A transition matrix is a square matrix describing the probabilities of moving from one state to another in a dynamic system. In each row there are the probabilities of moving, from the state represented by that row, to the other states. Thus each row of a transition matrix adds to one.

1. Anna Cornaglia
Risk Management
INTESA SANPAOLO
Transition matrices and
PD’s term structure
Credit Risk Management Forum
GLC, Wien, May 7-8, 2015

2. 2
Overview of transition matrices applications in Risk Management
Application Desiderata
New impairment model
(IFSR 9)
Lifetime EL (PD’s term structure on the basis of
internal data).
Pricing and Fair Value PD’s continuous term structure on the basis of
internal data.
Stress test model Historical series of internal one year transition
matrices, allowing us to estimate a relationship with
macro variables.
Backtesting Infra-annual transition matrices on the basis of
internal data.
Maturity treatment in
portfolio models
One year internal transition matrices.
PD’s term structure consistent with internal DRs.
Some points should be respected:
• at each bank level, the methodology should be the
same for different purposes and entities
• the proposed methodology should guarantee
consistency between matrices and cumulative PDs
Common ground: historical
series of internal matrices for
the different segments and
cumulative default
probabilities on multi-year
periods.
Methodology:
• Markov homogeneous chain /
non-homogeneous chains
• Vintage analysis
• Macro linkage

3. 3
What are transition matrices? What are PDs’ term structures?
Rating transition matrices show the probability of a company migrating from one rating
category to another during a certain period of time. They are based on historical data. The
rating categories can be either those used internally by the financial institution or those
produced by rating agencies such as Moody’s, S&P, or Fitch.
A transition matrix is a square matrix describing the probabilities of moving from one state
to another in a dynamic system. In each row there are the probabilities of moving, from the
state represented by that row, to the other states. Thus each row of a transition matrix adds
to one.
The term structure of default probabilities is the set of (cumulative or conditional) default
probabilities for future time periods.
Let pt denote the probability of default during period t: PD(t − 1, t) = pt . This is called the
conditional or marginal default probability since it is the probability that the firm defaults at
time t given that it has survived until t − 1.
The cumulative survival probability from now until period k is:
PS(0,k) = P[survival from 0 to k] = (1 − p1)(1 − p2)…(1 − pk)
The cumulative default probability until time k is the probability of defaulting at any point in
time until time k, or PD(0,k) = 1 − PS(0,k) .

4. 4
‘Well-behaved’ matrices
A ‘well-behaved’ matrix should respect some monotonicity requirements:
a) The worst rating classes should present higher default probabilities (data increasing in the
last column of the transition matrix).
b) Transition probabilities should decrease with the increase of the number of notches from
the initial rating class (decreasing probabilities from the principal diagonal to the extremes
of each row). This does not hold for the default event, which is normally more likely than a
downgrade to the worst ratings.
rating finale
Ratinginiziale
a)
+
-
b)
-
b)
c)
-
-
c)
c) Probability of migrating towards a
certain rating should be higher for
the nearest classes (transition
matrix columns decreasing from
the principal diagonal to the
extremes).
final rating
initialrating

5. 5
Markov’ transition matrices for different periods of time
Transition matrices can be used to calculate a
transition matrix for periods other than the basis.
The n-year transition matrix is in fact calculated as
the n-th power of the one year matrix. Not
surprisingly, the probability of a company keeping
the same credit rating over n years is much less
than it is over one year and default probabilities
over n years are much higher than over one year.
The credit rating change over a period less than a
year is not so easy to be calculated. For example,
estimating a transition matrix for six months
involves taking the square root of the one year
matrix; estimating the transition matrix for three
months involves taking the fourth root of the
matrix; and so on.
A stochastic process has the Markov
property if the conditional probability
distribution of future states of the
process depends only upon the present
state; that is, given the present, the
future does not depend on the past
(transitions are dependent only on the
values of the current state, and not on
the previous history of the system up
to that point).

6. 6
The generator of a transition matrix: the homogeneous case
Markov approaches to the estimate of PD’s term structure:
Markov
homogeneous
chain
,..)3,2,1()( ),(
)(
 kMDR DRrow
kk
R
)0())(exp( ),(
)(
 ttQDR DRrow
t
R continuous
If for a discrete time chain defined by a one
year migration matrix M the Q generator can
be found such that:
then the discrete time chain can be
incorporated into a continuous time chain.
)exp(QM 
Q can be defined as a
generator if:
jiq
iq
iq
ij
ii
N
j
ij



0
0
0
1
probability to migrate
from one to another state
does not depend on time
discrete
k-periods transition matrix
transition matrix from 0 to t

7. 7
Some considerations on matrices time homogeneity
The homogeneity hypothesis for transition matrices can be refused for a number of
reasons:
 rating models (especially the external agencies’ ones) tend to be not enough
dynamic: the worsening of credit quality is generally not immediately
acknowledged but it is taken into account with a time lag;
 Lando et al. observe a lower upgrade probability for counterparts that in
preceeding periods have suffered a downgrade;
 persistence of positive and negative phases of the economic cycle.
If the generator is estimated on internal data and used to build the cumulative
and forward probability curves, it’s fundamental to test homogeneity hypothesis.

8. 8
The generator of a transition matrix: the non-homogeneous case
A generator that approximates well on a one-year time
horizon can sometimes generate PDs term structures that
significantly deviate from frequencies observed on longer time
horizons.
homogeneity hypothesis is left apart
QtQt  )(time dependent generator
diagonal matrix in which each
element depends on 2 parameters
For each couple of parameters it’s possible to generate a term
structure of cumulated PDs calculating the migration matrices:
and optimising parameters in order to obtain a good fitting with the
observed term structures.
0)(t)exp(  tt tQM
Example of a term structure: homogeneity hypothesis
Example of a term structure: non-homogeneity hypothesis
C. Bluhm and L. Overbeck, “Calibration of PD Term Structures: To Be Markov Or Not To Be”, 2006

9. 9
Vintage analysis
Long term default curves can be derived on
observed data. Vintage curves, built on the basis
of the analysis of registered defaults divided by
cohort (time from origination), can trace portfolio
riskiness evolution and underline its changes,
linked to different credit politics and economic
cycle fluctuations.
Vintage curves need long historical time series,
but they are more precise than the markovian
approach.
Vintage curve for a mortgage
Time from origination
LT (lifetime) default curve
High Risk
Medium-High Risk
Medium Risk
Medium-Low Risk
Low Risk
A refinement of vintage analysis can be
realized differentiating the analysis by risk
class, in order to build the long period
default curves that we need for both the
IFSR 9 and the pricing model.
Marginal default curves are typically
strongly differentiated when risk increases.
AIFIRM NEWSLETTER RISK MANAGEMENT MAGAZINE year 8 n. 3-1

10. 10
Vintage analysis vs. Markov approach
• Markov approach implies a mean reversion
phenomenon such that, independently from the
starting rating class, long term PD tends to an
average: for higher rating classes PD increases in
time.
• In the Markov approach the conditional probability
to enter default at time t, being performing at time
t-1, depends only on the stochastic transition
process.
• For corporate portfolios, characterized by a higher
proportion of revolving credits and a plurality of
loans which insist on the same counterpart (thus
smoothing the vintage effect), Markov chains seem
to be more appropriate.
Vintage approach Markov approach
Time from origination Time from origination
Low risk curve
Medium-high risk curve
• In the vintage analysis the typical bell-shaped
form is generally maintained for all rating
classes.
• In the vintage approach the conditional
probability to enter default at time t, being
performing at time t-1, strongly depends on the
credit maturity at time t-1.
• The vintage approach captures risk dynamics
linked to the maturity, which is particularly
relevant for mortgages (and for the other
retail). Mortgages in fact, being both retail and
long term, show a different behaviour according
to the vintage of the loan.
Low risk curve
Medium-high risk curve
AIFIRM NEWSLETTER RISK MANAGEMENT MAGAZINE year 8 n. 3-1

11. 11
Credits
Objective impairment
evidence (loss event)
No objective
impairment evidence
(NO loss event)
Significant activity NON-significant
activity
Analytical
evaluation
Collective
evaluation
Analytical
evaluation
Possibility to evaluate by
homogenous risk classes
New impairment model
The credits impairment process in IAS 39

12. 12
New impairment model
The 3 buckets approach
At origination
credits are
evaluated on
the basis of one
year EL
In the case of «significant
deterioration» of credit
quality we shift to
Lifetime EL
Impairment
(decayed
credits): no
changes  One year EL can be treated
with Basel 2 methodologies,
conveniently integrated.
 Lifetime EL is
the actual value
of expected
losses until
credit
commitment.
 Its computation
requires a PD
term structure.

13. 13
Pricing model and full fair value
Outline of the models
Viceversa, starting from known Spread values (e.g. the market price of a CDS) referred to a certain
rating class and assuming that LGD is known, it’s possible to obtain the expected market return
(implicit Raroc), using the pricing model through the reverse engineering.
PRICING MODEL
INPUT:
PD, LGD,
Raroc (Ke)
OUTPUT:
CREDIT SPREAD
PRICING MODEL
OUTPUT:
IMPLICIT
RAROC
(market)
INPUT:
PD, LGD,
CREDIT SPREAD
Starting from known values of PD, LGD and expected revenue it is possible to apply the pricing
model in a direct way and obtain as an output the CREDIT SPREAD
market value
or

14. 14
Pricing model and full fair value
PD in pricing
Given the counterparty rating, a PD term structure is needed in order to determine Net Present
Value. Two (related) concepts of PD are required:
• Cumulated default probabilities = PD(t)= probability that the counterparty defaults before time t
• Forward default probability =PD(t|t-1)= probability that the counterparty defaults before time t
having survived until t-1.
The relationship between the two probabilities is described by
or
Starting from (monthly in our case) PDFwd matrices it’s possible to determine yearly (and monthly)
PDCum.
PDCum(t)1
PDCum(t)1)PDCum(t
1)tPDFwd(t,


 PDCum(t))1(1)tPDFwd(t,PDCum(t)1)PDCum(t 
 Monthly forward PDs are used in order to calculate each month Expected Loss for all the
operation lenght.
 Yearly forward PDs are used in the Regulatory Capital formula or in the Economic Capital
calculation, both having an yearly horizon.
 Monthly cumulated PD is used – as its opposite or survival probability (1 – PdCum) – as a
corrective factor in the actualization formula used to calculate Capital and Revenues

15. 15
Pricing model and full fair value
Full fair value evaluation
The pricing is used also to support FULL FAIR VALUE evaluation, which is a fair measure of
banking book assets without a market price.
The estimate applies the following formula, including the credit risk premium as an important
component of the actualization rate:
𝐹𝐹𝑉 =
𝑡=1
𝑛
CF𝑡
(1 + 𝑅𝐹𝑡 + 𝑅𝑃𝑡) 𝑡
where:
FFV = Full Fair Value;
CFt = (assumed) cash flow at time t
RFt = Risk Free market rate for lenght t
RPt= CREDIT RISK PREMIUM for lenght t
The Pricing model allows us to determine the RPt component of the FFV actualization rate,
that is Credit Spread.

16. 16
Pricing model and full fair value
A view on pricing model in ISP
 Corporate and Financial Standard & Poor’s matrices, calculated on 1980-2013 data, were used for the
estimate.
 Matrices were adjusted to eliminate the unrated column (i.e. credits that were set out of the ratings
sample, e.g. for mergers, debt repayment, etc.).
 The transition matrix logarithm was calculated and then a regolarization algorithm was applied,
imposing that negative elements out of the diagonal are equal to 0 and that the diagonal elements are
equal to the sum of the non diagonal elements of the row, with sign -.
 Parameter optimization was done so that the default column of multi-year transition matrices could
approximate the empirical term structure of S&P’s default rates.
 Internal PDs term structure was calculated through an interpolation among term structures referred to
different S&Ps ratings.
The generator calculated as the logarithm of an empirical matrix poses some problems:
• existence: there are no general conditions assuring that a generator exists and transition matrices
typically have properties precluding its existence (e.g. the presence of elements equal to zero)
• uniqueness: more than a generator can originate the same transition matrix
The matrix obtained as the logarithm of the starting transition matrix needs to be ‘regularized’ so that it
respects the structure a generator should have (sum of each row elements equal to zero and non-negative
out of diagonal numbers): some methods exist in practice to satisfy this purpose.
The same problem of existence and uniqueness exists in the discrete case if we raise the
matrix at a power less than one in order to obtain infra-annual matrices.

17. 17
ECONOMETRIC CREDIT
RISK MODEL DR
MACRO
VARIABLES
Migration Matrices
The model consists in the estimate of the
relationship betweeen the system default rates
(divided by sector) and the macro variables:
Credit quality change (PD) is determined through
establishing a link between the internal one year
migration matrix (differentiated by sector) and the
change in default rates subject to the stress.
Satellite Models
PD
PD MODEL
PD
TIME
TD
𝑇𝐷𝑡,𝑠 = 𝑓 𝑀𝑎𝑐𝑟𝑜𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑃𝐷𝑡,𝑠 = 𝑓 𝑇𝐷𝑡,𝑠
Stress test
The adopted approach
In the stress model punctual internal transition matrices are used, the last which are available
for the different segments.

18. 18
The starting point is the observed matrix (for the segments: Corporate, Mortgage, SME
retail) 𝑀𝑡−1,𝑡 . The matrix is transformed into 𝑀𝑡,𝑡+1
𝑆
guaranteeing that annual change in
portfolio default rate is equal to system default rate (forecast by the econometric model):
Departure matrix 𝑀𝑡−1,𝑡
A B C D E F D
A
B
C
D
E
F
Arrival matrix 𝑀𝑡,𝑡+1
𝑆
A B C D E F D
A
B
C
D
E
F
Change in the risk factor
common to all rating classes
k is optimized in order to satisfy condition
𝐷𝑅 𝑡+1
𝑝
𝐷𝑅 𝑡
𝑝 = 1 + ∆%𝐷𝑅𝑡+1
𝑆
Shock
Stress test
Transmission of Default Rates to Default Probabilities
The matrix stress is endogenous to the model and depends only indirectly on the
macroeconomic variables. The matrix in fact moves on the basis of a sole underlying factor,
such that percentage change in default rate is equal to the one in the stress scenario.

19. 19
Backtesting
How matrices can be used
Continuous transition matrices are useful to study rare events. If we use Markov chain in the continuous time,
it is possible to catch, inside an established time window (e.g. one year), the existence of indirect defaults
through a downgrade sequence (positive default rates also for investment grade).
0 1 year
D D
D
D
D
D
D
PD 6 months
Using an infra-annual migration matrix allows us to correctly check
the number of defaults which originate in each rating class.
6 months
PD 1 year

20. 20
In order to determine the component of economic capital which is due to longer than one
year maturity , in Intesa Sanpaolo:
 Migrations among internal rating classes were used.
 A smoothing procedure was applied, in order to guarantee the second requirement of a
‘well-behaved’ matrix, which is not respected by internal matrices. The third requirement,
not respected too, was on the contrary bypassed.
 To approximate migrations (as a function of the notching distance from the main
diagonal) an exponential function was used. Migration probabilities were then obtained
through minimization of the squared error with respect to the empirical matrix.
 In the process of smoothing the matrix the following limits were imposed:
 given an initial rating, the probability to remain in the same class at period end, the upgrade
probability and the downgrade probability were preserved
 default probabilities (set equal to the Master scale ones) are not modified
 On the basis of one year matrices cumulative default probabilities were calculated
through the application of an homogeneous Markov process. Then there is no guarantee
that these probabilities correctly approximate the cumulative default rates.
Maturity
Its treatment in Economic Capital (ISP case)

21. 21
For the various applications we would need:
 an historical time series of internal migration matrices, divided by segment
 cumulative default rates on multi-year periods, divided by segment
The available historical time series are relatively short, as:
- In the specific Intesa Sanpaolo case the merger caused a discontinuity in both rating
models and default definition (which is quite common in the banking context).
- The change of regulatory default definition also caused a break.
- The frequent (as it should be) revision or update of rating models implies a difficult
comparison along time. This can be more or less evident for the different rating
segments.
A note on internal availability of data

22. 22
The work on transition matrices is in progress. In this process some points deserve
attention:
 some methods should be found to reasonably combine internal and external
data, allowing us to use the maximum available information without neglecting
personalisation; this is useful also for the retail segment, that in principle should
be richer in data (but rating models changes and perimeter changes can pose
comparability problems);
 as the time horizon is generally quite long, it is important to forecast default
rates precisely, in order to avoid mistakes that can become relevant in a broad
period.
Some further work on matrices and term structures