changes of the full yield. Since the same relative changes are applied for all

yield curves, the results are the same for all yield curves. Again, the application

of volatility updating improves the predictive power and mean adjustment also

has a positive e¬ect.

3.9.3 Risk Factor: Spread over Benchmark Yield

The number of violations (see Table 3.12) is comparable to the latter two

variants. Volatility updating leads to better results, while the e¬ect of mean

3.9 Discussion of Simulation Results 79

VaR reliability plot

4

2

L/VaR quantiles

0

-2

-4

-4 -2 0 2 4

normal quantiles

Figure 3.10. Q-Q plot for volatility updating.

adjustment is only marginal. However, the mean squared deviations (see Ta-

ble 3.15) in the P-P plots are signi¬cantly larger than in the case, where the

risk factors are contained in the benchmark curve. This can be traced back to a

partly poor predictive power for intermediate con¬dence levels (see Figure 3.8).

Mean adjustment leads to larger errors in the P-P plots.

3.9.4 Conservative Approach

From Table 3.14 the conclusion can be drawn, that the conservative approach

signi¬cantly over-estimates the risk for all credit qualities. Table 3.17 indicates

the poor predictive power of the conservative approach over the full range of

con¬dence levels.

80 3 Quanti¬cation of Spread Risk by Means of Historical Simulation

full yield spread curve

Curve V1 V2 V3 V4 V1 V2 V3 V4

INAAA 0,87 0,28 0,50 0,14 8,13 22,19 8,14 16,15

INAA2 0,45 0,36 0,32 0,16 6,96 21,41 7,25 15,62

INAA3 0,54 0,41 0,43 0,23 7,91 21,98 7,97 15,89

INA1 0,71 0,27 0,41 0,13 7,90 15,32 8,10 8,39

INA2 0,50 0,39 0,42 0,17 9,16 15,15 9,51 6,19

INA3 0,81 0,24 0,58 0,24 9,53 12,96 9,61 7,09

INBBB1 0,71 0,29 0,54 0,13 9,59 15,71 9,65 11,13

INBBB2 0,33 0,34 0,26 0,12 11,82 14,58 11,59 10,72

INBBB3 0,35 0,59 0,40 0,34 7,52 11,49 7,78 6,32

INBB1 0,31 0,95 0,26 0,28 4,14 4,57 3,90 1,61

INBB2 0,52 0,49 0,36 0,19 6,03 3,63 5,89 2,12

INBB3 0,53 0,41 0,36 0,17 3,11 3,65 3,09 1,67

INB1 0,51 0,29 0,38 0,15 3,59 1,92 2,85 1,16

INB2 0,51 0,48 0,31 0,22 4,29 2,31 3,41 1,42

INB3 0,72 0,38 0,32 0,16 3,70 2,10 2,99 3,02

BNAAA 0,59 0,19 0,48 0,56 10,13 17,64 9,74 11,10

BNAA1/2 0,54 0,21 0,45 0,46 5,43 13,40 5,73 7,50

BNA1 0,31 0,12 0,29 0,25 8,65 17,19 8,09 8,21

BNA2 0,65 0,19 0,57 0,59 6,52 12,52 6,95 6,45

BNA3 0,31 0,19 0,32 0,29 6,62 9,62 6,59 3,80

Average 0,54 0,35 0,40 0,25 7,04 11,97 6,94 7,28

Table 3.15. MSD P-P Plot for the full yield and the spread

curve(—10 000)

The mean squared deviations are the worst of all approaches. Volatility updat-

ing and/or mean adjustment does not lead to any signi¬cant improvements.

3.9.5 Simultaneous Simulation

From Tables 3.14 and 3.17 it is apparent that simultaneous simulation leads to

much better results than the model with risk factors from the full yield curve,

when volatility updating is included. Again, the e¬ect of mean adjustment

does not in general lead to a signi¬cant improvement. These results lead to

the conclusion that general market risk and spread risk should be modeled

independently, i.e., that the yield curve of an instrument exposed to credit

risk should be modeled with two risk factors: benchmark changes and spread

changes.

3.10 XploRe for Internal Risk Models 81

Curve V1 V2 V3 V4

INAAA, INAA2, INAA3 0,49 0,23 0,26 0,12

INA1 0,48 0,23 0,26 0,12

INA2, INA3, INBBB1, INBBB2, INBBB3, 0,49 0,23 0,26 0,12

INBB1, INBB2

INBB3 0,47 0,23 0,25 0,12

INB1 0,49 0,23 0,26 0,12

INB2 0,47 0,23 0,25 0,12

INB3 0,48 0,23 0,26 0,12

BNAAA, BNAA1/2 0,42 0,18 0,25 0,33

BNA1 0,41 0,18 0,23 0,33

BNA2 0,42 0,18 0,25 0,33

BNA3 0,41 0,18 0,24 0,33

Average 0,47 0,22 0,25 0,17

Table 3.16. MSD P-P-Plot benchmark curve (—10 000)

3.10 XploRe for Internal Risk Models

In this contribution it is demonstrated that XploRe can be used as a tool in

the analysis of time series of market data and empirical loss distributions. The

focus of this contribution is on the analysis of spread risk. Yield spreads are

an indicator of an obligor™s credit risk. The distributions of spread changes are

leptokurtic with typical fat tails, which makes the application of conventional

variance-covariance risk models problematic. That is why in this contribution

we prefer the analysis of spread risk by means of historical simulation. Since

it is not a priori clear, how spread risk should be integrated in a risk model

for interest rate products and how it can be separated from general market

risk, we investigate several possibilities, which include modelling the full yield

curve (i.e., consideration of only one risk factor category, which covers both

benchmark and spread risk) as well as separately modelling spread risk and

benchmark risk. The aggregation of both risk categories is carried out in a

conservative way (addition of the risk measure for both risk categories) as well

as coherently (simultaneous simulation of spread and benchmark risk). More-

over, in addition to the basic historical simulation method we add additional

features like mean adjustment and volatility updating. Risk is quanti¬ed by

means of a quantile-based risk measure in this contribution - the VaR. We

demonstrate the di¬erences between the di¬erent methods by calculating the

VaR for a ¬ctive zero-bond.

82 3 Quanti¬cation of Spread Risk by Means of Historical Simulation

conservative approach simultaneous simulation

Curve V1 V2 V3 V4 V1 V2 V3 V4

INAAA 14,94 14,56 14,00 13,88 1,52 0,64 0,75 0,40

INAA2 13,65 13,51 14,29 14,31 0,79 0,38 0,40 0,23

INAA3 14,34 13,99 13,66 13,44 0,79 0,32 0,49 0,27

INA1 15,39 15,60 15,60 15,60 0,95 0,40 0,52 0,29

INA2 13,95 14,20 14,32 14,10 0,71 0,55 0,50 0,39

INA3 14,73 14,95 14,45 14,53 0,94 0,30 0,59 0,35

INBBB1 13,94 14,59 14,05 14,10 1,00 0,33 0,43 0,17

INBBB2 13,74 13,91 13,67 13,73 0,64 0,52 0,45 0,29

INBBB3 13,68 14,24 14,10 14,09 0,36 0,78 0,31 0,31

INBB1 19,19 20,68 18,93 19,40 0,73 1,37 0,52 0,70

INBB2 13,21 14,17 14,79 15,15 0,30 0,82 0,35 0,51

INBB3 15,19 16,47 15,40 15,67 0,55 0,65 0,15 0,21

INB1 15,47 15,64 15,29 15,51 0,53 0,44 0,19 0,26

INB2 14,47 14,93 15,46 15,77 0,24 0,55 0,24 0,24

INB3 14,78 14,67 16,77 17,03 0,38 0,44 0,27 0,22

BNAAA 14,80 15,30 16,30 16,64 1,13 0,33 0,99 0,96

BNAA1/2 13,06 13,45 14,97 15,43 0,73 0,16 0,57 0,50

BNA1 11,95 11,83 12,84 13,08 0,52 0,26 0,44 0,41

BNA2 13,04 12,58 14,31 14,56 0,78 0,13 0,51 0,58

BNA3 12,99 12,70 15,19 15,42 0,34 0,18 0,58 0,70

Average 14,33 14,60 14,92 15,07 0,70 0,48 0,46 0,40

Table 3.17. MSD P-P Plot for the conservative approach and the si-

multaneous simulation(—10 000)

The numerical results indicate, that the conservative approach over-estimates

the risk of our ¬ctive position, while the simulation results for the full yield as

single risk factor are quite convincing. The best result, however, is delivered

by a combination of simultaneous simulation of spread and benchmark risk

and volatility updating, which compensates for non-stationarity in the risk

factor time series. The conclusion from this contribution for model-builders

in the banking community is, that it should be checked, whether the full yield

curve or the simultaneous simulation with volatility updating yield satisfactory

results for the portfolio considered.

3.10 XploRe for Internal Risk Models 83

Bibliography

Bank for International Settlements (1998a). Amendment to the Capital Accord

to incorporate market risks, www.bis.org. (January 1996, updated to April

1998).

Bank for International Settlements (1998b). Overview of the Amendment to

the Capital Accord to incorporate market risk, www.bis.org. (January

1996, updated to April 1998).

Bundesaufsichtsamt f¨r das Kreditwesen (2001). Grundsatz I/Modellierung des

u

besonderen Kursrisikos, Rundschreiben 1/2001, www.bakred.de.

Gaumert, U. (1999). Zur Diskussion um die Modellierung besonderer

Kursrisiken in VaR-Modellen, Handbuch Bankenaufsicht und Interne

Risikosteuerungsmodelle, Sch¨¬er-Poeschel.

a

Hull, J. C. (1998). Integrating Volatility Updating into the Historical Simula-

tion Method for Value at Risk, Journal of Risk .

J.P. Morgan (1996). RiskMetrics, Technical report, J.P. Morgan, New York.

Kiesel, R., Perraudin, W. and Taylor, A. (1999). The Structure of Credit Risk.

Working Paper, London School of Economics.

Part II

Credit Risk

4 Rating Migrations

Ste¬ H¨se, Stefan Huschens and Robert Wania

o

The bond rating is one of the most important indicators of a corporation™s

credit quality and therefore its default probability. It was ¬rst developed by

Moody™s in 1914 and by Poor™s Corporation in 1922 and it is generally assigned

by external agencies to publicly traded debts. Apart from the external ratings

by independent rating agencies, there are internal ratings by banks and other

¬nancial institutions, Basel Committee on Banking Supervision (2001). Exter-

nal rating data by agencies are available for many years, in contrast to internal

ratings. Their short history in most cases does not exceed 5“10 years. Both

types of ratings are usually recorded on an ordinal scale and labeled alphabeti-

cally or numerically. For the construction of a rating system see Crouhy, Galai,

and Mark (2001).

A change in a rating re¬‚ects the assessment that the company™s credit quality

has improved (upgrade) or deteriorated (downgrade). Analyzing these rating

migrations including default is one of the preliminaries for credit risk models

in order to measure future credit loss. In such models, the matrix of rating

transition probabilities, the so called transition matrix, plays a crucial role. It

allows to calculate the joint distribution of future ratings for borrowers that

compose a portfolio, Gupton, Finger, and Bhatia (1997). An element of a

transition matrix gives the probability that an obligor with a certain initial

rating migrates to another rating by the risk horizon. For the econometric

analysis of transition data see Lancaster (1990).

In a study by Jarrow, Lando, and Turnbull (1997) rating transitions were mod-

eled as a time-homogeneous Markov chain, so future rating changes are not

a¬ected by the rating history (Markov property). The probability of chang-

ing from one rating to another is constant over time (homogeneous), which

is assumed solely for simplicity of estimation. Empirical evidence indicates

that transition probabilities are time-varying. Nickell, Perraudin, and Varotto

(2000) show that di¬erent transition matrices are identi¬ed across various fac-

88 4 Rating Migrations

tors such as the obligor™s domicile and industry and the stage of business cycle.

Rating migrations are reviewed from a statistical point of view throughout this

chapter using XploRe. The way from the observed data to the estimated one-

year transition probabilities is shown and estimates for the standard deviations

of the transition rates are given. In further extension, dependent rating migra-

tions are discussed. In particular, the modeling by a threshold normal model

is presented.

Time stability of transition matrices is one of the major issues for credit risk

estimation. Therefore, a chi-square test of homogeneity for the estimated rating

transition probabilities is applied. The test is illustrated by an example and

is compared to a simpler approach using standard errors. Further, assuming

time stability, multi-period rating transitions are discussed. An estimator for

multi-period transition matrices is given and its distribution is approximated

by bootstrapping. Finally, the change of the composition of a credit portfolio

caused by rating migrations is considered. The expected composition and its

variance is calculated for independent migrations.

4.1 Rating Transition Probabilities

In this section, the way from raw data to estimated rating transition prob-

abilities is described. First, migration events of the same kind are counted.

The resulting migration counts are transformed into migration rates, which are

used as estimates for the unknown transition probabilities. These estimates are

complemented with estimated standard errors for two cases, for independence

and for a special correlation structure.

4.1.1 From Credit Events to Migration Counts

We assume that credits or credit obligors are rated in d categories ranging from

1, the best rating category, to the category d containing defaulted credits. The

raw data consist of a collection of migration events. The n observed migration

events form a n — 2 matrix with rows

(ei1 , ei2 ) ∈ {1, . . . , d ’ 1} — {1, . . . , d}, i = 1, . . . , n.

Thereby, ei1 characterizes the rating of i-th credit at the beginning and ei2 the

rating at the end of the risk horizon, which is usually one year. Subsequently,

4.1 Rating Transition Probabilities 89

migration events of the same kind are aggregated in a (d ’ 1) — d matrix C of

migration counts, where the generic element

n

def

cjk = 1{(ei1 , ei2 ) = (j, k)}

i=1

is the number of migration events from j to k. Clearly, their total sum is

d’1 d

cjk = n.

j=1 k=1

4.1.2 Estimating Rating Transition Probabilities

We assume that each observation ei2 is a realization of a random variable ei2

˜

with conditional probability distribution

d

pjk = P(˜i2 = k|˜i1 = j),

e e pjk = 1,

k=1

where pjk is the probability that a credit migrates from an initial rating j to

rating k. These probabilities are the so called rating transition (or migration)

probabilities. Note that the indicator variable 1{˜i2 = k} conditional on ei1 = j

e ˜

is a Bernoulli distributed random variable with success parameter pjk ,

1{˜i2 = k} | ei1 = j ∼ Ber(pjk ).

e ˜ (4.1)

In order to estimate these rating transition probabilities we de¬ne the number

of migrations starting from rating j as

d

def

j = 1, . . . , d ’ 1

nj = cjk , (4.2)

k=1

and assume nj > 0 for j = 1, . . . , d ’ 1. Thus, (n1 , . . . , nd’1 ) is the composition

of the portfolio at the beginning of the period and

«

d’1 d’1

cj1 , . . . , cjd (4.3)

j=1 j=1

90 4 Rating Migrations

is the composition of the portfolio at the end of the period, where the last

element is the number of defaulted credits. The observed migration rate from

j to k,

def cjk

pjk =

ˆ , (4.4)

nj

is the natural estimate of the unknown transition probability pjk .

If the migration events are independent, i. e., the variables e12 , . . . , en2 are

˜ ˜

stochastically independent, cjk is the observed value of the binomially dis-

tributed random variable

cjk ∼ B(nj , pjk ),

˜

and therefore the standard deviation of pjk is

ˆ

pjk (1 ’ pjk )

σjk = ,

nj

which may be estimated by

pjk (1 ’ pjk )

ˆ ˆ

σjk =

ˆ . (4.5)

nj

The estimated standard errors must be carefully interpreted, because they are

based on the assumption of independence.

4.1.3 Dependent Migrations

The case of dependent rating migrations raises new problems. In this context,

cjk is distributed as sum of nj correlated Bernoulli variables, see (4.1), indicat-

˜

ing for each credit with initial rating j a migration to k by 1. If these Bernoulli

2

variables are pairwise correlated with correlation ρjk , then the variance σjk

of the unbiased estimator pjk for pjk is (Huschens and Locarek-Junge, 2000,

ˆ

p. 44)

pjk (1 ’ pjk ) nj ’ 1

2

ρjk pjk (1 ’ pjk ).

σjk = +

nj nj

The limit

2

lim σjk = ρjk pjk (1 ’ pjk )

nj ’∞

shows that the sequence pjk does not obey a law of large numbers for ρjk > 0.

ˆ

Generally, the failing of convergence in quadratic mean does not imply the

4.1 Rating Transition Probabilities 91

failing of convergence in probability. But in this case all moments of higher

order exist since the random variable pjk is bounded and so the convergence

ˆ

in probability implies the convergence in quadratic mean. For ρjk = 0 the law

of large numbers holds. Negative correlations can only be obtained for ¬nite

nj . The lower boundary for the correlation is given by ρjk ≥ ’ nj1 , which

’1

converges to zero when the number of credits nj grows to in¬nity.

The law of large numbers fails also if the correlations are di¬erent with ei-

ther a common positive lower bound, or non vanishing positive average cor-

relation or constant correlation blocks with positive correlations in each block

(Finger, 1998, p. 5). This failing of the law of large numbers may not sur-

prise a time series statistician, who is familiar with mixing conditions to ensure

mean ergodicity of stochastic processes (Davidson, 1994, chapter 14). In sta-

tistical words, in the case of non-zero correlation the relative frequency is not

a consistent estimator of the Bernoulli parameter.

The parameters ρjk may be modeled in consistent way in the framework of a

threshold normal model with a single parameter ρ (Basel Committee on Bank-

ing Supervision, 2001; Gupton et al., 1997; Kim, 1999). This model speci-

¬es a special dependence structure based on a standard multinormal distri-

bution for a vector (R1 , . . . , Rn ) with equicorrelation matrix (Mardia, Kent,

and Bibby, 1979, p. 461), where Ri (i = 1, . . . , n) is the standardized asset

return and n is the number of obligors. The parameter ρ > 0 may be inter-

preted as a mean asset return correlation. In this model each pair of variables

(X, Y ) = (Ri , Ri ) with i, i = 1, . . . , n and i = i is bivariate normally dis-

tributed with density function

x2 ’ 2ρxy + y 2

1

exp ’

•(x, y; ρ) = .

2(1 ’ ρ2 )

1 ’ ρ2

2π

The probability P[(X, Y ) ∈ (a, b)2 ] is given by

b b

β(a, b; ρ) = •(x, y; ρ) dx dy. (4.6)

a a

Thresholds for rating j are derived from pj1 , . . . , pj,d’1 by

def def def def

zj0 = ’∞, zj1 = ¦’1 (pj1 ), zj2 = ¦’1 (pj1 + pj2 ), . . . , zjd = +∞,

where ¦ is the distribution function of the standardized normal distribution

and ¦’1 it™s inverse. Each credit in category j is characterized by a normally

distributed variable Z which determines the migration events by

pjk = P(Z ∈ (zj,k’1 , zjk )) = ¦(zjk ) ’ ¦(zj,k’1 ).

92 4 Rating Migrations

The simultaneous transition probabilities of two credits i and i from category

j to k are given by

pjj:kk = P(˜i2 = ei 2 = k|˜i1 = ei 1 = j) = β(zj,k’1 , zjk ; ρ),

e ˜ e ˜

i.e., the probability of simultaneous default is

pjj:dd = β(zj,d’1 , zjd ; ρ).

For a detailed example see Saunders (1999, pp. 122-125). In the special case of

independence we have pjj:kk = p2 . De¬ning a migration from j to k as suc-

jk

cess we obtain correlated Bernoulli variables with common success parameter

pjk , with probability pjj:kk of a simultaneous success, and with the migration

correlation

pjj:kk ’ p2jk

ρjk = .

pjk (1 ’ pjk )

Note that ρjk = 0 if ρ = 0.

Given ρ ≥ 0 we can estimate the migration correlation ρjk ≥ 0 by the restricted

Maximum-Likelihood estimator

β(ˆj,k’1 , zjk ; ρ) ’ p2

z ˆ ˆjk

ρjk

ˆ = max 0; (4.7)

pjk (1 ’ pjk )

ˆ ˆ

with

k

’1

zjk = ¦

ˆ pji

ˆ . (4.8)

i=1

The estimate

pjk (1 ’ pjk ) nj ’ 1

ˆ ˆ

ρjk pjk (1 ’ pjk )

σjk =

ˆ + ˆˆ ˆ (4.9)

nj nj

of the standard deviation

pjk (1 ’ pjk ) nj ’ 1

ρjk pjk (1 ’ pjk )

σjk = +

nj nj

is used. The estimator in (4.9) generalizes (4.5), which results in the special

case ρ = 0.

4.1 Rating Transition Probabilities 93

4.1.4 Computation and Quantlets

counts = VaRRatMigCount (d, e)

computes migration counts from migration events

The quantlet VaRRatMigCount can be used to compute migration counts from

migration events, where d is the number of categories including default and e

is the n — 2 data matrix containing n migration events. The result is assigned

to the variable counts, which is the (d ’ 1) — d matrix of migration counts.

XFGRatMig1.xpl

b = VaRRatMigRate (c, rho, s)

computes migration rates and related estimated standard errors

The quantlet VaRRatMigRate computes migration rates and related estimated

standard errors for m periods from an input matrix of migration counts and

a given correlation parameter. Here, c is a (d ’ 1) — d — m array of m-period

migration counts and rho is a non-negative correlation parameter as used in

(4.6). For rho = 0 the independent case is computed.

The calculation uses stochastic integration in order to determine the probability

β from (4.6). The accuracy of the applied Monte Carlo procedure is controlled

by the input parameter s. For s > 0 the sample size is at least n ≥ (2s)’2 .

This guarantees that the user-speci¬ed value s is an upper bound for the stan-

dard deviation of the Monte Carlo estimator for β. Note that with increasing

accuracy (i. e. decreasing s) the computational e¬ort increases proportional to

n.

The result is assigned to the variable b, which is a list containing:

• b.nstart

the (d ’ 1) — 1 — m array of portfolio weights before migration

• b.nend

the d — 1 — m array portfolio weights after migration

• b.etp

the (d ’ 1) — d — m array of estimated transition probabilities

94 4 Rating Migrations

• b.etv

the (d ’ 1) — (d ’ 1) — m array of estimated threshold values

• b.emc

the (d ’ 1) — d — m array of estimated migration correlations

• b.esd

the (d ’ 1) — d — m array of estimated standard deviations

The matrices b.nstart and b.nend have components given by (4.2) and (4.3).

The matrices b.etp, b.emc, and b.esd contain the pjk , ρjk , and σjk from

ˆ ˆ ˆ

(4.4), (4.7), and (4.9) for j = 1, . . . , d ’ 1 and k = 1, . . . , d. The estimates ρjk

ˆ

are given only for pjk > 0. The matrix b.etv contains the zjk from (4.8) for

ˆ ˆ

j, k = 1, . . . , d ’ 1. Note that zj0 = ’∞ and zjd = +∞.

XFGRatMig2.xpl

4.2 Analyzing the Time-Stability of Transition

Probabilities

4.2.1 Aggregation over Periods

We assume that migration data are given for m periods. This data consist in m

matrices of migration counts C(t) for t = 1, . . . , m each of type (d ’ 1) — d. The

generic element cjk (t) of the matrix C(t) is the number of migrations from j to

k in period t. These matrices may be computed from m data sets of migration

events.

An obvious question in this context is whether the transition probabilities can

be assumed to be constant in time or not. A ¬rst approach to analyze the

time-stability of transition probabilities is to compare the estimated transition

probabilities per period for m periods with estimates from pooled data.

The aggregated migration counts from m periods are

m

def

c+ = cjk (t) (4.10)

jk

t=1

4.2 Analyzing the Time-Stability of Transition Probabilities 95

which are combined in the matrix

m

+ def

C = C(t)

t=1

of type (d ’ 1) — d. The migration rates computed per period

cjk (t)

def

pjk (t) =

ˆ , t = 1, . . . , m (4.11)

nj (t)

with

d

def

nj (t) = cjk (t)

k=1

have to be compared with the migration rates from the pooled data. Based on

the aggregated migration counts the estimated transition probabilities

c+

def jk

p+ =

ˆjk (4.12)

nj +

with

d m

def

n+ = c+ j = 1, . . . , d ’ 1

= nj (t),

j jk

t=1

k=1

can be computed.

4.2.2 Are the Transition Probabilities Stationary?

Under the assumption of independence for the migration events the vector

of migration counts (cj1 (t), . . . cjd (t)) starting from j is in each period t a

realization from a multinomial distributed random vector

(˜j1 (t), . . . , cjd (t)) ∼ Mult(nj (t); pj1 (t), . . . , pjd (t)),

c ˜

where pjk (t) denotes the transition probability from j to k in period t. For

¬xed j ∈ {1, . . . , d ’ 1} the hypothesis of homogeneity

H0 : pj1 (1) = . . . = pj1 (m), pj2 (1) = . . . = pj2 (m), . . . , pjd (1) = . . . = pjd (m)

may be tested with the statistic

2

cjk (t) ’ nj (t)ˆ+

d m ˜ pjk

2

Xj = . (4.13)

nj (t)ˆ+

pjk

k=1 t=1

96 4 Rating Migrations

This statistic is asymptotically χ2 -distributed with (d’1)(m’1) degrees of free-

dom under H0 . H0 is rejected with approximative level ± if the statistic com-

puted from the data is greater than the (1 ’ ±)-quantile of the χ2 -distribution

with (d ’ 1)(m ’ 1) degrees of freedom.

The combined hypothesis of homogeneity

t = 1, . . . , m ’ 1, j = 1, . . . , d ’ 1,

H0 : pjk (t) = pjk (m), k = 1, . . . , d

means that the matrix of transition probabilities is constant over time. There-

fore, the combined null hypothesis may equivalently be formulated as

H0 : P(1) = P(2) = . . . = P(m),

where P(t) denotes the transition matrix at t with generic element pjk (t). This

hypothesis may be tested using the statistic

d’1

2 2

X= Xj , (4.14)

j=1

which is under H0 asymptotically χ2 -distributed with (d’1)2 (m’1) degrees of

freedom. The combined null hypothesis is rejected with approximative level ± if

the computed statistic is greater than the (1’±)-quantile of the χ2 -distribution

with (d ’ 1)2 (m ’ 1) degrees of freedom (Bishop, Fienberg, and Holland, 1975,

p. 265).

This approach creates two problems. Firstly, the two tests are based on the as-

sumption of independence. Secondly, the test statistics are only asymptotically

χ2 -distributed. This means that su¬ciently large sample sizes are required. A

rule of thumb given in the literature is nj (t)ˆ+ ≥ 5 for all j and k which is

pjk

hardly ful¬lled in the context of credit migrations.

The two χ2 -statistics in (4.13) and (4.14) are of the Pearson type. Two other

frequently used and asymptotically equivalent statistics are the corresponding

χ2 -statistics of the Neyman type

2

cjk (t) ’ nj (t)ˆ+

d m d’1

˜ pjk

Yj2 2

Yj2

= , Y=

cjk (t)

˜

k=1 t=1 j=1

and the χ2 -statistics

d m d’1

cjk (t)

˜

G2 2

G2 ,

=2 cjk (t) ln

˜ , G=

j j

nj (t)ˆ+

pjk

k=1 t=1 j=1

4.2 Analyzing the Time-Stability of Transition Probabilities 97

which results from Wilks log-likelihood ratio.

Considering the strong assumptions on which these test procedures are based

on, one may prefer a simpler approach complementing the point estimates

pjk (t) by estimated standard errors

ˆ

pjk (t)(1 ’ pjk (t))

ˆ ˆ

σjk (t) =

ˆ

nj (t)

for each period t ∈ {1, . . . , m}. For correlated migrations the estimated stan-

dard deviation is computed analogously to (4.9). This may graphically be

visualized by showing

p+ , pjk (t) ± 2ˆjk (t),

ˆjk pjk (t),

ˆ ˆ σ t = 1, . . . , m (4.15)

simultaneously for j = 1, . . . , d ’ 1 and k = 1, . . . , d.

4.2.3 Computation and Quantlets

The quantlet XFGRatMig3.xpl computes aggregated migration counts,

estimated transition probabilities and χ2 -statistics. The call is out =

XFGRatMig3(c, rho, s), where c is a (d ’ 1) — d — m array of counts for

m periods and rho is a non-negative correlation parameter. For rho = 0 the

independent case is computed, compare Section 4.1.4. The last input parameter

s controls the accuracy of the computation, see Section 4.1.4.

The result is assigned to the variable out, which is a list containing:

• out.cagg

the (d ’ 1) — d matrix with aggregated counts

• out.etpagg

the (d ’ 1) — d matrix with estimated aggregated transition probabilities

• out.esdagg

the (d ’ 1) — d matrix with estimated aggregated standard deviations

• out.etp

the (d’1)—d—m array with estimated transition probabilities per period

• out.esd

the (d ’ 1) — d — m array with estimated standard deviations per period

98 4 Rating Migrations

• out.chi

the 3 — d matrix with χ2 -statistics, degrees of freedom and p-values

The matrices out.cagg, out.etpagg and out.etp have components given by

(4.10), (4.12) and (4.11). The elements of out.esdagg and out.esd result

by replacing pjk in (4.9) by p+ or pjk (t), respectively. The matrix out.chi

ˆ ˆjk ˆ

contains in the ¬rst row the statistics from (4.13) for j = 1, . . . , d ’ 1 and

(4.14). The second and third row gives the corresponding degrees of freedom

and p-values.

The quantlet XFGRatMig4.xpl (XFGRatMig4(etp, esd, etpagg)) graphs

migration rates per period with estimated standard deviations and migration

rates from pooled data. The inputs are:

• etp

the (d’1)—d—m array with estimated transition probabilities per period

• esd

the (d ’ 1) — d — m array with estimated standard deviations per period

• etpagg

the (d ’ 1) — d matrix with estimated aggregated transition probabilities

The output consists of (d ’ 1)d graphics for j = 1, . . . , d ’ 1 and k = 1, . . . , d.

Each graphic shows t = 1, . . . , m at the x-axis versus the four variables from

(4.15) at the y-axis.

4.2.4 Examples with Graphical Presentation

The following examples are based on transition matrices given by Nickell et al.

(2000, pp. 208, 213). The data set covers long-term bonds rated by Moody™s

in the period 1970“1997. Instead of the original matrices of type 8 — 9 we

use condensed matrices of type 3 — 4 by combining the original data in the

d = 4 basic rating categories A, B, C, and D, where D stands for the category

of defaulted credits.

The aggregated data for the full period from 1970 to 1997 are

® ®

21726 790 0 0 0.965 0.035 0 0

ˆ

C = ° 639 21484 139 421 » , P = ° 0.028 0.947 0.006 0.019 » ,

0 44 307 82 0 0.102 0.709 0.189

4.2 Analyzing the Time-Stability of Transition Probabilities 99

ˆ

where C is the matrix of migration counts and P is the corresponding matrix

of estimated transition probabilities. These matrices may be compared with

corresponding matrices for three alternative states of the business cycles:

® ®

7434 277 0 0 0.964 0.036 0 0

ˆ

° 273 7306 62 187 » , ° 0.035 0.933 0.008 0.024 » ,

C(1) = P(1) =

0 15 94 33 0 0.106 0.662 0.232

for the through of the business cycle,

® ®

7125 305 0 0 0.959 0.041 0 0

ˆ

C(2) = ° 177 6626 35 147 » , P(2) = ° 0.025 0.949 0.005 0.021 » ,

0 15 92 24 0 0.115 0.702 0.183

for the normal phase of the business cycle, and

® ®

7167 208 0 0 0.972 0.028 0 0

ˆ

° 189 7552 42 87 » , ° 0.024 0.960 0.005 0.011 » ,

C(3) = P(3) =

0 14 121 25 0 0.088 0.756 0.156

for the peak of the business cycle. The three categories depend on whether

real GDP growth in the country was in the upper, middle or lower third of the

growth rates recorded in the sample period (Nickell et al., 2000, Sec. 2.4).

In the following we use these matrices for illustrative purposes as if data from

m = 3 periods are given. Figure 4.1 gives a graphical presentation for d = 4

rating categories and m = 3 periods.

In order to illustrate the testing procedures presented in Section 4.2.2 in

the following the hypothesis is tested that the data from the three periods

came from the same theoretical transition probabilities. Clearly, from the

construction of the three periods we may expect, that the test rejects the null

hypothesis. The three χ2 -statistics with 6 = 3(3 ’ 1) degrees of freedom for

testing the equality of the rows of the transition matrices have p-values 0.994,

> 0.9999, and 0.303. Thus, the null hypothesis must be clearly rejected for

the ¬rst two rows at any usual level of con¬dence while the test for the last

row su¬ers from the limited sample size. Nevertheless, the χ2 -statistic for the

simultaneous test of the equality of the transition matrices has 18 = 32 · (3 ’ 1)

degrees of freedom and a p-value > 0.9999. Consequently, the null hypothesis

must be rejected at any usual level of con¬dence.

XFGRatMig3.xpl

ˆ

A second example is given by comparing the matrix P based on the whole data

ˆ

with the matrix P(2) based on the data of the normal phase of the business

100 4 Rating Migrations

25

1

1

25

20

0.5

0.5

20

0.95+Y*E-2

0.02+Y*E-2

15

Y

Y

0

0

15

10

-0.5

-0.5

10

5

-1

-1

5

1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Periods Periods Periods Periods

25

8

4

20

20

6

3

0.015+Y*E-2

0.002+Y*E-2

0.005+Y*E-2

0.92+Y*E-2

15

15

4

2

10

10

2

1

5

5

1 1.5 2 2.5 3

1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Periods Periods Periods Periods

1

25

25

15

0.5

20

20

0.55+Y*E-2

0.05+Y*E-2

Y*E-2

Y

15

0

15

10

10

-0.5

10

5

5

5

-1

1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Periods Periods Periods Periods

Figure 4.1. Example for XFGRatMig4.xpl

cycle. In this case a test possibly may not indicate that di¬erences between

P and P(2) are signi¬cant. Indeed, the χ2 -statistics for testing the equality

of the rows of the transition matrices with 3 degrees of freedom have p-values

0.85, 0.82, and 0.02. The statistic of the simultaneous test with 9 degrees of

freedom has a p-value of 0.69.

4.3 Multi-Period Transitions 101

4.3 Multi-Period Transitions

In the multi-period case, transitions in credit ratings are also characterized by

rating transition matrices. The m-period transition matrix is labeled P(m) .

(m)

Its generic element pjk gives the rating transition probability from rating

j to k over the m ≥ 1 periods. For the sake of simplicity the one-period

transition matrix P(1) is shortly denoted by P in the following. This transition

matrix is considered to be of type d — d. The last row contains (0, 0, . . . , 0, 1)

expressing the absorbing default state. Multi-period transition matrices can be

constructed from one-period transition matrices under the assumption of the

Markov property.

4.3.1 Time Homogeneous Markov Chain

Let {X(t)}t≥0 be a discrete-time stochastic process with countable state space.

It is called a ¬rst-order Markov chain if

P [(X(t + 1) = x(t + 1)|X(t) = x(t), . . . , X(0) = x(0)]

= P [X(t + 1) = x(t + 1)|X(t) = x(t)] (4.16)

whenever both sides are well-de¬ned. Further, the process is called a homoge-

neous ¬rst-order Markov chain if the right-hand side of (4.16) is independent

of t (Br´maud, 1999).

e

Transferred to rating transitions, homogeneity and the Markov property imply

constant one-period transition matrices P independent of the time t, i. e. P

obeys time-stability. Then the one-period d — d transition matrix P contains

the non-negative rating transition probabilities

pjk = P(X(t + 1) = k|X(t) = j).

They ful¬ll the conditions

d

pjk = 1

k=1

and

(pd1 , pd2 , . . . , pdd ) = (0, . . . , 0, 1).

The latter re¬‚ects the absorbing boundary of the transition matrix P.

102 4 Rating Migrations

The two-period transition matrix is then calculated by ordinary matrix mul-

tiplication, P(2) = PP. Qualitatively, the composition of the portfolio after

one period undergoes the same transitions again. Extended for m periods this

reads as

P(m) = P(m’1) P = Pm

with non-negative elements

d

(m) (m’1)

pjk = pji pik .

i=1

The recursive scheme can also be applied for non-homogeneous transitions, i.e.

for one-period transition matrices being not equal, which is the general case.

4.3.2 Bootstrapping Markov Chains

The one-period transition matrix P is unknown and must be estimated. The

ˆ

estimator P is associated with estimation errors which consequently in¬‚uence

the estimated multi-period transition matrices. The traditional approach to

quantify this in¬‚uence turns out to be tedious since it is di¬cult to obtain

ˆ

the distribution of (P ’ P), which could characterize the estimation errors.

ˆ (m) ’ P(m) ), with

Furthermore, the distribution of (P

ˆ (m) def Pm ,

=ˆ

P (4.17)

has to be discussed in order to address the sensitivity of the estimated tran-

sition matrix in the multi-period case. It might be more promising to apply

resampling methods like the bootstrap combined with Monte Carlo sampling.

For a representative review of resampling techniques see Efron and Tibshirani

(1993) and Shao and Tu (1995), for bootstrapping Markov chains see Athreya

and Fuh (1992) and H¨rdle, Horowitz, and Kreiss (2001).

a

Assuming a homogeneous ¬rst-order Markov chain {X(t)}t≥0 , the rating tran-

sitions are generated from the unknown transition matrix P. In the spirit of

the bootstrap method, the unknown transition matrix P is substituted by the

ˆ

estimated transition matrix P, containing transition rates. This then allows to

draw a bootstrap sample from the multinomial distribution assuming indepen-

dent rating migrations,

(˜— , . . . , c— ) ∼ Mult(nj ; pj1 , . . . , pjd ),

cj1 ˜jd ˆ ˆ (4.18)

4.3 Multi-Period Transitions 103

for all initial rating categories j = 1, . . . , d ’ 1. Here, c— denotes the bootstrap

˜jk

random variable of migration counts from j to k in one period and pjk is the ˆ

estimated one-period transition probability (transition rate) from j to k.

Then the bootstrap sample {c— }j=1,...,d’1,k=1,...,d is used to estimate a boot-

jk

—

ˆ

strap transition matrix P with generic elements p— according

ˆjk

c—

jk

p—

ˆjk = . (4.19)

nj

Obviously, defaulted credits can not upgrade. Therefore, the bootstrap is not

ˆ—

necessary for obtaining the last row of P , which is (ˆ— , . . . , p— ) = (0, . . . , 0, 1).

pd1 ˆdd

Then matrix multiplication gives the m-period transition matrix estimated

from the bootstrap sample,

ˆ —(m) = P—m ,

ˆ

P

—(m)

with generic elements pjk .

ˆ

ˆ —(m) by Monte Carlo sampling, e. g. B

We can now access the distribution of P

ˆ —(m) for b = 1, . . . , B. Then the distribution of

samples are drawn and labeled P b

—(m)

ˆ (m)

ˆ

P estimates the distribution of P . This is justi¬ed since the consistency

of this bootstrap estimator has been proven by Basawa, Green, McCormick,

ˆ —(m) , the

and Taylor (1990). In order to characterize the distribution of P

—(m) (m)

standard deviation Std pjk

ˆ which is the bootstrap estimator of Std pjk

ˆ ,

is estimated by

B 2

1

—(m) —(m) ˆ ˆ—(m)

pjk,b ’ E pjk

Std pjk

ˆ = ˆ (4.20)

B’1

b=1

with

B

ˆ ˆ—(m) = 1 —(m)

E pjk pjk,b

ˆ

B

b=1

—(m)

for all j = 1, . . . , d ’ 1 and k = 1, . . . , d. Here, pjk,b is the generic element of

ˆ

ˆ —(m)

the b-th m-period bootstrap sample Pb . So (4.20) estimates the unknown

(m)

standard deviation of the m-period transition rate Std pjk

ˆ using B Monte

Carlo samples.

104 4 Rating Migrations

4.3.3 Computation and Quantlets

For time homogeneity, the m-period rating transition matrices are obtained

by the quantlet XFGRatMig5.xpl (q = XFGRatMig5(p, m)). It computes all

t = 1, 2, . . . , m multi-period transition matrices given the one-period d—d matrix

p. Note that the output q is a d — d — m array, which can be directly visualized

by XFGRatMig6.xpl (XFGRatMig6(q)) returning a graphical output. To vi-

sualize t-period transition matrices each with d2 elements for t = 1, . . . , m, we

plot d2 aggregated values

k

(t)

j’1+ pjl , j, k = 1, . . . , d (4.21)

l=1

for all t = 1, . . . , m periods simultaneously.

A typical example is shown in Figure 4.2 for the one-year transition matrix

given in Nickell et al. (2000, p. 208), which uses Moody™s unsecured bond

ratings between 31/12/1970 and 31/12/1997. According (4.21), aggregated

values are plotted for t = 1, . . . , 10. Thereby, the transition matrix is condensed

for simplicity to 4 — 4 with only 4 basic rating categories, see the example in

Section 4.2.4. Again, the last category stands for defaulted credits. Estimation

errors are neglected in Figure 4.2.

out = VaRRatMigRateM (counts, m, B)

bootstraps m-period transition probabilities

Bootstrapping is performed by the quantlet VaRRatMigRateM. It takes as input

counts, the (d ’ 1) — d matrix of migration counts, from which the bootstrap

sample is generated. Further, m denotes the number of periods and B the

number of generated bootstrap samples. The result is assigned to the variable

out, which is a list of the following output:

• out.btm

the (d’1)—d—B array of bootstrapped m-period transition probabilities

• out.etm

the (d ’ 1) — d matrix of m-period transition rates

• out.stm

the (d ’ 1) — d matrix of estimated standard deviations of the m-period

transition rates

4.3 Multi-Period Transitions 105

4.5

4

3.5

3

Aggregations

2.5

2

1.5

1

0.5

0

2 4 6 8 10

Periods

Figure 4.2. Example for XFGRatMig6.xpl:

Aggregated values of multi-period transition matrices.

The components of the matrices out.btm are calculated according (4.18) and

(4.19). The matrices out.etm and out.stm have components given by (4.17)

and (4.20).

106 4 Rating Migrations

To k

From j 1 2 3 4 5 6 Default nj

1 0.51 0.40 0.09 0.00 0.00 0.00 0.00 35

2 0.08 0.62 0.19 0.08 0.02 0.01 0.00 103

3 0.00 0.08 0.69 0.17 0.06 0.00 0.00 226

4 0.01 0.01 0.10 0.64 0.21 0.03 0.00 222

5 0.00 0.01 0.02 0.19 0.66 0.12 0.00 137

6 0.00 0.00 0.00 0.02 0.16 0.70 0.12 58

Default 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0

Table 4.1. German rating transition matrix (d = 7) and the number of

migrations starting from rating j = 1, . . . , d

4.3.4 Rating Transitions of German Bank Borrowers

In the following the bootstrapping is illustrated in an example. As estimator

ˆ

P we use the 7 — 7 rating transition matrix of small and medium-sized German

bank borrowers from Machauer and Weber (1998, p. 1375), shown in Table 4.1.

The data cover the period from January 1992 to December 1996.

With the quantlet VaRRatMigRateM the m-period transition probabilities are

(m)

estimated by pjk and the bootstrap estimators of their standard deviations

ˆ

are calculated. This calculations are done for 1, 5 and 10 periods and B = 1000

Monte Carlo steps. A part of the resulting output is summarized in Table

4.2, only default probabilities are considered. Note that the probabilities in

Table 4.1 are rounded and the following computations are based on integer

migration counts cjk ≈ nj pjk .

XFGRatMig7.xpl

4.3.5 Portfolio Migration

Based on the techniques presented in the last sections we can now tackle the

problem of portfolio migration, i. e. we can assess the distribution of n(t) credits

over the d rating categories and its evolution over periods t ∈ {1, . . . m}. Here,

a stationary transition matrix P is assumed. The randomly changing number

of credits in category j at time t is labeled by nj (t) and allows to de¬ne non-

˜

4.3 Multi-Period Transitions 107

—(1) —(5) —(10)

(1) (5) (10)

From j pjd

ˆ Std pjd

ˆ pjd

ˆ Std pjd

ˆ pjd

ˆ Std pjd

ˆ

1 0.00 0.000 0.004 0.003 0.037 0.015

2 0.00 0.000 0.011 0.007 0.057 0.022

3 0.00 0.000 0.012 0.005 0.070 0.025

4 0.00 0.000 0.038 0.015 0.122 0.041

5 0.00 0.000 0.079 0.031 0.181 0.061

6 0.12 0.042 0.354 0.106 0.465 0.123

Table 4.2. Estimated m-period default probabilities and the bootstrap

estimator of their standard deviations for m = 1, 5, 10 periods

negative portfolio weights

nj (t)

˜

def

wj (t) =

˜ , j = 1, . . . , d,

n(t)

which are also random variables. They can be related to migration counts

cjk (t) of period t by

˜

d

1

wk (t + 1) =

˜ cjk (t)

˜ (4.22)

n(t) j=1

counting all migrations going from any category to the rating category k. Given

the weights wj (t) = wj (t) at t, the migration counts cjk (t) are binomially

˜ ˜

distributed

cjk (t)|wj (t) = wj (t) ∼ B (n(t) wj (t), pjk ) .

˜ ˜ (4.23)

The non-negative weights are aggregated in a row vector

w(t) = (w1 (t), . . . , wd (t))

˜ ˜ ˜