Credit Risk Modeling
✩1
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
✪
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
✩6
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
✩9
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
✩11
• 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
✩13
• 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)
✩14
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
✩15
• 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
✩16
• 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
✩19
(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.
✩20
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
✩21
✪
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
✩23
• 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
✩25
• 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.
✩26
✪
An Introduction to Credit Risk Modeling.pdf – hosted by Ofer Abarbanel online library