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Summer School in Financial Derivatives
Imperial College London, 3 June 2005
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Portfolio Credit Risk
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Mark DavisDepartment of MathematicsImperial College LondonLondon SW7 2AZ
www.ma.ic.ac.uk/∼mdavis
1
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
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
4
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
0%
5%
10%
15%
20%
25%
1 2 3 4 5 6 7 8 9 10 11Years
AaaA1Baa1Ba1B1
6
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’.
7
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.
8
1.1 Credit Default Swaps
Premium
Contingentpayments
Protectionseller
Protectionbuyer
Referencebond/issuer
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 byn∑
i=1
πp(0, ti)F (ti) = π × CV01 =n∑
i=1
(F (ti−1)− F (ti))(1−R)p(0, ti).
9
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.
10
1.2 Collateralized Debt Obligations (CDO)
Cash Flow CBO
Senior, L+x,(80%, AAA)
Mezzanine, L+y,(8%, BBB)
Equity, 12%residual receipts
SPVBondPortfolio
Investors subscribe $100 to SPV which purchases bond portfolio. SPV issues
rated notes to investors. Coupons paid in seniority order.
11
Synthetic CDO
12-100%x
3-12%y
0-3%z
SPVCounter-party
Single-nameCDSs
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’.
12
The design process for cash-flow CDOs
Cash Flow CBO
Senior, L+x,(80%, AAA)
Mezzanine, L+y,(8%, BBB)
Equity, 12%residual receipts
SPVBondPortfolio
• 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.
13
1.3 The rating process: Moody’s Binomial Expansion Technique
Start with a portfolio ofM 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, computed from the following
table.
14
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
15
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.
16
Binomial loss distributions
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Fraction of portfolio
d = 60d = 45d = 30
17
“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
1 + c+a2
1 + c+ ∙ ∙ ∙+
an
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
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.
18
Expected Loss in Senior Tranche
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
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
19
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.
20
• The return for obligor i is
ri = wi,0ri,0 +N∑
j=1
wi,jRj
where Rj is the normalized return for sector index Ij and ri,0 is an idiosyn-
cratic factor.
• The obligor correlations are
ρik = corr(ri, rk) =wTi Qwk√
(wTi Qwi + w2i,0)(w
TkQwi + w
2k,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)
21
BBB
BBB
BBB A AA
A
AA
BBB
Obligor 1
Obligor 2
This procedure gives the joint transition probabilities for all obligors. (In
the figure, Obligor 1 starts at BBB, obligor 2 at A.)
22
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) =∫ ∞
t
f(u)du.
The hazard rate is
h(t)dt =f(t)
G(t)dt ≈ P [τ ∈]t, t+ dt]|τ > t],
and there is a 1-1 relation between h and G in that
G(t) = e−∫ t0h(u)du.
Thus specifying h is equivalent to specifying f .
23
For random variables τ1, τ2 ≥ 0 with joint density f(t1, t2) the marginal and
joint distributions are
F1(t) =
∫ t
0
∫ ∞
0
f(u, v)dv du
F2(t) =
∫ ∞
0
∫ t
0
f(u, v)dv du
F (t1, t2) =
∫ t1
0
∫ t2
0
f(u, v)dv du,
while the survivor function is
G(t1, t2) =
∫ ∞
t1
∫ ∞
t2
f(u, v)dv du
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.
24
Let τmin = min(τ1, τ2), τmax = max(τ1, τ2). Then
P [τmin > t] = G(t, t).
The initial hazard rate is therefore (see picture)
h0(t) =1
G(t, t)
(∫ ∞
t
f(u, t)du +
∫ ∞
t
f(t, v)dv
)
t t+dt
t+dtt
25
Suppose τ1 occurs first. The conditional density of τ2 is then
f(τ1, t)∫∞τ1f(τ1, v)dv
for t ≥ τ1, so that the new hazard rate is
h2(t) =f(τ1, t)∫∞
tf(τ1, v)dv
,
while if τ2 occurs first the hazard rate switches to
h1(t) =f(t, τ2)∫∞
tf(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)
26
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.
27
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∑
i=1
πp(0, ti)F (ti) = π × CV01 =n∑
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 distri-
butions {Fθ, θ ∈ Rm} then we can determine the ‘implied default distribution’
Fθ̂.
28
A standard procedure is to take
Fθ(t) = exp
(
−∫ t
0
h(s)ds
)
where
h(s) =m∑
i=1
θi1]Ti−1,Ti](s).
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,EX2i = 1,EXiXj = ρij.
29
NB: Care must be taken that the covariance matrix
Σ =
1 ρ11 ρ12 . . .
ρ21 1 ρ23 . . .............
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
can simply take X1 = Z1, X2 = ρ12Z1 +√1− ρ212Z2.
(iv) Default times τi are given by
τi = F−1i,θi(Ui).
(There is an obvious algorithm for computing this when Fi,θ is as above.)
30
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’.
0
1
2
3A non-def
B non-def
A def
B def
A def, B non-def
A non-def, B def
h1 ah2
h2 ah1
31
• 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.
32
Marginal distributions
F1(t) = 1− e−(h1+h2)t −
h2e−ah1t
h1 + h2 − ah1
(1− e−(h1+h2−ah1)t
)
F2(t) = 1− e−(h1+h2)t −
h1e−ah2t
h1 + h2 − ah2
(1− e−(h1+h2−ah2)t
)
Double Default
FDD(t) = 1− e−(h1+h2)t −
h2e−ah1t
h1 + h2 − ah1
(1− e−(h1+h2−ah1)t
)
−h1e
−ah2t
h1 + h2 − ah2
(1− e−(h1+h2−ah2)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)
33
Continuous-time CDS premium πi on asset i is determined by
πi
∫ T
0
e−rt(1− Fi(t))dt = (1−R)∫ T
0
e−rtfi(t)dt,
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 ) = 1α(1− e−αT ))
I1 = h1(1− a)m(r + h1 + h2, T ) + h2m(r + ah1, T ),
I2 = (h1 + h2)h1(1− a)m(r + h1 + h2, T ) + ah1h2m(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.
34
Calibrated parameters h1, h2, and First-to-Default premium
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
1 2 3 4 5 6 7 8enhancement factor a
h1h2FTD
35
Generators and Backward Equations
The generator of a Markov process xt is an operator A acting on functions
D(A) such that
M ft = f(xt)− f(x0)−∫ t
0
Af(xs)ds
is a martingale for f ∈ D(A). The corresponding backward equation is
∂v
∂t+Av − βv = 0
v(T, x) = h(x)
The solution of the backward equation is
v(t, x) = Et,x
[e−
∫ Ttβ(xs)dsh(xT )
]
(as in Black-Scholes).
36
For the diamond model, xt ∈ {0, 1, 2, 3} and the generator can be expressed in
matrix form as
A =
−(hA + hB) hA hB 0
0 −βhB 0 βhB
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
dtp(t) = p(t)A
for the probability distribution of the process at time t (expressed as a row
vector)
Other, more complex, models are also easily computable.
37
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