Summer School in Financial Derivatives

Imperial College London, 3 June 2005

Portfolio Credit Risk

Mark Davis
Department of Mathematics
Imperial College London
London SW7 2AZ
www.ma.ic.ac.uk/∼mdavis

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 Plan for the day

I Reduced-form models of credit risk.

II Statistical analysis of default-time data.
III Stochastic network models for large portfolios.
IV Optimization of credit portfolios.

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 .

I. Reduced-form Models of Credit Risk

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 Agenda

• Credit Ratings

– Ratings and rating transitions

– CDS and CDO
– Moody’s Binomial Expansion Technique and application to CBOs.
– Credit Metrics

• Joint Distributions, Hazard Rates and Copulas

– Definitions of hazard rates and copulas

– Calibration
– The Diamond Default model

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1 Credit Rating

Rating agencies (Moody’s-KMV, S&P, Fitch) assign credit ratings (AAA, AA,. . .)
to firms and transactions on the basis of detailed case-by-case analysis. They
also compile statistics of changes of rating and defaults.
Charts show

• Cumulative default probabilities out to 10 years;

• Change of rating matrix

These are obtained by a ‘cohort analysis’: start with (say) all the AA-tated
firms on 1 January 1981 ..

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 Moody's Cumulative Default Probabilities

25%

20%

15% Aaa
A1
Baa1
Ba1
10% B1

5%

0%
1 2 3 4 5 6 7 8 9 10 11
Years

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S&P 1-year rating transition matrix

Current rating Rating in one year

AAA AA A BBB BB B CCC Default
AAA 87.74 10.93 0.45 0.63 0.12 0.10 0.02 0.02
AA 0.84 88.23 7.47 2.16 1.11 0.13 0.05 0.02
A 0.27 1.59 89.05 7.40 1.48 0.13 0.06 0.03
BBB 1.84 1.89 5.00 84.21 6.51 0.32 0.16 0.07
BB 0.08 2.91 3.29 5.53 74.68 8.05 4.14 1.32
B 0.21 0.36 9.25 8.29 2.31 63.89 10.13 5.58
CCC 0.06 0.25 1.85 2.06 12.34 24.86 39.97 18.60
For example, the probability that a bond rated BBB today will be rated instead
AA in one year, is equal to 1.89 %. Note: BBB is the minimum ‘investment
grade’.

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 Is the rating transition process Markovian? Let Mk denote the empirical
k-year transition matrix. If the process is Markov, we expect to find

Mk = (M1 )k .

In fact this is not at all accurate. There is a ‘momentum effect’: firms that have
been recently downgraded are more likely to be downgraded again than other
firms in the same rating category. David Lando suggests a Markov model with
additional ‘hyperstates’ A*, BBB* etc. Downgrade probabilities are higher in
A* than in A. A company that is downgraded moves first to A* and then, after
some time, to A.

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1.1 Credit Default Swaps

Reference
bond/issuer

Protection Premium Protection

buyer seller
Contingent
payments

Protection buyer pays regular premiums π until min(τ, T ) where T is the

contract expiry time and τ the default time of the Reference Bond.
Protection seller pays (1 − R)1(τ <T ) at next coupon date after τ , where R
is the recovery rate. If F is the risk-neutral survivor function of τ , the ‘fair
premium’ π is determined by
X n n
X
πp(0, ti )F (ti ) = π × CV01 = (F (ti−1 ) − F (ti ))(1 − R)p(0, ti ).
i=1 i=1

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 If we have CDS rates πk for maturities Tk , k = 1, . . . , m and a family of distri-
butions {Fθ , θ ∈ Rm } then we can determine the ‘implied default distribution’
Fθ̂ . Example: m = 1 and Fθ (t) = e−θt .
Moral: CDS rates determine the risk-neutral marginal default time distri-
bution for the reference issuer.
Note: Selling credit protection is (nearly) equivalent to buying the reference
bond with borrowed funds:

• Borrow $100 at Libor L.

• Buy bond at par for $100.

• Bond pays coupon L + x, so net payment is (x − losses).

• At maturity, sell bond and redeem loan.

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1.2 Collateralized Debt Obligations (CDO)

Cash Flow CBO

Senior, L+x,
Bond (80%, AAA)
Portfolio SPV
Mezzanine, L+y,
(8%, BBB)
Equity, 12%
residual receipts

Investors subscribe $100 to SPV which purchases bond portfolio. SPV issues
rated notes to investors. Coupons paid in seniority order.

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 Synthetic CDO

12-100%
Counter- x
party SPV
3-12%
y
0-3%
z
Single-name Tranche
CDSs CDSs

Here SPV sells credit protection to counterparty as individual-name CDS, buys

credit protection on tranches from investors with premiums x < y < z.
The joint default distribution is the key thing here.
New market product: iTraxx index – tranche quotes publicly available on
a standardised debt portfolio. Significance: market data directly related to
‘correlation’.

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The design process for cash-flow CDOs
Cash Flow CBO

Senior, L+x,
Bond (80%, AAA)
Portfolio SPV
Mezzanine, L+y,
(8%, BBB)
Equity, 12%
residual receipts

• Set the size of the senior tranche as big as possible while satisfying expected
loss constraints needed to secure AAA credit rating (see below).

• Similarly for mezzanine tranches.

• Size of equity tranche is set by risk appetite of investors.

• Pricing (i.e. spreads) on rated tranches are determined by market condi-

tions on launch date.

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1.3 The rating process: 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 0 ≤ M independent bonds, each having
notional value XM/M 0 . M 0 is the diversity score, computed from the following
table.

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Diversity score table:

No. of firms Diversity 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

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Diversity score example:
M = 60 bonds.

No. of firms in sector 1 2 3 4 5

No. of incidences 12 12 5 1 1
Diversity 12 18 10 2.3 2.6

Meaning: 12 cases where the firm is the only representative of its industry
sector, 12 pairs of firms in the same sector, etc.
Diversity score = 45.

The effect of reduced diversity is to push weight out into the tail of the loss
distribution, as the chart shows.

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 Binomial loss distributions

0.18

0.16
d = 60
0.14 d = 45
d = 30
0.12

0.1

0.08

0.06

0.04

0.02

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Fraction of portfolio

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“Loss” in rated tranches
Suppose the coupon in a rated tranche is c and the amounts actually received
are a1 , . . . , an . Then the loss is 1 − q where
a1 a2 an
q= + + ∙∙∙ + .
1+c 1+c 1+c
Note that q = 1 (loss zero) when ai = c, i < n and an = 1 + c.
Moody’s rates tranches on a threshold of expected loss.

Example: M = 60, p = 0.1.

Expected no. of defaults is μ = np = 6, standard deviation
p
σ = np(1 − p) = 2.32.

The senior tranche might have a loss threshold of μ + 3σ = 13.

Chart shows expected loss as function of diversity score. Expected loss in-
creases by a factor of 10 as diversity score is reduced from 60 to 30.

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 Expected Loss in Senior Tranche

0.05

0.045

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Diversity Score

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1.4 CreditMetrics

CreditMetrics is aimed at producing the Value-at-Risk over a 1-year time hori-

zon for – say – a bond portfolio. Assumptions:

• There is a fixed credit spread for each credit rating

• Change in value is due only to change in credit rating

• Change in credit rating follows the Moody’s 1-year transition matrix.

What about correlation?

• There are N industry sectors and each obligor i has a weight N -vector wi
such that wi,j represents the participation of obligor i in sector j.

• CreditMetrics estimates the equity return correlation for sector indices

I1 , . . . , IN , giving a correlation matrix Q.

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• The return for obligor i is
N
X
ri = wi,0 ri,0 + wi,j Rj
j=1

where Rj is the normalized return for sector index Ij and ri,0 is an idiosyn-
cratic factor.

• The obligor correlations are

wiT Qwk
ρik = corr(ri , rk ) = q
(wiT Qwi + wi,0
2 )(w T Qw + w 2 )
k i k,0

• Generate a normal M -vector X with mean 0 and covariance matrix A with

diagonal elements 1 and off-diagonal elements ρik .

• Choose quantiles in each coordinates direction so that Xi gives the correct

transition probabilities for obligor i (see picture)

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 AA

A Obligor 2

BBB

B BB BBB A AA

Obligor 1
This procedure gives the joint transition probabilities for all obligors. (In
the figure, Obligor 1 starts at BBB, obligor 2 at A.)

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2 Joint distributions, Hazard Rates and Copulas

Let τ ≥ 0 be a random variable with density f (t). The survivor function G

and distribution function F are
Z ∞
P [τ > t] = G(t) = 1 − F (t) = f (u)du.
t

The hazard rate is

f (t)
h(t)dt = dt ≈ P [τ ∈]t, t + dt]|τ > t],
G(t)
and there is a 1-1 relation between h and G in that
Rt
− h(u)du
G(t) = e 0 .

Thus specifying h is equivalent to specifying f .

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 For random variables τ1 , τ2 ≥ 0 with joint density f (t1 , t2 ) the marginal and
joint distributions are
Z tZ ∞
F1 (t) = f (u, v)dv du
0 0
Z ∞Z t
F2 (t) = f (u, v)dv du
0 0
Z t1 Z t2
F (t1 , t2 ) = f (u, v)dv du,
0 0
while the survivor function is
Z ∞Z ∞
G(t1 , t2 ) = f (u, v)dv du
t1 t2

The joint distribution F is related to the marginals by the copula function C

given by
F (t1 , t2 ) = C(F1 (t1 ), F2 (t2 )). (1)
Given marginal distributions F1 , F2 , formula (1) defines a bone fide joint distri-
bution for any choice of copula function C.

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 Let τmin = min(τ1 , τ2 ), τmax = max(τ1 , τ2 ). Then

P [τmin > t] = G(t, t).

The initial hazard rate is therefore (see picture)

Z ∞ Z ∞ 
1
h0 (t) = f (u, t)du + f (t, v)dv
G(t, t) t t

t+dt
t

t t+dt

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 Suppose τ1 occurs first. The conditional density of τ2 is then
f (τ1 , t)
R∞
τ1 f (τ1 , v)dv
for t ≥ τ1 , so that the new hazard rate is
f (τ1 , t)
h2 (t) = R ∞ ,
t f (τ1 , v)dv
while if τ2 occurs first the hazard rate switches to
f (t, τ2 )
h1 (t) = R ∞ .
t f (u, τ2 )du
The hazard rate process is therefore


h(t) = h0 (t)1(t<τmin ) + h2 (t)1(τmin =τ1 ) + h1 (t)1(τmin =τ2 ) 1(τmin ≤t<τmax )

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2.1 Copula-based calibration

Let τ1 , τ2 , . . . default times of issuers A, B, . . .. If F is the joint distribution

and Fi is the marginal distribution of τi then

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

for some copula function C (= a multivariate DF with uniform marginals).

(i) Back out marginal default distributions F1 (t), F2 (t), . . . from credit spreads
or CDS rates.

(ii) Choose your favourite copula function – say, Gaussian copula.

(iii) Take correlation parameter ρ = correlation of equity returns.

(iv) Define joint distribution F as above.

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In more detail:
(i) In a CDS contract, protection buyer pays regular premiums π until min(τ, T )
where T is the contract expiry time and τ the default time of the Reference
Bond. Protection seller pays (1 − R)1(τ <T ) at next coupon date after τ , where
R is the recovery rate. If F is the risk-neutral survivor function of τ , the ‘fair
premium’ π is determined by
n
X n
X
πp(0, ti )F (ti ) = π × CV01 = (F (ti−1 ) − F (ti ))(1 − R)p(0, ti ).
i=1 i=1

If we have CDS rates πk for maturities Tk , k = 1, . . . , m and a family of distri-

butions {Fθ , θ ∈ Rm } then we can determine the ‘implied default distribution’
Fθ̂ .

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 A standard procedure is to take
 Z t 
Fθ (t) = exp − h(s)ds
0

where m
X
h(s) = θi 1]Ti−1 ,Ti ] (s).
i=1
Then θi , θ2 , . . . are determined recursively using π1 , π2 , . . ..
Note: in a multivariate setting there is a different parametrization Fj,θj for
each issuer.

(ii),(iii) To generate a random vector U with uniform marginals and a gaussian

copula, take Ui = N −1 (Xi ) where X is a normal random vector with EXi =
0, EXi2 = 1, EXi Xj = ρij .

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 NB: Care must be taken that the covariance matrix
 
1 ρ11 ρ12 . . .
 
Σ= ρ
 21 1 ρ 23 . . . 

.. .. .. ..
. . . .

is actually non-negative definite. If the ρij are obtained from sample estimates,
it may not be!
To generate X, take X = AZ where A is the Cholesky factorization of Σ
(i.e. Σ = AAT ) and Zi are independent N (0, 1). In the 2-dimensional case we
p
can simply take X1 = Z1 , X2 = ρ12 Z1 + 1 − ρ212 Z2 .
(iv) Default times τi are given by
−1
τi = Fi,θ i
(Ui ).

(There is an obvious algorithm for computing this when Fi,θ is as above.)

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2.2 Simultaneous calibration: The Diamond Model

An obvious drawback of copula methods is: how do you choose the copula? An
alternative is to think in terms of ‘infection’.

A def, B non-def

1
h1 ah2
A non-def A def
0 3
B non-def B def
h2 ah1
2

A non-def, B def

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• Hazard rate of remaining issuer increases by a factor a after first default.

• If functions h1 , h2 are time-dependent and we replace ah1 , ah2 by general

h3 , h4 then we can represent any joint distribution this way.

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Marginal distributions
h2 e−ah1 t  
−(h1 +h2 )t −(h1 +h2 −ah1 )t
F1 (t) = 1 − e − 1−e
h1 + h2 − ah1
h1 e−ah2 t  
−(h1 +h2 )t −(h1 +h2 −ah2 )t
F2 (t) = 1 − e − 1−e
h1 + h2 − ah2
Double Default

h2 e−ah1 t  
−(h1 +h2 )t −(h1 +h2 −ah1 )t
FDD (t) = 1 − e − 1−e
h1 + h2 − ah1
h1 e−ah2 t  
−(h1 +h2 −ah2 )t
− 1−e
h1 + h2 − ah2

Calibration
Joint calibration to credit spreads/CDS rates for issuers A and B, for given
‘enhancement’ parameter a. (Time-varying h1 , h2 required for term structure
of credit spreads)

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 Continuous-time CDS premium πi on asset i is determined by
Z T Z T
−rt
πi e (1 − Fi (t))dt = (1 − R) e−rt fi (t)dt,
0 0

where r is the riskless rate and fi (t) = dFi (t)/dt. From the model, we find
I2
π1 = (1 − R)
I1
where (with m(α, T ) = α1 (1 − e−αT ))

I1 = h1 (1 − a)m(r + h1 + h2 , T ) + h2 m(r + ah1 , T ),

I2 = (h1 + h2 )h1 (1 − a)m(r + h1 + h2 , T ) + ah1 h2 m(r + ah1 , T ).

The first default time τmin = τ1 ∧ τ2 is exponential with rate (h1 + h2 ). Hence
the FTD premium is
πFTD = (1 − R)(h1 + h2 ).

Chart shows calibrated h1 , h2 when CDS rates are π1 = 75bp, π2 =200bp.

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 Calibrated parameters h1, h2, and First-to-Default premium

0.035

0.03

0.025
h1
h2
0.02 FTD

0.015

0.01

0.005

0
1 2 3 4 5 6 7 8
enhancement factor a

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Generators and Backward Equations

The generator of a Markov process xt is an operator A acting on functions

D(A) such that Z t
Mtf = f (xt ) − f (x0 ) − Af (xs )ds
0
is a martingale for f ∈ D(A). The corresponding backward equation is
∂v
+ Av − βv = 0
∂t
v(T, x) = h(x)

The solution of the backward equation is

h RT i
− t β(xs )ds
v(t, x) = Et,x e h(xT )

(as in Black-Scholes).

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For the diamond model, xt ∈ {0, 1, 2, 3} and the generator can be expressed in
matrix form as
 
−(hA + hB ) hA hB 0
 
 0 −βhB 0 βhB 
 
A= .
 0 0 −αhA αhA 
 
0 0 0 0

Solving the backward equation amounts to computing the matrix exponential

eAt . We can also solve the forward equation
d
p(t) = p(t)A
dt
for the probability distribution of the process at time t (expressed as a row
vector)
Other, more complex, models are also easily computable.

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