Risk Management

Topic Two – Default correlation and industrial correlation models

2.1 Mixture models for modeling default correlation

2.2 CreditRisk+

2.3 CreditMetrics and copula functions

1

2.1 Mixture models for modeling default correlation

Why are we concerned about dependence between default events and/or

credit quality changes?

• It affects the distribution of loan portfolio losses – critical in deter-

mining quantities or other risk measures used for allocating capital for

solvency purposes.

• Clustering phenomena: simultaneous defaults could affect the stability

of the financial system with profound effects on the entire economy.

The Occurrence of disproportionately many joint defaults is termed

the “extreme credit risk”.

2

Difficulties

• Parameter specification for default dependence models is often some-

what arbitrary.

Issues

• The “marginal” credit risk of each issuer in a pool is usually “well”

determined. Modeling various correlation structures that work with

the given marginal characteristics is the major challenge.

• Counterparty risk: The counterparty of a financial contract cannot

honor the contractual specification. For example, in a credit default

swap, the counterparty risk is high when the credit quality of the

protection seller is correlated with that of the underlying reference

securities.

3

Bernuolli mixture model

• The default indicator variable 1{default} equals 0 or 1, corresponding

to “no default”or “default” of an obligor, respectively. This is a

discrete Bernuolli random variable with only two possible values.

• We describe the loss of a portfolio from a loss statistics L = (L1, · · · , Lm)

with Bernuolli variables Li ∼ B(1;Pi). Here, B(m; p) denotes the bino-

mial distribution with m independent trials and stationary probability

of success p. More generally, we allow the loss probabilities (over a

given time horizon) to be a vector of random variables

P = (P1, · · · , Pm) ∼ F

for some distribution function F with support in [0,1]m.

• In a mixture model, the default probability Pi of obligor i is assumed

to depend on a set of common economic factors. This is how default

correlation among the obligors arise.

4

Conditional independence

Two events R and B are said to be conditionally independent given the

third event G if and only if

P [R ∩B|G] = P [R|G]P [B|G].

The last relation can be replaced by the following equivalent relation:

P [R|B ∩G] = P [R|G].

That is, given that G occurs, the conditional probability that R occurs is

unchanged by whether or not B occurs.

5

Two random variables X and Y are conditionally independent given an

event G if they are independent in their conditional probability distribution

given G. In terms of density functions:

fX(x|G)fY (y|G) = fX,Y (x, y|G),

or in terms of distribution functions:

FX(x|G)FY (y|G) = FX,Y (x, y|G).

We say that X and Y are statistically independent given G. In short hand,

we write

X ⊥ Y |G.

6

In the context of the mixture model, we assume that conditional on the

realization P = (P1, · · · , Pm) of the vector of loss probabilities (random)

P , the Bernuolli default indicator variables L1, · · · , Lm are independent.

That is, given the default probabilities, defaults of different obligors are

independent.

The (unconditional) joint distribution of Li’s is obtained by integrating

over the probability distribution of F :

P[L1 = ℓ1, · · · , Lm = ℓm] =∫[0,1]m

m∏i=1

Pℓii (1− Pi)

1−ℓi dF (P1, · · · , Pm),

where ℓi ∈ {0,1}. Here “0” denotes “no default” and “1” denotes “de-

fault”. For example,

P[L1 = 1, L2 = 0, L3 = 1] =∫[0,1]3

P1(1− P2)P3 dF (P1, P2, P3).

7

First and second order moments of Li The first and second order mo-

ments of the single loss indicator variable Li are computed as follows.

Conditional on the random probability P , we have

E[Li|P ] = 1× Pi +0× (1− Pi) = Pi,

By the tower rule in conditional expectation, we obtain

E[Li] = E[E[Li|P ]] = E[Pi].

The dependence between defaults stems from the dependence of the

default probabilities on a set of common random factors.

8

Recall: E[L2i |P ] = 12 × Pi +02 × (1− Pi) = Pi. Also,

E[LiLj] = E[E[LiLj|P ]] = E[PiPj].

By the conditional variance formula, we obtain

var(Li) = var(E[Li|P ]) + E[var(Li|P )]

= var(Pi) + E[E[L2i |P ]− E[Li|P ]2]

= var(Pi) + E[Pi(1− Pi)] = E[Pi](1− E[Pi]).

The covariance between a pair of losses

cov(Li, Lj) = E[LiLj]− E[Li]E[Lj].

9

Note that

E[LiLj] = P(Li = 1, Lj = 1)× 1× 1+ P(Li = 1, Lj = 0)× 1× 0

+ P(Li = 0, Lj = 1)× 0× 1+ P(Li = 0, Lj = 0)× 0× 0

= P(Li = 1, Lj = 1)

=∫ 1

0

∫ 1

0PiPj dF (Pi, Pj) by conditional independence

= E[PiPj].

Hence,

cov(Li, Lj) = E[PiPj]− E[Pi]E[Pj] = cov(Pi, Pj),

so that the default correlation in a Bernuolli mixture model is

corr(Li, Lj) =cov(Pi, Pj)√

E[Pi](1− E[Pi])√E[Pj](1− E[Pj])

.

The covariance between a pair of losses in the portfolio is fully captured

by the covariance structure of the multivariate distribution F of the vector

of random loss probabilities P .

10

Correlation of defaults of a pair of risky assets

Consider two obligors A and B and a fixed time horizon T .

Define 1{A} =

{1 A defaults before T0 otherwise

and similar definition for 1{B}.

pA = probability of default of A before T = E[1{A}]

pB = probability of default of B before T = E[1{B}]

pAB = joint default probability that A and B default before T

pA|B = probability that A defaults before T , given that B has

defaulted before T

pA|B =pAB

pB, pB|A =

pAB

pA, and recall

ρAB = linear correlation coefficient between default events

=E[1{A}1{B}]− E[1{A}]E[1{B}]

σ1{A}σ1{B}

=pAB − pApB√

pA(1− pA)pB(1− pB).

11

Note that pAB can be expressed in terms of pA, pB and ρAB. When there

are 2 obligors, we can compute the probabilities of all elementary events

by using ρAB. Since default probabilities are very small, the correlation

ρAB can have a much larger effect on the joint risk of a position. Observe

that

pAB = pApB + ρAB

√pA(1− pA)pB(1− pB)

pA|B = pA + ρAB

√pApB

(1− pA)(1− pB) and

pB|A = pB + ρAB

√pBpA

(1− pA)(1− pB).

Write ρAB = ρ, which is assumed to be not small; and let pA = pB = p ≪1, then

pAB ≈ p2 + ρp ≈ ρp

pA|B ≈ ρ.

The joint default probability and the conditional default probability are

dominated by the correlation coefficient ρ.

12

Limitations of pairwise correlation coefficient of default events

• With 3 obligors, we have 8 elementary events but only 7 restrictions (3

individual probabilities, 3 correlations and sum of probabilities). The

probability of the joint default of all 3 obligors cannot be determined

by the 3 pairs of correlation.

• For N obligors, we have N(N −1)/2 correlations, N individual default

probabilities. Yet we have 2N possible joint default events. The

correlation matrix only gives the bivariate marginal distributions while

the full distribution remains undetermined.

• For any pair of random variables X and Y , we have

cov(1−X,1− Y ) = cov(X,Y )

so that the linear correlation coefficient between survival events is the

same as that between default events. Hence, the covariance measure

is unable to capture the fact that risky assets exhibit greater tendency

to crash together than to boom together.

13

One-factor Bernuolli mixture model

Retail banking portfolios and portfolios of smaller banks are often quite

homogeneous. Assuming Li ∼ B(1; p) with a common single random

default probability p ∼ F , where F is a distribution function with support

in [0,1]. As the mixture distribution is dependent on the single distribution

F (p), this leads to the one-factor Bernuolli mixture model.

The joint distribution of the loss variables Li’s:

P[L1 = ℓ1, · · · , Lm = ℓm] =∫ 1

0pk(1− p)m−k dF (p)

where k =m∑

i=1

ℓi and ℓi ∈ {0,1}, i = 1,2, · · · ,m. Here, k counts the number

of defaults among m obligors in the portfolio.

14

• We write L as the random number of defaults. Since there are Cmk

possible combinations of defaults among the m obligors that lead to

a total of k defaults, the probability that exactly k defaults occur is

P[L = k] = Cmk

∫ 1

0pk(1− p)m−k dF (p).

This is the mixture of the binomial probabilities with the mixing dis-

tribution F .

• The uniform default probability of any obligor in the homogenous

portfolio is given by

p = P[Li = 1] = E[Li] =∫ 1

0p dF (p).

15

• Note that E[LiLj] = P[Li = 1, Lj = 1]. The uniform default correla-

tion of a pair of obligors is

ρ = corr(Li, Lj) =P[Li = 1, Lj = 1]− p2

p(1− p)

=

∫ 10 p2 dF (p)− p2

p(1− p)=

var(p)

p(1− p).

• Intuitively, with a higher var(p), we have a higher corr(Li, Lj).

• Recall that the variance of the Bernuolli variable that assumes value

either 0 or 1, and has parameter p is

E[X2]− E[X]2 = p− p2 = p(1− p).

Note that ρ = 1 if and only if var(p) = p(1 − p). In order that this

occurs, the common random default probability p must be Bernuolli

with parameter p.

16

Remarks

1. Since var(p) ≥ 0, so corr(Li, Lj) ≥ 0. The non-negativity of default

correlation is obvious since Li and Lj are dependent on the common

mixture variable p. In other words, we cannot implement negative

dependencies between the default events of obligors under this one-

factor binomial mixture model.

2. corr(Li, Lj) = 0 if and only if var(p) = 0, implying no randomness with

regard to p. In this case, p assumes the single value p. The random

portfolio loss L follows a binomial distribution with constant default

probability p. Correspondingly, the default events are independent.

That is, corr(Li, Lj) = 0 ⇒ independence of default events. The

converse statement is obvious since independence of Li and Lj implies

corr(Li, Lj) = 0. We then have corr(Li, Lj) = 0 ⇔ independence of

default events.

17

3. corr(Li, Lj) = 1 implies a “rigid” behavior of losses in the portfolio.

This corresponds to p = 1 with probability p and p = 0 with probability

1− p, where the distribution F of the random default probability p is

a Bernoulli distribution. Financially speaking, when an external event

occurs with probability p, all obligors in the portfolio default and the

total portfolio is lost. Otherwise, with probability 1 − p, all obligors

survive.

• A non-financial analogy is the death events of all passengers in an

aeroplane, either there is no crash with probability 1−p (all passengers

survive) or an accident occurs with probability p (all passengers die).

18

Fractional losses under the one-factor binomial mixture model

Define Dn =∑n

i=1Li, which is the total number of defaults in the port-

folio. We then have

E[Dn] =n∑

i=1

E[Li] = np.

Recall the following formulas:

var(Li) = E[Pi](1− E[Pi]) = p(1− p).

var(Dn) =n∑

i=1

var(Li) +n∑

i=1

n∑j = 1j = i

cov(Li, Lj).

Under the assumption of a common random default probability, we have

var(Dn) = np(1− p) + n(n− 1)(E[p2]− E[p]2).

19

Therefore, we have

var(Dn

n

)=

p(1− p)

n+

n(n− 1)

n2var(p) −→ var(p) as n → ∞.

When considering the fractional loss for n large, the only remaining vari-

ance is that of the distribution of p.

• One can obtain any default correlation in [0,1] through appropriate

choices of var(p) and E[p]. Note that the correlation of a pair default

of events depends only on the first and second order moments of F .

However, the distribution of Dn can be quite different for different

distribution F .

20

Large portfolio approximation

We perform the sampling of default events among a population of n

obligors that share a common random default probability p. By the law

of large numbers, the sample mean converges to the population mean.

In the current context, we have

Dn

n→ p as n → ∞

where p denotes the realized default probability. The conditional proba-

bility distribution of Dnn is given by

P[Dn

n≤ θ

∣∣∣∣∣p = p

]n↑∞−→

{0 if θ < p1 if θ ≥ p

.

21

The fractional loss distribution is obtained by integrating over the distri-

bution of p, that is,

P[Dn

n≤ θ

]n↑∞−→

∫ 1

01{θ≥p}f(p) dp

=∫ θ

0f(p) dp = F (θ).

As n → ∞, it is the probability distribution of the random default proba-

bility p that determines the fractional loss distribution.

22

Example – beta distribution for the common random default probability

A beta distribution for p gives us a flexible class of distributions in the

interval [0,1]. The beta distribution is characterized by the two positive

parameters α, β. The mean and variance of the distribution are

E[p] =α

α+ β, var(p) =

αβ

(α+ β)2(α+ β +1).

If we look at various combinations of the two parameters for which αα+β =

p for a given level of expected default probability p, the variance of the

distribution decreases as we increase α [see Hw 3, Qn 3(b)].

It is the common dependence on the mixture variable p that induces

the correlation in the default events. The more variability that is in the

mixture distribution, the stronger correlation of default events and more

weight there in the tails of the loss distribution.

23

Two beta distributions are shown, both with the common mean 0.1 but

with different variances. The density corresponds to α = 1, β = 9 (the

curve that always slopes downwards) observes a higher probability of see-

ing smaller losses and larger losses, which then gives a higher variance in

the default-event correlation.

24

Comparison with the case of independence of defaults

• The binomial distribution for independent defaults has a very thin

tail, thus not representing the possibility of a large number of defaults

realistically.

• Taking N = 100 obligors

Default Prob. (%) 1 2 3 4 5 6 7 8 9 1099.0% VaR Level 5 7 9 11 13 14 16 17 19 20

What is VaRα(X)? It is the maximum loss which is not exceeded with

a given probability (or confidence level α).

V aRα(X) = inf{x ≥ 0|P [X ≤ x] ≥ α}.

Take p = 5%, the probability with 13 defaults or less is at least

99%, that is, 99% confidence level. VaRα(X) is seen to increase with

default probability.

25

The distribution of the number of defaults among 50 issuers in the case

of a pure binomial model with default probability 0.1 and in cases with

beta distributions (α = 1, β = 9) as mixture distributions over the default

probability. The later case exhibits a higher probability of seeing a large

number of default losses (commonly known as the thick tail) and a small

number of losses also.

26

Moody’s binomial expansion method

• For a binomial distribution with independent obligors, the tail with

fewer (smaller value of n) obligors is “fatter” than the tail with many

(larger value of n) independent obligors. Actually, when n → ∞, the

variance of fractional loss decreases as O(1n

)since var

(Dnn

)= p(1−p)

n .

• Moody is aware that a pure binomial distribution with independent

defaults is unrealistic. They make the tails of distribution fatter by

assuming a smaller number of obligors (the diversity score). For

example, adjustment is made for industry concentration

The idea is to approximate the loss on a portfolio of n positively correlated

loans with the loss on a smaller number of independent loans with larger

face value.

27

Moody’s binomial expansion technique (BET)

The two parameters in a binomial experiment are number of trials n and

probability of success p.

• Diversity score, weighted average rating factor (default probability)

and binomial expansion technique.

• Generate the loss distribution.

To build a hypothetical pool of uncorrelated and homogeneous assets

that mimic the default behavior of the original pool of correlated and

inhomogeneous assets.

28

Criteria of an idealized comparison portfolio

The diversity score of a given pool of participations is the number n

of bonds in an idealized comparison portfolio that meets the following

criteria:

• Comparison portfolio and collateral pool have the same face value.

• Bonds in the comparison portfolio have equal face values.

• Comparison bonds are equally likely to default, and their defaults are

independent.

• Comparison bonds are of the same average default probability as that

of the participations of the collateral pool.

• According to some measure of risk (say variance of the loss principal),

the comparison portfolio has the same total risk as does the collateral

pool.

29

Binomial approximation using diversity scores

Seek the reduction of the problem of generating the distribution of mul-

tiple defaults to binomial distributions.

If the n loans each with equal face value are independent and they have

the same default probability, then the distribution of the portfolio loss is

a binomial distribution with n as the number of trials.

Let Fi be the face value of each bond, pi be the probability of default with-

in the relevant time horizon, and ρij be the linear correlation coefficient

of default events.

Assuming zero recovery, the loss variable Li (in dollar amount) associated

with bond i with face value Fi is given by

Li = Fi1{Di},

where Di is the default event of bond i.

30

With n bonds, the total notional principal of the portfolio isn∑

i=1

Fi. The

mean and variance of the loss principal P (in dollar amount) is

E[P ] = E[L1 + · · ·+ Ln] =n∑

i=1

FiE[1{Di}] =n∑

i=1

piFi

var(P ) =n∑

i=1

n∑j=1

E[LiLj]− E[Li]E[Lj]

=n∑

i=1

n∑j=1

FiFj(E[1{Di}1{Dj}]− E[1{Dj}]E[1{Dj}])

=n∑

i=1

n∑j=1

FiFjρij√pi(1− pi)pj(1− pj).

Here, ρij is specified rather than as a quantity calculated based on the

information on joint defaults.

31

• We construct an approximating portfolio consisting D independent

loans, each with the same face value F and the same default proba-

bility p.

To determine the binomial distribution of the loss amount of the com-

parison portfolio, we need to specify the constant default probability p,

number of obligors D and the common face value of the bonds. These

lead to the following system of equations:

n∑i=1

Fi = DF

E[P ] =n∑

i=1

piFi = DFp

var(P ) = F2Dp(1− p).

32

Note that

FD(1− p) =n∑

i=1

(1− pi)pi

and

D var(P ) = FDp[FD(1− p)].

Solving the equations, we obtain

p =

∑ni=1 piFi∑ni=1 Fi

D =

∑ni=1 piFi

∑ni=1(1− pi)Fi∑n

i=1∑n

j=1 FiFjρij√pi(1− pi)pj(1− pj)

F =n∑

i=1

Fi

/D.

Here, D is called the diversity score.

33

Contagion models

Contagion means that once a firm defaults, it may bring down other

firms with it. Define Yij to be an “infection” variable. Both Xi and Yijare Bernuolli variables assuming values either 0 or 1.

Xi is the default indicator of firm i due to its firm specific causes

Yij =

{1 if default of firm i brings down firm j0 if default of firm i does not bring down firm j

.

Assuming homogeneous property on the parameters:

E[Xi] = p and E[Yij] = q.

Also, Xi and Yij, for all i and j, are assumed to be independent.

34

The default indicator of firm i is

Zi = Xi + (1−Xi)

1−∏j =i

(1−XjYji)

.Note that Zi equals one either when there is a direct default of firm i

or if there is no direct default and∏j =i

(1 − XjYji) = 0. The latter case

occurs when at least one of the factor XjYji is 1, which happens when

firm j defaults and infects firm i. Define Dn = Z1 + · · · + Zn, Davis and

Lo (2001) find that [see Qn (5) in Hw 3]

E[Dn] = n[1− (1− p)(1− pq)n−1]

var(Dn) = n(n− 1)βpqn − (E[Dn])

2

cov(Zi, Zj) = βpqn − (E[Dn/n])

2,

where

βpqn = p2 +2p(1− p)[1− (1− q)(1− pq)n−2]

+ (1− p)2[1− 2(1− pq)n−2 + (1− 2pq + pq2)n−2].

35

1. When there is no infection, q = 0, so zero contagion gives a pure

binomial model. In this case, E[Dn] becomes np, which is the same

as that of the binomial distribution.

2. Increasing the contagion brings more mass to high and low default

numbers. To preserve the mean, we must compensate for an increase

in the infection parameter by decreasing the probability of direct de-

fault. Note that

E[Zi] = E

1− (1−Xi)∏j =i

(1−XjYji)

= 1− (1− p)(1− pq)n−1.

By equating the probability of no default with and without infection

effect, we find p(q) such that

[1− p(q)][1− p(q)q]n−1 = 1− p.

We have p(0) = p and p(q) < p for q > 0. While the mean is preserved,

the variance is increased.

36

Probability mass function of Dn

Let F (k;n, p, q) denote the probability mass function of Dn, where

F (k;n, p, q) = P[Dn = k],

then F (k;n, p, q) = Cnkα

pqnk, where

αpqnk = pk(1− p)n−k(1− q)k(n−k)

+k−1∑i=1

Cki p

i(1− p)n−i[1− (1− q)i]k−i(1− q)i(n−k).

• It is necessary to consider separately the special case where all of the

k defaulting bonds do not infect any other non-defaulting bonds in

the portfolio. This is captured by the first term in αpqnk.

37

Recall the following results:

• (1− q)i = probability that all i self-defaulting bonds do not

affect a particular non-defaulting bond• 1−(1−q)i = probability that at least one of the self-defaulting

bonds affect a particular non-defaulting bond

• The k bonds that are defaulting can be chosen in Cnk combinations.

Write αpgnk as the probability that out of the n bonds, k (≤ n) particular

bonds default. Consider the two cases:

(i) All these k bonds are self-defaulting and they do not infect any of the

remaining n− k bonds;

(ii) Of the k bonds defaulting, i of them are self-defaulting while k − i

of them are infected by the first of these i self-defaulting bonds,

i = 1,2, · · · , k − 1.

38

• With k bonds that are defaulting, i of them are self-defaulting and

k− i of them are infected by the first i of these self-defaulting bonds.

For i = 1,2, · · · , k − 1, consider

0 0 0 0 0 0 0 0 0 0 0 0

first k bonds default

i of these

defaulting bonds

are self-defaulting

k - i bonds

defaulted since

they are

infected by

the first i

self-defaulting bonds

n – k bonds remain non-defaulting

Probability of occurrence for given value of i is pi(1 − p)n−i[1 − (1 −q)i]k−i(1− q)i(n−k). There are Ck

i combinations to choose these i bonds

from k bonds, and subsequently, we sum from i = 1 to i = k − 1.

39

2.2 CreditRisk+

This is an industrial code that applies actuarial mathematics to calculate

the loss distribution of a portfolio

• It requires a limited amount of input data and assumptions. It uses

as basic input the same data as required by the Basel II internal rating

system.

A loan is understood as a Bernuolli random variable (default or no

default over a given time horizon). No profits or losses from rating

migrations are considered.

40

• It provides an analytic-based portfolio approach for rapid and unam-

biguous calculations of the loss distribution (no simulations are need-

ed). Efficient numerical techniques, like the Fast Fourier transform,

and analytic approximation method, like the saddle point approxima-

tion method, can be applied to hasten the computation.

– Unambiguity of the distribution is important (in particular, infor-

mation on the tail of the distribution) since simulation based port-

folio approaches usually fail to give agreeing answers (due to low

probabilities of default).

– Rapid calculations are helpful when comparative statistics (“what

if” analyzes) are performed: performing a large number of scenario

tests under different scenarios for the parameters.

The resulting portfolio loss distribution can be expressed as a sum of

independent compound negative binomial random variables.

41

Limitations

• CreditRisk+ takes the Poisson approximation to the Bernuolli type

default events, which implicitly allow multiple defaults. This is a rea-

sonable approximation, given that the expected default probabilities

are small. Though highly unlikely, possible paradoxes may be created,

say, calculated losses > sum of exposures since multiple defaults of

single obligor is allowed.

• Limited range of default correlation: produces only positive and usu-

ally moderate levels of default correlation among 2 obligors sharing a

common risk driver.

• Recovery rates are absorbed in the exposure upon default and they

are assumed to be independent of default events.

42

Fundamentals of CreditRisk+ – mixture Poisson distribution

• No financial modeling for the default event: The reason for default

of an obligor is not the subject of the model. Instead the default is

described as a purely random event, characterized by a probability of

default.

• Stochastic probability of default: The probability of default of an

obligor is not seen as a constant, but a randomly varying quantity,

driven by one or more (systematic) risk factors. This feature is used

to generate default correlation among the obligors. The distribution

of the default intensity is usually assumed to be a gamma distribution.

43

• Conditional independence: Given the risk factors, the default of oblig-

ors are independent.

• Only implicit correlation via the risk drivers: Default correlation be-

tween obligors are not explicit, but arise only implicitly due to the

common risk factors which drive the probability of defaults. A lin-

ear relationship between the systematic risk factors and the default

probabilities is assumed.

• Using exposure bands: Group the exposures in the portfolio into var-

ious bands. Replace each exposure amount LA of obligor A by the

nearest integer multiple of L, where L is the basic unit of exposure.

The distribution of losses is derived from the probability generating func-

tions, assuming that the losses (either as the count of defaulting obligors

or dollar amount) take discrete values.

44

Probability generating functions GK(z)

The probability generating function (pgf) of a discrete non-negative integer-

valued random variable K is a function of the auxiliary variable z such that

the probability that K = k is given by the coefficient of zk in the polyno-

mial expansion of the probability generating function. We have

GK(z) = E[zK] =∞∑

k=0

P [K = k]zk,

where z can be complex with |z| ≤ 1 (technical condition that guarantees

convergence of the infinite series).

The pgf of the Poisson random variable N with parameter α is

GN(z) = eα(z−1) = e−α∞∑

n=0

(αz)n

n!,

which observes the following probability mass function of N :

P [N = n] =e−ααn

n!.

45

For a Bernuolli random variable Y with probability p, the corresponding

pgf is

GY (z) = (1− p)z0 + pz = 1+ p(z − 1).

The pgf contains all the information to calculate the probability mass

function of the associated non-negative integer valued random variable

N , where

P [N = n] =1

n!G(n)N (0).

Conditional pgf: GN(z|·) = E[zN |·].

Given GN(z|x) where x has distribution F (x), then the (unconditional)

pgf is given by

GN(z) =∫

GN(z|x) dF (x).

46

More properties on pgf

• Let K1 and K2 be a pair of independent non-negative integer-valued

random variables. The pgf of the sum K1 +K2 is simply the product

of the two pgf’s since E[zK1+K2] = E[zK1]E[zK2]. This stems from

the property that for any given pair of independent random variables

X1 and X2, we have

E[f1(X1)f2(X2)] = E[f1(X1)]E[f2(X2)]

for any functions f1 and f2. Here, we choose f1 and f2 to be expo-

nential functions.

• GnY (z) = GY (zn) for any natural number n.

For example, take n = 3, we have

G3Y (z) = P [3Y = 0]z0 + P [3Y = 3]z3 + P [3Y = 6]z6 + · · ·= P [Y = 0]z0 + P [Y = 1]z3 + P [Y = 2]z6 + · · ·

=∞∑

k=0

P [Y = k]z3k.

47

Input data for the model

• For an obligor A in a loan portfolio, we designate PA to be the ex-

pected probability of default of A (say, from the output of a rating

process).

• Write vA as the potential (dollar) loss for A upon default. The ex-

pected (dollar) loss for A is given by

ϵA = PAvA.

• To work with discretized losses, we fix a loss unit L and choose a

positive integer vA as a rounded version of vA/L. To compensate for

the error due to rounding, we adjust the expected default probability

PA =ϵAvAL

=

(vAvAL

)PA.

We then work with PA and the integer-valued loss vA.

48

For example, suppose we take vA = $160,000 and PA = 0.2%, then

ϵA = $160,000× 0.2% = $320.

We take L = $100,000 so that the rounded exposure in unit of L is 2.

The adjusted expected default probability is

PA =$160,000× 0.2%

2× $100,000= 0.16%.

49

Example Rounding of exposure to units of L

Obligor A Exposure ($) Exposure Round-off Band j(potential loss (in $100,000) exposuregiven default) (in $100,000)

vA vA/L vA1 150,000 1.5 2 22 460,000 4.6 5 53 435,000 4.35∗ 5 54 370,000 3.7 4 45 190,000 1.9 2 26 480,000 4.8 5 5

Out of the 6 obligors, 2 to Band 2, 1 to Band 4, and 3 to Band 5.

∗ Rounding up to the nearest integer is adopted so that the round-off

exposure as an integer multiple of L always starts at 1 for the sake of

convenience.

50

Allocation of 350 obligors into 6 bands

vj number of obligors ϵj µj1 30 1.5 1.52 40 8 43 50 6 24 70 25.2 6.35 100 35 76 60 14.4 2.4

vj = common exposure in band j in units of L

ϵj = expected dollar loss in band j in units of L

µj = expected number of defaults in band j

We have ϵj = vjµj. Say, µ4 = 6.3 means the expected number of defaults

among 70 obligors in band 4 is 6.3 over a given time horizon, and the

potential dollar loss in band 4 is E4 = 4× 6.3 = 25.2 (in units of L).

51

Pgf of the number of default in a portfolio

Since a default event is considered as a Bernuolli event, the pgf for a

single obligor A is

FA(z) = (1− PA) + PAz.

For the whole portfolio of obligors, we have

F (z) =∞∑

n=0

P [n defaults]zn.

As a consequence of independence between default events, the probability

generating function for the whole portfolio is

F (z) =∏A

FA(z) =∏A

[1 + PA(z − 1)].

52

Taking logarithm on both sides, and observing

ln[1 + PA(z − 1)] ≈ PA(z − 1),

we have

lnF (z) ≈∑A

PA(z − 1).

As a Poisson approximation, we take

F (z) ≈ e∑

APA(z−1) = eµ(z−1) =∞∑

n=0

e−µµn

n!zn,

where µ =∑A

E[1A] =∑A

PA =m∑

j=1

µj = expected number of default

events over a given time horizon from the whole portfolio. Here, 1A is

the indicator default variable for obligor A.

53

If the probabilities of individual defaults are small, though not necessarily

equal, the probability of realizing n default events in the portfolio over a

given time horizon is

P[n defaults] =e−µµn

n!.

This is the well known Poisson distribution with single parameter µ.

• The distribution has only one parameter, namely, the expected num-

ber of defaults µ of the whole portfolio. The distribution does not

depend on the exposures or individual probabilities of default, provided

that they are uniformly small.

• Up to this point, we assume the default probabilities to be fixed con-

stants. However, there is no necessity for the obligors to have equal

probabilities of default.

54

Remark

Suppose the Poisson approximation is not taken, then the probability

generating function is given by

F (z) =n∏

i=1

[1 + Pi(z − 1)],

where Pi is the expected default probability of the ith obligor in a portfolio

of n obligors. It would be very cumbersome to find the mass probability

function of the random number of the defaults in the portfolio involving

products of Pi (default cases) and 1− Pj (non-default cases).

55

For example,

P [no of defaults = 1] =n∑

i=1

[Pi

n∏j = ij = 1

(1− Pj)],

P [no of defaults = 2] =n∑

i=1

n∑i =j

[PiPj

n∏k = ik = j

(1− Pk)].

Using the Poisson approximation, letting µ =∑n

i=1 Pi, we have

P [no of defaults = 1] = e−µµ,

P [no of defaults = 2] = e−µµ2

2 .

56

Pgf of the loss amount from the whole portfolio in units of L

Likelihood of suffering a given level of loss amount for band j is captured

by:

Probability generating function of the distribution of loss amounts for band j

= Gj(z) =∞∑

n=0

P [loss = nvj]zn =

∞∑n=0

P [n defaults]znvj

≈∞∑

n=0

e−µjµnj

n!znvj = e−µj

∞∑n=0

(µjzvj)n

n!= exp(−µj + µjz

vj).

The dollar loss amount in units of L in band j assumes integer values:

0, vj,2vj, · · · , corresponding to no default, one default, two defaults, etc.

in that band.

57

The default events in the bands are independent so that the pgf of the

loss amount of the whole portfolio with m bands is given by the product

of the pgf’s of the loss amount of individual band. That is,

G(z) =m∏

j=1

exp(−µj + µjzvj).

We then have

P [dollar loss ofnL] =1

n!

dnG(z)

dzn

∣∣∣∣∣z=0

.

Remark

Even without the independence assumption, we still obtain a closed for-

m loss distribution when correlated defaults are assumed (see the later

section on sector analysis).

58

Poisson approximation of the pgf of loss amount (in units of L)

From G(z) = exp

− m∑j=1

µj +m∑

j=1

µjzvj

, µ =m∑

j=1

µj,

if we express the variability of exposure bands within the portfolio via the

following polynomial

p(z) =

∑mj=1 µjz

vj

µ=

∑mj=1

ϵjvjzvj∑m

j=1ϵjvj

=m∑

j=1

P [V = vj]zvj ,

then

G(z) = eµ[p(z)−1].

Here,

P [V = vj] = µj/µ.

Given that an obligor defaults, P [V = vj] gives the probability that the

loss amount is vjL, or equivalently, the defaulting obligor falls in band j.

59

The functional form for G(z) expresses mathematically the compounding

of two sources of uncertainty arising from

(i) Poisson randomness of the incidence of default events, and

(ii) variability of exposure bands within the portfolio.

Indeed, G(z) is a compound Poisson distribution with

G(z) = Gµ(GV (z)),

where Gµ(z) = eµ(z−1) and GV (z) is the pgf of a random variable V taking

values in {v1, · · · , vm} with distribution:

GV (z) = p(z) = P [V = v1]zv1 + · · ·+ P [V = vm]zvm.

60

Numerical example

Consider the following portfolio with 5 obligors:

P1 = 0.2%, v1 = 3; P2 = 0.5%, v2 = 6; P3 = 0.4%, v3 = 7;

P4 = 0.1%, v4 = 8; P5 = 0.9%, v5 = 10.

We have the mean number of defaults

µ = 0.2%+ 0.5%+ 0.4%+ 0.1%+ 0.9% = 0.021,

while the pgf for the variability of exposure is given by

GV (z) =0.2

2.1z3 +

0.5

2.1z6 +

0.4

2.1z7 +

0.1

2.1z8 +

0.9

2.1z10.

The pgf of the compound Poisson distribution is

G(z) = Gµ(GV (z)) = e0.021[(0.22.1z

3+0.52.1z

6+0.42.1z

7+0.12.1z

8+0.92.1z

10)−1].

61

Correlation in defaults

• The observed default probabilities are volatile over time, even for

obligors having comparable credit quality. The variability of default

probabilities can be related to the underlying variability in a number

of background factors, like the state of the economy.

• Two obligors are sensitive to the same set of background factors (with

differing weights), so their default probabilities will move together.

These co-movements in default probabilities give rise to correlation

in defaults.

• CreditRisk+ considers default rates as continuous random variables

and incorporates the volatility of default rates in order to capture

the uncertainty in the level of default rates. It does not attempt to

model default correlation explicitly but captures the same concentra-

tion effects through the use of the default rate volatilities and sector

analysis.

62

The standard deviation of the Poisson distribution with mean µ is√µ.

Historical evidence of the standard deviation of default event frequencies

is invariably much larger thanõ. Thus, the assumption of fixed de-

fault rates cannot account for what is observed in the data. With the

correlation of defaults becomes higher, we expect to observe higher prob-

abilities of seeing both “small number of defaults” and “large number of

defaults”.

63

Sector analysis – independent sector risk factors

• We consider n different sectors S1, · · · , Sn, with the corresponding

positive random risk factor xk, k = 1, · · · , n, whose mean is µk and

standard deviation is σk. The sector risk factors are assumed to be

independent. Sectors could be constructed with respect to industries,

countries or rating classes. Increasing the number of sectors reduces

the correlation (the highest correlation is resulted if all obligors are

allocated to single sector).

• Let Xk denote the discrete non-negative integer random variable that

equals the number of defaults in sector k, whose distribution is a

Poisson mixture distribution with random default intensity xk. Each

(say, industrial) sector is driven by a single underlying risk factor xk of

the kth industry, which models the variability over time in the mean

total number of defaults over a given time horizon measured for that

sector. Here, E[xk] = µk =∑

A∈SkPA.

64

Single-factor setting

The whole portfolio is parameterized by the n factors, one factor for each

sector. In the one-factor setting, we assume that the random default

rate xA of an individual obligor A is influenced by single factor. Later,

we generalize to the scenario where xA depends on multiple sector risk

factors.

Distribution assumption of the risk factors

Unlike the usual Poisson distribution whose parameter is a constant (the

Poisson parameter equals the mean and also the variance of the Poisson

distribution), the random default intensity xk is assumed to be gamma

distributed. The gamma distribution is chosen since it is an analytical-

ly tractable two-parameter distribution. It is a skew distribution which

approximates the normal distribution when its mean is large.

65

Gamma distribution

The density function of the gamma-distributed xk with parameters α and

β is given by

P [x ≤ xk ≤ x+ dx] = f(x) dx =1

βαγ(α)e−x/βxα−1 dx,

where γ(α) = (α−1)! =∫ ∞

0e−xxα−1 dx. The mean and standard deviation

of xk are

µ = αβ and σ2 = αβ2.

For sector k, the parameters αk and βk are related to µk and σk by

αk = µ2k/σ2k and βk = σ2k/µk.

Once µk and σk are specified, the two parameters αk and βk in the gamma

distribution can be determined.

66

How to specify µk and σk?

• The underlying factor influences the sector through the total number

of defaults in that sector, which is modeled as a gamma-mixed Poisson

distribution with random default intensity xk whose mean is µk and

standard deviation is σk.

• We assume that the standard deviation σA (besides the expected

default probability PA) has been assigned for the default rate for each

obligor in the sector.

• We take the mean of factor xk by

µk =∑

A∈Sk

ϵAvA

=∑

A∈Sk

PA =∑

A∈Sk

E[1A],

where the summation extends over all obligors in sector k.

• We randomize the default probability of obligor A, and write it as xA.

Its mean and standard deviation are PA and σA, respectively.

67

Linear relation between xk and xA

For any obligor xA, we assume a linear relationship between the random-

ized default probability xA and random default intensity xk, where

xAxk

=PA

µk.

As a check for consistency, we take the sum:

∑A∈Sk

xA =

∑A∈Sk

PA

µkxk = xk;

and the respective means:

xA =PA

µkxk = PA.

68

Following the linear relation between xA and xk, where xA =ϵAvA

xkµk

, we

take σA =ϵAvA

σkµk

so that for any pair of obligors A1 and A2

σA1

xA1

=σA2

xA2

=σkxk

and∑A

σA =∑A

ϵAvA

σkµk

=

∑APA

µkσk = σk.

For the obligors in the same sector k, we observeσA1

µA1

=σA2

µA2

= · · · ; and

they are all equal toσkµk

so that

σkµk

=

∑A σA∑A PA

=∑

A∈Sk

wAσAPA

, where wA =PA

ΣAPA.

The ratio of σk to the mean µk is a weighted average of the ratio of

standard deviation to mean for each obligor, weighted with respect to

the expected default probabilities.

69

Numerical example

The obligors’ expected defaulted probabilities over a given time period in

3 different industrial sectors are listed below:

{1%,2%,3%}, {0.5%,1.5%,2.5%,3.5%},{0.3%,0.6%,0.9%,1.2%,1.5%,1.8%}.

We then have

µ1 = mean number of defaults in sector 1 = 0.06, µ2 = 0.08, µ3 = 0.063.

Let xi, i = 1,2,3, denote the random default intensity of sector i. Suppose

x1 assumes the realized value y, then

P [no of defaults in sector 1 = n] =e−yyn

n!.

We also have

x1 = µ1 = 0.03, x2 = µ2 = 0.08, x3 = µ3 = 0.063.

70

Let xji denote the random default probability of the jth obligor in the ith

sector. We have

x11 =1

6x1, x21 =

2

6x1 and x31 =

3

6x3;

x11 = 1%, x21 = 2% and x31 = 3%;

µ1 = x1 = x11 + x21 + x31.

For the standard deviations of these random variables, we adopt the linear

relations where

σ1x1

=σ11x11

=σ21x21

=σ31x31

or σ11 =1

6σ1, σ21 =

2

6σ1, σ31 =

3

6σ1,

so

σ1 = σ11 + σ21 + σ31 orσ1x1

=1

6

σ11x11

+2

6

σ21x21

+3

6

σ31x31

.

71

Sector default distribution

The random number of defaults in sector k is gamma-mixed Poisson with

mixture variable xk. Suppose the sector risk factors are assumed to be

independent, then the pgf for the whole portfolio F (z) can be written as

the product of the pgf of the individual sectors. This gives

F (z) =n∏

k=1

Fk(z).

The pgf for the distribution of default events conditional on xk = x is

given by

Fk(z)

∣∣∣∣∣xk=x

= ex(z−1).

The probability generating function for Xk, which equals the random

number of default events in one sector is the average of the conditional

probability generating function over all possible values assumed by xk.

72

Let fk(x) denote the probability density of xk, we have

P [n defaults] =∫ ∞

0P [n defaults|x]fk(x) dx

and∞∑

n=0

znP [n defaults|x] = Fk(z)|z=x = ex(z−1)

The pgf of xk is given by

Fk(z) =∞∑

n=0

P [n defaults]zn

=∫ ∞

0

∞∑n=0

znP [n defaults|x]fk(x) dx

=∫ ∞

0ex(z−1)fk(x) dx

73

=∫ ∞

0ex(z−1)e

−x/βkxαk−1

βαkk γ(αk)

dx [set y = (1

βk+1− z)x]

=1

βαk γ(αk)

∫ ∞

0

y1βk

+1− z

αk−1

e−y

/(1

βk+1− z

)dy

=γ(αk)

βαkk γ(αk)

(1βk

+1− z)αk

=1

βαkk

(1βk

+1− z)αk

=

[1

1+ βk(1− z)

]αk

=

(1− Pk

1− Pkz

)αk

,

where Pk =βk

1+ βk, 1− Pk =

1

1+ βk, 1− Pkz =

1+ βk(1− z)

1 + βk.

74

Recall the binomial expansion with negative exponent:

(1− Pkz)−αk = 1+ (−αk)(−Pkz) +

(−αk)(−αk − 1)

2!(−Pkz)

2

+(−αk)(−αk − 1)(−αk − 2)

3!(−Pkz)

3 + · · · .

Expanding Fk(z) in its Taylor series, we obtain

Fk(z) = (1− Pk)αk

1+∞∑

n=1

(n+ αk − 1

n

)Pnk z

n

so that

P [n defaults] =

(n+ αk − 1

n

)Pnk (1− Pk)

αk.

= probability mass function of (modified) negative binomial

distribution with parameters αk and Pk.

= probability of having n failures before the αthk success wi-

-th probability of success of each trial equals 1− Pk.

Note that αk in general is NOT an integer.

75

Summary

The portfolio has been divided into n sectors with the random default

intensity xk distributed according to the Gamma distribution

Γ(αk, βk), h = 1,2, . . . , n.

The pgf for the random number of default events from the whole portfolio

is

F (z) =n∏

k=1

Fk(z) =n∏

k=1

(1− Pk

1− Pkz

)αk

where αk = µ2k/σ2k , βk = σ2k/µk and Pk =

βk1+ βk

=σ2k

σ2k + µk.

Note that F (z) is not Negative Binomial but it is a product of the Negative

Binomial sector distributions.

76

Default losses with variable default rates

We have obtained the pgf for the number of default events. In order

to pass from default events to default losses, this distribution must be

compounded with the information about the distribution of exposures

within each sector.

Define the pgf of the distribution of loss amount by

G(z) =∞∑

n=0

P [aggregate losses = nL]zn.

Invoking the sector independence assumption, we have

G(z) =n∏

k=1

Gk(z),

where Gk(z) is the loss probability generating function for sector k,1 ≤k ≤ n.

77

We link default events with losses for each sector via the compound

distribution

Gk(z) = Fk(Pk(z))

by defining the polynomial Pk(z) that captures the distribution of expo-

sures of dollar losses of the obligors in the kth sector by

Pk(z) =

∑m(k)j=1

ϵ(k)j

v(k)j

zv(k)j

∑m(k)j=1

ϵ(k)j

v(k)j

=1

µk

m(k)∑j=1

ϵ(k)j

v(k)j

zv(k)j , 1 ≤ k ≤ n.

Here, m(k) is the number of obligors in the kth sector, and

µk =m(k)∑j=1

ϵ(k)j

v(k)j

,

which is the expected total number of defaults µk over the sector k.

78

Recall that

P [V (k) = v(k)j ] =

P(k)j

µk=

ϵ(k)j /v

(k)j

µk.

We may rewrite Pk(z) as a sum over individual obligors belonging to sector

k, where

Pk(z) =1

µk

∑A∈Sk

ϵAvA

zvA.

When we deal with random default intensity xk and random default prob-

ability xA, we take the assumptionxkxA

=µkPA

and integrate out the distri-

bution with respect to xk.

79

Using the linear relation on default probabilities: xA =ϵAvA

xkµk

, we have

exp

− ∑A∈Sk

xA +∑

A∈Sk

xAzvA

= e

∑A∈Sk

xA(zvA−1)

= exkµk

∑A∈Sk

ϵAvA

(zvA−1)= exk[Pk(z)−1],

which gives the probability generating function of the distribution of losses

where each obligor A has default rate xA.

In a similar manner as shown in the calculation of Fk(z), we have

Gk(z) =∞∑

n=0

∫ ∞

0P [loss of nL|xk]fk(xk) dxk

=∫ ∞

0exp

∑A∈Sk

xA(zvA − 1)

fk(xk) dxk

=∫ ∞

0exk[Pk(z)−1]fk(xk) dxk.

80

By performing a similar integration procedure as that of F (z) (see P.71),

except that z is replaced by Pk(z), we obtain the following closed form

expression for the pgf of the distribution of loss amount:

G(z) =n∏

k=1

Gk(z) =n∏

k=1

[1− Pk

1− PkPk(z)

]αk

=n∏

k=1

1− Pk

1− Pkµk

∑m(k)j=1

ϵ(k)j

v(k)j

zv(k)j

αk

,

where Pk =σ2k

σ2k + µkand Pk(z) is the pgf of the variability of exposure in

the kth sector.

Summary

In the single-factor setting, it is assumed that the portfolio is divided into

sectors, each of which is a subset of the set of obligors. In other words,

obligors fall into classes, each of which is driven by one factor. These

random factors (random default intensity) are mutually independent.

81

General sector analysis

It is not necessary that the default rate of an individual obligor depends

on only one of the factors. Recall that

G(z) =n∏

k=1

Gk(z) =n∏

k=1

∫ ∞

0exk[Pk(z)−1]fk(xk) dxk

=∫ ∞

0· · ·

∫ ∞

0e∑n

k=1 xk[Pk(z)−1]n∏

k=1

fk(xk) dx1 · · · dxn.

The probability generating function is integrated over the space of all

possible states represented by (x1 x2 · · · xn) and weighted by their asso-

ciated probability density functions.

82

Consider the exponent term

n∑k=1

xk[Pk(z)− 1] =n∑

k=1

∑A∈Sk

xkµk

ϵAvA

(zvA − 1)

=n∑

k=1

∑A

δAkxkµk

ϵAvA

(zvA − 1), where δAk =

{0 A ∈ k1 A ∈ k

.

Here,∑

A means summation over all obligors within the portfolio.

• The reduction to the single-factor setting is captured via the use of

δAk so that the two different methods of summation∑

A δAk and∑

A∈Sk

are equivalent.

83

Generalization

To allow each obligor to be influenced by more than one factor xk, we

replace the delta function δAk with an allocation of the obligors among

sectors by choosing for each obligor A

θAk :n∑

k=1

θAk = 1.

Here, θAk represents the extent to which the default probability of obligor

A is affected by the random factor xk underlying sector k.

For example, with n = 3, we may choose

θA1 =1

2, θA2 =

1

6and θA3 =

1

3, where θA1 + θA2 + θA3 = 1.

This would mean the relative dependence of the random default rate of

obligor A on the sector risk factors x1, x2 and x3 is in the ratio 3 : 1 : 2.

84

If obligor A is affected by only one sector xk, then we take θAk = δAk. In

general, the exponent term is given by

n∑k=1

xk[Pk(z)− 1] =n∑

k=1

∑A

θAkxkµk

ϵAvA

(zvA − 1)

=∑A

ϵAvA

n∑k=1

θAkxkµk

(zvA − 1).

• Each obligor contributes the following term to the pgf G(z)

exp(xA(zvA − 1)) where xA =

ϵAvA

n∑k=1

θAkxkµk

.

• The variability of the exposures of the obligors for sector k is captured

by

Pk(z) =1

µk

∑A

θAkϵAvA

zvA where µk =∑A

θAkϵAvA

.

• In a similar manner, we deduce that

σk =∑A

θAkσA.

85

2.3 CreditMetrics and Copula functions

Key features

• It is based on credit migration analysis, using the probability of moving

from one credit quality to another within a given horizon as input data.

• It models the full forward distribution of the values of any bond or loan

portfolio, where the changes in values are related to credit migration

only, while interest rates are assumed to evolve in a deterministic

fashion.

• Credit-VaR of a portfolio is derived as the percentile of the distribution

corresponding to the desired confidence level.

86

Reference

M. Crouhy, D. Galai, R. Mark, “A comparative analysis of current risk

models,” Journal of Banking and Finance, vol.24 (2000) p.59-117.

Challenging difficulties

• The portfolio distribution is far from being normal (actually it is highly

skewed and fat-tailed).

• The correlation in credit quality changes for all pairs of obligors are not

directly observable. The correlation coefficient in the joint probability

of asset returns is used as a proxy for the correlation in credit quality

changes.

87

Credit-VaR for a bond

1. Specify a rating system

Rating categories are combined with the probabilities of migrating

from one credit quality to another over the credit risk horizon. Say,

we adopt Moody’s or S&P’s or a proprietary rating system internal to

the bank.

Assumption: all obligors are credit-homogeneous within the same rat-

ing class.

2. Specify the forward discount curve at the risk horizon(s) for each

credit category, and the recovery rate.

Translate the above information into the forward distribution of the changes

in portfolio value consecutive to credit migration.

88

Specification of the transition matrix

• The transition probabilities are based on more than 20 years of history

of firms across all industries.

• Actual transition and default probabilities vary quite substantially over

the years, depending whether the economy is in recession, or in ex-

pansion.

• Many banks prefer to rely on their own statistics which relate more

closely to the composition of their loan and bond portfolios. They may

have to adjust historical values to be consistent with one’s assessment

of current environment.

89

Transition matrix, probabilities of credit rating migrating from one rat-ing quality to another, within one year.

InitialRating

Rating at year-end (%)

AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0 0.22 1.30 2.38 1.24 64.86 19.79

90

Specify the one-year forward discount curve

Category Year 1 Year 2 Year 3 Year 4

AAA 3.60 4.17 4.73 5.12

AA 3.65 4.22 4.78 5.17

A 3.72 4.32 4.93 5.32

BBB 4.10 4.67 5.25 5.63∗

BB 5.55 6.02 6.78 7.27

B 6.05 7.02 8.03 8.52

CCC 15.05 15.02 14.03 13.52

One year forward zero curves for each credit rating (%)

⋆ For example, the discount rate applied to the cash flow to be received

at the end of the fifth year from now (4 years from one year forward)

for a BBB-rated bond is 5.63%.

91

Spread of high (low) investment grade bonds increases (decreases) with

maturity.

92

Specify the forward pricing model

One-year forward price of the BBB bond if it remains to be in the BBB

rating class:

VBBB = 6+6

1.041+

6

(1.0467)2+

6

(1.0525)3+

106

(1.0563)4= 107.55

93

Specify the recovery rate

Seniority Class Mean (%) Standard Deviation (%)

Senior Secured 53.80 26.86

Senior Unsecured 51.13 25.45

Senior subordinated 38.52 23.81

Subordinated 32.74 20.18

Junior subordinated 17.09 10.90

Source: Carty & Lieberman [1996]

Recovery rates by seniority class (% of face value, i.e., “par”)

94

One-year forward values for a BBB bond

Year-end rating Value($)AAA 109.37

AA 109.19

A 108.66

BBB 107.55

BB 102.02

B 98.10

CCC 83.64

Default∗ 51.13

⋆ The bond is senior unsecured and the corresponding mean recovery

rate is 51.13%.

95

Derive the forward distribution of the changes in bond value

Year-end Probability Forward Change in

rating of state: price: V($) value: ∆V

p (%) ($)

AAA 0.02 109.37 1.82

AA 0.33 109.19 1.64

A 5.95 108.66 1.11

BBB 86.93 107.55 0

BB 5.30 102.02 -5.53

B 1.17 98.10 -9.45

CCC 0.12 83.64 -23.91

Default 0.18 51.13 -56.42

Source: CreditMetrics, J.P. Morgan

Distribution of the bond values, and changes in value of a BBB bond, in

one year. Note the long downside tail and the limited upside gain.

96

86.93

Frequency

5.955.30

Probabilityof State

(%)

1.17

.33

.18

.12

.02

......

Default CCC B BB BBB A AA AAA

51.13

-56.42

83.64

-23.91

98.10

-9.45

102.2

-5.53

107.55

0

109.37

1.82

...

...

Forward Price: V

Change in value: ∆V

Histogram of the 1-year forward prices and changes in value of a BBB bond.

97

Mean (∆V ) = m =∑i

Pi∆Vi = −0.46

Variance (∆V ) = σ2 =∑i

Pi(∆Vi −mean) = 8.95

First percentile of a normal distribution = m− 2.33σ = −7.43.

• Recall that 98% of observations lie between 2.33σ and −2.33σ from

the mean for normal distributions.

98

Joint migration probabilities (%) with zero correlation for 2 issuers

rated BB and A

Obligor #2 (A)

Obligor #1 AAA AA A BBB BB B CCC Default

(BB) 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

AAA 0.03 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00

AA 0.14 0.00 0.00 0.13 0.01 0.00 0.00 0.00 0.00

A 0.67 0.00 0.02 0.61 0.40 0.00 0.00 0.00 0.00

BBB 7.73 0.01 0.28 7.04 0.43 0.06 0.02 0.00 0.00

BB 80.53 0.07 1.83 73.32 4.45 0.60 0.20 0.01 0.05

B 8.84 0.01 0.02 8.05 0.49 0.07 0.02 0.00 0.00

CCC 1.00 0.00 0.02 8.05 0.49 0.07 0.02 0.00 0.00

Default 1.06 0.00 0.02 0.97 0.06 0.01 0.00 0.00 0.00

Assuming zero correlation, each entry is given by the product of the

transition probabilities for each obligor. For example, the joint probability

that Obligor #1 remains to be BB and Obligor #2 remains to be A is

given by

73.32% = 80.53%× 91.05%.

99

Correlation between changes in credit quality

1. Correlation is expected to be higher for firms within the same industry

or in the same region.

2. Correlations vary with the relative state of the economy in the business

cycle. For example, when the economy is performing well, default

correlations go down. However, when the economy is deteriorating,

most firms are dragged down.

3. Default and migration probabilities should not stay stationary over

time.

CreditMetrics have chosen the equity price as a proxy for the asset value

of the firm that is not directly observed.

100

Derive the credit quality thresholds for each credit rating

101

• We slice the distribution of asset returns into bands in such a way that

is we draw randomly from this distribution, we reproduce exactly the

migration frequencies shown in the transition matrix. For example,

P [−1.23 < rBB < 1.37] = 80.53%

P [−∞ < rBB < −2.30] = 1.06%, etc.

• Zccc is the threshold point to trigger default; Zccc is commonly called

the “distance-to-default”.

• CreditMetrics estimates the correlations between the equity returns

of various obligors, then the model infers the correlations between

changes in credit quality directly from the joint distribution of equal-

ity returns. The use of equity returns as a proxy is based on the

assumption that firm’s activities are all equity financed.

• The normalized log-returns on both assets follow a joint normal dis-

tribution:

f(rBB, rA; ρ) =1

2π√1− ρ2

exp

(−r2BB − 2ρrBBrA + r2A

2(1− ρ2)

).

102

Transition probabilities and credit quality thresholds for BB and A ratedobligors

Rated-A obligor Rated-BB obligor

Rating in 1 year Prob(%) Z(σ) Prob(%) Z(σ)

AAA 0.09 3.12 0.03 3.43

AA 2.27 1.98 0.14 2.93

A 91.05 -1.51 0.67 2.39

BBB 5.52 -2.30 7.73 1.37

BB 0.74 -2.72 80.53 -1.23

B 0.26 -3.19 8.84 -2.03

CCC 0.01 -3.24 1.00 -2.30

Default 0.06 1.06

103

Calculation of the joint rating probabilities

P(−1.23 < rBB < 1.37,−1.51 < rA < 1.98)

=∫ 1.37

−1.23

∫ 1.98

−1.51f(rBB, rA; ρ) drBBdrA = 0.7365

Rating of second company (A)

Rating

of first

company AAA AA A BBB BB B CCC Def Total

(BB)

AAA 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.03

AA 0.00 0.01 0.13 0.00 0.00 0.00 0.00 0.00 0.14

A 0.00 0.04 0.61 0.01 0.00 0.00 0.00 0.00 0.67

BBB 0.02 0.35 7.10 0.20 0.02 0.01 0.00 0.00 7.73

BB 0.07 1.79 73.65 4.24 0.56 0.18 0.01 0.04 80.53

B 0.00 0.08 7.80 0.79 0.13 0.05 0.00 0.00 8.84

CCC 0.00 0.01 0.85 0.11 0.02 0.01 0.00 0.00 1.00

Def 0.00 0.01 0.90 0.13 0.02 0.01 0.00 0.00 1.06

Total 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06 100

Joint rating probabilities (%) for BB and A rated obligors when

correlation between asset returns is 20%.

104

Correlation coefficient of joint defaults

corr(DEF1, DEF2) =P (DEF1, DEF2)− P1 · P2√

P1(1− P1) · P2(1− P2)

P (DEF1, DEF2) = P [r1 ≤ d12, r2 ≤ d22] = N2(d12, d

22; ρ)

Probability of joint defaults as a function of asset return correlation

105

Sample calculation

Take the correlation coefficient of the joint log-returns of the two assets

to be ρ = 20%,

P (DEF1, DEF2) = N2(d12, d

22; ρ) = N2(−3.24,−2.30; 0.20) = 0.000054.

P1 = 0.06% and P2 = 1.06%; so

corr(DEF1, DEF2)

=0.000054− 0.06%× 1.06%√

0.06%× 99.94%× 1.06%× 98.94%= 0.019 = 1.9%.

Note that ρ and corr(DEF1, DEF2) refer to two different types of corre-

lation.

106

Credit diversification

Implement a Monte Carlo simulation to generate the full distribution of

the portfolio values:

1. Derivation of the asset return thresholds for each rating categories.

2. Estimation of the correlation between each pair of obligor’s asset

returns.

3. Generation of asset return scenarios according to their joint normal

distribution (using the Cholesky decomposition). Each scenario is

characterized by n standardized asset returns, one for each obligor.

4. Given the spread curves which apply for each rating, the portfolio is

revalued.

5. Repeat the procedure a large number of times and plot the distribution

of the portfolio values.

107

Remark

Suppose the covariance matrix of the log-returns of the assets is the

symmetric positive definite matrix∑, we perform the Cholesky decompo-

sition:∑

= MMT . Let e be the vector of n independent standard normal

random variables, then e′ = Me gives the vector of n correlated standard

normal random variables with covariance structure as specified by∑.

Write e′ = (e′1 e′2 · · · e′n)T . As an illustration, let the first asset be A-

rated and the second asset be BB-rated. Suppose the sampled outcomes

are e′1 = −2.1, e′2 = 2.60, then the A-rated asset migrates to the rating

class BBB and the BB-rated asset migrates to the rating class A (based

on the thresholds shown in the table on P.100).

108

Calculation of economic capital

Economic capital stands as a cushion to absorb unexpected losses related

to credit events.

V (p) = value of the portfolio in the worst case scenario at the p% confi-

dence level

VF = forward value of the portfolio = V0(1 +RP )

where RP = promised return on the portfolio and V0 = current mark-to-

market value of the portfolio

E[V ] = expected value of the portfolio = V0(1 + E[R])

where E[R] = expected return on the portfolio

EL = expected loss = VF −E[V ]. The expected loss does not contribute

to the capital allocation. The capital charge comes only as a protection

against unexpected losses.

Capital = E[V ]− V (p)

109

Credit-VaR and calculation of economic capital.

110

One may query the following assumptions in CreditMetrics

1. Firms within the same rating class are assumed to have the same

default rate.

2. The actual default rate (migration probabilities) are set equal to the

historical default rate (migration frequencies).

3. Default is only defined in a statistical sense (non-firm specific) without

explicit reference to the process which leads to default.

111

Empirical studies show:

• Historical average default rate and transition probabilities can deviate

significantly from the actual rates.

• Substantial differences in default rates may exist within the same bond

rating class.

Some other credit risk models attempt to make improvements on

• Defaults rates vary with current economic and financial conditions of

the firm.

• Default rates change continuously (ratings are adjusted in a discrete

fashion).

• Microeconomic approach to default: a firm is in default when it cannot

meet its financial obligations.

112

Copula approach

Two aspects of modeling the default times of several obligors.

1. Default dynamics of a single obligor.

2. Models the dependence structure of defaults between the obligors.

• Knowing the joint distribution of random variables allows us to derive

the marginal distributions and the correlation structure among the

random variables but not vice versa.

• A copula function links univariate marginals to their full multivariate

distribution and introduce the dependence structure to generate the

joint probabilities.

113

Proposition

If a one-dimensional continuous random variable X has distribution func-

tion F , that is, F (x) = P [X ≤ x], then the distribution of the random

variable U = F (X) is a uniform distribution on [0,1].

Proof

P [U ≤ u] = P [F (X) ≤ u] = P [X ≤ F−1(u)] =∫ F−1(u)

−∞f(s) ds

where f(x) = F ′(x) is the density function of X.

Let y = F (s), then dy = f(s) ds and

P [U ≤ u] =∫ F (F−1(u))

F (−∞)dy =

∫ u

0dy.

Conversely, if U is a random variable with uniform distribution on [0,1],

then X = F−1(U) has the distribution function F .

114

Remark

To simulate an outcome of X, one may simulate an outcome u from a

uniform distribution, Then let the outcome of X be x = F−1(u).

For example, suppose X is the standard normal random variables with

distribution function

F (x) = N(x) =1√2π

∫ x

−∞e−t2/2dt.

The random variable U = N(X) has a uniform distribution on [0,1]. To

simulate X, we simulate an outcome u from the uniform distribution,

and x = N−1(u) is a simulated outcome of the standard normal random

variable X.

115

Definition of a copula

A n-dimensional copula is a distribution function on [0,1]n with standard

marginal distributions. In other words, there are uniform random variables

U1, U2, . . . , Un taking values in [0,1] such that the copula C is their joint

distribution function. A function C: [0,1]n → [0,1] is a copula if

C(u1, u2, . . . , un) = P [U1 ≤ u1, U2 ≤ u2, . . . , Un ≤ un],

where ui ∈ [0,1], i = 1,2, . . . , n.

It is seen that C has uniform marginal distributions. For all i ≤ n, ui ∈[0,1], we have

C(1, . . . ,1, ui,1, . . . ,1) = ui.

In the analysis of dependency with copula, the joint distribution can be

separated into two parts, namely, the marginal distribution functions of

the random variables (marginals) and the dependence structure between

the random variables which is described by the copula.

116

Construction of the multivariate distribution functions

Given the copula C and the set of univariate marginal distribution function

F1(x1), F2(x2), . . . , Fn(xn), the function which is defined using a copula C:

C(F1(x1), F2(x2), . . . , Fn(xn)) = F (x1, x2, . . . , xn)

results in a multivariate distribution function with univariate marginal

distributions specified as F1(x1), F2(x2), . . . , Fn(xn).

Proof

C(F1(x1), . . . , Fn(xn)) = P [U1 ≤ F1(x1), . . . , Un ≤ Fn(xn)]

= P [F−11 (U1) ≤ x1, . . . , F

−1n (Un) ≤ xn]

= P [X1 ≤ x1, . . . , Xn ≤ xn]

= F (x1, . . . , xn).

The marginal distribution of Xi = F−1i (Ui) is

C(F1(∞), . . . , Fi(xi), . . . , Fn(∞))

= P [X1 < ∞, . . . , Xi ≤ xi, . . . , Xn < ∞]

= P [Xi ≤ xi] = Fi(xi).

117

As a converse of the above result, any multivariate distribution function

F can be written in the form of a copula.

Sklar’s theorem

If F (x1, x2, . . . , xn) is a joint multivariate distribution function with univari-

ate marginal distribution functions F1(x1), . . . , Fn(xn), then there exists a

copula C(u1, u2, . . . , un) such that

F (x1, x2, . . . , xn) = C(F1(x1), F2(x2), . . . , Fn(xn)).

If each Fi is continuous, then C is unique.

Remark

Going through all copula functions gives us all the possible types of de-

pendence structures that are compatible with the given one-dimensional

marginal distributions. The above theorem is just an existence proof.

The actual finding of the copula for a given joint distribution can be very

cumbersome.

118

CreditMetrics and normal copula

• CreditMetrics uses the normal copula in its default correlation formula

even though it does not use the concept of a copula explicitly.

• CreditMetrics calculates the joint default probability of two credits A

and B using the following steps:

(i) Let qA and qB denote the one-year default probabilities for A and

B, respectively. Let ZA and ZB denote the credit index of A and

B, respectively, both are assumed to be standard normal random

variables.

qA = P [ZA ≤ zA] and qB = P [ZB ≤ zB].

(ii) Let ρ denote the asset correlation, the joint default probability for

credits A and B is given by

P [ZA ≤ zA, ZB ≤ zB] =∫ zA

−∞

∫ zB

−∞n2(x, y; ρ) dxdy = N2(zA, zB; ρ). (A)

119

Bivariate normal copula

C(u1, u2) = N2(N−1(u1), N

−1(u2); γ), − 1 ≤ γ ≤ 1.

Suppose we use a bivariate normal copula with a correlation parameter

γ, and denote the default times for credits A and B as TA and TB.

We observe that the one-year default probabilities are given by

qi = P [Ti ≤ 1] = Fi(1) = N(zi) and zi = N−1(qi) for i = A,B.

The one-year joint default probabilities are given by

P [TA ≤ 1, TB ≤ 1] = C(FA(1), FB(1)) = N2(N−1(FA(1)), N

−1(FB(1)); γ) (B)

where FA(1) = P [TA ≤ 1] and FB(1) = P [TB ≤ 1] are the distribution

functions for the default times TA and TB.

Eqs. (A) and (B) are equivalent if we have ρ = γ. Note that this

correlation parameter is not the correlation coefficient between the two

default times TA and TB. The two stochastic state variables, Ti and Zi,

i = A,B, provide different approaches to characterize a default event.

120

Simulation of the default times of a basket of obligors

Let Fi(t) denote the distribution function of the random default time Ti.

Assume that for each credit i in the portfolio, we have constructed a

credit curve so that Fi(t) can be calibrated.

Using a copula C, we obtain the joint distribution of the default times as

follows:

F (t1, t2, . . . , tn) = C(F1(t1), F2(t2), . . . , Fn(tn)).

Suppose we adopt the normal copula, where

F (t1, t2, . . . , tn) = Nn(N−1(F1(t1)), N

−1(F2(t2)), . . . , N−1(Fn(tn))).

The normal credit index variable Yi and Ti are related by Yi = N−1(Fi(Ti)),

i = 1, . . . , n. To simulate the correlated default times, we introduce the

set of standard normal random variables:

Y1 = N−1(F1(T1)), Y2 = N−1(F2(T2)), . . . , Yn = N−1(Fn(Tn)).

121

There is a one-to-one mapping between the random default time T and

the credit index Y .

Simulation scheme

• Simulate the set of normally distributed credit indexes Y1, Y2, . . . , Yn

from an n-dimensional standard normal distribution with correlation

coefficient matrix∑.

• Obtain the simulated sample of the random default times T1, T2, . . . , Tn

using Ti = F−1i (N(Yi)), i = 1,2, . . . , n.

With each simulation run, we generate the default times for all the credits

in the portfolio. A portfolio credit derivative is a financial instrument

whose payoff depends on the number of defaults in the portfolio by the

maturity date of the derivative. With the information on the arrival times

of defaults, we can value any credit derivative payoff structure written on

the portfolio.

122

Generating the set of correlated default times via the copula approach

(here we assume F1 = F2). Note that N(ri) is U(0,1).

Term structure of the cumulative default probability(DP): F(1)=0.0071, F(2)=0.0180, etc.year 1 2 3 4 5 6 7 8 9 10DP 0.0071 0.0180 0.0320 0.0484 0.0666 0.0859 0.1060 0.1264 0.1469 0.1672

123

Exponential model for dependent defaults

Reference

Kay Giesecke, “A simple exponential model for dependent defaults,” Jour-

nal of Fixed Income, vol.13 (3), p.74-83 (2003).

Poisson process Let λ be the mean rate of arrivals (intensity) of a Pois-

son process N(t). The probability mass function of the random number

of arrivals within the time interval [0, t] is given by

P [N(t) = n] =e−λt(λt)n

n!, n = 0,1,2, . . . .

• A firm’s default is driven by idiosyncratic as well as other region-

al, sectoral or economy-wide shocks, whose arrivals are modeled by

independent Poisson processes. Here, the Poisson approximation is

adopted again for nice analytic tractability.

• Default times are assumed to be jointly exponentially distributed. In

this case, the exponential copula arises naturally.

124

Bivariate version of the exponential models

Suppose there are Poisson processes N1, N2 and N with respective inten-

sities λ1, λ2 and λ. Here, λi is the idiosyncratic shock intensity of firm

i and λ is the intensity of a macro-economic shock affecting both firms

simultaneously. Define the default time τi of firm i by

τi = inf{t ≥ 0 : Ni(t) +N(t) > 0}, i = 1,2.

That is, a default occurs unexpectedly if either an idiosyncratic or a sys-

tematic shock strikes the firm for the first time. Assuming independence

of the two shocks, firm i defaults with intensity λi+λ so that its survival

function is

Si(t) = P [τi > t] = P [Ni(t) +N(t) = 0] = e−(λi+λ)t.

The joint survival probability is found to be

S(t1, t2) = P [τ1 > t1, τ2 > t2]

= P [N1(t1) = 0, N2(t2) = 0, N(t1 ∨ t2 = 0)]

= e−λ1t1−λ2t2−λ(t1∨t2) = e−(λ1+λ)t1−(λ2+λ)t2+λ(t1∧t2)

= S1(t1)S2(t2)min(eλt1, eλt2).

125

Survival copula

There exists a unique solution Cτ : [0,1]2 → [0,1], called the survival

copula of the default time vector (τ1, τ2) such that the joint distribution

of the survival probabilities can be represented by

S(t1, t2) = Cτ(S1(t1), S2(t2)).

The copula Cτ describes the complete non-linear default time dependence

structure.

Define ui = Si(ti) ∈ [0,1], and θi =λ

λi + λ, i = 1,2, we obtain

Cτ(u1, u2) = S(S−11 (u1), S

−12 (u2)) = min(u2u

1−θ11 , u1u

1−θ22 ).

The parameter vector θ = (θ1, θ2) controls the degree of dependence

between the default times. Note that θi is the ratio of the intensity of

the economy wide shock to the total intensity of all shocks that affect

firm i.

126

1. Firms default independently of each other (λ = 0 or λ1, λ2 → ∞)

θ1 = θ2 = 0, Cτθ (u1, u2) = u1u2 (product copula)

2. Firms’ defaults are perfectly correlated (firms default simultaneously,

λ → ∞ or λ1 = λ2 = 0)

θ1 = θ2 = 1 and Cτθ (u1, u2) = u1 ∧ u2.

It can be shown that

u1u2 ≤ Cτθ (u1, u2) ≤ u1 ∧ u2, (θ1, θ2) ∈ [0,1]2, u1 ∈ [0,1], u2 ∈ [0,1].

Under this exponential model, the default events can only be positively

correlated. The positive default correlation is driven by the common

macro-economic stock, which always enhances the chance of default of

both firms.

127

Default copula

Similarly, we define the default copula by

Kτ(P1(t1), P2(t2)) = P [τ1 ≤ t1, τ2 ≤ t2] = P (t1, t2),

where Pi(t) = P [τi ≤ t] = 1− Si(t), i = 1,2.

Since the survival function and default functions are related by

S(t1, t2) = 1− P1(t1)− P2(t2) + P (t1, t2).

Setting ui = Pi(ti) = 1 − Si(ti), i = 1,2, the survival copula and default

copula are related by

Kτ(u1, u2) = Cτ(1− u1,1− u2) + u1 + u2 − 1

= min((1− u2)(1− u1)1−θ1, (1− u1)(1− u2)

1−θ2)

+ u1 + u2 − 1.

128

Multivariate extension

Assume that there are n ≥ 2 firms. The default of an individual firm

is driven by its individual idiosyncratic shock as well as other sectoral,

industry, country-specific or economy-wide shocks.

Define m =n∑

k=1

nCk = 2n − 1. Also, define a matrix (aij)n×m, where

aij = 1 if shock j ∈ {1,2, · · · ,m} modeled through the Poisson process Nj

with intensity λj, leads to a default of firm i ∈ {1,2, · · · , n} and aij = 0

otherwise. Taking n = 3 and m = 7, we consider the following 3 × 7

matrix:

(aij) =

1 0 0 1 1 0 10 1 0 1 0 1 10 0 1 0 1 1 1

.

129

• The fourth column indicates that the fourth shock affects Firms 1

and 2 only.

• The seventh shock is the economy wide shock.

• Suppose the economy-wide shock events are excluded, one then set

ai7 = 0 for i = 1,2,3. This reduces to bivariate dependence only,

where one shock can affect at most 2 firms.

• Firm 1 defaults by time t if

N1(t) +N4(t) +N5(t) +N7(t) =7∑

k=1

a1kNk(t) > 0.

130

Joint survival function

Under this exponential model, the random default time of the ith firm is

given by

τi = inf

t ≥ 0 :m∑

k=1

aikNk(t) > 0

.

Assuming independence of the shocks, this indicates that firm i defaults

with intensitym∑

k=1

aikλk and the survival function of the ith firm is given

by

Si(t) = P [τi > t] = P [m∑

k=1

aikNk(t) = 0] = exp

− m∑k=1

aikλkt

.

The joint survival function is given by

S(t1, . . . , tn) = P [τ1 > t1, . . . , τn > tn].

These n firms all survive up to the maximum value of survival times

max(t1, . . . , tn), provided that all these m shocks do not affect any firm.

131

Example (assuming the arrivals of shocks to be independent events)

Taking n = 3, the joint survival function is

S(t1, t2, t3) = exp(−λ1t1 − λ2t2 − λ3t3

− λ4max(t1, t2)− λ5max(t1, t3)

− λ6max(t2, t3)− λ7max(t1, t2, t3)).

• The first term e−λ1t1 gives the probability that the first shock has not

arrived up to time t1.

• The fourth term e−λ4max(t1,t2) gives the probability that the fourth

shock has not arrived up to max(t1, t2), so both the first and second

firms have not been affected by the fourth shock up to max(t1, t2).

• The seventh term e−λ7max(t1,t2,t3) gives the probability that the sev-

enth shock has not arrived up to max(t1, t2, t3), so all firms have not

been affected by the seventh shock up to max(t1, t2, t3).

132

General case

Consider the kth shock, k = 1,2, . . . ,m, it influences firm i only when

aik = 1. The condition that the kth shock does not affect any firm up to

max(t1, t2, . . . , tn) is given by

Nk(a1kt1) = 0, . . . , Nk(anktn) = 0

or equivalent to

Nk(max(a1kt1, . . . , anktn)) = 0.

Since the m shocks are independent, the joint survival function is given

by

S(t1, . . . , tn) = P [N1(max(a11t1, . . . , an1tn)) = 0, . . . ,

Nm(max(a1mt1, . . . , anmtn)) = 0]

=m∏

k=1

exp(−λkmax(a1kt1, . . . , anktn))

= exp

− m∑k=1

λkmax(a1kt1, . . . , anktn)

.

133

Survival copula function

The exponential survival copula associated with the survival function

S(t1, . . . , tn) can be found via Cτ(u1, . . . , un) = S(S−11 (u1), . . . , S

−1n (un)).

Fixing some i, j ∈ {1,2, . . . , n} with i = j, the two-dimensional marginal

copula is given by

Cτ(ui, uj) = Cτ(1, . . . ,1, ui,1, . . . ,1, uj,1, . . . ,1)

= S(0, . . . ,0, ti,0, . . . ,0, tj,0, . . . ,0)

= exp

− m∑k=1

λkmax(aikti, ajktj)

.

The survival function for firm i and firm j are, respectively, given by

ui = Si(ti) = exp(−(m∑

k=1

aikλk)ti),

uj = Sj(tj) = exp(−(m∑

k=1

ajkλk)tj).

134

In terms of Si(ti) and Sj(tj), the marginal copula can be expressed as

Cτ(ui, uj) = exp(−m∑

k=1

aikλkti)exp(−m∑

k=1

ajkλktj)

min(exp(m∑

k=1

aikajkti), exp(m∑

k=1

aikajktj))

= Si(ti)Sj(tj)min(exp(m∑

k=1

aikajkti), exp(m∑

k=1

aikajktj)).

We define, analogously to the bivariate case, the dependence parameters

in the bivariate marginal copula by

θi =

∑mk=1 aikajkλk∑mk=1 aikλk

, θj =

∑mk=1 aikajkλk∑mk=1 ajkλk

.

Note that the kth shock affects both firm i and firm j, provided that

aikajk = 1. Here, θi is the ratio of the sum of intensities due to shocks

that affect both firms i and j to the sum of intensities due to shocks that

affect firm i only. We then obtain

Cτ(ui, uj) = min(uju1−θii , uiu

1−θjj ).

135

Simulation of correlated default arrival times

The following three-step algorithm generates the default arrival times

with an exponential dependence structure Cτ while allowing for arbitrary

marginal default time distributions.

1. Simulate an m-vector (t1, . . . , tm) of independent exponential shock

arrival times with given parameter vector (λ1, . . . , λm), where λk > 0.

This is done by drawing, for k ∈ {1,2, . . . ,m}, an independent standard

uniform random variate Uk and setting

tk = −1

λklnUk.

Indeed, P [tk > T ] = P [− lnUk/λk > T ] = P [Uk ≤ e−λkT ] = e−λkT .

2. Simulate an n-vector (T1, . . . , Tn) of joint exponential default times

by considering, for each firm i ∈ {1,2, . . . , n}, the minimum of the

relevant shock arrival times:

Ti = min{tk : 1 ≤ k ≤ m,aik = 1}.

Only those shocks that correspond to aik = 1 lead to default of firm

i.136

3. Generate a sample (v1, . . . , vn) from the (survival) default time copula

Cτ by setting, for i ∈ {1,2, . . . , n},

vi = Si(Ti) = exp

−Ti

m∑k=1

aikλk

.

Let the marginal distribution of the random default time Zi of firm i be

denoted by qi(Zi). In order to generate an n-vector Z = (Z1, . . . , Zn) of

correlated default arrival times with given marginal survival function qi,

we set

Zi = q−1i (vi), i = 1,2, . . . , n;

provided that the inverse q−1i exists.

The above algorithm then generates correlated default times Zi with the

given marginal distribution qi and exponential dependence structure. The

joint survival copula is given by

P [Z1 > t1, . . . , Zn > tn] = Cτ(q1(t1), . . . , qn(tn)).

137

Mathematical Appendices

Understanding “conditional independence”

Example Hemophilia is a hereditary disease. If a mother has

it, then with probability 12, any of her sons independently will inherit it.

Let H,H1 and H2 denote the events that the mother, the first son, and

the second son are hemophilic, respectively. Note that H1 and H2 are

conditionally independent given H. However H1 and H2 are not indepen-

dent. This is because if we know that one son is hemophilic, the mother is

hemophilic. With probability 1/2, the other son is also hemophilic. Note

that

P [H1|H] = P [H1|H ∩H2] =1

2

P [H2|H] = P [H2|H ∩H1] =1

2;

P [H1 ∩H2|H] =1

2×

1

2=

1

4.

138

Conditional Variance

Just as we have defined the conditional expectation of X given the value

of Y , we can also define the conditional variance of X, given that Y = y,

as follows:

var(X|Y ) = E[(X − E[X|Y ])2|Y ].

That is, var(X|Y ) is equal to the (conditional) expected square of the

difference between X and its (conditional) mean when the value of Y

is given. In other words, var(X|Y ) is exactly analogous to the usual

definition of variance, but now all expectations are conditional on the

fact that Y is known.

Formula

var(X) = E[var(X|Y )] + var(E[X|Y ]).

139

To obtain the unconditional variance of X, we add the expectation of

conditional variance of X given Y to the variance of expectation of X

given Y .

Proof

We start from

var(X|Y ) = E[X2|Y ]− (E[X|Y ])2

so

E[var(X|Y )] = E[E[X2|Y ]]− E[E[X|Y ])2]

= E[X2]− E[(E[X|Y ])2]. (i)

Treating E[X|Y ] as a random variable, and observing E[E[X|Y ]] = E[X],

var(E[X|Y ]) = E[(E[X|Y ])2]− (E[X])2. (ii)

By adding eqs. (i) and (ii), we arrive at the formula.

140

Negative binomial distribution

• The number X of Bernuolli trials required to obtain r successes on

the set {r, r+1, r+2, · · · }. Out of the (k−1) earlier trials, there have

been r− 1 successes, where k = r, r+1, r+2, · · · . The distribution is

given by

Pr[X = k] =

(k − 1r − 1

)pr(1− p)k−r, k = r, r +1, r +2, · · · .

• The number Y = X − r of failures before the rth success, supported

on the set {0,1,2, · · · }. The modified negative binomial distribution

is given by

Pr[Y = k] =

(k + r − 1

r − 1

)pr(1− p)k

=

(k + r − 1

k

)pr(1− p)k, k = 0,1,2, · · · .

141