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Quantitative Methods for RiskManagement
Paul Embrechts and Johanna Neslehova
ETH Zurich
www.math.ethz.ch/∼embrecht
BaFin 2006
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L. Operational Risk and Insurance Analytics
1. A New Risk Class
2. Insurance Analytics Toolkit
3. The Capital Charge Problem
4. Marginal VaR Estimation
5. Global VaR Estimation
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L1. A New Risk Class
The New Accord (Basel II)
• 1988: Basel Accord (Basel I): minimal capital requirements against
credit risk, one standardised approach, Cooke ratio
• 1996: Amendment to Basel I: market risk, internal models, netting
• 1999: Several Consultative Papers on the New Accord (Basel II)
• to date: CP3: Third Consultative Paper on the
New Basel Capital Accord (www.bis.org/bcbs/)
• 2007+: full implementation of Basel II
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Basel II: What is new?
• Rationale for the New Accord: More flexibility and risk sensitivity
• Structure of the New Accord: Three-pillar framework:
Ê Pillar 1: minimal capital requirements (risk measurement)
Ë Pillar 2: supervisory review of capital adequacy
Ì Pillar 3: public disclosure
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Basel II: What is new? (cont’d)
• Two options for the measurement of credit risk:
− Standard approach
− Internal rating based approach (IRB)
• Pillar 1 sets out the minimum capital requirements
(Cooke Ratio, McDonough Ratio):
total amount of capital
risk-weighted assets≥ 8%
• MRC (minimum regulatory capital)def= 8% of risk-weighted assets
• Explicit treatment of operational risk
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Operational Risk
Definition:
The risk of losses resulting from inadequate or failed internal
processes, people and systems, or external events.
Remark:
This definition includes legal risk, but excludes strategic and
reputational risk.
Note:
Solvency 2
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Operational Risk (cont’d)
• Notation: COP: capital charge for operational risk
• Target: COP ≈ 12% of MRC (down from initial 20%)
• Estimated total losses in the US (2001): $50b
• Some examples
− 1977: Credit Suisse Chiasso-affair
− 1995: Nick Leeson/Barings Bank, £1.3b
− 2001: September 11
− 2001: Enron (largest US bankruptcy so far)
− 2002: Allied Irish, £450m
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Risk Measurement Methods for Operational Risk
Pillar 1 regulatory minimal capital requirements for operational risk:
Three distinct approaches:
1. Basic Indicator Approach
2. Standardised Approach
3. Advanced Measurement Approach (AMA)
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Basic Indicator Approach
• Capital charge:
CBIAOP = α×GI
• CBIAOP : capital charge under the Basic Indicator Approach
• GI: average annual gross income over the previous three years
• α = 15% (set by the Committee based on CISs)
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Standardised Approach
• Similar to the BIA, but on the level of each business line:
CSAOP =
8∑i=1
βi ×GIi
βi ∈ [12%, 18%], i = 1, 2, . . . , 8 and 3-year averaging
• 8 business lines:
Corporate finance (18%) Payment & Settlement (18%)
Trading & sales (18%) Agency Services (15%)
Retail banking (12%) Asset management (12%)
Commercial banking(15%) Retail brokerage (12%)
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Advanced Measurement Approach (AMA)
• Allows banks to use their internally generated risk estimates
• Preconditions: Bank must meet qualitative and quantitative
standards before being allowed to use the AMA
• Risk mitigation via insurance possible (≤ 20% of CSAOP)
• Incorporation of risk diversification benefits allowed
• “Given the continuing evolution of analytical approaches for
operational risk, the Committee is not specifying the approach
or distributional assumptions used to generate the operational risk
measures for regulatory capital purposes.”
• Example: Loss distribution approach
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Internal Measurement Approach
• Capital charge (similar to Basel II model for Credit Risk):
C IMAOP =
8∑i=1
7∑k=1
γik eik (first attempt)
eik: expected loss for business line i, risk type k
γik: scaling factor
• 7 loss types: Internal fraud
External fraud
Employment practices and workplace safety
Clients, products & business practices
Damage to physical assets
Business disruption and system failures
Execution, delivery & process management
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Loss Distribution Approach (LDA)
BL8
rrr
rrr
r r r r r r
BLi
BL1
RT1 RTk RT7
LT+1
LT+1i,k
����
����
��
����
����
���'
&$%
type 2
1992 1994 1996 1998 2000 2002
05
1015
20
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LDA: continued
• For each business line/loss type cell (i, k) one models
LT+1i,k : OP risk loss for business line i, loss type k
over the future (one year, say) period [T, T + 1]
LT+1i,k =
NT+1i,k∑
`=1
X`i,k (next period’s loss for cell (i, k))
Note that X`i,k is truncated from below
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LDA: continued
Remark: Look at the structure of the loss random variable LT+1
LT+1 =8∑
i=1
7∑k=1
LT+1i,k (next period’s total loss)
=8∑
i=1
7∑k=1
NT+1i,k∑
`=1
X`i,k
=8∑
i=1
LT+1i (often used decomposition)
• Check again the overall complexity of the (BL, RT) matrix
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L2. Insurance Analytics: an Essential Toolkit
Total Loss Amount
Denote by N(t) the (random) number of losses over a fixed period
[0, t] and write X1, X2, . . . for the individual losses. The aggregate
loss is
SN(t) =N(t)∑k=1
Xk
Remarks:
• FSN(t)(x) = P (SN(t) ≤ x) is called the total loss df. If t is fixed,
we write SN and FSNinstead
• The random variable SN(t) is also referred to as random sum
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Compound Sums
Assume:
1. (Xk) are iid with common df G, G(0) = 0
2. N and (Xk) are independent
SN is then referred to as a compound sum. The pdf of N is denoted
by pN(k) = P (N = k), k = 0, 1, . . . and N is called a compounding
rv.
Proposition 1:
Let SN be a compound sum and the above assumptions hold. Then
FSN(x) =
∞∑k=0
pN(k)G∗k(x), x ≥ 0
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Proposition 2:
Let SN be a compound sum and the above assumptions hold. Then
the Laplace-Stieltjes transform of SN satisfies
FSN(s) =
∫ ∞
0
e−sxdFSN(x) =
∞∑k=0
pN(k)Gk(s) = MN(G(s)), s ≥ 0
where MN denotes the moment-generating function of N .
Proposition 3:
Let SN be a compound sum and the above assumptions hold. If
E(N2) <∞ and E(X21) <∞, we have that
E(SN) = E(N)E(X1)
var(SN) = var(N)(E(X1))2 + E(N)var(X1)
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Compound Poisson Distribution
Example 1:
Consider N ∼ Poi(λ). Then SN is referred to as a compound
Poisson rv.
• The moment-generating function of N satisfies
MN(s) = exp(−λ(1− s)) and hence
FSN(s) = exp(−λ(1− G(s)))
• Notation: SN ∼ CPoi(λ,G)
• If E(X21) <∞, the moments of SN are by Proposition 3
E(SN) = λE(X1) and var(SN) = λE(X21)
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Aggregation of Compound Poisson rvs
Suppose that the compound sums SNi∼ CPoi(λi, Gi), i = 1, . . . , d
and that these rvs are independent. Then
SN :=d∑
i=1
SNi=
d∑i=1
Ni∑k=1
Xi,k
is again a compound Poisson rv, SN ∼ CPoi(λ,G) where
λ =d∑
i=1
λi and G =d∑
i=1
λi
λGi
G is hence a mixture distribution. A simulation from G can be done
in two steps: first draw i, i ∈ {1, . . . , d} with probability λi/λ and
then draw a loss with df Gi.
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Binomial Loss Model
Example 2:
Suppose N ∼ Bin(n, p). SN is then called the (individual risk)
binomial model.
Consider a time interval [0, 1] and let Nn denote the total number of
losses in [0, 1] for a fixed n. Suppose further that we have a number
of potential loss generators that can produce, with probability pn, a
loss in each small subinterval ((k − 1)/n, k/n], k = 1, . . . , n.
Moreover, the occurrence of a loss in any particular subinterval is
not influenced by the occurrence of losses in other intervals and
npn → λ for a λ > 0 as n→∞.
For a fixed severity distribution, SNn is then a binomial model with
Nn ∼ Bin(n, pn) and converges in law to a compound Poisson rv
with parameter λ as n→∞.
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Over-dispersion
For compound Poisson rvs with N ∼ Poi(λ) we have that
E(N) = var(N) = λ
• Count data however often exhibit over-dispersion meaning that
they indicate E(N) < var(N)
• This can be achieved by mixing, i.e. by randomizing the parameter
λ
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Randomization
Other examples of mixing include:
• from Black-Scholes to stochastic volatility models (randomize σ)
• from Nd(µ,Σ) to elliptical distributions (randomize Σ); i.e. the
multivariate t distribution. Randomization of both µ and Σ leads
to generalized hyperbolic distributions
• mixing models for credit risk (randomizing the default probability)
• credibility theory in insurance (randomizing the underlying risk
parameter)
• Bayesian inference ...
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Poisson Mixtures
Definition: Let Λ be a positive rv with distribution function FΛ. A
rv N given by
P (N = k) =∫ ∞
0
P (N = k|Λ = λ)dFΛ(λ) =∫ ∞
0
e−λ λk
k!dFΛ(λ)
is called a mixed Poisson rv with structure or mixing distribution FΛ.
A compound sum with a mixed Poisson rv as the compounding rv is
called a compound mixed Poisson rv.
Lemma: Suppose that N is mixed Poisson with structure df FΛ.
Then E(N) = E(Λ) and var(N) = E(Λ) + var(Λ), i.e. for Λnon-degenerate, N is over-dispersed.
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Negative Binomial Distribution
Example 3: For Λ ∼ Ga(α, β), the mixed Poisson rv is negative
binomial, N ∼ NB(α, β/(β + 1)):
P (N = k) =(
β
β + 1
)α( 1β + 1
)k Γ(α+ k)(β + 1)α+k
Further,
E(N) =α
β= E(Λ)
var(N) =α(β + 1)
β2=α
β+α
β2= E(Λ) + var(Λ)
Compounding leads to compound mixed Poisson rvs.
• Many more interesting models exist.
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Approximations
• The distribution of SN is generally intractable
Normal approximation for CPoi(λ,G):
FSN(x) ≈ Φ
(x− E(N)E(X1)√
var(N)(E(X1))2 + E(N)var(X1)
)However, the skewness of SN is positive:
E(SN − E(SN))3
(var(SN))3/2=E(X3
1)√λ
> 0
Translated-gamma approximation for CPoi(λ,G):Approximate SN by k + Y where k is a translation parameter and
Y ∼ Ga(α, β). The parameters k, α, β are found by matching the
mean, variance and skewness.
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Example: Light-tailed Severities
Simulated CPoi(100,Exp(1)) data together with normal and
translated gamma approximations. The 99.9% quantile estimates are
also given.
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Example: Heavy-tailed Severities
Simulated CPoi(100,Pa(4, 1)) data together with normal and
translated gamma approximations. GPD approximation based on the
POT method is also performed.
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Panjer Class
• Recursive method for approximating SN in case the severity
distribution G is discrete and N satisfies a specific condition.
Panjer ClassThe probability mass distribution of N belongs to the Panjer(a, b)class for some a, b ∈ R if pN(k) = (a+ (b/k))pN(k − 1) for k ≥ 1
The only nondegenerate examples of distributions belonging to a
Panjer(a, b) class are
• binomial B(n, p) with a = −p/(1− p) and b = (n+ 1)p/(1− p)
• Poisson Poi(λ) with a = 0 and b = λ
• Negative binomial NB(α, p) with a = 1− p and b = (α− 1)(1− p)
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Panjer Recursion
For a discrete severity rv X1 we denote
gi := P (X1 = i) and si := P (SN = i)
Theorem: Suppose N satisfies the Panjer(a, b) class condition and
g0 = 0. Then s0 = pN(0) = 0 and, for k ≥ 1,
sk =k∑
i=1
(a+
bi
k
)gisk−i.
• For continuous severity distributions, discretization necessary
• Correction for g0 > 0 possible
• Estimation of sk far in the tail is more tricky
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Example
Simulated CPoi(100, LN(1, 1)) data together with the Panjer
recursion approximation. Normal, translated gamma and GPD
approximations are also performed.
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Further Topics
• SN can be looked upon as a process in time, SN(t). Instead of N
we then have the process {N(t), t ≥ 0} counting the number of
events in [0, t]. Interesting examples for N(t) are
− homogeneous Poisson process
− non-homogeneous Poisson process
− Cox or doubly stochastic Poisson process
• Of further interest is the surplus process Ct = u+ ct− SN(t) and
the corresponding ruin probability
ψ(u, T ) = Pu{ inf0≤t≤T
Ct ≤ 0}
• Rare event simulation
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Homogeneous Poisson Process
Ten realizations of a homogeneous Poisson process with λ = 100.
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Mixed Poisson Process
Ten realizations of a mixed Poisson process with Λ ∼ Ga(100, 1).
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Ruin Probability
Recall ruin results for
ψ(u) := ψ(u,∞) = Pu{ inf0≤t≤∞
Ct < 0}
• Cramer-Lundberg: “small claims”
ψ(u) < e−Ru, ∀u > 0
ψ(u) ∼ ε1e−Ru, u→∞
• Embrechts-Veraverbeke: “large claims” with df G
ψ(u) ∼ ε2
∫ ∞
u
G(t)dt, u→∞
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Subexponential Distributions
Definition:
For X1, . . . , Xn positive iid random variables with common
distribution function FX, denote Sn =∑n
k=1Xk and
Mn = max(X1, . . . , Xn). The distribution function FX is called
subexponential (denoted by FX ∈ S) for some (and then for all)
n ≥ 2 if
limx→∞
P (Sn > x)P (Mn > x)
= 1
Examples:
• Pareto, Generalized Pareto, Lognormal, Loggamma, ...
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Ruin Process with Exponential Claims
20 simulations from the ruin process Ct, 0 ≤ t ≤ 1, with
(N(t)) ∼ HPois(100t) and X1 ∼ Exp(1).
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Ruin Process with Pareto Claims
20 simulations from the ruin process Ct, 0 ≤ t ≤ 1, with
(N(t)) ∼ HPois(100t) and X1 ∼ Pareto(2, 1).
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Ruin Process with Exponential Claims
20 simulations from the ruin process Ct, 0 ≤ t ≤ 1, with (N(t)) a doubly
stochastic Poisson process with a two-state Markov intensity process: HPois(10t)
and HPois(100t) with mean holding times 5 and 0.2, respectively, and
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Ruin Process with Pareto Claims
20 simulations from the ruin process Ct, 0 ≤ t ≤ 1, with (N(t)) a doubly
stochastic Poisson process with a two-state Markov intensity process: HPois(10t)
and HPois(100t) with mean holding times 5 and 0.2, respectively, and
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L3. The Capital Charge Problem within LDA
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Loss Distribution Approach (cont’d)
Choose:
• Period T
• Distribution of LT+1i,k for each cell i, k
• Interdependence between cells
• Confidence level α ∈ (0, 1), α ≈ 1
• Risk measure gα
Capital charge for:
• Each cell: CT+1,ORi,k = gα(LT+1
i,k )
• Total OR loss: CT+1,OR based on CT+1,ORi,k
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Basel II proposal
• Period: one year
• Distribution: should be based on
− internal data/models− external data− expert opinion
• Confidence level: α = 99.9%, for economic capital purposes evenα = 99.95% or α = 99.97%
• Risk measure: VaRα
• Total capital charge:
CT+1,OR =∑i,k
VaRα(LT+1i,k )
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Basel II Proposal: Summary
• Marginal VaR calculations
VaR1α, . . . ,VaRl
α
• Global VaR estimate
VaR+α = VaR1
α + · · ·+ VaRlα
• Reduction because of “correlation effects”
VaRα < VaR+α
• Further possibilities: insurance, pooling, ...
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Coherence and VaR
VaRα is in general not coherent:
1. skewness
2. special dependence
3. very heavy-tailed losses
VaRα is coherent for:
• elliptical distributions
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Skewness
• 100 iid loans: 2%-coupon, 100 face value, 1% default probability
(period: 1 year):
Xi ={
−2 with probability 99%
100 with probability 1% (loss)
• Two portfolios L1 =100∑i=1
Xi, L2 = 100X1
• VaR95%(L1)︸ ︷︷ ︸VaR95%
100Pi=1
Xi
! > VaR95%(100X1)︸ ︷︷ ︸100Pi=1
VaR95%(Xi)
(!)
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Special Dependence
• Given rvs X1, . . . , Xn with marginal dfs F1, . . . , Fn, then one can
always find a copula C so that for the joint model
F (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn))
VaRα is superadditive:
VaRα
(n∑
k=1
Xk
)>
n∑k=1
VaRα(Xk)
• In particular, take the “nice” case
F1 = · · · = Fn = N(0, 1)
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Special Dependence
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Very Heavy-tailedness
• Pareto: take X1, X2 independent with P (Xi > x) = x−1/2, x ≥ 1then for x > 2
P (X1 +X2 > x) =2√x− 1x
> P (2X > x)
so that
VaRα(X1 +X2) > VaRα(2X1) = VaRα(X1) + VaRα(X2)
• Pareto-type: similar result holds for X1, X2 independent with
P (Xi > x) = x−1/ξL(x),
where ξ > 1, L slowly varying
• For ξ < 1, we obtain subadditivity - WHY?
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Several reasons:
• (Marcinkiewicz-Zygmund) Strong Law of Large Numbers
• Argument based on stable distributions
• Main reason however comes from functional analysis
In the spaces Lp, 0 < p < 1, there exist no convexopen sets other than the empty set and Lp itself.
Hence as a consequence 0 is the only continuous linear functional
on Lp; this is in violent contrast to Lp, p ≥ 1
• Discussion:
− no reasonable risk measures exist
− diversification goes the wrong way
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Definition:An Rd-valued random vector X is said to be regularly varying if
there exists a sequence (an), 0 < an ↑ ∞, µ 6= 0 Radon measure on
B(Rd\{0}
)with µ(Rd\R) = 0, so that for n→∞,
nP (a−1n X ∈ ·) → µ(·) on B
(Rd\{0}
).
Note that:
• (an) ∈ RV ξ for some ξ > 0
• µ(uB) = u−1/ξµ(B) for B ∈ B(Rd\{0}
)Theorem: (several versions – Samorodnitsky)
If (X1, X2)′ ∈ RV −1/ξ, ξ < 1, then for α sufficiently close to 1,
VaRα is subadditive.
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Phase Transition of Value-at-Risk
Theorem:Assume that X = (X1, X2) is a rv such that
• Xi ∼ F for all i where F is continuous and F ∈ RV −β, β > 0
• −X has an Archimedean copula with generator ψ which is regularly
varying at 0 with index −δ < 0
then there exists a constant q2(δ, β)such that
VaRα(X1 +X2) ∼ (q2(δ, β))1/βVaRα(X1), α→ 1
• Behavior of q2(δ, β) with respect to β and δ, respectively
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L4. Marginal VaR Estimation
LDA revisited
• Recall: VaRi,kα is a Value-at-Risk of a compound sum
LT+1i,k =
NT+1i,k∑
l=1
X li,k
• Tasks:
− Suitable model for the severity Xi,k
− Suitable model for the frequency NT+1i,k
− Estimation of VaRi,kα
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Some OpRisk Data
type 1
1992 1994 1996 1998 2000 2002
010
2030
40
type 2
1992 1994 1996 1998 2000 2002
05
1015
20
type 3
1992 1994 1996 1998 2000 2002
02
46
8
pooled operational losses
1992 1994 1996 1998 2000 2002
010
2030
40
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•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
•••••••••••••••••••
•••••••• ••••
••••••• ••
•••••• • • • •
•• • • •
•
• •
•
•
•
•
•
0 2 4 6 8
24
68
1012
14
Threshold
Mea
n E
xces
s
pooled operational losses: mean excess plot
• P (L > x) ∼ x−1/ξL(x)
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Stylized Facts
• Stylized facts about OpRisk losses:
− Loss amounts show extremes
− Loss occurence times are irregularly spaced in time
(reporting bias, economic cycles, regulation, management interactions,
structural changes, . . . )
− Non-stationarity (frequency(!), severity(?))
• Large losses are of main concern
• Repetitive versus non-repetitive losses
• However: severity is of key importance
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Peaks-over-Threshold Method
• Distribution of the exceedances
• Distribution of the inter-arrival times
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Peaks-over-Threshold Method (POT)
X1, . . . , Xn iid with distribution function F satisfying
F (x) = x−1/ξL(x), ξ > 0 and L slowly varying
• Excess distribution: asymptotically Generalized Pareto (GPD)
P (X − u > y|X > u) ∼(
1 + ξy
β(u)
)−1/ξ
, u→∞
ß POT-MLE estimation of tail probabilities and risk measures
F (x) =Nu
n
(1 + ξ
x− u
β
)−1/ξ
, x > u
VaRα = u+β
ξ
(( nNu
(1− α))−ξ − 1
), α “close” to 1
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Threshold Choice
Application of POT based estimates requires a choice of an
appropriate threshold u
• Rates of convergence: very tricky
- No generally valid convergence rate
- Convergence rate depends on F , in particular on the slowly
varying function L, in a complicated way and may be very slow
- L is not visible from data directly
ß Threshold choice is very difficult. Trade-off between bias and
variance usually takes place.
• Diagnostic tools:
- Graphical tools (mean excess plot, shape plot,...)
- Bootstrap and other methods requiring extra conditions on L
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Basel II QIS 2002 Data
• POT Analysis of Severities: P (Li > x) = x−1/ξiLi(x)
Business line ξiCorporate finance 1.19 (*)
Trading & sales 1.17
Retail banking 1.01
Commercial banking 1.39 (*)
Payment & settlement 1.23
Agency services 1.22 (*)
Asset management 0.85
Retail brokerage 0.98
* means significant at 95% level
ξi > 1: infinite mean
• Remark: different picture at level of individual banks
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Issues Regarding Infinite Mean Models
• Reason for ξ > 1?
• Potentially:
− wrong analysis
− EVT conditions not fulfilled
− contamination, mixtures
• We concentrate on the latter:
Two examples:
Ê Contamination above a high threshold
Ë Mixture models
• Main aim: show through examples how certain data-structures can
lead to infinite mean models
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Contamination Above a High Threshold
Example 1: Consider the model
FX(x) =
1−(1 + ξ1x
β1
)−1/ξ1
if x ≤ v,
1−(1 + ξ2(x−v∗)
β2
)−1/ξ2
if x > v,
where 0 < ξ1 < ξ2 and β1, β2 > 0.
• v∗ is a constant depending on the model parameters in a way that
FX is continuous
• VaR can be calculated explicitly:
VaRα(X) =
{1ξ1β1
((1− α)−ξ1 − 1
)if α ≤ FX(v),
v∗ + 1ξ2β2
((1− α)−ξ2 − 1
)if α > FX(v).
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Shape Plots
0 100 200 300 400 500 600
−1
01
23
xdata
rep(
xi1,
leng
th(x
data
))
0 100 200 300 400 500 600
−1
01
23
xdatare
p(xi
1, le
ngth
(xda
ta))
Easy case: v low Hard case: v high
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Finite Mean Case
0 100 200 300 400 500 600
−2
−1
01
23
xdata
rep(
xi1,
leng
th(x
data
))
Careful: similar picture for v low and ξ1 � ξ2 < 1
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Contamination above a high threshold (cont’d)
• Easy case: v low
− Change of behavior typically visible in the mean excess plot
• Hard case: v high
− Typically only few observations above v
− Mean excess plot may not reveal anything
− Classical POT analysis easily yields incorrect resuls
− Vast overestimation of VaR possible
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Mixture Models
Example 2: Consider
FX = (1− p)F1 + pF2,
with Fi exact Pareto, i.e. Fi(x) = 1− x−1/ξi for x ≥ 1 and
0 < ξ1 < ξ2.
• Asymptotically, the tail index of FX is ξ2
• VaRα can be obtained numerically and furthermore
− does not correspond to VaRα of a Pareto distribution with
tail-index ξ∗
− equals VaRα∗ corresponding to F2 at a level α∗ lower than α
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• Classical POT analysis can be very misleading:
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●● ●
●●●
● ●
●●
●
●
●
●
●
0 50 100 150 200
010
020
030
0
Threshold
Mea
n E
xces
s
500 433 366 299 232 165 98 48
0.0
0.5
1.0
1.5
0.936 1.220 1.740 2.920 4.640 13.500
Exceedances
Sha
pe (
xi)
(CI,
p =
0.9
5)
Threshold
0 100 200 300 400 500 600
−2
−1
01
23
xdata
rep(
xi1,
leng
th(x
data
))
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VaR for Mixture Models
α VaRα(FX) VaRα(Pareto(ξ2)) ξ∗
0.9 6.39 46.42 0.8
0.95 12.06 147.36 0.83
0.99 71.48 2154.43 0.93
0.999 2222.77 105 1.12
0.9999 105 4.64 · 106 1.27
0.99999 4.64 · 106 2.15 · 108 1.33
Value-at-Risk for mixture models with p = 0.1, ξ1 = 0.7 and
ξ2 = 1.6.
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Including Frequencies
The POT method can be embedded into a wider framework based
on Point processes
• iid case: exceedance times follow asymptotically a homogeneous
Poisson Process
• Extensions: several possibilities
− Including the severities: marked Poisson process
− Non-stationarity: non-homogeneous Poisson processes
− Over-dispersion: doubly stochastic processes
− Short-range dependence: clustering
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VaR Estimation for Compound Sums
Proposition:
Let X,X1, X2, . . . be iid with FX ∈ S. If for some ε > 0,∑∞n=1(1 + ε)nP (N = n) <∞ (satisfied for instance in the
important binomial, Poisson and negative binomial cases), then
limx→∞
P (∑N
i=1Xi > x)1− FX(x)
= E(N).
Approximation of VaR:
VaRα
(N∑
i=1
Xi
)∼ VaRα∗(X), α∗ = 1− 1− α
EN, α→ 1
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Marginal VaR Estimation
Approximative method
1. Estimate the excess distribution of severities using POT
2. Calculate
VaRα(LT+1i,k ) = u+
β
ξ
( n(1− α)NuE(NT+1
i )
)−ξ
− 1
Monte Carlo methods
1. Choose a distribution for severities and a process for frequencies
(evt. jointly)
2. After a large number of simulations, estimate the VaR of the
compound sum via the POT-MLE method
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POT-MLE VaR Estimate
VaR99.9% of a GPD(ξ, 1) rv. as a function of ξ ∈ (0, 1.5)
• Small changes in ξ lead to considerable changes in VaR
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Marginal VaR Estimate: Issues
• VaRα is an exponential function of ξ and therefore very sensitive
to ξ
• Confidence intervals for VaRα widen rapidly with increasing α and
decreasing sample size
• Fazit 1: for very high levels (99.9% or 99.97%) there is typically
substantial uncertainty and variability in the VaR estimate due to
the lack of data
• Fazit 2: Issues far in the tail call for judgement
• ξ > 1 is an issue!
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L5. Global VaR Estimation
Recall:
• Global VaR estimate
VaR+α = VaR1
α + · · ·+ VaRlα
• Reduction because of “correlation effects”
VaRα < VaR+α
• In general, VaR+α is not the upper bound!
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Bounds on VaR
Find optimal bounds for
VaRT+1l,α ≤ VaRT+1
α
(d∑
k=1
LT+1k
)≤ VaRT+1
u,α
given marginal VaR’s and dependence information
Solution:
• Frechet Problem
• Mass Transportation Problem
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Example 1: Comonotonic Case
Recall:
LT+11 , . . . , LT+1
d are comonotonic if there exists a rv Z and
increasing functions fT+11 , . . . , fT+1
d , so that
LT+1i = fT+1
i (Z), i = 1, . . . , d
• If LT+11 , . . . , LT+1
d are comonotonic VaR is additive
VaRT+1α
(d∑
k=1
LT+1k
)=
d∑k=1
VaRT+1k,α
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Example 2: No Dependence Information
Take LT+1i = Li, i = 1, . . . , d = 8 and
FL1 = · · · = FLd= Pareto(1, 1.5)
• Comonotonic case:
V aR0.999
(8∑
i=1
Li
)=
8∑i=1
VaR0.999(Li) = 0.79
• Unconstrained upper bound:
VaR0.999
(8∑
i=1
Li
)≤ 1.93
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Example 3: No Dependence Information
FL1 6= · · · 6= FLdis more difficult
Bounds on VaR using the OpRisk portfolio given in Moscadelli(2004)
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Correlation
Correlation:
Correlation (linear, rank, tail) is one-number summary: ρ, τ , ρS ...
• Careful: linear correlation does not exist for ξ > 0.5
• Linear correlation is typically small for heavy tailed rvs
• Knowledge of correlation (linear, rank, tail...) is sufficient for
individual models, but totally insufficient in general
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Correlation
Upper and lower bound on linear correlation ρ(L1, L2) for L1 ∼ Pareto(2.5) (left)
and L1 ∼ Pareto(2.05) (right) and L2 ∼ Pareto(β)
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Copulas
Copula:
With LT+1i ∼ Fi the joint distribution can be written as
P (LT+11 ≤ l1, . . . , L
T+1d ≤ ld) = C(F1(l1), . . . , Fd(ld))
The function C is known as copula and is a joint distribution on
[0, 1]d with uniform marginals
• A copula and marginal distributions determine the joint model
completely
• However: there are not enough OpRisk data: one year of loss data
comprises to a single observation of (LT+11 , . . . , LT+1
d )
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Dynamic Dependence Models
In order to use the data at hand, we need a dynamic model for the
compound processes.
Consider:
d = 2 : Lk =Nk(T )∑i=1
Xi,k, k = 1, 2
and
1. make (Xi,1) and (Xi,2) dependent, i ≥ 1
2. make {N1(t) : t ≤ T} and {N2(t) : t ≤ T} dependent
3. combination of both
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Dependent Counting Processes
• Various models for dependent counting processes {N1(t) : t ≤ T}and {N2(t) : t ≤ T} exist:
− Common shock models
− Point process models
− Mixed Poisson processes with dependent mixing rvs.
− Levy Copulas
• So far, there is no general dependence concept
• It is not clear how to quantify dependence between processes
• It is less transparent how the dependence structure of the frequency
processes affects the dependence structure of the compound rvs
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One Loss Causes Ruin Problem
Question: how do the marginal severities affect the global loss?
• based on Lorenz curve in economics
− 20 – 80 rule for 1/ξ = 1.4
− 0.1 – 95 rule for 1/ξ = 1.01
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Proposition 1
For L1, . . . , Ld iid and subexponential we have for
L = L1 + · · ·+ Ld that
P (L > x) ∼ dP (L1 > x), x→∞Proposition 2
Suppose in addition that Li =∑Ni
k=1Xi,k where Ni ∼ Poi(λi) are
independent. Furthermore, for i = 1, . . . , d, Xi,k’s are iid with
P (Xi > x) = x−1/ξihi(x), hi slowly varying
If ξ1 > · · · > ξd, we have that
P (L > x) ∼ cP (X1 > x)
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Discussion
• Tail issues:
− robust statistics
− scaling
− mixtures
• Infinite mean: industry occasionally uses
− constrained estimation to ξ < 1
− estimate under the condition of a finite upper limit
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Discussion (cont’d)
• Aggregation issues:
− adding risk measures across a 7× 8 table
− reduction because of “correlation effects”
• Data issues:
− impact of pooling
− incorporation of external data and expert opinion
− credibility theory
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References
[1] Embrechts, P., Kluppelberg, C., and Mikosch, T. (1997). Model-ling Extremal Events for Insurance and Finance. Springer.
[2] McNeil, A.J., Frey, R., and Embrechts, P. (2005) QuantitativeRisk Management: Concepts, Techniques and Tools. Princeton
University Press.
[3] Moscadelli, M. (2004) The Modelling of Operational Risk:
Experience with the Analysis of the Data, Collected by the
Basel Committee, Banaca d’Italia, report 517-July 2004
[4] Neslehova, J., Embrechts, P. and Chavez-Demoulin, V. (2006)
Infinite mean models and the LDA for operational risk. Journal
of Operational Risk, 1(1), 3-25.
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