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

Credit Risk Modeling

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Credit Risk Modeling

Credit Risk Modeling Ofer Abarbanel online library

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

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

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

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

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)

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))

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

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

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.

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.

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.

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)

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)

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)

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.

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

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

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

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

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

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

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

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