An Introduction to Credit Risk Modeling – hosted by Ofer Abarbanel online library

Credit Risk Modeling

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Credit Risk Modeling

References:
• An Introduction to Credit Risk Modeling by Bluhm, Overbeck and Wagner,
Chapman & Hall, 2003
• Credit Risk by Duffie and Singleton, New Age International Publishers, 2005
• Credit Risk Modeling and Valuation: An Introduction, by Kay Giesecke, http://www.stanford.edu/dept/MSandE/people/faculty/giesecke/introduction.pdf,
2004
• Options, Futures, and Other Derivatives, Hull, Prentice Hall India

Ofer Abarbanel – Online Library

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Credit Risk Modeling ✩2
The Basics of Credit Risk Management

• Loss Variable

L˜ = EAD × SEV × L

• Exposure at Default (EAD) = OUT ST + γ COMM
Basel Committee on banking supervision: 75% of off-balance sheet amount. Ex. Committed line of one billion, current outstandings 600 million,
EAD = 600 + 75% × 400 = 900.
• Loss Given Default (LGD) = E[SEV ] – Quality of collateral
– Seniority of claim
• L = 1D, P (D) = DP : Probability of Default
– Calibration from market data, Ex. KMV Corp.
– Calibration from ratings, Ex. Moodys, S & P, Fitch, CRISIL : Statistical tools
+ Soft factors
– Ratings DP: Fit “curve” to RR vs average DP plot
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Credit Risk Modeling ✩3

• Expected Loss (EL) E[L˜] = EAD × LttD × DP
• Unexpected Loss (UL) = .V (L˜)
= EAD × √V ( SEV ) × DP 2 + LttD2 × DP (1 − DP )

Portfolio:

L˜PF = Σm

EADi × SEVi × Li

• ELPF = Σm

EADi × LttDi × DPi

• UL2

m i,j=1

EADi × EADj × Cov(SEVi × Li, SEVj × Lj)

• Constant Severities
Σ .

=
i,j=1

EADi × EADj × LttDi × LttDj ×

DPi(1 − DPi)DPj(1 − DPj) ρij

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Credit Risk Modeling ✩4
• Value at Risk (VaR): qα
qα : inf{q > 0 : P [L˜PF ≤ q] ≥ α}
• Economic Capital (ECα) = qα − ELPF
• Expected Shortfall:
T CEα = E[L˜PF | L˜PF ≥qα] • Economic Capital based on Shortfall Risk: T CEα − ELPF
• Loss Distribution
– Monte-Carlo Simulation
– Analytical Approximation: Credit Risk+
• Today’s Industry Models
– Credit Metrics and KMV-Model
– Credit Risk+
– CreditPortfolio View
– Dynamic Intensity Models
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Credit Risk Modeling ✩5
Credit Metrics and the KMV-Model

• Asset Price Process: At
• Valuation Horizon: T

Li = 1 ( )

∼ B(1; P (A < Ci))

{AT
. A(i) Σ

<Ci} T

ri = log

T
(i) 0

= RiΦi + si, i = 1, 2,… ,m

• Firm’s composite factor Φi is a superposition of systematic influences
(industry and country indices)
• si : Firm specific or idiosyncratic part

• R2

= portion of the volatility in ri explained by volatility in Φi
ri ∼ N (0, 1); Φi ∼ N (0, 1); si ∼ N (0, 1 − R2)

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Credit Risk Modeling

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Global Correlation Model

Industry and Country Indices: Ψj = ΣN bj,nΓn + δn, j = 1,… ,J
Independent Global Facors: Γn
J
Φi = wijΨj
j=1

Li = 1{ri<ci} ∼ B(1,P (ri < ci)) ri < ci ≡ si < ci − RiΦi
pi = P (ri < ci) ⇒ ci = N −1(pi)

pi(Φi) =

Σ N −1(pi) − RiΦi Σ

Σ N −1(pi) − Ri ΣJ

wijΨj Σ

• Simulate a realization of Ψj → Simulate realization of Li
→ One realization of the loss
• Loss Distribution ✪

Credit Risk Modeling ✩7

• KMV tool GCorr computes asset correlations
• KMV provides the weights and asset correlations to its customers
• Can use these correlations with heavy tailed copulas to obtain stronger tail dependencies:
– Fn Univariate t−distribution with n d.f.
– Fn,Γ Multivariate t−distribution with n d.f. and correlation matrix Γ.
– Cn,Γ(u1,… , um) = Fn,Γ(Fn−1(u1),… , Fn−1(um))
– Φn,Γ(x1,… , xm) = Cn,Γ(N (x1),… , N (xm))

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Credit Risk Modeling ✩8

Two Differences Between KMV-Model and Credit Metrics

• Credit Metrics uses equity price correlations, whereas KMV carries out the complicated translation from equity and market information to asset values
• Credit Metrics uses indices referring to a combination of some industry in some particular country, whereas KMV considers industries and countries separately

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Credit Risk Modeling

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CreditPortfolio View

• Default and rating migrations are subject to random fluctuations that depend
on the economic cycle
• Unconditional migration matrix M¯ = (m¯ ij), i, j = 1,… ,K :
rating categories

• m¯ iK : one year historic probability of default in rating category i

• S risk segments that react differently to the economic conditions

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Credit Risk Modeling ✩10
1. Simulate a segment specific conditional default probability ps, s = 1,… , S.
Aggregated Second Level Scenario
2. Define the risk index
r = ps
s p¯s
p¯s unconditional default probability of segment s
3. Conditional migration matrix M (s) :

s = αij(rs − 1) + m¯ ij
The shift matrix (αij) satisfying j αij = 0 must be calibrated by the user
αij ≥ 0, i < j, αij ≤ 0, i > j
M (s) applies to all obligors in segment s. Some entries may turn out to be negative. Set equal to 0 and renormalize.

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Credit Risk Modeling

s = αij(rs − 1) + m¯ ij

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• rs < 1 : expanding economy, lower possibility of downgrades and higher number of upgrades • rs = 1 : average macroeconomic scenario • rs > 1 : recession, downgrades more likely
For each realization of the default probabilities, simulate the defaults and loss. Repeat simulation several times to generate the loss distribution.

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Credit Risk Modeling ✩12
CPV supports two modes of calibration:
• CPV Macro: default and rating migrations are explained by a macroeconomic
regression model. Macroeconomic model is calibrated by means of times series of empirical data.

Ys,t = ws,0 + Σ ws,jXs,j,t + ss,t, ss,t ∼ N (0, σ2 )
j=1

t0
Xs,j,t = θj,0 + θj,kXs,j,t−k + ηs,j,t
k=1

ps,t =

1
1+ exp(−ys,t)

• CPV Direct: ps drawn from a gamma distribution. Need only to calibrate the
two parameters of the gamma distribution for each s. ps can turn out to be larger than 1.

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Credit Risk Modeling

Dynamic Intensity Models

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• Basic Affine or Intensity Process
dλ(t) = κ(θ − λ(t)) dt + σ√λ(t) dB(t)+ ∆J (t)
• J (t) : pure jump process independent of the BM B(t) with jumps arriving according to a Poisson process with rate α and jump sizes ∆J (t) ∼ exp(µ)
• κ = mean-reversion rate; σ = diffusive volatility;

m¯ = θ + αµ/κ long-run mean

Σ − R t λ(u) duΣ

• Unconditional Default Probability q(t) = E e 0
• Correlated defaults λi = Xc + Xi
Xc, Xi basic affine processes with parameters (κ, θc, σ, µ, αc) and (κ, θi, σ, µ, αi)
representing the common performance aspects and the idiosyncratic risk
• λi : basic affine process with parameters (κ, θc + θi, σ, µ, αc + αi)
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Credit Risk Modeling

dXp(t) = κ(θp − Xp(t)) dt + σ.Xp(t) dBp(t)+ ∆Jp(t), p = c, i
d(Xc + Xi)(t) = κ((θc + θi) − (Xc + Xi)(t)) dt
+σ(√Xc(t) dBc(t)+ √Xi(t) dBi(t)) + ∆(Jc + Ji)(t)

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dWt =

c t
Xc + Xi

dBc(t)+

i t
Xc + Xi

dBi(t)

t t t t

d(Xc + Xi)(t) = κ((θc + θi) − (Xc + Xi)(t)) dt + σ (Xc + Xi)(t) dW (t)
+∆(Jc + Ji)(t)
Conditioned on a realization of λi(t), 0 ≤ t ≤ T , the default time of obligor
i is the first arrival in a non-homogenous Poisson process with rate λi(·)

Conditional Probability of No Default = exp(− ∫ T

λ(s)ds)

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Credit Risk Modeling

The Credit Risk+ Model

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• Introduced in 1997 by CSFB
• Actuarial Model
• One of the most widely used credit portfolio models
• Advantages:
– Loss Distribution can be computed analytically
– Requires no Monte-Carlo Simulations
– Explicit Formulas for Obligor Risk Contributions
• Numerically stable computational procedure (Giese, 2003)

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Credit Risk Modeling

The Standard CR+ Model

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• Choose a suitable basic unit of currency ∆L
• Adjusted exposure of obligor A, νA = |EA/∆L∫
• Smaller number of Exposure Bands
• pA expected default probability
• The total portfolio loss L = A νA NA.
• NA ∈ Z+ Default of obligor A
• PGF of the Loss Distribution tt(z) = Σ∞n=0 P (L = n) zn.

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Credit Risk Modeling ✩17
•• ApApppoorrttionion OObbligoligorr RisRiskk amongamong KK SSeectoctorrss ((IndustrIndustryy,, CountryCountry)) byby cchhoooossiinngg
numbers gA such that ΣK A = 1.

• Sectoral Default Rates represented by non-negative variables γk
E(γk) = 1, Cov(γk, γl) = σkl k = 1, …., K.
• Standard CR+ Model assumes σkl = 0, k ƒ= l
• Relating Obligor default rates to sectoral default rates
K
A
k
k=1
• pA(γ) default rate conditional on the sector default rates γ = (γ1,… , γK).
• Specific Sector: γ0 ≡ 1. Captures Idosyncratic Risk

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Credit Risk Modeling ✩18
• Conditional on γ default variables NA assumed to be independent Poisson
• Main Criticism of CR+ Model. Not Fair
– pA = 0.1 → P (NA = 2) = 0.0045
– Need not assume NA is Poisson, but Bernoulli
• Conditional PGF
K
ttγ(z) = exp( γk Pk(z)),
k=1
Pk(z) = Σ gApA(zνA − 1)

A

=
m=1

{νAΣ=m}

gApA (zm − 1)

• Number of defaults in any exposure band is Poisson

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Credit Risk Modeling

• Default correlation between obligors arise only through their dependence on the common set of sector default rates
• Unconditional PGF of Loss Distribution
K
tt(z) = Eγ(exp( γk Pk(z))) = Mγ(T = P (z))
k=1
• MGF of Univariate Gamma Distribution with Mean 1 and Variance σkk is

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(1 − σkktk

)− σkk
ttCR+(z) = exp .−

Σk=1

1

σkk

log(1 − σkkPk(z))Σ

• Giese(2003): Numerically Stable Fast Recursion Scheme

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Credit Risk Modeling

The Compound Gamma CR+ Model (Giese, 2003)

• Introduce sectoral correlations via common scaling factor S
• Conditional on S γK is Gamma distributed with shape parameter
αˆk(S) = Sαk, αk > 0, and constant scale parameter βk.
• S follows Gamma with E[S] = 1 and V ar(S) = σˆ2.
• 1 = Eγk = αkβk
• σkl = δklβk + σˆ2
• Uniform Level of Cross Covariance ⇒ Distortion of Correlation Structure.

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MCG(T ) = exp

• Calibration Problems

1

σˆ2

log

Σ1+ σˆ2

kΣ=1

β log(1 − βktk)ΣΣ

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Credit Risk Modeling

The Two Stage CR+ Model (SKI, AD)

• Y1,… , YN : Common set of Uncorrelated Risk Drivers
N
γk = aki Yi
i=1
• Yi ∼ Gamma with mean 1 and variance Vii
• Principle Component Analysis of Macroeconomic Variables
• Factor Analysis

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Credit Risk Modeling ✩22

K K N
tt(z) = Eγ(exp(Σ γk Pk(z))) = EY (exp(Σ(Σ aki Yi) Pk(z)))

k=1
N K

k=1

i=1

= EY (exp(Σ(Σ aki Pk(z)) Yi))
i=1 k=1
N
= EY (exp( Yi Qi(z))) = MY (T = Q(z))
i=1

K
Qi(z) = aki Pk(z)
k=1

tt(z) = exp .−

Σi=1

1

σii

log(1 − σiiQi(z))Σ

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Credit Risk Modeling

Model Comparison

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• Giese (2003) had pointed out deficiencies in the earlier attempt to incorporate correlations due to Burgisser et al
• We compare the compound gamma and the two stage gamma models
• Test portfolio made up of K = 12 sectors, each containing 3,000 obligors
• Obligors in sectors 1 to 10 belong in equal parts to one of three classes with adjusted exposures E1 = 1, E2 = 2.5, and E3 = 5 monetary units and respective default probabilities p1 = 5.5%, p2 = .8%, p3 = .2%.
• For the three obligor classes in sectors 11 and 12, we assume the same default rates but twice as large exposures (E1 = 2, E2 = 5, E3 = 10)
• σkk = 0.04,k = 1,… , 10 σ11,11 = σ12,12 = 0.49
• Correlation between sectors 11 and 12 is 0.5 whereas correlations between all the other sectors are set equal to 0
• γi = Yi, i = 1,… , 11, γ12 = 0.5(Y11 + Y12), with V ar(Y11) = 0.49
V ar(Y12) = 1.47, and Var(Yi) = 0.04 for i = 1,… , 10

Credit Risk Modeling ✩24

Standard CR+ Compound Gamma Model Two-Stage Model
Expected Loss 1% 1% 1%
Std Deviation 0.15% 0.17% 0.17%
99% Quantile 1.42% 1.48% 1.53%
99.5% Quantile 1.49% 1.55% 1.62%
99.9% Quantile 1.64% 1.71% 1.84%
Table 1: Comparison of the loss distributions from the standard CR+, compound gamma and two stage models for the test portfolio in example 1. All loss statistics are quoted as percentage of the total adjusted exposure.
• σˆ2 = 0.013. This translates to a much lower correlation of 0.0265 (instead of
0.5) between sectors 11 and 12

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Credit Risk Modeling

Risk Contributions

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• Value at Risk VAR αq
• Economic Capital αq − E[L] • Expected Shortfall E[L|L ≥ αq] • Quantile Contribution QCA

QCA = νAE(NA|L = αq) = pAνA

K
k=1

gk D(Aq−νA)ttk(z)
D(Aq)tt(z)

ttk(z) = ∂ MY (T = Q(z))

ttk

N
(z) = tt(z)
i=1
N

aki
1 − σiiQi(z)

= tt(z) ( ak,iexp(−log(1 − σiiQi(z)))).
i=1

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Credit Risk Modeling

Sector CR+ Compound Gamma Model Two-Stage Model
1, 2 24.25% 21.71% 27.42%
3,… , 10 0.37% 1.64 % 0.2 %
11, 12 24.25 % 21.71% 21.59 %

Table 2: Aggregated risk contributions (in percent). Contributions to the loss variance for the risk-adjusted breakdown of VaR (on a 99.9% confidence level).
• Compound gamma model can’t pick up differing correlations among sectors that
are otherwise similar.

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An Introduction to Credit Risk Modeling.pdf – hosted by Ofer Abarbanel online library