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.
2
.
I. Reduced-form Models of Credit Risk
3
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 ..
5
Moody's Cumulative Default Probabilities
25%
20%
Aaa
A1
Baa1
Ba1
B1
15%
10%
5%
0%
1
2
3
4
5
6
7
Years
6
8
9
10
11
S&P 1-year rating transition matrix
Current rating
Rating in one year
AAA
AAA
AA
A BBB
BB
B CCC Default
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
buyer
Premium
Protection
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
n
n
X
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 distributions {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 distribution 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
Bond
Portfolio
Senior, L+x,
(80%, AAA)
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
Counterparty
12-100%
x
SPV
3-12%
y
0-3%
z
Single-name
CDSs
Tranche
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
Bond
Portfolio
Senior, L+x,
(80%, AAA)
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 conditions 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
d = 45
d = 30
0.14
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
Fraction of portfolio
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0.35
0.4
0.45
0.5
“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
q=
a1
a2
an
+
+ ∙∙∙ +
.
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 increases 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 horizon 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
ri = wi,0 ri,0 +
N
X
wi,j Rj
j=1
where Rj is the normalized return for sector index Ij and ri,0 is an idiosyncratic factor.
• The obligor correlations are
wiT Qwk
ρik = corr(ri , rk ) = q
2 )(w T Qw + w 2 )
(wiT Qwi + wi,0
i
k
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
Obligor 2
A
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
P [τ > t] = G(t) = 1 − F (t) =
Z
∞
f (u)du.
t
The hazard rate is
h(t)dt =
f (t)
dt ≈ P [τ ∈]t, t + dt]|τ > t],
G(t)
and there is a 1-1 relation between h and G in that
G(t) = e
−
Rt
0
h(u)du
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
F1 (t) =
F2 (t) =
F (t1 , t2 ) =
while the survivor function is
G(t1 , t2 ) =
Z tZ
Z
Z
Z
0
0
0
∞
f (u, v)dv du
0
∞Z t
f (u, v)dv du
0
t1 Z t2
f (u, v)dv du,
0
∞Z ∞
t1
f (u, v)dv du
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 distribution 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
f (u, t)du +
f (t, v)dv
h0 (t) =
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
R∞
τ1
f (τ1 , t)
f (τ1 , v)dv
for t ≥ τ1 , so that the new hazard rate is
h2 (t) = R ∞
t
f (τ1 , t)
,
f (τ1 , v)dv
while if τ2 occurs first the hazard rate switches to
h1 (t) = R ∞
t
f (t, τ2 )
.
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
i=1
πp(0, ti )F (ti ) = π × CV01 =
n
X
i=1
(F (ti−1 ) − F (ti ))(1 − R)p(0, ti ).
If we have CDS rates πk for maturities Tk , k = 1, . . . , m and a family of distributions {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
h(s) =
m
X
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
1
ρ
.
.
.
23
 21

..
..
..
..
.
.
.
.
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
(Ui ).
τi = Fi,θ
i
(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
A non-def
B non-def
ah2
h1
3
0
h2
ah1
2
A non-def, B def
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A def
B def
• 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 −ah1 )t
F1 (t) = 1 − e
−
1−e
h1 + h2 − ah1
h1 e−ah2 t
−(h1 +h2 )t
−(h1 +h2 −ah2 )t
−
1−e
F2 (t) = 1 − e
h1 + h2 − ah2
−(h1 +h2 )t
Double Default
h2 e−ah1 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
−(h1 +h2 )t
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
π1 = (1 − R)
I2
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
FTD
0.02
0.015
0.01
0.005
0
1
2
3
4
5
enhancement factor a
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6
7
8
Generators and Backward Equations
The generator of a Markov process xt is an operator A acting on functions
D(A) such that
Mtf
= f (xt ) − f (x0 ) −
Z
t
0
Af (xs )ds
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
h(xT )
v(t, x) = Et,x e
(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




A=


−(hA + hB )
0
0
hA
hB
−βhB
0
0
0
0
−αhA
0
0


βhB 

.
αhA 

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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