival
2
RISK PROFESSIONAL A U G U S T 2 0 1 2
( ) with a time-dependent hazard rate ( ) is
(4a)
(4b)
www.garp.org
(4c)
Forma
where
⃗( )
(3)
⃗
( ) is the time-dependent hazard rate matrix (HRM) for rating transitions.
In the limit as
, the HRM
transition matrix for the time period
specifically, the ij element of
has the meaning that
Q U A N T P EisRtheSrating
PECTIVES
where
is the identity matrix. More
is the probability that a bond in rating level j
will transition to rating level i in the time period
. We specify a negative sign in
equation (3) to maintain consonance with equation (1). As a result, the components
of
satisfy
ing transition dynamics have treated this one-year RTM as the
J. M. Pimbley
analytical starting point and then created a “generator matrix”
12
similar to the HRM Q of this article.
(4b)
The primary
difference
between these
earlier
ma-in less
Our HRM
differs from
the generator
matrices
of generator
past studies
trices
and
our
HRM
is
our
interpretation
of
Q
as
the
agent
of
(4c)
significant
terms as well. The HRM has a preceding negative algebraic 13sign for
time evolution of the rating probability vector in equation (3).
consistency We
withconsider
the scalar
rate.guiding
To accommodate
rating
probability
∑
. (4d)
thedefault
HRMhazard
to be the
concept and
relegate
the RTM
to an ancillary
column vectors
in equations
(3), (5),role.
and (6), our hazard rate matrix indices are
The special The
designation
in (4a) for rating
N category
reflects the
condition that
a HRM differs from the generator matrices of past studspecial designation
in (4a)category
for rating
N reflects
Our
carries
transposed from the convention of past studies. That is, the component
the condition
a bond in
defaulttransition
cannot experience
a rating
in less significant terms as well. The HRM has a preceding
bond in default
cannot that
experience
a rating
to leave this
default ies
state.
transition to leave this default state. Equation (4d) imposes
the negative
algebraic of
sign
for migration
consistency
with
the scalar
information
on the probability
rating
from
rating
level j default
to rating level
Equation (4d) imposes the requirement that the sum of probabilities of all rating
requirement that the sum of probabilities of all rating
levels
hazard
rate.
To
accommodate
rating
probability
column
veci rather than from rating level i to rating level j.
levels must
be be
constant
(at the
value
1). 1).
Thus,
equations
(4a-d)
show
us thattors is
must
constant
(at the
value
Thus,
equations
(4a-d)
show
in equations (3), (5) and (6), our hazard rate matrix indices
that Q in
is an
N x column
N matrixNiniswhich
N is filled
withPast
are
transposed
fromhave
the convention
past studies.
That is,
the
research
projects
applied a ofhazard
rate matrix
approach
to
an N x Nus matrix
which
filled column
with zeroes,
all other
diagonal
zeroes, all other diagonal elements are positive, all other off- component Q ij carries information on the probability of rating
problems
than
time evolution of credit ratings. Takada and Sumita (2011)
elements diagonal
are positive,
all other
and in
theother
sum
of thefrom
elements
areoff-diagonal
negative andelements
the sumare
of negative,
the elements
migration
rating level j to rating level i rather than from
studied financial
default
based
eachincolumn
is zero.
rating level
i to risk
rating
level on
j. obligor industry and macro-economic
the elements
each column
is zero.
Just as we can solve the scalar hazard rate equation (1)
to
get
Past
research
projects
have
a hazard rateMarkov
matrix apconsiderations.
Singer and Spilerman applied
(1976) investigated
processes
J. M.
Pimbley
Just
we can time
solveS(t)
theofscalar
hazard
get the survival
theassurvival
equation
(2),rate
theequation
solution(1)
to to
equation
proach to problems other than the time evolution of credit ratrelevant to ings.
sociology
such as
immigration
and juvenile
J. M. Pimbley
Takadatopics
and Sumita
(2011)
studied financial
defaultdelinquency
risk
time ( )(3)
of is
equation (2), the solution to equation (3) is
(
) ⃗⃗⃗⃗ (5)
⃗( )
based
on
obligor
industry
and
macro-economic
considerations.
recidivism. Keilson and Kester (1974) provided mathematical background and
Singer and Spilerman (1976) investigated Markov processes
(
) ⃗⃗⃗⃗ (5) 5
⃗( )
properties
(5) for hazard rate matrices in Markov processes.
special case in which
has no time dependence. In equation (5),
the
relevant to sociology topics such as immigration and juvenile
which
has
time
equation
(5), the everyIn delinquency recidivism. Keilson and Kester (1974) provided
for the special
hasIn“no
time
dependence.
tl case
of thein exponential
is no
an case
N x independence.
N which
matrixQwith
” multiplying
Properties background
of the Hazard
Matrix
equation
argument
of the
is anevery
N Necessary
x N ma- mathematical
and Rate
properties
for hazard rate mahe of
exponential
is an(5),
N the
x N
matrix with
“ exponential
multiplying
ent
.10 The initial
rating
probability
vector
is”⃗⃗⃗⃗.
For a time-dependent
11
trix,
with
“–t”
multiplying
every
component
of
Q.
The
initial
trices
in
Markov
processes.
.10it The
initial
rating
vector
is ⃗⃗⃗⃗.a time-dependent
For atotime-dependent
( ),
is not
possible
toprobability
determine
a solution
equation HRM
(5) unless
rating
probability
vector
is r→
Q(t),We list here and discuss briefly a collection of properties of the HRM in
o. Forsimilar
addition
of equations
(4a) – (4d).
itisisofnot
possible
determine
a solution
similar(5)
to unless
equation
(5), to those
Necessary
Properties
of the Hazard Rate Matrix
is not possible
tothe
determine
aa solution
to multiplied
equation
dependence
form ofto
constant similar
matrix
by
an arbitrary
unless the time dependence is of the form of a constant matrix We list here and discuss briefly a collection of properties of the
dence is of( the
form
of awith
constant( matrix
multiplied
by an arbitraryrating
)
( scalar
), thefunction
unction
Thus,
time-dependent
a zeroHRM
eigenvalue
A).multiplied
by an arbitrary
g(t). Thus,has
with
Q in addition to those of equations (4a) – (4d).
→
(
).
(
)
(
),
nty vector ⃗(Q(t)=Ag(t),
Thus,
therating
time-dependent
rating r(t) is
probability vector
) is withthe time-dependent
Since column N of
Q has a zero eigenvaluehas all zero entries, this matrix is singular. All
ctor ⃗( ) is
singular
have at
one
zero entries,
eigenvalue.
Oneis singular.
determines the
Since column
N ofleast
Q has
all zero
this matrix
(6) matrices
( )] ⃗⃗⃗⃗ (6)
⃗( )
[
∫
All
singular
matrices
have
at
least
one
zero
eigenvalue.
Oneall zero
corresponding eigenvector with little effort to be the column vector with
( )] ⃗⃗⃗⃗ (6)
⃗( )
[
∫
Unless
stated
otherwise
for
the
remainder
of
this
study,
we
determines
the
corresponding
eigenvector,
with
little
effort,
to
Unless stated otherwise for the remainder of this study, we take the HRM
setentries,
to 1. Writing
⃗ as this
entries
thecolumnelement
take the HRM Q to be constant, so that equation (5)
is thesavebeforthe
vector which
with allwe
zero
save for theand
Nth elestated otherwise
for the remainder
of this study,expression
we take thefor
HRM
→
onstant
so that equation
(5) is the
the rating
relevant expression
for therelevant
rating probability vector.
It
would and
ment,
which werespectively,
set to 1. Writing
this eigenvalue
N and N as
eigenvalue
eigenvector,
these 
statements
become
tty so
that
equation
(5)
is
the
relevant
expression
for
the
rating
(
)
vector. be
It would
be worthwhile
a time-dependent
in
worthwhile
to employtoaemploy
time-dependent
Q(t) in modeling
and eigenvector, respectively, these statements become
exercises
for
which
one
desires
a
rating
volatility
that
increases
(
)
ctor.
It
would
be
worthwhile
to
employ
a
time-dependent
in
g exercises for which one desires a rating volatility that increases or
or decreases with a prescribed g(t).
(7)
( ) . (7)
⃗
⃗
⃗
(4a)
2012
rcises
which one
( ).desires a rating volatility that increases or
s with afor
prescribed
( ).
a prescribed
Standard
Rating Transition Matrix
(5) permits
us to write the RTM for any future time t
rd RatingEquation
Transition
Matrix
as
exp(–Qt).
The
most
typical period is one year, so the one-year
ating Transition Matrix
quation (5)RTM
permits
us be
to simply
write exp(–Q).
the RTMMany
for any
futurestudies
time of ratas
would
previous
on). (5)
us toperiod
write isthe
anyone-year
future time
as be
Thepermits
most typical
oneRTM
year, for
so the
RTM would
he (most). typical
period is studies
one year,
so thetransition
one-yeardynamics
RTM would
Many previous
of rating
havebetreated
). Many
studies ofstarting
rating transition
have atreated
year
RTMprevious
as the analytical
point anddynamics
then created
“generator
www.garp.org
RTM astothe
starting
point 11
andThe
thenprimary
createddifference
a “generator
similar
theanalytical
HRM of
this article.
between
11
All other eigenvalues of Q are real and positive
For all other eigenvalues i and eigenvectors →
 i with i=1,...,N-1,
it would be highly desirable if the eigenvalues were real
and positive. While the matrix Q is real, it is not symmet7
A U G U S T 2 0 1 2 RISK PROFESSIONAL
3
⃗ with
For all eigenvalues and eigenvectors, we can write
⃗
J. M. Pimbley
. Excluding the last eigenvalue/eigenvector pair, we know
012
All other eigenvalues of
are real and positive
. We
Q U A N T define
P EtheRrow
S Pvector
E C⃗ Tto Ihave
V rank
E SN with all elements equal to 1.
Pre-
multiplying the HRM
by ⃗ gives another rank N row vector in which all
, it
elements are zero (because the sum of each column of is zero). This observation
would be highly desirable if the eigenvalues were real and positive. While the
means that
⃗ ⃗
. The pre-multiplication of ⃗ by ⃗ simply sums the
matrix is real, it is not symmetric. It is in principle possible for some eigenvalues
elements of ⃗ . Since
, the sum of the elements of each eigenvector ⃗ must
to be complex even with the properties of equations (4a) – (4d).13 Therefore, we
be zero. Note that this zero-sum condition does not apply to the last eigenvector ⃗
impose the constraint on the elements
of that all eigenvalues be real. Our
→
ric. It is in principle possible for some eigenvalues
sinceto be .elements of each eigenvector i must be zero. Note that this
14
motivation for this
requirement
the the
behavior
of solutions
of equation
that we zero-sum condition does not apply to the last eigenvector →N,
complex
even iswith
properties
of equations
(4a) (5)
– (4d).
The
eigenvectors of the rating transition matrix are readily
Therefore, we impose the constraint on the elements
Qij eigenvalues
of Q sinceand
N =0.
discuss later.
For all other eigenvalues
and eigenvectors ⃗ with
that all eigenvalues be real. Our motivation for thisdetermined
requirement
is
the
behavior
of
solutions
of
equation
(5),
which
we The eigenvalues and eigenvectors of the rating transition matrix are readily
Considering the sub-matrix of
that excludes the
row and column,
discuss later.
determinedearlier that the one-year RTM is
( ) and the more
Wetoobserved
Gerschgorin’s Circle Theorem (GCT) and equations (4b) - (4d) are sufficient
Considering the sub-matrix of Q that excludes the Nth row We observed earlier that the one-year RTM is exp(–Q) and the
14
(
). Working with the latter, this RTM is a
general RTM
for time is
show that all eigenvalues
of the
sub-matrix are
greater
than zero
(i.e.,
and column,
Gerschgorin’s
Circle
Theorem
(GCT)
and equa-). more general RTM for time t is exp(–Qt). Working with the latwith
seriesis representation
(4b) - (4d)ofarethe
sufficient
to show
all eigenvalues
of the
ter,infinite
this RTM
a matrix with the infinite series representation
Note that thesetions
“eigenvalues
sub-matrix”
arethat
simply
thematrix
remaining
the sub-matrix are greater than zero (i.e., i>0).15 Note that
Equations (4b) – (4d) are sufficient
eigenvalues
these “eigenvalues ofofthe .sub-matrix”
are simply the remain(
)
(9) .
(9)
..., N-1 ofdiagonally
support for the assertion
that the 
sub-matrix
dominant.
Since
ing eigenvalues
Q. Equations
(4b) – (4d)
i with i=1,is strictly
are sufficient
support
for the
assertion
that the
sub-matrix
is We’ve
We’ve
written
the scalar
thescalar
left oft to
thethe
terms
with
of thepowers
HRM of. The
writtentothe
left of
thepowers
terms with
all diagonal elements
are positive,
the radii
of the
Gerschgorin
circles
of the GCT
2012
M.
Pimbley
strictly diagonally dominant. Since all diagonal elements are the HRM Q. The notation Qk simply means that J.
the
matrix
Q isSince
notation
simply means that the matrix is multiplied
by itself kk times.
→
do not permit zero
or negative
eigenvalues.
→
positive,
the radii
of the Gerschgorin circles of the GCT
Q Pimbley
→i=ik →i,
2012do multiplied by itself k times. Since Q i = i i implies
J. M.
not permit zero or negative eigenvalues.
⃗ we
implies
⃗ eigenvector
, we see that
⃗ is alsowith
an eigenvector of
⃗
see that →⃗i is also an
of exp(–Qt),
For convenience, we number the eigenvalues from largest to smallest.
For convenience, we number the eigenvalues from largest
to
(
)
(
)
⃗
⃗
( 2012 ) with
To be published in Risk Professional
J. M. Pimbley
Incorporating thesmallest.
possibility
that two orthe
more
eigenvalues
mayorhave
same
(
)⃗
(
) ⃗
Incorporating
possibility
that two
morethe
eigenvalue, we write values may have the same value, we write
. (8)
The matrix
[(
(8)
hasThe
a set
of N Q
linearly
independent
matrix
has a set
of N linearlyeigenvectors
independent eigenvectors
)⃗
(
) (() )(
[
(⃗
[ (
9
→ the eigenvectors of the RTM
eigenvectorThus,
N Thus, the eigenvectors
)
] ⃗ ] )⃗ ⃗
)(10)
⃗ ( (10)
.) .
(10)
] ⃗
are simply
the eigenvectors
of the
the
of( the )RTM
exp(–Qt)
are simply
We’ve already shown in equation (7) the
(
) are simply the eigenvectors of the eiThus,
the eigenvectors
of the RTM
We’ve already
shown in to
equation
(7) the eigenvector
⃗ Q
corresponding
⃗ eigenvalues
(10) .
corresponding
the eigenvalue
 =0. While
has N–1 HRM
ad-to . In terms of the eigenvalues of the HRM, the RTM
are
.
N
genvectors of the HRM Q. In terms of the eigenvalues of the
t
HRMas
. In
of the
eigenvalues
of the
HRM, theare
RTM
are
.
ditional
eigenvalues
real and
positive,
assumed
or
. While
has that are
additional
eigenvalues
that
areterms
realThus,
and
HRM,
the vector
RTM
eigenvalues
e-ieigenvalues
. of eigenvectors
) are
the
eigenvectors
of the
simply
the eigenvectors of the
The rating
probability
⃗( )RTM
is a linear( combination
proven earlier, it is possible that Q does not have a full set of
.
HRM vector
. In terms
of
of the HRM,of
theeigenvectors
RTM eigenvalues are
) istheaeigenvalues
The rating probability
⃗(
linear
combination
Since
matrix
has
a set
of N→
linearly
independent
eigenvectors
⃗ with
N linearly independent eigenvectors.
The the
rating
probability
vector
r(t)
is a linear
combination
of eigenvectors
As an example, imagine there are only four rating states (including the default state) so that
. Let
) isNa rating
If the N–1 positive eigenvalues of equation (8) are distinct,
The rating, probability
vector
linear
of independent
eigenvectors
wethe
can matrix
write
the ⃗(
rank
probability
vector
⃗( ) as a timehas
set
ofcombination
N linearly
eigenSince the matrixSince
has →a set
of NQlinearly
independent
eigenvectors
⃗ with
...
then
Q
would
have
this
complete
set
of
eigenvectors.
But
there
vectors

with
i=1,
,
N,
we
can
write
the
rank
N
rating
i
dependent,
weighted
sum
of
the
eigenvectors:
). Of the four eigenvalues of this matrix, two are real (0 and Since the matrix has a set of N linearly independent eigenvectors ⃗ probe HRM be (
with
, weconcan write
rank →r(t)
N rating
probability vectorweighted
⃗( ) as sum
a timeis no assurance of distinct eigenvalues. As with the earlier
abilitythevector
as a time-dependent,
of the
∑
) ⃗ probability
⃗( ) the rank
. (11)
, we can write
N (rating
vector ⃗( ) as a timecern(0.252
for complex
eigenvalues,
we add
a constraint
on weighted
the
per-off-sum
097) and two are complex
+ i 0.048 and
0.252 – i 0.048).
Relatively
small
changes
to some
eigenvectors:
dependent,
of the eigenvectors:
dependent,
weighted
sum
of
the
eigenvectors:
agonal matrix elementsmitted
produceelements
four real eigenvalues.
Qij of Q that there exists a set of N linearlyInin-equation (11), the coefficients are time-dependent while the eigenvectors ⃗
→
See, for example, Weisstein (2012). In our application
of this theorem,
we consider the sum of column,
dependent
eigenvectors
 i with
i=1,..., N.
With this condition,
)
∑
( ) ∑⃗ . ((11)
aresince
constant.
equation
(11)
) ⃗ . (11)
⃗( ) (3),
ther than row, element absolute values for the Gerschgorin
radius. This substitution is permissible
a ⃗(Applying
the
matrix
Q
is
diagonalizable.
atrix and its transpose have the same eigenvalues.
equation
(11), the coefficients
are time-dependent
while
the eigenvectors
⃗
In equation (11), Inthe
coefficients
are the
time-dependent
while the
eigenvectors
⃗while
In⃗ equation
⃗ (11),
∑
⃗coefficients
∑ i⃗ are time-dependent
∑
⃗
→
are
constant.
Applying
equation
(3),
The elements of each of the8first N-1 eigenvectors sum
to
zero
the
eigenvectors
are constant.→ Applying
equation
(3), i are constant. Applying equation (3),
→
For all eigenvalues and eigenvectors, we can write Q i = i i
) ⃗
( )
∑ (
. (12)
with i=1,..., N. Excluding the last eigenvalue/eigenvector pair,
⃗
∑
⃗
∑
⃗
∑
⃗
⃗
⃗ rankWeNuse⃗ the notation
∑
⃗to denote the
∑
⃗
∑ value ⃗of .
we know i0. We define the row vector →yT to have
(time-independent)
initial
(12)
with all elements equal to 1. Pre-multiplying the HRM Q by
) ⃗ the probability
( vector
)
∑ ( (11) and (12),
. (12)
With equations
⃗( ) becomes
y→T gives another rank N row vector in which all elements are
)
)o
∑ ob( use the notation
. (12)
We
denote
(time-independent)
value of .
)notation
∑ the(
⃗( to
⃗ . (13a) initial
zero (because the sum of each column of Q is zero). This
We use⃗ the
i to denote the (time-independent)
→
→T →
servation means that i y i=0. The pre-multiplication of The
i by
initial
value
of(11)
vector
. With
equations
(11)vector
and
(12),
the. probability
iand
)of
) is initial
With
equations
(12),
⃗(
becomes
rating
probability
⃗ the
⃗( probability
We use the notationinitial
to
denote
the
(time-independent)
value
y→T simply sums the elements of →i. Since i0, the sum of the vector →r(t) becomes
the eigenvalue
⃗)
⃗ . ⃗(13b). (13a)
∑
With equations (11) and (12), ⃗(the
probability
vector ⃗( ) becomes
4
RISK PROFESSIONAL A U G U S T 2 0 1 2
∑
Since
the eigenvectors
⃗ are known,
(13b) to determine the
) is
⃗( equation
The initial
rating probability
vector ⃗we use
⃗( )
∑
given the choice of ⃗ .
The initial rating probability vector ⃗
⃗
∑
⃗( ) is
⃗
. (13a)
⃗
. (13b)
www.garp.org
Since the eigenvectors ⃗ are known, we use equation (13b) to determine the
given the choice of ⃗ .
11),(11),
the the
coefficients
time-dependent
while
the the
eigenvectors
⃗ ⃗
on
coefficients are are
time-dependent
while
eigenvectors
Applying
equation
(3), (3),
ant.
Applying
equation
⃗
∑ ∑(
⃗
(
⃗ ∑∑ ⃗
) ⃗) ⃗
⃗
∑∑ ⃗
( )( )
⃗
QU
A N⃗ T P E R S P E C T I V E S
∑
∑
⃗
2012
2012
. (12)
. (12)
J. M. Pimbley
J. M. Pimbley
otation
the (time-independent)
initial
value
of vector
and all other
zero. From
equation
one with
as theone ascomponent
he
notation to denote
to denote
the (time-independent)
initial
value
of. . with
component
and elements
all other elements
zero.
From equation
vector
the
(7), this is(7),
alsothis
theis alsoeigenvector
⃗ . Stated⃗ symbolically,
equations
(11)(11)
and and
(12),(12),
the probability
vector
⃗( )⃗(becomes
the
eigenvector
. Stated symbolically,
) becomes
ith equations
the probability
vector
⃗( )⃗( ) ∑ ∑
⃗
⃗. (13a)
. (13a)
ting
probability
vector
⃗ ⃗ ⃗( )⃗(is ) is
→
l rating
probability
vector
The initial
rating probability vector ro =
r(0)→ is
(13a)
⃗( )
(
⃗( ))
⃗ )
(
⃗
. (14)
. (14)
(14)
Byofinspection
of(13a),
equation
(13a),
we can improve
the equaBy(13b)
inspection
equation
we(13a),
can we
improve
the equation
(14)
Bytion
inspection
of
equation
can
improve
thelargest
equation
(14)
(14) asymptotic expression by retaining the next
→
asymptotic
expression
by
retaining
the
next
largest
term:
the eigenvectors
i are known,
use equation
to term:
asymptotic
expression by retaining the next largest term:
envectors
⃗Since
are
known,
we we
use use
equation
(13b)
towe
determine
the (13b)
eigenvectors
⃗ are
known,
equation
(13b)
→ to determine the
determine the io given the choice of ro.
ice
of ⃗of. ⃗ .
⃗( )
⃗⃗( )
⃗
. (15) . (15)
choice
(15)
⃗
⃗
⃗
⃗ ∑ ∑
⃗
⃗. (13b)
. (13b)
→
The probability character of the vector r (t) is preserved The ordering of eigenvalues of equation (8) motivates this expression. The
→
The all
ordering
of eigenvalues
of equation
(8) motivates
this expression.
The
To interpret the vector r (t) as a “rating probability” vector,
The ordering
of eigenvalues
of equation
(8) motivates
this exand
we
designate
this
as
the
“penultimate
smallest
positive
eigenvalue
is
its elements must be non-negative and sum to one.
Clearly,
one
pression.
The
smallest
positive
eigenvalue
is

and
we
desigand we designate thisN-1
as the “penultimate
smallest positive eigenvalue is
→
would choose an
probability vector r o that
satisfies theseWe nate
this N-1 as
the
“penultimate
eigenvalue.”
We
consider
N
eigenvalue”.
consider
(
)
to
be
the
“last
eigenvalue”
and
makeand
the
10initial
10
eigenvalue”. We consider
(
) to be the “last eigenvalue”
make the
conditions. From equation (13b), we see requiring the sum of (=0) to be the “last eigenvalue” and make the assumption that
→
assumption
that
the
antepenultimate
eigenvalue,
,
is
strictly
greater
than
the
assumption
the antepenultimate
eigenvalue,
, isgreater
strictlythan
greater
the elements of r o to be one implies that No=1.16 With this
as- thethat
antepenultimate
eigenvalue,
N-2, is strictly
the than the
penultimate
eigenvalue.
All
other
terms
of
equation
(13a)
are
exponentially
smaller
signment, equation (13a) then shows that the sum
of elements
penultimate
eigenvalue.
All
other
terms
of
equation
(13a)
are
penultimate eigenvalue. All other terms of equation (13a) are exponentially smaller
→
of r (t) for all t >0 is also one (since N =0).
exponentially
smaller
thetotwo
terms of equation (15) as
than
the
two
terms
of equation
(15)
as timethan
goes
infinity.
→
than
the two
terms
of to
equation
To show that all elements of r (t) remain non-negative
protime
t goes
infinity.(15) as time goes to infinity.
→
vided they are non-negative at time t, consider r (t+∆t).AFor
suf- feature
A striking
feature (15)
of equation
(15)“penultimate
is that the “penultimate
striking
equation
is that the
eigenvector”
A strikingoffeature
→
→
→ of equation (15) is that the “penultimate eigenvector”
ficiently small ∆t, r (t+∆t)[I–Q∆t]r (t). Given that Q has the eigenvector” N-1 provides a fixed distribution of rating probprovides
aprovides
fixed distribution
of rating probabilities
above theabove
defaultthestate
(N). state (N).
→ ⃗
a fixed
distribution
of rating
probabilities
default
⃗
properties (4a)-(4d) and that the elements of r (t) are non-negaabilities
above
the default
state (N).
That is, imagine
that
the
That
is,
imagine
that
the
probability
of
each
rating
level
above
D
is
specified
as
tive, the observation that all elements of the matrix I–Q∆t
probability
rating level
above
D is
specified
as
Thatareis, imagine
that of
theeach
probability
of each
rating
level
above
D being
is specified
as
→
→
positive implies that all elements of r (t+∆t) arebeing
non-negative.
proportional
to each corresponding
element
ofThen,
N-1. Then,
as
.
as
time
proportional
to
each
corresponding
element
of
⃗
Then,
being proportional
to each
corresponding
element
of ⃗ of .each
Applying these time steps ∆t repeatedly shows that all elements
time increases,
the relative
probability
“amount”
rat- as time
→
increases,
the
relative
probability
“amount”
of
each
rating
above
D
remains
of r (t) remain non-negative as time t increases.
D remains
the “amount”
same evenofthough
the absolute
increases,ing
theabove
relative
probability
each rating
above probD the
remains the
-N-1t
It is worth noting that this property of persistent
abilities
of each
rating are
diminishing
withdiminishing
the e
exponensame non-negaevensame
though
the
absolute
probabilities
of
each
rating
are
with
the
even though the absolute probabilities of each rating are diminishing with the
→
tivity for the elements of r (t) does not arise from a non-nega- tial of equation (15).
exponential
of equation
If the
financial
world deliberately
added new
→
of (15).
equation
(15).
If the financial
added new
tivity property of the elements of the eigenvectors i (i=1,..., N) exponential
If the financial
world
deliberately
addedworld
newdeliberately
bonds to re→
to bonds
replaceto place
bonds
that
default,
then
one
could
interpret
the
penultimate
of the HRM Q. Other than the eigenvector Nbonds
corresponding
bonds
that
default,
then
one
could
interpret
the
penultireplace bonds that default, then one could interpret the penultimate
→
to the zero eigenvalue, all the i will have at least
one negative
mate eigenvector
as rating
providing
a steady-state
rating
distribution
eigenvector
as
providing
a steady-state
distribution
above D.
In reality,
eigenvector
as providing
a steady-state
rating
distribution
above
D. of
Inthere
reality, there
element (given that the sum of vector elements is zero and
the above
D. In reality,
there is no
replacement
mechanism
this
→we will think of ⃗
as
providing
is
no
replacement
mechanism
of
this
type,
so
elements are not all zero).
type, so we
will think
as so
providing
as providing
is no replacement
mechanism
of of
thisN-1
type,
we will merely
think ofa ⃗“natural
distribution.”
Regardless
of
the
initial
rating state, the
merely a merely
“naturala rating
rating
distribution”.
Regardless
of
the
initial
“natural rating distribution”. Regardless of the initial rating state, the
Asymptotic Behavior of Credit Ratings for Large rating probability vector will evolve over time to the shape of
rating probability
vector
will
evolve
timeover
to the
shape
of shape
this “natural
rating probability
vector
willover
evolve
time
to the
of this rating
“natural rating
Time
this “natural
rating
distribution.”
→
Just as we know that the survival probability S(t)distribution”.
for the simple
Given
this
meaning
of

,
we
conclude
that
each
of
the
N-1
distribution”.
hazard rate model of equation (2) must approach zero as time first N-1 elements of this penultimate eigenvector must either
, we
that each that
of the
firstofN-1
Given thisGiven
⃗
,sign
we conclude
each
the elements
firstthese
N-1 elements
meaning
of ⃗conclude
t goes to infinity, we have a similar expectation for the
rating meaning
havethis
theof
same
algebraic
or be zero.
To
add clarity,
→
of this penultimate
eigenvector
must
have
the
same
algebraic
sign
or
be
zero.
Tozero. To
probability vector r (t) as t→ .
N-1
elements
cannot
consist
of
both
positive
and
negative
enof this penultimate eigenvector must have the same algebraic sign or be
→
Regardless of the initial rating state r o, all bonds eventually tries. This statement is not an additional requirement to place
default. Hence, all probability will accumulate in the default on the HRM. Rather, the properties we developed earlier are
state. The probability vector will approach the column vector sufficient to prove this12behavior.
12
→
with one as the Nth component and all other elements zero.
Since r (t) maintains its character as a probability vector, then
→
From equation (7), this is also the Nth eigenvector N. Symboli- it must do so in the asymptotic limit expressed in equation (15).
→
cally, this is depicted as follows:
If the first N-1 elements of N-1 are non-negative, (non-posi-
www.garp.org
A U G U S T 2 0 1 2 RISK PROFESSIONAL
5
QUANT PERSPECTIVES
tive), then the constant N-1o is positive (negative). One can
also show that the Nth element of the penultimate eigenvector
must have the opposite algebraic sign to that of the first N-1
elements.
Equations (13a) and (15) also show why complex eigenvalues are undesirable. Earlier, we announced a restriction on the
HRM to ensure that all eigenvalues are real. In the absence
of this restriction, complex eigenvalues would be possible and
they would occur in complex conjugate pairs.
If the penultimate eigenvalue N-1 were one of a complex
conjugate pair, then we would need to include its conjugate,
the antepenultimate eigenvalue N-2, in the asymptotic expression of equation (15), since both N-1 and N-2 would have the
same real part. This inclusion is certainly feasible mathematically, but it produces an asymptotic rating probability vector
→
r (t) that oscillates like a cosine function. Such oscillation is not
permitted within the context of rating transition to default,
so this is the basis for our forbidding complex eigenvalues.
We should add that our “anti-complexity argument” strictly
applies only to the smallest non-zero eigenvalues N-1 and N-2,
but intuition suggests a ban on all complex eigenvalues is most
reasonable.
rating state.17 Hence, since it is not possible to transition out
of the default state D, the last column is filled with zeroes. The
bottom row consists of default hazard rates, because the values
connote transition into the default state D.
To create the example of Figure 1, we first conjured the diagonal entries in a manner that is loosely consistent with values
of Moody’s (2011). We imposed the behavior of the diagonal
value monotonically increasing from the highest (AAA) category to the lowest (CCC) category above default, because the
CRAs often describe their ratings as being less volatile in the
higher categories. For off-diagonal entries, we imposed plausible default hazard values in the last row, geometric decreases
in absolute values moving up and down from the diagonal, and
the zero-sum condition for each column (equation (4d)).
To compute eigenvalues and eigenvectors of the Figure 1
and other matrices, we apply the methods and algorithms of
Press, Flannery, Teukolsky and Vetterling (1992).18 For this
first example, we list the eigenvalues and associated eigenvecPimbley
tors 2012
in Figure 2 (below) with all values rounded toJ. M.
the
nearest
0.001.
Practical Result - Examples
As we related earlier, we do not gather empirical data from the
2012
J. M. Pimbley
CRAs to test this hazard rate matrix model for rating transitions since rating agency downgrades and upgrades are markPractical
Result - Examples
edly
non-Markovian.
Rather, we create and study an idealized
rating transition
model.
As a first example, consider the HRM
As we related earlier, we do not gather empirical data from the CRAs to test
ofthisFigure
1
(below)
with
all values rounded to the nearest
hazard rate matrix model for rating transitions since rating agency downgrades
0.0001.
and upgrades are markedly non-Markovian. Rather, we create and study an
Figure 2: Eigenvalues and Eigenvectors for the HRM of Figure 1
idealized rating transition model.
Figure 1: Example of Hazard Rate Matrix with N = 8
(0.0665)
0.2100
(0.0739)
(0.0370)
(0.0185)
(0.0092)
(0.0046)
(0.0002)
(0.0334)
(0.0668)
0.2400
(0.0743)
(0.0371)
(0.0186)
(0.0093)
(0.0005)
(0.0181)
(0.0361)
(0.0723)
0.2700
(0.0803)
(0.0402)
(0.0201)
(0.0030)
(0.0099)
(0.0198)
(0.0395)
(0.0791)
0.3000
(0.0878)
(0.0439)
(0.0200)
(0.0057)
(0.0115)
(0.0230)
(0.0459)
(0.0918)
0.3300
(0.1021)
(0.0500)
(0.0022)
(0.0044)
(0.0089)
(0.0178)
(0.0356)
(0.0711)
0.3600
(0.2200)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
With N = 8, the rating categories are {AAA, AA, A, BBB, BB, B, CCC, D}. Each
With N = 8, the rating categories are {AAA, AA, A, BBB,
off-diagonal entry is related to a probability of transition from the column rating
BB,
B, CCC, D}. Each off-diagonal entry is related to a probstate to the row rating state.16 Hence, the last column is filled with zeroes since it is
ability
of transition from the column rating state to the row
not possible to transition out of the default state D. The bottom row consists of
default hazard rates since the values connote transition into the default state D.
To create the example of Figure 1, we first conjured the diagonal entries in
6
Figure
2: Eigenvalues
andwe list
Eigenvectors
for the
the eigenvalues and associated
Vetterling
(1992). For this first example,
HRMeigenvectors
of Figure
in Figure 21with all values rounded to the nearest 0.001:
17
Eigenvalues
1
0.440
Eigenvector
1
(0.000)
Formatted: Page2break(0.000)
before
Components
3
(0.001)
4
(0.009)
5
0.137
6
(0.341)
7
0.363
8
(0.149)
2
0.384
3
0.335
4
0.293
5
0.238
6
0.148
7
0.021
8
0.000
(0.000)
(0.001)
(0.026)
0.184
(0.326)
0.042
0.274
(0.147)
0.001
(0.048)
0.219
(0.256)
(0.072)
0.105
0.174
(0.124)
(0.057)
0.237
(0.185)
(0.142)
0.025
0.110
0.128
(0.115)
0.175
(0.169)
(0.169)
(0.025)
0.089
0.121
0.114
(0.138)
(0.210)
(0.086)
0.031
0.111
0.139
0.123
0.096
(0.204)
0.092
0.095
0.092
0.080
0.064
0.046
0.031
(0.500)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
As a first example, consider the HRM of
Figure
1: Example of Hazard Rate Matrix with
Figure 1 below with all values rounded to the nearest 0.0001:
N=8
0.1500
(0.0761)
(0.0381)
(0.0190)
(0.0095)
(0.0048)
(0.0024)
(0.0001)
To compute eigenvalues and eigenvectors of the Figure 1 and other
matrices, we apply the methods and algorithms of Press, Flannery, Teukolsky, and
a manner that is loosely consistent with values of Moody’s (2011). We imposed
RISKthe
PROFESSIONAL
A Uvalue
G U monotonically
S T 2 0 1 increasing
2
behavior of the diagonal
from the highest
(AAA) category to the lowest (CCC) category above default since the CRAs often
Since an eigenvector multiplied by any non-zero constant remains an eigenvector
of the
same
eigenvalue, we aremultiplied
free to choose a normalizing
In Figure
2
Since
an
eigenvector
by anycondition.
non-zero
constant
and an
othereigenvector
representations of eigenvectors,
our normalization
is that thewe
sumare
of thefree to
remains
of the same
eigenvalue,
values of the elements is one and the last element of each eigenvector is
chooseabsolute
a normalizing
condition. In Figure 2 and other reprenegative (with the exception of the last eigenvector).
sentations of eigenvectors, our normalization is that the sum
Consistent with equation (8), we’ve ordered the eigenvalues in Figure 2
of the absolute
values of the elements is one and that the last
. The
from largest to smallest. The last and smallest eigenvalue
element
of
each
eigenvector is negative (with the exception of
eigenvector for this last eigenvalue, ⃗ , is shown in the last column of Figure 2 to
the lastconsist
eigenvector).
of all zeroes for the first seven elements and 1 for the last element as we
Consistent
with
equation
(8), iswe’ve ordered
thepenultimate
eigenvalues
and the
expect.
The
penultimate
eigenvalue
eigenvector
lies in largest
the columnto
beneath
this eigenvalue.
of the
in Figure
2 from
smallest.
The The
lastcomponents
and smallest
ei→
penultimate
above the D rating
the eigenvalue,
“natural rating  , is
genvalue
8=0.eigenvector
The eigenvector
for represent
this last
8
for the HRM of Figure 1. These seven components that begin with
showndistribution”
in the last
column of Figure 2 to consist of all zeroes
for the
first seven elements and 1 for the last element, as we
More specifically, we reduced the HRM to upper Hessenberg form by a series of similarity
17
transformations. We then applied an iterative “QR algorithm” to determine the eigenvalues of the upper
Hessenberg matrix (which are also the eigenvalues of the original HRM). With these eigenvalues, we next
calculated all eigenvectors by inverse iteration.
15
www.garp.org
QUANT PERSPECTIVES
expect.
For this second example, we list the eigenvalues and associThe penultimate eigenvalue is 7=0.021 and the penulti- ated eigenvectors
in Figure 4, with all values rounded
2012
J. M. Pimbley to the
mate eigenvector lies in the column beneath this eigenvalue. nearest 0.001.
Standard & Poor’s (2008). The diagonal values differ markedly between the two
The components of the penultimate eigenvector above the D
HRM examples.
rating represent the “natural rating distribution” for the HRM Figure 4:ForEigenvalues
and Eigenvectors for
this first example, we list the eigenvalues and associated eigenvectors in
of Figure 1. These seven components, which begin with 0.092 the HRM
ofallFigure
Figure 2 with
values rounded2
to the nearest 0.001:
and 0.095 and end with 0.031, show this natural distribution
Figure 4: Eigenvalues and Eigenvectors for the HRM of Figure 2
8
7
6
5
4
3
2
1
is primarily (and somewhat uniformly) in the AAA, AA and A
0.449
0.314
0.214
0.173
0.113
0.061
0.006
0.000
Eigenvalues
ratings, and then trails off at lower ratings.
Eigenvectors
1
(0.000)
0.001
(0.001)
0.013
0.134
(0.373)
0.185
0.000
Based on equations (13a) and (13b), a group of bonds that
Components
2
(0.000)
0.004
(0.007)
0.159
(0.412)
0.086
0.110
0.000
3
(0.001)
0.020
0.148
(0.422)
0.019
0.115
0.074
0.000
all have initial AA ratings will have a rating probability vector
4
(0.004)
0.103
(0.412)
0.043
0.132
0.130
0.061
0.000
2012
J. M. Pimbley
that
we write as a linear combination of the eigenvectors
of
5
(0.033) (0.434)
0.106
0.110
0.095
0.079
0.035
0.000
6
(0.323)
0.285
0.178
0.126
0.085
0.063
0.025
0.000
the HRM. Assuming the HRM of Figure 1, the eigenvalues of
7
0.500
0.087
0.068
0.050
0.035
0.027
0.011
0.000
0.092 and 0.095 and end with 0.031 show this natural distribution is primarily (and
8
(0.137) (0.066) (0.080) (0.078) (0.088) (0.127) (0.500)
1.000
the top row of Figure 2 give the rate of exponential decay of
somewhat uniformly) in the AAA, AA, and A ratings and then trails off at lower
each eigenvector component. Since the penultimate eigenvalratings.
Our qualitative observations for the eigenvalues and eigenvectors of Figure
ue 7=0.021, the “time constant” for decay of the penultimate
2 (the first example) also hold for this second example. The last and penultimate
Basedcomponent
on equations (13a)
(13b),
a group ofrating
bonds that
all have initial
eigenvector
(i.e.,andthe
“natural
distribution
Our eigenvectors
qualitative
observations
forthe the
eigenvalues
and eigenare of the
form we expect. Here
penultimate
and antepenultimate
AA ratings will
have a rating
probability
thatofwe0.021).
write asThus,
a linear vectors eigenvalues
are smaller
than first
those ofexample)
the first examples
which
implies
the this
time second
component”)
is roughly
48 years
(the vector
inverse
of Figure
2 (the
also
hold
for
constants for decay of eigenvector components to the initial rating probability
of theiseigenvectors
of the HRM. Assuming the HRM of Figure 1, the example.
thiscombination
component
long-lived.
The last and penultimate eigenvectors are of the
vector will be longer. The natural rating distribution (proportional to penultimate
eigenvalues
the top (antepenultimate)
row of Figure 2 give theeigenvalue
rate of exponential
of each form we expect. Here, the penultimate and antepenultimate
The
next of
largest
is decay
6=0.148,
eigenvector) is more skewed to AAA in this second example which may be due to
the “time eigenvalues
eigenvector
component.
Since the penultimate
eigenvalue
which
implies
a much shorter
time constant
of just about seven
than
ofof the
example, which
the smallare
AAA smaller
diagonal element
(0.04)those
of the HRM
Figure first
3.
years.
Hence,
afterofroughly
seveneigenvector
years, the
rating distribution
timewe constants
for indecay
of eigenvector
constant”
for decay
the penultimate
component
(i.e., the “natural implies the Finally,
present a third example
which the HRM,
like that of Figure 1, compomotivated
by therating
Moody’s (2011)
rating transition
data. Now
add longer.
in the
of the
undefaulted
bonds
that(the
were
initially
rated
initial
probability
vector
willwe be
The
ratingremaining,
distribution component”)
is roughly
48 years
inverse
of 0.021).
Thus, nents tois the
in which(proportional
all letter ratings from AA
to CCC have eigenAAthis
willcomponent
be approximately
proportional
to the natural
rating is naturalcustomary
ratingsub-categories
distribution
todown
penultimate
is long-lived. The
next largest (antepenultimate)
eigenvalue
distinct rating levels (described by “+” and “-“ signs or by numerical
distribution (components
of the
penultimate
eigenvector
of vector) three
is more
skewed to AAA in this second example which
which implies a much
shorter
time constant of
just about 7 years.
Figure
2).
may
be
due
to
the
small AAA diagonal element (0.04) of the
Hence, after roughly 7 years the rating distribution of the remaining, undefaulted
We emphasize again that we are not attempting here to infer hazard rate matrices from rating agency
data. This is inappropriate because the amount of data is insufficient to generate the precision we ascribe to
As
a
second
example,
consider
the
alternative
HRM
of
FigHRM
of
Figure
3.
our matrices and because credit ratings in practice are not Markovian. Our intent in referencing past data
bonds that were initially rated AA will be approximately proportional to the natural
from Moody’s Investors Service and Standard & Poor’s is simply to show that the numerical values we use
are similarwe
to real-world
credit rating
ure rating
3 (below),
with
all values
rounded
toeigenvector
the nearest
0.0001.
Finally,
present
a transition
thirddata.example in which the HRM, like
distribution
(components
of the
penultimate
of Figure
2).
17 by the Moody’s (2011) rating
that ofFormatted:
Figure
1,
is
motivated
No page break before
As a second example, consider the alternative HRM of Figure 3 below with
Figure 3: Example of Hazard Rate Matrix with transition data. Now we add in the customary sub-categories
all values rounded to the nearest 0.0001:
N=
8
in which all letter ratings from AA down to CCC have three
Figure 3: Example of Hazard Rate Matrix with N = 8
distinct rating levels (described by “+” and “-” signs or by numerical modifiers or by “high” and “low” designations).
0.0400 (0.0316) (0.0212) (0.0115) (0.0087) (0.0048) (0.0040) 0.0000
(0.0203) 0.0900 (0.0425) (0.0230) (0.0175) (0.0095) (0.0079) 0.0000
Instead of N=8, this example has N=20 rating levels. Rather
(0.0101) (0.0301) 0.1400 (0.0460) (0.0350) (0.0190) (0.0159) 0.0000
than
showing all 400 elements of this HRM, we describe it as
(0.0051) (0.0150) (0.0404) 0.1600 (0.0699) (0.0381) (0.0317) 0.0000
(0.0025) (0.0075) (0.0202) (0.0438) 0.2400 (0.0761) (0.0635) 0.0000
Figure 1, with interpolation along the bottom row (transition
(0.0013) (0.0038) (0.0101) (0.0219) (0.0666) 0.2600 (0.1270) 0.0000
to default) and along the diagonals with one modification: the
(0.0006) (0.0019) (0.0051) (0.0109) (0.0333) (0.0725) 0.4000
0.0000
diagonal elements need to increase by at least 50% relative
(0.0001) (0.0002) (0.0005) (0.0030) (0.0090) (0.0400) (0.1500) 0.0000
to the Figure 1 diagonal values. This increase in the diagonal
elements is sensible, because increasing the number of rating
Whereas the HRM of Figure 1 is an approximate, idealized version of data from
Whereas the HRM of Figure 1 is an approximate, ideal- levels should necessarily increase the number of rating transiMoody’s (2011), Figure 3 is an idealized and approximate rendering of data in
ized version of data from Moody’s (2011), Figure 3 is an ideal- tions. If we do not increase the diagonal values in this manner,
ized and approximate rendering of data in Standard & Poor’s then the HRM will have complex pairs of eigenvalues.
(2008). The diagonal values differ markedly between the two
This HRM for 20 rating levels has 20 eigenvalues and 20
HRM examples.19
corresponding eigenvectors. The eigenvalues are all real and
18
18
16
www.garp.org
A U G U S T 2 0 1 2 RISK PROFESSIONAL
7
QUANT PERSPECTIVES
distinct, and range from a high of 0.662 for 1 down to 0.013
and 0.000 (zero) for 19 and 20, respectively. Figure 5 (below)
shows the largest two eigenvalues with associated eigenvectors
and the smallest three eigenvalues with eigenvectors (with all
values rounded to the nearest 0.001).
Figure
5: Five of the Eigenvalues and Eigenvec2012
J. M. Pimbley
tors for the HRM with N = 20
Figure 5: Five of the Eigenvalues and Eigenvectors for the HRM with N = 20
Eigenvalues
Eigenvectors
Components
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20
0.662
19
0.632
3
0.080
2
0.013
1
0.000
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
0.002
(0.017)
0.075
(0.196)
0.310
(0.222)
0.113
(0.065)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
(0.001)
0.007
(0.038)
0.116
(0.191)
0.122
0.103
(0.170)
0.152
(0.101)
(0.050)
(0.056)
(0.045)
(0.032)
(0.015)
0.003
0.020
0.034
0.045
0.052
0.056
0.057
0.054
0.049
0.042
0.034
0.026
0.017
0.012
(0.303)
0.035
0.044
0.045
0.045
0.044
0.042
0.039
0.036
0.032
0.028
0.025
0.021
0.017
0.014
0.011
0.009
0.006
0.004
0.003
(0.500)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
The two columns furthest to the right of Figure 5 show that the last and
penultimate
eigenvalues and
eigenvectors
haveright
the properties
we derived
and that
The
two columns
furthest
to the
of Figure
5 show
earlier.
In particular, eigenvalues
the penultimate eigenvector
gives the shape have
of the the
the described
last and
penultimate
and eigenvectors
natural
rating
distribution.
properties we derived and described earlier. In particular, the
penultimate eigenvector gives the shape of the natural rating
distribution.
Parting Thoughts
This article considers the well-known credit rating transition
dynamic and re-formulates the problem in continuous time
with a hazard rate matrix. As an extension and analogy with
the hazard rate model for bond
default, we create the rating
19
probability vector and show how the hazard rate matrix governs its evolution in time. We explain the mathematical properties that this matrix must satisfy (both by imposition and by
proof) and also deduce analytical insights of the hazard rate
matrix (HRM).
The eigenvalues and eigenvectors of this HRM provide the
best view of the behavior of bond credit ratings at large time
8
RISK PROFESSIONAL A U G U S T 2 0 1 2
from initial rating. In particular, the penultimate eigenvector
gives the shape of the “natural rating distribution.” We believe
this study is the first to identify and derive this rating distribution that prevails long after inception.
Our analysis applies only to credit rating migration that is
Markovian. We do not attempt to fit or explain actual credit
rating agency data, since rating changes of the large public rating agencies are decidedly not Markovian. Rating accuracy,
however, requires Markovian evolution. As investors, banks or
asset managers create their own (internal) rating scales and procedures, it is our hope that the framework, analysis and results
of this article will be helpful.
Footnotes
1. The author is Principal of Maxwell Consulting (http://www.maxwell-consulting.com/) and welcomes followers on Twitter (http://
www.twitter.com/JoePimbley).
2. There exist 10 to 20 CRAs worldwide that have some significant
form of governmental, regulatory or market recognition. The largest
four are Moody’s Investors Service, Standard & Poor’s, Fitch Ratings
and DBRS.
3. See, for example, Moody’s (2011), Standard & Poor’s (2008), Fitch
(2012) and DBRS (2012).
4. The published default tables are often associated with the structured finance ratings of the CRAs, with no clear statement that these
tables also apply to the ratings of non-structured finance debt such as
sovereigns or banks. Further, Moody’s Investors Service describes its
ratings in terms of “expected loss” rather than “default probability,”
but the conversion between these two risk measures is straightforward.
5. See, for example, Moody’s (2011), Standard & Poor’s (2008), Fitch
(2012) and DBRS (2012).
6. Alternatively, the CRA may define its transition matrix such that
the default probabilities occupy the last column rather than the last
row.
7. See, for example: Israel, Rosenthal and Wei (2001), and Gupton,
Finger and Bhatia (2007).
8. See, for example, Lando and Skødeberg (2002), Altman and Kao
(1992), and Carty and Fons (1993).
9. See the Moody’s Investors Service “Special Comment” of Mann
and Metz (2011).
10. See, for example, Löffler and Posch (2011).
11. One interprets the exponential function of a matrix argument
as the MacLaurin series expansion of the exponential with scalar
product powers of the matrix argument.
www.garp.org
QUANT PERSPECTIVES
12. See, for example: Israel, Rosenthal and Wei (2001); Lando and
Skødeberg (2002); and Jarrow, Lando and Turnbull (1997).
13. Jarrow, Lando and Turnbull (1997) also wrote a “Kolmogorov
equation” similar to equation (3) to relate the “generator matrix” to
the time evolution of the RTM. Our work differs from this study in
our focus on the HRM and the rating probability vector.
14. As an example, imagine there are only four rating states (includ0.15 -0.17 -0.01 0
ing the default state), so that N=4. Let the HRM be -0.01
0.20 -0.02 0.
(
-0.13 -0.01 0.25
-0.01 -0.02 -0.22
0
0
)
Of the four eigenvalues of this matrix, two are real (0 and 0.097) and
two are complex (0.252 + i 0.048 and 0.252 – i 0.048). Relatively
small changes to some off-diagonal matrix elements produce four
real eigenvalues.
15. See, for example, Weisstein (2012). In our application of this
theorem, we consider the sum of column (rather than row) element
absolute values for the Gerschgorin radius. This substitution is permissible since a matrix and its transpose have the same eigenvalues.
16. Recall that the sum of all elements of each eigenvector →
vi is zero
for iN and that the sum of the elements of →
vN is one from equation
(7).
17. As stated previously, the HRM Q has the meaning that I – Q∆t
is the rating transition matrix for the time period ∆t,where I is the
identity matrix. Thus, the ij element of I – Q∆t is the probability
that a bond in rating level j will transition to rating level i in the time
period ∆t.
18. More specifically, we reduced the HRM to upper Hessenberg
form by a series of similarity transformations. We then applied an
iterative “QR algorithm” to determine the eigenvalues of the upper
Hessenberg matrix (which are also the eigenvalues of the original
HRM). With these eigenvalues, we next calculated all eigenvectors
by inverse iteration.
19. We emphasize again that we are not attempting here to infer
hazard rate matrices from rating agency data. This is inappropriate
because the amount of data is insufficient to generate the precision
we ascribe to our matrices and because credit ratings in practice are
not Markovian. Our intent in referencing past data from Moody’s
Investors Service and Standard & Poor’s is simply to show that the
numerical values we use are similar to real-world credit rating transition data.
Joe Pimbley (PhD) is Principal of Maxwell Consulting, a consulting firm he founded
in 2010, and a member of Risk Professional’s editorial board. His recent and current
engagements include financial risk management advisory, underwriting for structured and
other financial instruments, and litigation testimony and consultation. In a prominent
engagement from 2009 to 2010, Joe served as a lead investigator for the Examiner appointed by the Lehman bankruptcy court to resolve numerous issues pertaining to history’s
largest bankruptcy.
www.garp.org
References
E. Altman and D. Kao. “The implications of corporate bond rating
drift,” Financial Analysts Journal, 64-75, 1992.
B. Belkin, S. Suchower and L. Forest Jr. “A one-parameter representation of credit risk and transition matrices,” Credit Metrics Monitor, 1998.
L. Carty and J. Fons. “Measuring changes in corporate credit quality,” Moody’s Special Report, 1993.
J. Christensen, E. Hansen and D. Lando. “Confidence sets for continuous-time rating transition probabilities,” J. Banking & Finance 28,
2575-2602, 2004.
DBRS Industry Study. “2011 DBRS Corporate Rating Transition
and Default Study,” March 2012.
Fitch Ratings Special Report. “Fitch Ratings Global Corporate Finance 2011 Transition and Default Study,” March 2012.
G. M. Gupton, C. C. Finger and M. Bhatia. CreditMetrics – Technical Document, RiskMetrics Group, Inc., 2007.
R. B. Israel, J. S. Rosenthal and J. Z. Wei. “Finding generators for
Markov chains via empirical transition matrices with applications to
credit ratings,” Mathematical Finance 11(2), 245-265, 2001.
R. A. Jarrow, D. Lando, and S. M. Turnbull. “A Markov model for
the term structure of credit risk spreads,” Review of Financial Studies 10,
481-523, 1997.
J. Keilson and A. Kester. “Monotone matrices and monotone Markov
processes,” Technical Report No. 79 for National Science Foundation
Grant GP-32326X, Stanford University, 1974.
D. Lando and T. M. Skødeberg. “Analyzing rating transitions and
rating drift with continuous observations,” J. Banking & Finance 26,
423-444, 2002.
G. Löffler and P. N. Posch. Credit risk modeling using Excel and VBA, Second Edition, John Wiley & Sons, Ltd., 2011.
J. Mann and A. Metz. “Measuring the Performance of Credit Ratings,” Moody’s Investors Service Special Comment, November 2011.
Moody’s Investors Service Special Comment. “Corporate Default
and Recovery Rates, 1920-2010,” February 2011.
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling.
Numerical Recipes in C: The Art of Scientific Computing, 2nd Edition, Cambridge University Press, 1992.
B. Singer and S. Spilerman. “The representation of social processes
by Markov models,” American Journal of Sociology 82, 1-54, 1976.
Standard & Poor’s Ratings Direct. “2007 Annual Global Corporate
Default Study and Rating Transitions,” February 2008.
H. Takada and U. Sumita. “Credit risk model with contagious default
dependencies affected by macro-economic condition,” European Journal of Operational Research, 2011.
E. W. Weisstein. “Gerschgorin Circle Theorem,” from MathWorld
– a Wolfram Web Resource: http://mathworld.wolfram.com/GershgorinCircleTheorem.html, 2012.
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