Operational Risk and Pareto L evy Copulas Claudia Kl uppelberg - - PowerPoint PPT Presentation

Operational Risk and Pareto L´ evy Copulas

Claudia Kl¨ uppelberg Technische Universit¨ at M¨ unchen email: cklu@ma.tum.de http://www-m4.ma.tum.de References:

• B¨
• cker, K. and Kl¨

uppelberg, C. (2005) Operational VaR - a closed form approximation. Risk, December 2005.

• B¨
• cker, K. and Kl¨

uppelberg, C. (2008) First order approximations to operational risk - dependence and consequences. To appear in: G.N. Gregoriou (ed.) Operational Risk Toward Basel III, Best Practices and Issues in Modeling, Management and Regulation. Wiley, New York.

Contents (1) Basel II (2) The Loss Distribution Approach (LDA) for the Single Cell (3) Modelling Dependence of L´ evy Processes (4) The Multivariate Subexponential Compound Poisson (SCP) Model (5) Estimating Total OpVar

(1) Basel II www.bis.org/bcbs/bcbscp3.htmcp3 Structure of Risk Management:

• Pillar 1: minimal capital requirements
• Pillar 2: supervisory review of capital adequacy
• Pillar 3: public disclosure

Definition of Operational Risk: The risk of losses resulting from inadequate or failed internal processes, people and systems, or external events.

Examples: 1995 Barings Bank: Nick Leeson (1.3b British Pounds) 2001 Enron (largest US bankruptcy ever) 2005 Brokerhouse Mizuho Securities: instead of selling 1 share for 610 000 Yen a trader wrote 610 000 shares for 1 Yen each (190 Mio Euro) 2008 Societ´ e G´ en´ erale: Jerome Kerviel (4.9b Euro) Basel II distinguishes 7 loss types and 8 business types

(2) The Loss Distribution Approach (LDA) for the Single Cell Subexponential compound Poisson (SCP) model (1) The severities (Xk)k∈N are positive iid random variables with subexponential distribution function F. (2) The frequency process N(t) of loss events in the time interval [0, t] for t ≥ 0 constitutes a homogenous Poisson process with intensity λ > 0. In particular, P(N(t) = n) = pt(n) = e−λt (λt)n n! , n ∈ N0 . (3) The severity process and the frequency process are independent. (4) The aggregate loss process is given by S(t) =

N(t)

• k=1

Xk , t ≥ 0 .

Let (Xk)k∈N be iid random variables with distribution function F. Then F is said to be subexponential (F ∈ S) if lim

x→∞

P(X1 + · · · + Xn > x) P(max(X1, . . . , Xn) > x) = 1 for some (all) n ≥ 2. Denote F(x) := 1 − F(x) = P(X > x) for x > 0 the distribution tail of F. If for some α ≥ 0, lim

t→∞

F(xt) F(t) = x−α , x > 0 , then F is called regularly varying with index −α, denoted by F ∈ R−α. The quantity α is also called the tail index of F or X. See Embrechts, Kl¨ uppelberg and Mikosch (1997) for details.

The aggregate loss distribution function: Gt(x) = P(S(t) ≤ x) =

∞

• n=0

pt(n) P(S(t) ≤ x|N(t) = n) =

∞

• n=0

pt(n) F n∗(x), x ≥ 0, t ≥ 0 , where F is the distribution function of Xk, and F n∗(·) = P(n

i=1 Xi ≤ ·) with F 1∗ = F and F 0∗ = I[0,∞).

Operational VaR up to time t at confidence level κ: VaRt(κ) = G←

t (κ) = inf{x ∈ R : Gt(x) ≥ κ} ,

κ ∈ (0, 1) , is the (left continuous) generalized inverse of Gt. If Gt is strictly increasing and continuous, then VaRt(κ) = G−1

t (κ).

Theorem [Embrechts, Kl¨ uppelberg and Mikosch, Theorem 1.3.9] Consider the standard LDA S(t) = N(t)

n=0 Xi, t ≥ 0. Assume that the severities

Xi have distribution function F ∈ S. Fix t > 0 and define the frequency distribution by P(N(t) = n) = pt(n) for n ∈ N0. Then, the aggregate loss distribution is given by Gt(x) =

∞

• n=0

pt(n) F n∗(x) , x ≥ 0, t ≥ 0 . Assume that for some ε > 0,

∞

• n=0

(1 + ε)npt(n) < ∞ . Then for fixed t > 0, Gt(x) = EN(t)F(x)(1 + o(1)) ∼ EN(t)F(x) , x → ∞ .

Theorem [Analytical OpVaR] Consider the SCP model for fixed t > 0 and a subexponential severity with distribution function F. Then as κ ↑ 1, VaRt(κ) = F ←

• 1 − 1 − κ

λ t (1 + o(1))

• =

1 F ← λ t 1 − κ(1 + o(1))

• .

Proposition [Bingham, Goldie and Teugels (1987)] Let α ∈ (0, ∞). (1) F ∈ R−α if and only if (1/F)← ∈ R1/α. (2) G(x) ∼ cF(x) as x → ∞ iff (1/F)←(z) ∼ (1/G)←(cz) as z → ∞. (3) F(x) = x−αL(x) for x ≥ 0 if and only if (1/F)←(z) = z1/α L(z) for z ≥ 0, where L and L are slowly varying functions.

Corollary Assume that F ∈ R−α for α ∈ (0, ∞). Then by (2), as κ ↑ 1, VaRt(κ) ∼ 1 F ← λt 1 − κ

• ∼ t1/α

1 F ← λ 1 − κ

• =

t1/αVaR1(κ) ∼ λ t 1 − κ 1/α

• L

λ t 1 − κ

• .
• Remark If 1/F ∈ Rα for α ∈ (0, ∞]

(α = ∞ means that F(t)/F(xt) → ∞ for x > 1 and F(t)/F(xt) → 0 for x < 1), then VaRt(κ) ∼ 1 F ← λ t 1 − κ

• ,

κ ↑ 1 .

Popular subexponential severity distributions Name Distribution function Parameters Lognormal F(x) = Φ ln x − µ σ

• µ ∈ R, σ > 0

Weibull F(x) = 1 − e−(x/θ)τ θ > 0, 0 < τ < 1 Pareto F(x) = 1 −

• 1 + x

θ −α α, θ > 0

Approximated VaR (dashed line) and simulated VaR (solid line) for the Pareto-Poisson LDA with θ = 1.

0.998 0.9985 0.999 0.9995 Confidence Level 2000 3000 4000 5000 VaR alpha = 1.5 0.998 0.9985 0.999 0.9995 Confidence Level 10000 20000 30000 40000 VaR alpha = 1.1

(3) Modelling Dependence of L´ evy Processes Invoking the copula idea: A d-dimensional copula C is a distribution function on [0, 1]d with standard uniform marginals. Theorem [Sklar’s Theorem] Let F be a joint distribution function with marginals F1, . . . , Fd. Then there exists a copula C : [0, 1]d → [0, 1] such that for all x1, . . . , xd ∈ R = [−∞, ∞] F(x1, . . . , xd) = C(F1(x1), . . . , Fd(xd)). (1) If the marginals are continuous, then C is unique. Otherwise it is unique on Ran F1 × · · · × Ran Fd. Conversely, if C is a copula and F1, . . . , Fd are distribution functions, then the function F as defined in (1) is a joint distribution function with marginals F1, . . . , Fd.

Question Can we use copulas to model dependence of L´ evy processes? Our LDA model is a compound Poisson process with positive jumps. Problems The law of a L´ evy process X is completely determined by the distribution of X at time t for any t > 0. The copula Ct of (X1(t), . . . , Xd(t)) may depend on t. In general, Cs cannot be calculated from Ct, because Cs depends also on the marginal distributions. For given infinitely divisible marginal distributions it is unclear, which copulas Ct yield multivariate infinite divisible distributions. (Copulas are invariant under strictly increasing transformations, infinite divisibility is not!) Introduce a L´ evy copula [Cont & Tankov (2004), Kallsen & Tankov (2004), Barndorff-Nielsen & Lindner (2004).

Problem L´ evy measures may have a non-integrable singularity at 0. Define E := [0, ∞]d \ {0}. Let X be a spectrally positive L´ evy process in Rd with a L´ evy measure Γ, which has standard 1-stable one-dimensional marginals. Then we call Γ a Pareto L´ evy measure and the associated tail measure Γ(x) = Γ([x1, ∞) × · · · × [xd, ∞)) =: C(x1, . . . , xd) , x ∈ E , is called Pareto L´ evy copula C. Remark Extension to general L´ evy processes by quadrantwise definition (singularity in 0!).

Lemma Let X be a spectrally positive L´ evy process in Rd with L´ evy measure Π on E and continuous marginal tail measures Π1, . . . , Πd, where Πi(x) := Π([x, ∞)) for i = 1, . . . , d. Then Π(x) = Π([x1, ∞] × · · · × [xd, ∞]) = C

• 1

Π1(x1), . . . , 1 Πd(xd)

• ,

x ∈ E , and C is a Pareto L´ evy copula.

Theorem [Sklar’s Theorem for Pareto L´ evy copulas] Let Π be the tail measure of a d-dimensional spectrally positive L´ evy process with marginal tail measures Π1, . . . , Πd. Then there exists a Pareto L´ evy copula

• C : E → [0, ∞] such that

Π(x) = C

• 1

Π1(x1), . . . , 1 Πd(xd)

• .

(2) If the marginal tail measures are continuous on [0, ∞], then C is unique. Otherwise, it is unique on Ran

• 1

Π1

• × · · · × Ran
• 1

Πd

• .

Conversely, if C is a Pareto L´ evy copula and Π1, . . . , Πd are marginal tail measures, then Π as defined in (2) is a joint tail measure with marginals Π1, . . . , Πd.

Examples of Pareto L´ evy copulas Example Clayton Pareto L´ evy copula (special Archimedian Pareto L´ evy copula)

• Cϑ(x1, . . . , xd) =
• x1

ϑ + · · · + xd ϑ−1/ϑ

Note: limϑ→∞ Cθ(x1, . . . , xd) = C(x1, . . . , xd) complete positive dependence, limϑ→0 Cϑ(x1, . . . , xd) = C⊥(x1, . . . , xd) independence. A Pareto L´ evy copula C is homogeneous (of order 1), if for all t > 0

• C(x1, . . . , xd) = t

C(tx1, . . . , txd) , (x1, . . . , xd) ∈ E.

Translate results for stable processes Let 0 < α < 2. The L´ evy process X in Rd is an α-stable L´ evy process, if its L´ evy measure has the representation ν(B) =

• S

λ(dξ) ∞ 1B(rξ) dr r1+α , B ∈ B(Rd) , where S = {x ∈ R : |x| = 1} denotes the unit sphere with respect to any norm | · | in Rd. Theorem [Kallsen and Tankov (2006)] For 0 < α < 2 let X be a L´ evy process in Rd. The process X is α-stable if and only if it has α-stable one-dimensional marginals and it has a Pareto L´ evy copula, which is homogeneous of order 1.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12

t X, Y 1/2−stable severities, Clayton−Levy copula: θ =0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6

t ∆ X Jumps: θ=0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8

t ∆ Y

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5

t X, Y 1/2−stable severities, Clayton−Levy copula: θ =2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

t ∆ X Jumps: θ=2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2

t ∆ Y

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

t X, Y 1/2−stable severities, Clayton−Levy copula: θ =10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

t ∆ X Jumps: θ=10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

t ∆ Y

(4) The Multivariate SCP Model (1) All operational risk cells, indexed by i = 1, . . . , d, are described by an SCP model with aggregate loss process Si, continuous subexponential severity distribution function Fi and Poisson parameter λi > 0. (2) Dependence between different cells is modelled by a Pareto L´ evy copula: Let Πi : [0, ∞) → [0, ∞) be the tail measure to Si, i.e. Πi(·) = λi F i(·), and let C : E → [0, ∞] be a Pareto L´ evy copula. Then Π(x1, . . . , xd) = C

• 1

λ1F 1(x1), . . . , 1 λdF d(xd)

• defines the tail measure of the compound Poisson process S = (S1, . . . , Sd).

(3) The bank’s total aggregate operational loss process is defined as S+(t) = S1(t) + S2(t) + · · · + Sd(t) , t ≥ 0 , with tail measure Π

+(z) = Π({(x1, . . . , xd) ∈ [0, ∞)d : d i=1 xi ≥ z}) ,

z ≥ 0 .

Proposition Consider the multivariate SCP model. Its total aggregate loss process S+ is compound Poisson with intensity λ+ = limz↓0 Π

+(z)

and severity distribution F

+(z) = 1 − F +(z) = Π +(z)

λ+ , z ≥ 0 .

• Consider the multivariate SCP model with total aggregate loss S+(t) at time

t > 0 and denote G+

t (·) = P(S+(t) ≤ ·).

Total Operational VaR up to time t at confidence level κ is defined as VaR+

t (κ) = G+← t

(κ) , κ ∈ (0, 1) , with G+←

t

(κ) = inf{z ∈ R : G+

t (z) ≥ κ} for 0 < κ < 1.

2 4 6 8 10

x1

2 4 6 8 10

x2

Figure 1: Decomposition of the domain of the tail measure Π

+(z) for z = 6 into

a simultaneous loss part Π

+ (z) (orange area) and independent parts Π⊥1(z)

(solid black line) and Π⊥2(z) (dashed black line).

(5) Estimating Total OpVar One cell dominant Theorem For fixed t > 0 let Si(t) for i = 1, . . . , d have compound Poisson

• distributions. Assume that F 1 ∈ R−α for α > 0. Let ρ > α and suppose that

E[(Xi)ρ] < ∞ for i = 2, . . . , d. Then regardless of the dependence structure between (S1(t), . . . , Sd(t)), P(S1(t) + · · · + Sd(t) > x) ∼ EN1(t) P(X1 > x) , x → ∞ , VaR+

t (κ)

∼ F ←

1

• 1 − 1 − κ

EN1(t)

• = VaR1

t(κ) ,

κ ↑ 1 .

Multivariate SCP model with completely dependent cells

• All cell processes jump together

= ⇒ λ := λ1 = · · · = λd.

• The mass of the L´

evy measure is concentrated on {(x1, . . . , xd) ∈ (0, ∞)d : Π1(x1) = · · · = Πd(xd)} = {(x1, . . . , xd) ∈ (0, ∞)d : F1(x1) = · · · = Fd(xd)} . Let Fi be strictly increasing and continuous: F −1

i

(q) exists for all q ∈ [0, 1). Then Π

+(z)

= Π({(x1, . . . , xd) ∈ (0, ∞)d :

d

• i=1

xi ≥ z}) = Π1({x1 ∈ (0, ∞) : x1 +

d

• i=2

F −1

i

(F1(x1)) ≥ z}) , z > 0 .

Set H(x1) := x1 + d

i=2 F −1 i

(F1(x1)) for x1 ∈ (0, ∞) and note that it is

• invertible. Thus

Π

+(z) = Π1({x1 ∈ (0, ∞) : x1 ≥ H−1(z)}) = Π1

• H−1(z)
• ,

z > 0 . Theorem Consider a multivariate SCP model with completely dependent cell processes S1, . . . , Sd and strictly increasing and continuous severity distributions Fi. Then, S+ is compound Poisson with parameters λ+ = λ and F

+(z) = F 1

• H−1(z)
• .

If F + ∈ S ∩ R−α for α ∈ (0, ∞], then VaR+

t (κ) ∼ d

• i=1

VaRi

t(κ) ,

κ ↑ 1 , where VaRi

t(·) denotes the stand alone OpVaR of cell i.

Corollary Assume that the conditions of the Theorem hold and that F 1 ∈ R−α for α ∈ (0, ∞) and lim

x→∞

F i(x) F 1(x) = ci ∈ [0, ∞) . Assume that ci = 0 for i = 1, . . . , b ≤ d and ci = 0 for i = b + 1, . . . , d. Then VaR+

t (κ) ∼ b

• i=1

c1/α

i

VaR1

t(κ) ,

κ ↑ 1 .

Multivariate SCP model with independent cells

• Not two cell processes ever jump together.
• The mass of the L´

evy measure is concentrated on the axes. Π

+(z) = Π1(z1) + · · · + Πd(zd) .

Theorem Assume S1, . . . , Sd are independent. Then S+ defines a one-dimensional SCP model with parameters λ+ = λ1 + · · · + λd and F

+(z) = 1 λ+

• λ1F 1(z) + · · · + λdF d(z)
• ,

z ≥ 0 . If F1 ∈ R−α for α ∈ (0, ∞) and for all i = 2, . . . , d, limx→∞ F i(x) F 1(x) = ci ∈ [0, ∞) , then, setting Cλ = λ1 + c2λ2 + · · · + cdλd, VaR+

t (κ) ∼

1 F 1 ← Cλ t 1 − κ

• = VaR1

t

Cλ t 1 − κ

• ,

κ ↑ 1 .

Example [Pareto distributed severities] For i = 1, . . . , d let Fi be Pareto distributions with tail index α and scale parameters θi. Then VaR+

⊥(κ)

VaR+

(κ) =

d

i=1 θα i

1/α d

i=1 θi

     < 1 , α > 1 = 1 , α = 1 > 1 , α < 1 . Let d = 2 and θ1 = θ2 = 1. Each cell has stand alone VaR of EUR 100. α VaR+

• VaR+

⊥

1.2 178.2 1.1 187.8 1.0 200.0 0.9 200.0 216.0 0.8 237.8 0.7 269.2

Multivariate SCP model with regularly varying L´ evy measure (a) Let Π be a L´ evy measure of a spectrally positive L´ evy process in Rd. Assume that there exists a Radon measure ν on E such that for x ∈ E lim

u→∞

Π({y : y1 > ux1 or · · · or yd > uxd} Π1(u) = ν({y : y1 > x1 or · · · or yd > xd}) =: ν([0, x]c) . Then we call Π multivariate regularly varying. (b) The measure ν has a scaling property: there exists some α > 0 such that for every s > 0 ν([0, sx]c) = s−αν([0, x]c) , and Π is called multivariate regularly varying with index −α.

Theorem Consider an SCP model with multivariate regularly varying cell processes (S1, . . . , Sd) with index −α and limit measure ν. Assume further that the severity distributions Fi for i = 1, . . . , d are strictly increasing and

• continuous. Then, S+ is compound Poisson with parameters

λ+F

+(x) ∼ ν+(1, ∞)λ1F 1(x) ,

x → ∞ . where ν+(z, ∞] = ν{x : d

i=1 xi > z} for z > 0.

Furthermore, λ+F

+(·) ∈ R−α and total OpVaR is asymptotically given by

VaRt(κ) ∼ F ←

1

• 1 −

1 − κ t λ1 ν+(1, ∞]

• ,

κ ↑ 1 .

Example [Clayton Pareto L´ evy copula] Assume that F 2(x)/F 1(x) → c as x → ∞. Set c := (λ2/λ1) c . ν+(1, ∞] = 1 + c1/αE

• c1/α + Y −1/α

ϑ

)α−1 , where Yϑ is a positive random variable with density g(s) = (1 + sϑ)−1/ϑ−1. For α = 1 we have ν+(1, ∞] = 1 + c, independent of ϑ. Consequently, total OpVar is for all 0 < ϑ < ∞ asymptotically equal to the independent OpVar. If αϑ = 1, then ν+(1, ∞] = c1+1/α − 1

• c1/α − 1