[PPT] - Modelling Dependent Credit Risks Advanced Mathematical Methods for PowerPoint Presentation

SLIDE 1

Modelling Dependent Credit Risks with Extensions of CreditRisk+ and Application to Operational Risk

Workshop on Risk Analysis and Management April 23–25, 2006, Side, Antalya, Turkey

Prof. Dr. Uwe Schmock

Dipl.-Ing. Richard Warnung Financial and Actuarial Mathematics Vienna University of Technology, Austria www.fam.tuwien.ac.at/~schmock www.fam.tuwien.ac.at/~rwarnung Workshop and Mid-Term Conference on Advanced Mathematical Methods for Finance (Mid-Term AMaMeF Meeting 2007) September 18–23, 2007, Vienna, Austria Organized by Research Group for Financial and Actuarial Mathematics and CD-Laboratory for Portfolio Risk Management Institute for Mathematical Methods in Economics Vienna University of Technology A-1040 Vienna, Austria www.fam.tuwien.ac.at/amamef2007

c April 24, 2006, U. Schmock, FAM, TU Vienna 2

Program of the Workshop Saturday, April 23, 2006 8.45 Opening Ceremony 9.00 Credit Rating

Prof. Dr. Ralf Korn

(Kaiserslautern University, Germany) 11.00 Coffee break 11.30 Credit Rating (cont.) 13.25 Lunch 15.15 Credit Rating (cont.) 16.30 Coffee break 17.00 Sessions/Diskussions 18:00 End of first day

c April 24, 2006, U. Schmock, FAM, TU Vienna 3

Program of the Workshop (Cont., 2. Day) Sunday, April 24, 2006 9.00 CreditRisk+ and Extensions

Prof. Dr. Uwe Schmock

(Vienna University of Technology, Austria) 11.00 Coffee break 11.30 CreditRisk+ and Extensions (cont.) 13.25 Lunch 15.15 Default Risk

Prof. Dr. Monique Jeanblanc (Evry University)

16.30 Coffee break 17.00 Default Risk (cont.) 19:00 End of second day

c April 24, 2006, U. Schmock, FAM, TU Vienna 4

SLIDE 2

Program of the Workshop (Cont., 3. Day) Monday, April 25, 2006 9.00 CreditRisk+ and Extensions

Prof. Dr. Uwe Schmock (TU Vienna)

11.00 Coffee break 11.30 Sessions/Diskussions 13.25 Lunch 15.15 Default Risk

Prof. Dr. Monique Jeanblanc (Evry University)

16.30 Coffee break 17.00 Sessions/Diskussions 19:00 Closing Ceremony

c April 24, 2006, U. Schmock, FAM, TU Vienna 5

Introduction to Credit Risk Modelling

Prof. Dr. Uwe Schmock

Director of PRisMa Lab Financial and Actuarial Mathematics Vienna University of Technology, Austria www.fam.tuwien.ac.at/∼schmock www.prismalab.at

c April 24, 2006, U. Schmock, FAM, TU Vienna 6

Components of Credit Risk

Arrival risk: Uncertainty whether a default will
ccur or not. Measured by the probability of default,

within a given time horizon, usually one year.

Timing risk: Uncertainty about the time of default.
Exposure risk: Relatively clear for loans or bonds

(face value, market value), greater uncertainty in the credit reinsurance business as primary insurers might have successfully decreased credit lines in advance.

Recovery risk: Uncertainty about the size of the loss

w.r.t. the exposure. Historical data show a large variability of recovery rate, depending on collateral, seniority of the bond, etc. Specified by conditional distribution of recovery rate given default occurred.

c April 24, 2006, U. Schmock, FAM, TU Vienna 7

Components of Credit Risk (Cont.)

Rating transition risk: Risk of changing market price
f a defaultable security due to a changed perception
f the market towards the timing or recovery risk

(without an actual default already happening). It often happens together with an up- or down-rating

f the creditworthiness by a rating agency.
Default correlation risk: Risk of several obligors

defaulting together; leads to substantial losses even in well diversified portfolios. Defaults in investment- grade rating classes are rare, hence it is hard to collect data to estimate the dependence of defaults.

c April 24, 2006, U. Schmock, FAM, TU Vienna 8

SLIDE 3

Time Series versus Default Modelling Time series modelling (e.g. exchange rates)

Collect data for a long time (e.g. CHF/US-$).
Assume stationarity of stochastic behaviour,

fit a suitable model (random walk, GARCH, etc.) and make predictions about the future. Default modelling

Observing a firm until today doesn’t give a default
bservation (→ observation bias).
Solution: Observe a group of firms,

draw conclusions for a specific firm.

Problems: Relevance of data for the specific firm?

When are firms similar w.r.t. creditworthiness?

c April 24, 2006, U. Schmock, FAM, TU Vienna 9

Credit Ratings for Bonds A credit rating is a current opinion of an obligor’s

verall financial capacity (its creditworthiness) to pay

its financial obligations. Standard & Poor’s Investor Services Investment grade: AAA, AA, A, BBB Speculative: BB, B, CCC, CC, C (D = Default) AA–B: + = above, − = below average in rating class Moody’s Investor Services Investment grade: Aaa, Aa, A Speculative: Baa, Ba, B, Caa, Ca, C Aa–B: 1 = above, 2 = at, 3 = below average in rating class

c April 24, 2006, U. Schmock, FAM, TU Vienna 10

Some Offered Bonds July 2004 (in CHF)

Matur- Rating Company Coupon ity Price S & P Moody’s Date WestLB∗ float 28.7.06 100.060 AA Aa2 22.7. GECC† float 23.7.07 100.070 AAA Aaa 20.7. Commonwealth Bank of Australia 1.750 4.9.07 100.085 AA– Aa3 30.7. Principal Financial Global Funding 2.750 12.7.10 100.190 AA Aa3 8.7. Sigma Finance 2.375 29.7.11 100.225 AAA Aaa 21.7. Hypo Tirol Bank 3.000 20.11.12 101.600 AAA 5.8.

∗3-month Libor flat †3-month Libor + 2 bp Neue Z¨ urcher Zeitung (NZZ, Vol. 225), July 7, 2004 c April 24, 2006, U. Schmock, FAM, TU Vienna 11

Some Current Bonds (in USD)

Matur- Rating Company Coupon ity Price S & P Moody’s YTM Daimler Chr. NA 6.400 15.5.06 100.194 BBB A3 6.292 AT & T 7.500 1.6.06 100.193 A Baa2 5.398 Wal Mart Stores 5.450 1.8.06 100.231 AA Aa2 4.506 Time Warner 8.110 15.8.06 100.794 BBB+ Baa2 5.384 Citigroup Inc. 5.500 8.9.06 100.271 AA− Aa1 4.486 HP Co. 5.750 15.12.06 100.334 A− A3 5.2

Source: www.bonddesk.com April 21, 2006 c April 24, 2006, U. Schmock, FAM, TU Vienna 12

SLIDE 4

Classification of Credit Risk Models

Firm-value (or structural) models

Pioneered by Black & Scholes (1973), Merton (1974) Industry models: Portfolio Manager (by KMV), CreditMetrics (RiskMetrics Group)

Intensity-based (or reduced-form) models

Jarrow & Turnbull (1995), Jarrow, Lando & Turnbull (1997), Lando (1996, 1998), Duffie & Singleton (1999)

Actuarial models

Mixture models, CreditRisk+ (CS Financial Products)

Macroeconomic models

Industry: CreditPortfolioView (McKinsey & Company)

c April 24, 2006, U. Schmock, FAM, TU Vienna 13

1 2 3 4 50 100 150 200

Exponential Brownian motion At = 200 exp(0.3Wt). Default occures when At < L.

Assets of the firm Liabilities of L = 100 Years At

Merton Model

c April 24, 2006, U. Schmock, FAM, TU Vienna 14

Classification of Credit Risk Models

Firm-value (or structural) models

Pioneered by Black & Scholes (1973), Merton (1974) Industry models: Portfolio Manager (by KMV), CreditMetrics (RiskMetrics Group)

Intensity-based (or reduced-form) models

Jarrow & Turnbull (1995), Jarrow, Lando & Turnbull (1997), Lando (1996, 1998), Duffie & Singleton (1999)

Actuarial models

Mixture models, CreditRisk+ (CS Financial Products)

Macroeconomic models

Industry: CreditPortfolioView (McKinsey & Company)

c April 24, 2006, U. Schmock, FAM, TU Vienna 15

Discrete-Time Motivation for Intensity Models Time 1 . . . t t + 1 t + 2 . . . T Notation X promised (but defaultable) payout at time T ht conditional probability at time t for default during the period [t, t + 1] rt continuously compounded, default-free interest rate for the period [t, t + 1] ϕt+1 random recovery at t + 1 in case

f default during [t, t + 1]

EQ[ ·|Ft] conditional expectation under Q given all the information Ft up to time t

c April 24, 2006, U. Schmock, FAM, TU Vienna 16

SLIDE 5

Evolution of Market Value Vt Time t Vt ht 1 − ht ϕt+1 (recovery) Vt+1 Time t + 1 Recursion formula Vt = ht e−rt EQ[ϕt+1|Ft] + (1 − ht) e−rt EQ[Vt+1|Ft] with terminal value VT = X. An explicit formula for V0 by backward induction is available but complicated to evaluate.

c April 24, 2006, U. Schmock, FAM, TU Vienna 17

Different Assumptions for the Recovery Recovery of face value: ϕt = 1 − Lt The creditor receives a (possible random) fraction of the face value 1 immediately upon default. Recovery of treasury: ϕt = (1 − Lt)P(t, T) The creditor receives a (possible random) fraction of a corresponding default-free government bond. Recovery of market value (RMV): EQ[ϕt+1|Ft] = (1 − Lt) EQ[Vt+1|Ft] The expected recovery is a (random) fraction of the expected market value in case of no default.

c April 24, 2006, U. Schmock, FAM, TU Vienna 18

Bank Loans Senior Secured Bonds Senior Unsecured Bonds Subordinated Bonds Junior Subordinated Bonds 10 20 30 40 50 60 70 80 90 100

Distribution of Recovery by Seniority

90% quantile median 10% quantile central 50%

Based on Moody’s data 1974–1997

Percentage of face value in secondary market after default

c April 24, 2006, U. Schmock, FAM, TU Vienna 19

Transition to Hazard and Loss Rates With recovery of market value Vt = {(1 − ht) e−rt + ht e−rt(1 − Lt)}

=: e−Rt

EQ[Vt+1|Ft] with e−Rt = (1 − htLt)e−rt ≈ e−(rt+htLt), because

1 − ht

n Lt n → e−htLt n → ∞. n is the number of subdivisions per period. Hence V0 = EQ

e−(R0+···+RT −1)X
.

c April 24, 2006, U. Schmock, FAM, TU Vienna 20

SLIDE 6

RiskLab Project: Intensity-Based Non-Parametric Default Model for Residential Mortgage Portfolios Swiss banks hold over 500 billion CHF in mortgages. Data for 73 683 obligors of Credit Suisse used. Default intensity tested for dependence on – Regional unemployment rates, – Fixed- or variable-rate mortgage product, – Interest-rate changes, – Divorce rates, – Regional real estate price indices, – Time-lags until default. Reference: Paper (38 pages) by Enrico De Giorgi http://www.risklab.ch/Papers.html#RMSRMMLP

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Modelling Dependence of Defaults

Firm-value (or structural) models

(asset correlations, macroeconomic factors)

Intensity-based (or reduced-form) models

(macroeconomic factors, business cycle)

Actuarial models, mixture models

(random intensities, random default probabilities)

Infectious defaults (M. Davis & V. Lo, 1999)
Looping defaults, primary/secondary firms

(R. Jarrow & Fan Yu, 2001)

Hindering defaults

→ Subject of current research

c April 24, 2006, U. Schmock, FAM, TU Vienna 22

Introduction to CreditRisk+, Features

Developed by Credit Suisse Financial Products.
Actuarial model for the aggregation of credit risks.
Based on the Poisson approximation of individual de-

faults and the divisibility of the Poisson distribution.

Takes exposures/recovery rates into account.
Several independent risk factor for dependence of

default frequencies can be considered.

Probability generating function ϕL of the credit

portfolio loss L is available in closed form.

c April 24, 2006, U. Schmock, FAM, TU Vienna 23

Extensions of CreditRisk+

Stochastic losses of individual obligors are allowed,

distribition may depend on the risk factor.

Risk groups with dependent stochastic losses given

default are possible.

Risk factors for default frequencies may be dependent.
Risk contributions of obligors can be calculated.
Even with all the extensions, the probability

generating function ϕL of the credit portfolio loss L is available in closed form.

Distribution of L and risk contributions can be calcu-

lated from ϕγ,L with a numerically stable algorithm.

c April 24, 2006, U. Schmock, FAM, TU Vienna 24

SLIDE 7

Project CreditRisk+ and Extensions

Background: 2. Basel Capital Accord of the Basel

Committee on Banking Supervision (“Basel II”)

Research and development cooperation of

– Research Group Financial and Actuarial Mathematics – Austrian Central Bank (OeNB) – Austrian Financial Market Authority (FMA)

Aim: Supervision of credit risk in the portfolio of all

(≥ 900) Austrian banks

Large single credit risks are reported individually
Efficient and numerically stable algorithm

to calculate risk of credit portfolio

Java code (Mag. Severin Resch, Dipl.-Ing. R. Warnung)

c April 24, 2006, U. Schmock, FAM, TU Vienna 25

Motivation: Bernoulli Model for Defaults

Bernoulli loss indicators

Ni = 1 if obligor i defaults (within one year),

therwise.
Default probability pi = P(Ni = 1) for i = 1, . . . , m.
Random number of defaults N = N1 + · · · + Nm.
Probability distribution for n ∈ {0, . . . , m}

P(N = n) =

I⊂{1,...,m}

|I|=n

P

Ni = 1I(i) for i = 1, . . . , m
if ind.

= (

i∈I pi) i∈{1,...,m}\I(1−pi)

m = 1000, n = 100 = ⇒ 1000

100

≈ 6.4 × 10139 terms

c April 24, 2006, U. Schmock, FAM, TU Vienna 26

Expectation and Variances of Portfolio Defaults

Expectation

E[N] =

m

i=1

E[Ni] =

m

i=1

pi

Variance

Var(N) =

m

i=1

Var(Ni) +

i,j∈{1,...,m}

i=j

Cov(Ni, Nj) with Var(Ni) = E[N 2

i ] − (E[Ni])2 = pi(1 − pi)

because N 2

i = Ni.

c April 24, 2006, U. Schmock, FAM, TU Vienna 27

Motivation: General Bernoulli Mixture Model

(P1, . . . , Pm) ∼ F random default probabilities
Assume P(Ni = 1|P1, . . . , Pm)

a.s.

= P(Ni = 1|Pi)

a.s.

= Pi

Assume independence of N1, . . . , Nm given P1, . . . , Pm.
Then for all n1, . . . , nm ∈ {0, 1}

P(N1 = n1, . . . , Nm = nm) = E

P(N1 = n1, . . . , Nm = nm |P1, . . . , Pm)
= E

m

i=1

P ni

i (1 − Pi)1−ni

=
[0,1]m

m

i=1

pni

i (1 − pi)1−ni F(dp1, . . . , dpm)

c April 24, 2006, U. Schmock, FAM, TU Vienna 28

SLIDE 8

Variances and Covariances in Mixture Models Lemma: If X and Y are square integrable random variables on a probability space (Ω, A, P) and B ⊂ A a sub-σ-algebra, then Cov(X, Y ) = E[Cov(X, Y |B)] + Cov(E[X |B], E[Y |B]) and Var(X) = E[Var(X |B)] + Var(E[X |B])

c April 24, 2006, U. Schmock, FAM, TU Vienna 29

Observations . . .

Already the Bernoulli model with independent loss

indicators has far too many terms for the calculation

f the portfolio loss distribution in the general case.
In the general Bernoulli mixture model, individual

terms are too complicated to compute numerically.

Different exposures and recovery rates are not even

considered. . . . and Conclusions

Simplifying assumptions are necessary.
Approximations need to be considered.

c April 24, 2006, U. Schmock, FAM, TU Vienna 30

Divisibility/Additivity of the Poisson Distribution Let X ∼ Poisson(λ), Y ∼ Poisson(µ) be independent. Then X +Y ∼ Poisson(λ+µ) because for every n ∈ N0 P(X + Y = n) =

n

k=0

P(X + Y = n|Y = k)

= P(X=n−k|Y =k) = P(X=n−k)

P(Y = k) =

n

k=0

e−λ λn−k (n−k)! e−µ µk k! = e−(λ+µ) n!

n

k=0

n k

λn−kµk
= (λ+µ)n

.

More on divisibility: F.R. Steutel, K.V. Harn: Infinite Divisibility

f Probability Distributions on the Real Line. Pure and Applied

Mathematics, Vol. 259 (2004), Dekker, ISBN 0-8247-0724-9.

c April 24, 2006, U. Schmock, FAM, TU Vienna 31

Simple Poisson Model for Defaults

Number Ni of defaults of obligor i ∈ {1, . . . , m}
Assume Ni ∼ Poisson(λi) for all i ∈ {1, . . . , m}

(several defaults of an obligor possible).

Assume independence of N1, . . . , Nm.
Random number of defaults N = N1 + · · · + Nm.
N ∼ Poisson(λ) with λ = λ1 + · · · + λm, i.e.,

P(N = n) = λn n! e−λ for all n ∈ N0.

m = 20, λi = 0.2 =

⇒ P(N > 20) ≤ 2 × 10−9.

c April 24, 2006, U. Schmock, FAM, TU Vienna 32

SLIDE 9

Poisson Approximation

X1, . . . , Xm independent default 0-1-indicators
Intensity λ = m

i=1 pi with pi = P(Xi = 1)

Number of default events W = m

i=1 Xi

Total variation distance

dTV(µ, ν) = sup

A⊂N0

|µ(A) − ν(A)| Quality of Poisson approximation (Barbour/Hall, 1984): dTV

L(W), Poisson(λ)
≤ 1 − e−λ

λ

m

i=1

p2

i

For full proof with Stein–Chen method, see e.g. Barbour, Holst and Janson: Poisson Approximation, Clarendon Press (1992).

c April 24, 2006, U. Schmock, FAM, TU Vienna 33

Proof of Poisson Approximation via Coupling Weaker claim (Le Cam, 1960): 1 2

k∈N0
P(W = k) − λk

k! e−λ

≤

m

i=1

p2

i

Proof: For i ∈ {1, . . . , m} define Ωi = {−1} ∪ N0, Pi({k}) = ⎧ ⎪ ⎨ ⎪ ⎩ e−pi − (1 − pi) ≥ 0 for k = −1, 1 − pi for k = 0, pk

i e−pi/k!

for k ∈ N, and the product space Ω = Ω1 × · · · × Ωm with the product measure P = P1 ⊗ · · · ⊗ Pm.

c April 24, 2006, U. Schmock, FAM, TU Vienna 34

For i ∈ {1, . . . , m} and ω = (ω1, . . . , ωm) ∈ Ω define Ni(ω) = if ωi ∈ {−1, 0}, ωi if ωi ≥ 1, and Xi(ω) = if ωi = 0, 1

therwise.

X1, . . . , Xm are independent with P(Xi = 1) = pi. N1, . . . , Nm are independent with Ni ∼ Poisson(pi). P(Xi = Ni) = 1 − P(Xi = Ni) = 1 − Pi({0, 1}) = 1 − (1−pi +pie−pi) = pi(1−e−pi

≤pi

) ≤ p2

i

c April 24, 2006, U. Schmock, FAM, TU Vienna 35

Note that N = N1 + · · · + Nm ∼ Poisson(λ). With error ek = P(W = k) − P(N = k) the total variation distance is attained on M = {k ∈ N0 | ek > 0}: 1 2

k∈N0

|ek| = 1 2

k∈M

ek − 1 2

k∈N0\M

ek =

k∈M

ek − 1 2

k∈N0

ek

= 0

= P(W ∈ M) − P(N ∈ M) = P(W ∈ M, W = N

⊂ {W =N}

) + P(W ∈ M, W = N

⊂ {N∈M}

) − P(N ∈ M) ≤ P(W = N) ≤

m

i=1

P(Xi = Ni)

≤p2

i

≤

m

i=1

p2

i

c

April 24, 2006, U. Schmock, FAM, TU Vienna 36

SLIDE 10

Poisson Approximation using Stein–Chen Method For W = m

i=1 Xi and N ∼ Poisson(λ) with λ = E[W]

we claim sup

A⊂N0

P(W ∈ A) − P(N ∈ A)
≤ 1 − e−λ

λ

m

i=1

p2

i .

For A ⊂ N0 and l ∈ N0 define fA(l) = 1A(l) − P(N ∈ A). Then fN0 = 0, E[fA(N)] = 0 and E[fA(W)] = P(W ∈ A) − P(N ∈ A).

c April 24, 2006, U. Schmock, FAM, TU Vienna 37

Stein Equation for fA and Its Solution For A ⊂ N0 and l ∈ N0 define gA(0) = 0 and gA(l + 1) = P(N ≤ l, N ∈ A) − P(N ≤ l) P(N ∈ A) λ P(N = l) . Since λ P(N = l − 1) = e−λλl/(l − 1)! = l P(N = l), gA solves the Stein equation λgA(l + 1) − lgA(l) = P(N = l, N ∈ A) − P(N = l) P(N ∈ A) P(N = l) = fA(l). Exercise: gA is bounded and unique.

c April 24, 2006, U. Schmock, FAM, TU Vienna 38

Key Estimate Using the Stein Equation With W = X1 + · · · + Xm and Wi = W − Xi we get E[fA(W)] = E[λgA(W +1) − WgA(W)] = λ

= m

i=1 pi

E[gA(W +1)] −

m

i=1

E[XigA(Wi + Xi)|Xi = 1]

= E[gA(Wi+1)] by indep. of Wi, Xi

pi =

m

i=1

pi E[gA(W +1)

= gA(Wi+1) if Xi = 0, and gA(Wi+2) otherwise

−gA(Wi +1)|Xi = 1] pi. Hence

E[fA(W)]
≤ sup

l∈N

gA(l + 1) − gA(l)
=: ΔgA(l)
m
i=1

p2

i.

c April 24, 2006, U. Schmock, FAM, TU Vienna 39

Estimate of Increments ΔgA of the Stein Solution Set gk = g{k}. Since Δgk(l) ≤ 0 for k = l (exercise), we have ΔgA(l) =

k∈A Δgk(l) ≤ Δgl(l).

Since fA + fAc = fN0 = 0, we have gA = −gAc, hence −ΔgA(l) = ΔgAc(l) ≤ Δgl(l). Finally, Δgl(l) = gl(l + 1) − gl(l) =

= P(N>l)

1 − P(N ≤ l)

λ +

≤ l P(1≤N≤l)/λ

P(N ≤ l − 1) P(N = l)

λ P(N = l −1)

= l P(N=l)

≤ P(N ≥ 1) λ = 1 − e−λ λ .

c April 24, 2006, U. Schmock, FAM, TU Vienna 40

SLIDE 11

General Poisson Mixture Model

Λ1, . . . , Λm random default intensities
Assume for every i ∈ {1, . . . , m} and ni ∈ N0

P(Ni = ni |Λ1, . . . , Λm)

a.s.

= P(Ni = ni |Λi)

a.s.

= Λni

i e−Λi/ni!

Assume independence of N1, . . . , Nm given Λ1, . . . , Λm.
Then for the number N = N1 + · · · + Nm of defaults

L(N |Λ1, . . . , Λm) = Poisson(Λ1 + · · · + Λm), i.e., for all n ∈ N0, P(N = n) = E (Λ1 + · · · + Λm)n n! e−(Λ1+···+Λm) .

c April 24, 2006, U. Schmock, FAM, TU Vienna 41

Gamma-Poisson Mixture Model

Assume Λi = λiΛ for all obligors i ∈ {1, . . . , m},

where Λ is gamma distributed with α, β > 0. Density fΛ(x) = βα Γ(α)xα−1e−βx, x ≥ 0.

Define λ = λ1 + · · · + λm and p = β/(β + λ).

Then for every n ∈ N0 P(N = n) = E (λΛ)n n! e−λΛ = n+α−1 n

pα(1−p)n.

= ⇒ No. of defaults has negative binomial distribution!

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Input Parameters of CreditRisk+ (Extended Version)

Number of obligors m ∈ N.
Basic loss unit E > 0.
Number K ∈ N0 of risk factors or non-idiosyncratic,

independent default causes.

Relative default variances σ2

k > 0 of risk factors

k ∈ {1, . . . , K}.

Collection G of nonempty subsets of all obligors

{1, . . . , m}, called risk groups.

c April 24, 2006, U. Schmock, FAM, TU Vienna 43

Input Parameters of CreditRisk+ (Cont.) For every group g ∈ G we need

the (one year) default probability pg∈ [0, 1],
the susceptibility wg,0∈ [0, 1] to idiosyncratic risk,
the susceptibilities wg,k∈ [0, 1] to risk factors k ∈

{1, . . . , K},

the multivariate probability distributions

Qg,k = {qg,k,µ}µ∈Ng

0 on Ng

0 describing the stochastic

losses of all the obligors i ∈ g in multiples of the basic loss unit E in case the risk group g defaults due to risk k ∈ {0, . . . , K}.

c April 24, 2006, U. Schmock, FAM, TU Vienna 44

SLIDE 12

Further Assumptions, Notation

We assume that every obligor i ∈ {1, . . . , m} belongs

to at least one group g ∈ G.

Let Gi := {g ∈ G | i ∈ g} denote the set of all risk

groups to which obligor i ∈ {1, . . . , m} belongs, by assumption Gi = ∅.

We assume that for each group the susceptibilities

(also called weights) exhaustively describe the risk

factors. That is, for all g ∈ G,

K

k=0

wg,k = 1.

c April 24, 2006, U. Schmock, FAM, TU Vienna 45

Notation for Default Events of Risk Groups Number of defaults for every risk group g ∈ G:

Ng,0 due to idiosyncratic risk,
Ng,k due to risk k ∈ {1, . . . , K},
Ng := K

k=0 Ng,k total.

Notation for Default Events of Individual Obligors Number of defaults for every obligor i ∈ {1, . . . , m}

Ni,0 :=

g∈Gi Ng,0 due to idiosyncratic risk,

Ni,k :=

g∈Gi Ng,k due to risk k ∈ {1, . . . , K},

Ni := K

k=0 Ni,k = g∈Gi Ng total.

c April 24, 2006, U. Schmock, FAM, TU Vienna 46

Notation for Stochastic Losses Loss at default number n ∈ N of risk group g ∈ G due to risk factor k ∈ {1, . . . , K} or idiosyncratic risk k = 0

Lg,i,k,n part attributed to obligor i ∈ g
Lg,k,n :=

i∈g Lg,i,k,n loss of entire group

Summation over default numbers, risks and groups:

Lg,k := Ng,k

n=1 Lg,k,n total loss of the group for risk k

Lg := K

k=0 Lg,k total of the risk group

L =

g∈G Lg portfolio loss

c April 24, 2006, U. Schmock, FAM, TU Vienna 47

Loss Attributed to Obligor i ∈ {1, . . . , m}

Due to group g ∈ Gi and risk k ∈ {0, . . . , K}

Lg,i,k :=

Ng,k

n=1

Lg,i,k,n.

Due to risk k ∈ {0, . . . , K}

Li,k :=

g∈Gi

Lg,i,k.

Total attributed loss

Li :=

K

k=0

Li,k.

c April 24, 2006, U. Schmock, FAM, TU Vienna 48

SLIDE 13

Probabilistic Assumptions for the Extended Version of CreditRisk+

For every group g ∈ G and every risk k ∈ {0, . . . , K},

the sequence of Ng

0-valued random vectors (Lg,i,k,n)i∈g

with n ∈ N is i.i.d. and independent of all other ran- dom variables, with distribution P(Lg,i,k,1 = µi for all i ∈ g) = qg,k,µ, µ ∈ Ng

0.

For each group g ∈ G, the number Ng,0 of idiosyn-

catic defaults is Poisson distributed according to the Poisson intensity λg and the susceptibility wg,0, i.e., Ng,0 ∼ Poisson(λgwg,0) for every g ∈ G.

c April 24, 2006, U. Schmock, FAM, TU Vienna 49

Probabilistic Assumptions (Cont.)

The group default numbers {Ng,0}g∈G due to id-

iosyncratic risk are independent from one another and from all other random variables.

The risks factors Λ1, . . . , ΛK are independent,

each one gamma distributed with E[Λk] = 1 and Var(Λk) = σ2

k > 0, i.e., αk = βk = 1/σ2 k.

For all groups g ∈ G and risks k ∈ {1, . . . , K},

L (Ng,k| Λ1, . . . , ΛK)

a.s.

= L (Ng,k| Λk)

a.s.

= Poisson(λgwg,kΛk) .

Conditionally on Λ1, . . . , ΛK, the risk factor based de-

faults

Ng,k
g ∈ G, k ∈ {1, . . . , K}
are independent.

c April 24, 2006, U. Schmock, FAM, TU Vienna 50

Default Numbers: Expectation and Variance

Expected number of defaults of obligor i ∈ {1, . . . , m}

E[Ni] =

g∈Gi

λg

Corresponding variance

Var(Ni) =

g∈Gi

λg +

K

k=1

g∈Gi

λgwg,k 2 Var(Λk)

=σ2

k

Covariance of default numbers of obligors i = j

Cov(Ni, Nj) =

g∈Gi∩Gj

λg+

K

k=1

σ2

k g∈Gi

λgwg,k

g′∈Gj

λg′wg′,k

c

April 24, 2006, U. Schmock, FAM, TU Vienna 51

Expectation, Variance and Covariance in Standard Version of CreditRisk+

Expected number of defaults of obligor i ∈ {1, . . . , m}

E[Ni] = λi

Corresponding variance

Var(Ni) = λi + λ2

i K

k=1

wi,kσ2

k

Covariance of default numbers for obligors i = j

Cov(Ni, Nj) = λiλj

K

k=1

σ2

kwi,kwj,k

c April 24, 2006, U. Schmock, FAM, TU Vienna 52

SLIDE 14

Weighted Probability Generating Function In order to calculate terms needed for the risk contri- butions we will need what we call weighted probability generating functions. Definition: For L : Ω → N0 and an integrable random variable X : Ω → R, we define the X-weighted probability generating function by ϕL,X(s) = E

XsL

=

∞

n=0

E

X1{L=n}
sn,

which is meaningful at least for all s ∈ C with |s| ≤ 1.

c April 24, 2006, U. Schmock, FAM, TU Vienna 53

Weighted Probability Generating Function (Cont.) We will need expressions of the form E[Λk1{L=n}] for k ∈ {1, . . . , K} and n ∈ N0, which can be derived by ϕ(n)

L,Λk(0) = n! E

Λk1{L=n}
.

Unifying approach for the γ-weighted probability gener- ating function of the loss: Fix γ = (γ1, . . . , γK) ∈ [0, ∞)K and define ϕL,γ(s) := E

Λγ1

1 . . . ΛγK K sL

, |s| ≤ 1, for the risk factors Λ1, . . . , ΛK and the total loss L. γ = 0 gives the probability generating function ϕL of L.

c April 24, 2006, U. Schmock, FAM, TU Vienna 54

WPGF Expansion Details ϕL,γ(s) = E

Λγ1

1 . . . ΛγK K sL

=

g∈G

E

sLg,0 K
k=1

E

Λγk

k s

g∈G Lg,k

=

g∈G

exp

λgwg,0(ϕLg,0,1(s) − 1)
×

K

k=1
E[Λγk

k ]

1−σ2

k

g∈G

λgwg,k(ϕLg,k,1(s)−1)

−(

1 σ2 k

+γk)

Inversion similar to: H. Haaf, O. Reiß, J. Schoenmakers, Numerically Stable Computation of CreditRisk+, In: M. Gundlach and F. Lehrbass (eds.), CreditRisk+ in the Banking Industry, Springer, 2003, 69–77.

c April 24, 2006, U. Schmock, FAM, TU Vienna 55

The Closed Form of the WPGF Using the probablistic assumptions and properties of the Poisson and the Gamma distribution we arrive at ϕL,γ(s) = Cγ exp

ψ0(s)−

K

k=1

1 σ2

k

+γk

log
1−σ2

kψk(s)

,

where Cγ := K

k=1 E[Λγk k ] = 1 if all γk ∈ {0, 1}, which

are the cases which we will consider later on and ψk(s) :=

g∈G

λg wg,k(ϕLg,k,1(s) − 1) for all k ∈ {0, . . . , K}.

c April 24, 2006, U. Schmock, FAM, TU Vienna 56

SLIDE 15

10 20 30 40 50 0.01 0.02 0.03 Loss distribution in a credit portfolio of 25 exposures calculated with CreditRisk+ and basic loss unit 100 000. 95% Probability Portfolio loss in millions 25 Exposures Total: 130 513 072 Largest: 20 238 895 with p = 7.5% 15 410 906 with p = 10% 7 727 651 with p = 1.6% Smallest: 358 475 with p = 30% No loss with p = 5.76%

c April 24, 2006, U. Schmock, FAM, TU Vienna 57

20 40 60 80 100 120 0.0005 0.001 0.0015 Loss distribution in a credit portfolio of 100 exposures calculated with CreditRisk+ and basic loss unit 100 000. 99% Probability Portfolio loss in millions

c April 24, 2006, U. Schmock, FAM, TU Vienna 58

200 400 600 800 1000 0.0001 0.0002 Portfolio loss in millions Loss distribution in a credit portfolio of 1000 exposures calculated with CreditRisk+ and basic loss unit 100 000. 99% Probability

c April 24, 2006, U. Schmock, FAM, TU Vienna 59

Measuring Risk by Quantiles Let X be a loss variable and δ ∈ (0, 1) a level. Definition: Lower δ-quantile of X qδ(X) := min{x ∈ R | P(X ≤ x) ≥ δ}. Remark: Quantiles are used as value-at-risk, they have bad properties concerning diversification. Properties: qδ(X) can jump when

the level δ varies slightly,
the loss variable X varies slightly.

c April 24, 2006, U. Schmock, FAM, TU Vienna 60

SLIDE 16

Measuring Risk by Expected Shortfall Let X be a loss variable and δ ∈ (0, 1) a level. Definition: The expected shortfall is defined as ESδ[X] := E[X1{X>qδ(X)}] + qδ(X)(P(X ≤ qδ(X)) − δ) 1 − δ . Remark: If P(X ≤ qδ(X)) = δ, in particular if the distribution function R ∋ x → P(X ≤ x) of X is also left-continuous at x = qδ(X), then ESδ[X] = E[X |X > qδ(X)].

c April 24, 2006, U. Schmock, FAM, TU Vienna 61

Properties of Expected Shortfall

Positive homogeneity: If α > 0, then

ESδ[αX] = α ESδ[X].

Translation (or cash) invariance: If a ∈ R, then

ESδ[X + a] = ESδ[X] + a.

Scenario representation:

(i) ESδ[X] = sup

f∈Fδ,X

E[Xf], (ii) if E[X+] < ∞, then ESδ[X] = sup

f∈Fδ

E[Xf].

c April 24, 2006, U. Schmock, FAM, TU Vienna 62

Properties of Expected Shortfall (Cont.)

Sub-additivity:

ESδ[X + Y ] ≤ ESδ[X] + ESδ[Y ].

Monotonicity: If X ≤ Y , then

ESδ[X] ≤ ESδ[Y ].

Convexity: If α ∈ (0, 1), then

ESδ[αX + (1 − α)Y ] ≤ α ESδ[X] + (1 − α) ESδ[Y ].

Bounds:

qδ(X) ≤ ESδ[X] ≤ E[X+] 1 − δ .

c April 24, 2006, U. Schmock, FAM, TU Vienna 63

Properties of Expected Shortfall (Cont.)

Quantile representation:

ESδ[X] = 1 1 − δ

[δ,1)

qu(X) du.

Let {Xn}n∈N be bounded below, i.e., Xn ≥ −a for

some a ∈ [0, ∞) all n ∈ N. Then ESδ

lim inf

n→∞ Xn

≤ lim inf

n→∞ ESδ[Xn].

Let {Xn}n∈N be bounded below and Xn

P

→ X. Then ESδ[X] ≤ lim inf

n→∞ ESδ[Xn].

c April 24, 2006, U. Schmock, FAM, TU Vienna 64

SLIDE 17

Calculation of Expected Shortfall in CreditRisk+

Credit portfolio loss L is a discrete random variable,

→ More complicated definition has to be used.

The lower quantile qδ(L) and P(L ≤ qδ(L)) can be

calculated using the CreditRisk+ algorithm.

Furthermore E[L1{L>qδ(L)}] = E[L]−E[L1{L≤qδ(L)}]

with E[L] =

g∈G

K

k=0

λg wg,kE[Lg,k,1] and E

L1{L≤qδ(L)}
=

qδ(L)

l=1

l P(L = l).

c April 24, 2006, U. Schmock, FAM, TU Vienna 65

Contributions to Expected Shortfall – Definition Definition: For a subportfolio loss X ∈ L1(P) within a portfolio loss Y ∈ L1(P) define the expected shortfall contribution at level δ ∈ (0, 1) of X to Y by ESδ[X, Y ] = E[X1{Y >qδ(Y )}] + βY E[X1{Y =qδ(Y )}] 1 − δ where βY = P(Y ≤ qδ(Y )) − δ P(Y = qδ(Y )) if P(Y = qδ(Y )) > 0 and 0 otherwise. Remark: If P(Y ≤ qδ(Y )) = δ, then βY = 0 and ESδ[X, Y ] = E[X|Y > qδ(Y )].

c April 24, 2006, U. Schmock, FAM, TU Vienna 66

Contributions to Expected Shortfall – Properties For X, Y, Z ∈ L1(P) we have the following properties:

Consistency with expected shortfall:

ESδ[X, X] = ESδ[X].

Diversification:

ESδ[X, Y ] ≤ ESδ[X, X].

Linearity: For all α, β ∈ R,

ESδ[αX + βY, Z] = α ESδ[X, Z] + β ESδ[Y, Z].

Monotoncity: If X ≤ Y , then

ESδ[X, Z] ≤ ESδ[Y, Z].

c April 24, 2006, U. Schmock, FAM, TU Vienna 67

Contributions to Expected Shortfall – Properties (2)

Independence: If X and Y are independent, then

ESδ[X, Y ] = E[X] .

Invariance of portfolio scale: For all α > 0

ESδ[X, αY ] = ESδ[X, Y ].

Subportfolio continuity:
ESδ[X, Z] − ESδ[Y, Z]
≤ ESδ[|X − Y |, Z] ≤ E[|X − Y |]

1 − δ .

c April 24, 2006, U. Schmock, FAM, TU Vienna 68

SLIDE 18

Contributions to Expected Shortfall – Properties (3)

Portfolio continuity:

If P(Y ≤ qδ(Y )) = δ, then capital allocation by expected shortfall at level δ is continuous at Y , i.e., for every {Yn}n∈N ⊂ L1(P) with Yn

P

→ Y , lim

n→∞ ESδ[X, Yn] = ESδ[X, Y ].

Representation of expected shortfall contributions

by directional derivatives: If capital allocation by expected shortfall is continu-

us at Y as specified by portfolio continuity, then

ESδ[X, Y ] = lim

ε→0

ESδ[Y + εX] − ESδ[Y ] ε .

c April 24, 2006, U. Schmock, FAM, TU Vienna 69

Contributions to Expected Shortfall – Calculation in Extended CreditRisk+ By definition, ESδ[Lg,i,k, L] = E

Lg,i,k1{L>qδ(L)}
+ βL E
Lg,i,k1{L=qδ(L)}
1 − δ

is the contribution attributed to obligor i ∈ {1, . . . , m} due to group g ∈ Gi and risk k ∈ {0, . . . , K} to the expected shortfall ESδ[L].

c April 24, 2006, U. Schmock, FAM, TU Vienna 70

Contributions to Expected Shortfall – Calculation in Extended CreditRisk+ (Cont.) By consistency and linearity of the allocation ESδ[L] = ESδ[L, L] =

m

i=1
g∈Gi

K

k=0

ESδ[Lg,i,k, L]. Since E

Lg,i,k1{L>qδ(L)}
= E[Lg,i,k]
= λg wg,k E[Lg,i,k,1]

− E

Lg,i,k1{L≤qδ(L)}
,

we will compute E[Lg,i,k1{L=l}] for l ∈ {1, . . . , qδ(L)}. This can be done using the following lemma.

c April 24, 2006, U. Schmock, FAM, TU Vienna 71

Lemma on Risk Contributions in CreditRisk+ For every obligor i ∈ {1, . . . , m}, every group g ∈ Gi and total loss l ∈ N0, E[Lg,i,01{L=l}] = λgwg,0

l

ν=1

E

Lg,i,0,11{Lg,0,1=ν}
P(L = l − ν)

and, for every risk k ∈ {1, . . . , K}, E[Lg,i,k1{L=l}] = λgwg,k

l

ν=1

E[Lg,i,k,11{Lg,k,1=ν}] E[Λk1{L=l−ν}].

c April 24, 2006, U. Schmock, FAM, TU Vienna 72

SLIDE 19

Introduction to Operational Risk: The Regulatory Framework and the Data Quantification of operational risk of financial institutions gained importance due to the Basel II accord (column 1) for capital requirements. Famous Examples for Operational Losses

Bankruptcy of the British Barings Bank in 1995
The terror attacks on the World Trade Center

in New York City on September 11th, 2001.

c April 24, 2006, U. Schmock, FAM, TU Vienna 73

The Nature of Operational Losses

Operational losses may occur frequently

with low impact

Also rare events with high impact occur,

this calls for the application of methods from extreme value theory.

Difference between operational risk and credit or

market risk: there is no chance for profit.

Operational risk comes along with any process
f a bank’s business despite of all efforts

to avoid malfunctions.

c April 24, 2006, U. Schmock, FAM, TU Vienna 74

Possible Approaches to Operational Risk Modelling The Basel committee defined three approaches towards the quantification of operational risk. The two simple

nes define concrete formulae for the risk capital,

namely

Basic indicator approach (BIA).
Standardized approach (SA).

To reduce supervisory capital needs, an individual

Advanced measurement approach (AMA)

can be chosen.

c April 24, 2006, U. Schmock, FAM, TU Vienna 75

Business Lines for Operational Risk

Eight business lines in the standardized approach:

(5) Payment & settlement, (1) Corporate finance, (6) Agency services, (2) Trading & sales, (7) Asset management, (3) Retail banking, (8) Retail brokerage. (4) Commercial banking,

These business lines also serve as categories

for an advanced measurement approach.

c April 24, 2006, U. Schmock, FAM, TU Vienna 76

SLIDE 20

Seven Loss-Types Distinguished for the Advanved Measurement Approach

Internal fraud
External fraud
Employment practices & workplace safety
Clients, products & business practice
Damage to physical assets
Business disruption & system failures
Execution, delivery & process management

c April 24, 2006, U. Schmock, FAM, TU Vienna 77

Remarks on Operational Loss Data

Usually little data on operational risk is available.
Estimation of frequent losses

can probably be managed using internal data.

For rare events causing high losses,
ften external data has to be used.
Attention: observed reporting bias

(increasing frequency of losses) coming from the rising awareness of the importance to collect operational risk data.

c April 24, 2006, U. Schmock, FAM, TU Vienna 78

Moscadelli’s Study of Operational Loss Data Moscadelli1 found that

severity distributions are heavy-tailed,
six business lines (among the eight mentioned

before) yield estimations of distributions with infinite mean. This causes problems when calculating risk measures (expected shortfall gives infinity).

1M. Moscadelli, The modelling of operational risk: experience with

the analysis of the data collected by the Basel Comittee, Temi di Discussione del Servizio Studi 517 (2004), available on the home- page of Banca d’Italia www.bancaditalia.it/ricerca/consultazioni/temidi.

c April 24, 2006, U. Schmock, FAM, TU Vienna 79

Application of Extended CreditRisk+ Methodology to Operational Risk: Reinterpretation of the Credit Risk Notation

Number m of obligors → number of business lines

(m = 8 for the ones given in the Basel commitee’s document is an appropriate choice).

Basic loss unit E stays the same (E = 10 000).
Number K of non-ideosyncratic risk factors

→ number of loss types (K = 7 for above list).

Numbers σ2

k > 0 denote the relative variance of

ccurrences of losses of type k ∈ {1, . . . , K}.
G contains the subsets of all business lines which

can incur a loss due to the same event.

c April 24, 2006, U. Schmock, FAM, TU Vienna 80

SLIDE 21

Notation for Business Lines and Risk Groups We need for every risk group g ∈ G of business lines

the (one year) intensity λg ≥ 0 for being hit by an
perational loss event,
the conditional probabilty wg,0 ∈ [0, 1] for an

idiosyncratic operational loss event not to belong to the types in {1, . . . , K}, of course wg,0 = 0 is a possible choice,

the conditional probabilities wg,k ∈ [0, 1] for an
perational loss event to be of type k ∈ {1, . . . , K},

c April 24, 2006, U. Schmock, FAM, TU Vienna 81

the multivariate probability distribution Qg,k =

{qg,k,µ}µ∈Ng

0 on Ng

0 describing the severity of the

stochastic losses of the business lines i ∈ g in case an operational loss event of type k ∈ {0, . . . , K} hits the group g of business lines. Operational Risk Management With the adoption of the extended CreditRisk+ model for operational risk, a risk manager can

calculate the distribution of the operational loss,
calculate risk measures such as value-at-risk and

expected shortfall (might be infinity)

and identify risky business lines and groups by their

risk contribution (in case of finite expected shortfall).

c April 24, 2006, U. Schmock, FAM, TU Vienna 82

Extensions of CreditRisk+: Dependent Risk Factors

Hidden gamma distributed risk factor groups:

A collection H of nonempty h ⊂ {1, . . . , K}, each factor k belonging to at least on h ∈ H, and independent {Λh}h∈H with Λh ∼ Γ(αh, 1). Define Λk = 1 βk

h∈H,h∋k

Λh ∼ Γ(αk, βk) with αk = βk =

h∈H,h∋k αh.

Compound gamma distribution:

Multiply the shape parameter of the gamma dis- tributed Λ1, . . . , ΛK by an independent gamma distributed S with E[S] = 1 and Var(S) = σ2.

c April 24, 2006, U. Schmock, FAM, TU Vienna 83

Some Literature on Credit Risk Modelling

Tomasz R. Bielecki and Marek Rutkowski, Credit Risk:

Modeling, Valuation and Hedging, Springer (2003).

Ch. Bluhm, L. Overbeck and Ch. Wagner, An Introduction

to Credit Risk Modeling, Chapman & Hall/CRC (2003).

Georg Bol et al. (Eds.), Credit Risk, Measurement,

Evaluation and Management, Physica-Verlag, (2003).

D. Duffie and K.J. Singleton, Credit Risk: Pricing, Measure-

ment and Management, Princeton Univ. Press (2003).

Bernd Schmid, Pricing Credit Linked Financial Instruments:

Theory and Empirical Evidence, Lecture Notes in Economics and Math. Systems, Vol. 516, Springer (2002).

Philipp J. Sch¨
nbucher, Credit Derivatives Pricing Models.

Models, Pricing and Implementation, Wiley (2003).

M. Gundlach and F. Lehrbass (Eds.), CreditRisk+ in the

Banking Industry, Springer (2004).

c April 24, 2006, U. Schmock, FAM, TU Vienna 84