Anna Cornaglia

Risk Management

INTESA SANPAOLO

Transition matrices and PD’s term structure

Credit Risk Management Forum

GLC, Wien, May 7-8, 2015

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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

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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) .

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‘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

Rat

ing

iniz

iale

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

initi

al ra

ting

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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).

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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

kkR

)0( ))(exp( ),()( ttQDR DRrowt

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

jij

0

0

01

probability to migrate from one to another state does not depend on time

discrete

k-periods transition matrix

transition matrix from 0 to t

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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.

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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

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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

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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

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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

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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.

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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 MODELINPUT:

PD, LGD, Raroc (Ke)

OUTPUT:

CREDIT SPREAD

PRICING MODELOUTPUT:

IMPLICITRAROC (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

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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(t1)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

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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:

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.

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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.

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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 (𝑇 𝐷𝑡 ,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.

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The starting point is the observed matrix (for the segments: Corporate, Mortgage, SME retail) . The matrix is transformed into guaranteeing that annual change in portfolio default rate is equal to system default rate (forecast by the econometric model):

D

A B C D E F D

A

B

C

D

E

F

A

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

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.

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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 DD

DD

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

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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)

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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

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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