The GARP Credit and

Counterparty Risk Summit

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Calculating Default Probabilities: Accurate Measurement,

Models and Default Correlations

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Mark Davis

Imperial College London

BroadStreet Group, NY/London/Tokyo

www.ma.ic.ac.uk/∼mdavis

1

Agenda

• Structural and Reduced Form models

• Moody’s Diversity Score analysis

• Infectious defaults

• An ‘enhanced risk’ stochastic model

• Application to CBO equity tranches

2

Calibration of default probabilities

Market data: Credit spreads for bonds of different maturites

Maturity Spread

T1 s1

T2 s2

T3 s3

Define implied survivor function for default time τ

F (t) = P [τ > t] = exp(−a(t)),

where

a(t) = a0 + a11t>T1+ a21t>T2

.

Determine a0, a1, a2 by requiring that bond price = expected discounted

value of cash flows.

Alternative: calibrate from CDS spreads.

3

Structural Form models (Merton, KMV)

Model value of firm Vt as log-normal process

dVt = µVtdt+ σVtdwt.

Firm defaults on obligation at time T if VT < K, where K is debt repay-

ment due at T . By Black-Scholes

P [VT < K] = 1−N(d2).

Alternative: firm defaults at τ = inf{t ≤ T : Vt ≤ K}. (Analytic formula

in 1 dimension.) For n firms with values V 1t , . . . , V

nt we need the pairwise

correlations ρij such that

E[dwidwj] = ρijdt.

4

Estimating ρij

1. Estimate correlation of asset returns from stock price historical data.

2. (CreditMetrics) Define “participation vectors” βi such that

dSi =M∑1

βijdYj + dZ i

where dY j are returns of standard country/industry indices and dZ i is

an idiosyncratic (issuer-specific) component. Now ρij can be computed

from the CreditMetrics covariance matrix for dY .

The probability distribution of the number of defaults is now a multivariate

normal probability. For large portfolios: Monte Carlo with importance

sampling.

5

-4 -2 0 2 4-4

-2

0

2

4

Hull-White model (Journal of Derivatives, 2000)

Take a Brownian motion process B(t) as proxy for the firm value. Take a

time-varying barrier f(t) and define

τ = min{t : B(t) ≤ f(t)}.

Now calibrate the barrier function such that

P [τ > t] = F (t)

where F (t) is the desired survivor function.

For multiple issuers, take correlation of Brownian motions = correlation of

equity returns.

7

Level-crossing barriers for Hull-White model

-5

-4

-3

-2

-1

0

1

2

0 1 2 3 4 5 6

BBBBAA

Reduced Form Models (Jarrow-Turnbull, Duffie-Singleton,..)

If τ is the default time of an issuer, we specify a hazard rate process h(t)

such that

P (τ ∈]t, t+ dt]| τ > t) = h(t)dt+ o(dt).

More precisely, let N(t) = 1(t≥τ). Then h is the hazard rate if

M(t) = N(t)−∫ t

0

h(s)ds

is a martingale. Simplest case: h(t) ≡ λ (constant). Then

P [τ > t] = e−λt

Under general conditions for stochastic h

η(s, t) = P [τ > t|Fs]1(τ>s)

= E[e−

∫ tsh(u)du

∣∣∣Fs]1(τ>s)

(η(s, t) is the conditional survival probability.)

9

Choice of h: select convenient process with

• η(0, t) matches estimated default distribution (calibration)

• volatility is ‘appropriate’.

What about correlation between the default times τ 1 and τ 2 of two is-

suers? Key point: this is not determined by the joint probability law of

(h1(·), h2(·)).Simulation of default times

1. Compute η(s, t) analytically and draw samples from the distribution

function Fs(t) = η(s, s+ t). (Feasible if h is Hull-White or CIR model)

2. Simulate the process h(t), take an independent U ∼ U [0, 1] and let

τ = inf

(t : exp

(−∫ t

0

h(s)ds

)< U

).

10

Simulation of default time with stochastic hazard rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

time

U

tau

For two issuers, simulate (h1(·), U 1), (h2(·), U 2). Here U i ∼ U [0, 1] for

i = 1, 2, but we have to specify the joint distribution of (U 1, U 2). This is

called a copula (see Embrechts, McNeil & Straumann, Risk 1999). There

are many possibilities, e.g.

• (U 1, U 2) independent: C(u1, u2) = u1u2

• U 1 = U 2: C(u1, u2) = min(u1, u2)

• U 1 = 1− U 2: C(u1, u2) = max(u1 + u2 − 1, 0)

Chart shows

P [max(τ1, τ2) ≤ t]

for these 3 cases when

h1(t) = h2(t) = λ.

12

Two-Default Distribution

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

Time, years

Prob

equalindependentanti-sym

Moody’s Binomial Expansion Technique

Start with a portfolio of M bonds, each (for simplicity) having the same

notional value X. Each issuer is classified into one of 32 industry classes.

The portfolio is deemed equivalent to a portfolio of M ′ ≤ M independent

bonds, each having notional value XM/M ′. M ′ is the diversity score,

determined from the following table:

14

No of firms in Diversity

same industry Score

1 1.0

2 1.5

3 2.0

4 2.3

5 2.6

6 3.0

7 3.2

8 3.5

9 3.7

10 4.0

≥ 11 evaluated

case by case

15

Example: Portfolio of 60 bonds

No of issuers in sector 1 2 3 4 5

No of incidences 2 7 6 4 2

Diversity 2 10.5 18 9.2 5.2

Meaning: 2 cases where issuer is sole representative of industry sector, 7

cases where there are pairs of issuers in same sector, ..

Diversity score = 45.

• Default probability: Individual default probability p determined by

credit rating. Number of defaulting bonds N has binomial distribution

P [N = n] = CM ′n pn(1− p)M ′−n

• Expected loss: suppose Lk is the loss incurred when k defaults occur

in the equivalent portfolio (i.e. fraction k/M ′ default). Expected loss

isM∑k=0

pkLk.

16

Example: M = 60, p = 0.1

With diversity 60, expected loss is µ = Mp = 6, standard deviation is

σ =√Mp(1− p) = 2.32. In CBO structures, losses in senior tranche

occur in extreme cases — represent by call option payoff with “strike”

K = µ+ 3σ = 13.

Charts show loss distribution, expected loss and probability of loss as

diversity is reduced.

• Problems:

1. Probabilistic basis for diversity score?

2. What about default timing?

17

Loss Histogram for Different Diversity Scores

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Equivalent Number Defaulting

Prob

abili

ty

Div = 60Div = 55Div = 50Div = 45Div = 40

0

20

40

60

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Default Fraction

Loss

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

60 55 50 45 40

Diversity Score

Exp LossLoss Prob

The Infectious Defaults Model Consider n identical bonds (same no-

tional and credit rating). Let (Zi, i = 1, . . . , n) be random variables such

that Zi = 1 if bond i defaults and Zi = 0 otherwise. Thus the number

defaulting is

N = Z1 + Z2 + · · ·+ Zn.

The value of Zi is determined as follows. For i = 1, . . . , n and j = 1, . . . , n

with j 6= i let Xi, Yij be independent Bernoulli random variables with

P [Xi = 1] = p

P [Yij = 1] = q.

Then

Zi = Xi + (1−Xi)

1−∏j 6=i

(1−XjYji)

.

Bond i may default ‘directly’ (Xi = 1), or may be ‘infected’ by default of

bond j (Yji = 1 for some j).

21

Default Distribution: Let F (n, k, p, q) denote the probability mass distri-

bution of the random variable N with Zi defined above, i.e.

F (n, k, p, q) = P [N = k].

This is given by

F (n, k, p, q) = Cnkα

pqnk,

where

αpqnk = pk(1− p)n−k(1− q)k(n−k)+

k−1∑i=2

Cki p

k−i(1− p)n−k+i(1− (1− q)k−i)(1− q)(n−k)(k−i).

The expected value is

E[N ] = n(1− (1− p)(1− pq)n−1

).

There is also an explicit expression for the variance.

22

Example

As an example, consider a portfolio of n = 50 bonds with, initially, p = 0.5.

Compute the distribution for q > 0, keeping the expected number of defaults

constant by reducing p as q increases.

Small infection probability has a dramatic effect in increasing the weight

in the tails of the distribution.

q Implied p Standard

Deviation

0 0.5 3.54

0.05 0.194 6.05

0.1 0.116 7.70

0.2 0.064 10.32

23

Default distributions for Infectious Defaults model

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35 40 45 50

Number

Prob

abili

ty

Series1q=0.05q=0.1q=0.2

Bonds in Different Industry Sectors

Calculating the Default Probabilities: Portfolio of n = n1 + n2 + · · · +

nm bonds in m different industry sectors. Assume that different industry

sectors are independent, infection within each industry sector.

The probability of ki defaults in sector i for i = 1, . . . ,m is

m∏i=1

F (ki, ni, pi, qi),

where pi, qi are the infection model parameters for sector i. The probability

of exactly k defaults for the portfolio as a whole is therefore∑a∈Am(k)

m∏i=1

F (ki, ni, pi, qi),

where Am(k) is the set of arrangements a = {k1, . . . , km}, of k defaults in

the m industry sectors.

25

Example: 60-bond portfolio as before

No of issuers in sector 1 2 3 4 5

No of incidences 2 7 6 4 2

Diversity 2 10.5 18 9.2 5.2

This has diversity score 45. Charts show that in terms of expected loss the

equivalent infection parameter is q = 0.2 (assuming same infection in all

sectors)

26

Distributions with Infection

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 5 10 15 20 25 30

No of Defaults

q = 0q = 0.05q = 0.1q = 0.15

Expected Loss and Loss Probability as Functions of Infection

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0 0.05 0.1 0.15 0.2

Infection q

Exptd LossLoss Prob

A dynamic model: Enhanced Risk

Diversity score and infection models are static: they give default distribu-

tion over one period. In CBO applications, timing is important. For this

we need a stochastic process model.

Independent case: n independent bonds, default times exponential, param-

eter λ. Let Nt be the number of defaults in [0, t]. Then

Mt = Nt −∫ t

0

λ(n−Ns)ds

is a martingale (hazard rate ∝ number still alive). Thus m(t) = ENt

satisfies

m(t) = nλt−∫ t

0

λm(s)ds,

so that

m(t) = n(1− e−λt)in agreement with the binomial distribution.

29

Macroeconomic “shock”

Consider a model in which

• Initially, all bonds have hazard rate λ

• When one bond defaults, remaining bonds become more risky: hazard

rate increases to aλ, where a ≥ 1 is the ‘risk enhancement’ parameter

• After an exponentially-distributed time (parameter µ), hazard rates

return to normal level λ.

This models a situation where default occurs in ‘bursts’, triggered off by an

actual default or some other external event. Mathematically, it is a Markov

process Xt on the state space E = {(i, j) : i ∈ {0, 1}, j ∈ {0, 1, . . . , n}}.(i = 0, 1 for normal and enhanced risk respectively; j is number of unde-

faulted bonds. Initial point X0 = (0, n).

30

Possible transitions are

• (0, j)→ (1, j − 1), j > 0

rate jλ

• (1, j)→ (1, j − 1), j > 0

rate ajλ

• (1, j)→ (0, j)

rate µ

• • n

• •• • j

• • j − 1

• •• •

i = 0 i = 1

31

Time-t distributions can be computed by solving the backward equation

∂

∂tv(t, x) +Av(t, x) = 0, (t, x) ∈ [0, T ]× E

v(T, x) = l(x), x ∈ Ewhere A is the differential generator of the process Xt. The solution is

v(t, x) = Et,xl(XT ),

so if we take for example, with x = (i, j),

l((0, n− k)) = l((1, n− k)) = 1,

l(x) = 0, j 6= n− kthen

v(0, (0, n)) = P [k defaults in [0, T ]].

Since the state space E is finite, the backward equation is an ordinary

differential equation in dimension 2n: solve by Runge-Kutta.

Example: n = 60, µ = 0.2, (1 − e−λT ) = 0.1 (consistent with infectious

defaults model).32

Distributions with Enhanced Risk

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 5 10 15 20 25 30

No of Defaults

A = 1A = 1.5A = 2A = 2.5A = 3

PDP Model

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

1 1.5 2 2.5 3

Enhanced Risk Factor

Exptd LossLoss Prob

CBO Equity Tranche

In a conventional CBO structure the equity tranche has no guaranteed

coupon but receives all collateral receipts after coupons are paid to senior

and mezzanine noteholders. Typically the equity tranche comprises 10-15%

of the issue. Thus most of the risk is concentrated in the equity tranche.

Chart shows (for a typical structure) estimate of Risk/Return characteris-

tics of equity tranche as function of diversity (or risk enhancement).

Conclusion: reduced diversity implies increased volatility for equivalent

returns.

35

Risk/return of CBO Equity Tranche

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

0% 5% 10% 15% 20% 25%

Volatility

Mea

n IR

R

a=0

a=4

Summary

• Correlation or ‘concentration risk’ can’t be ignored

• Concentration risk ≡ heavy-tailed default distribution

• Any model to cope with concentration risk must say something about

the mechanism that links default events for different issuers. Modelling

default rates isn’t enough. Using correlation of equity returns as a

proxy is questionable.

• Data is sparse, so there’s no point in introducing models with zillions

of extra parameters.

• Infectious Default and Enhanced Risk models have just one parameter

beyond the single-issuer default probability. This parameter could be

(as here) calibrated to Moody’s Diversity score, or determined by one

further empirical statistic (for example, the variance of the defaults in

successive periods.

37