slides loisel-charpentier-ottawa

Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas

On some classes of hierarchicalArchimedean copulas and their use in

actuarial science

Arthur Charpentier and Stephane Loisel

29/05/08

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Outline

Introduction : some actuarial issuesSolvency II and ruin probabilitiesA quick look at insurance dataDependence models for Solvency II

Archimdedean copulasPairwise Archimedean copulasMultivariate extension and link with frailty modelsDrawbacks of Archimedean copulas

Nested Archimedean copulasNested Archimedean copulasNesting copulas of the same familyNesting copulas of different families

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Outline




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Risk aggregation in Solvency II

• Dependence between losses in different lines of business(Liability, motor insurance, fire insurance, ...) and newsolvency regulations in Europe : Solvency II.

• Typical problem : for some α ∈ [0.95, 0.999], computeV aRα(X1 + · · ·+X25) or TV aRα(X1 + · · ·+X25).

• The Tail-Value-at-Risk is TV aRα(X) = 11−α

∫ 1α V aRq(X)dq.

• If FX is continuous, then

TV aRα(X) = E [X | X > V aRα(X)] .

• Why using TV aR instead of V aR ?TV aRα is sub-additive (and V aRα is not),comonotonic-additive and takes diversification effect intoaccount : the worst case is when the Xk’s are comonotonic.

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Ruin probabilities with dependent claims

• Dependence between successive claim amounts in risk theory.• Typical problem : compute or approximate the probability of

ruin in finite or infinite time with large initial reserve, whenclaim amounts are no longer independent.

A typical risk process.5 / 22


A quick look at very basic insurance data

Some very basic insurance data : empirical Kendall’s τ of lossratios of different lines of business.

The above risks are far from being exchangeable !

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Hierarchical structure of Solvency II

The standard formula for the Solvency Capital Requirement (SCR)uses a so-called bottom-up approach.

Hierarchical bottom-up approach in the standard formula.

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Hierarchical structure of Solvency II

The standard formula for the Solvency Capital Requirement (SCR)uses a so-called bottom-up approach. Each class of risks is

subdivided into sub-classes of risks with a hierarchical model.

Fig.: QIS 4 correlationparameters in non-lifeinsurance

Fig.: QIS 4 correlationparameters in life insurance

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Classical tools

Average and large insurance companies currently develop internalmodels or partial internal models. To model dependence between

risks, classical tools that are often used include :

• Gaussian copulas (quite easy to implement and flexibility ofthe correlation matrix, but often not adapted to real-worlddata)

• Other elliptical copulas (Student for example)

• Archimedean copulas (not adapted as risks arenon-exchangeable)

• Vine copulas (trees and conditional distributions).

An alternative solution (among many others) in the spirit of thehierarchical standard approach :

nested (or hierarchical) Archimedean copulas.

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Outline




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2-dimensional Archimedean copulas

Definition : An Archimedean copula C with generatorϕ : [0, 1]→ [0,+∞] is defined by

∀(u, v) ∈ [0, 1]2, C(u, v) = ϕ[−1] (ϕ(u) + ϕ(v)) ,

• where ϕ is convex, decreasing, and such that ϕ(1) = 0,

• and where ϕ[−1] : [0,+∞]→ [0, 1] is either• the reciprocal bijection ϕ−1 of ϕ if ϕ(0) = +∞

(strict Archimedean copulas),• or defined by

• ϕ[−1](x) = ϕ−1(x) for x ≤ ϕ(0),• ϕ[−1](x) = 0 for x > ϕ(0)

if ϕ(0) < +∞ (non-strict Archimedean copulas).

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Examples of Archimedean copulas

Classical examples can be found in Nelsen (1999) :21 parametric families, including

Family ϕθ(t) Dom(θ) strictClayton 1

θ

(t−θ − 1

)[−1,+∞) \ {0} if θ > 0

Gumbel (− ln t)θ [−1,+∞) \ {0} yes

Frank − ln(e−θt−1e−θ−1

)[−1,+∞) \ {0} yes

AMH ln(

1−θ(1−t)t

)[−1,+∞) \ {0} yes

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Extensions to dimension n ≥ 3

• In dimension n ≥ 3, define an Archimedean copula Cn withgenerator ϕ by

∀(u1, . . . , un) ∈ [0, 1]n, C(u1, . . . , un) = ϕ[−1] (ϕ(u1) + · · ·+ ϕ(un)) .

• Notation : setLn = {f s.t. ∀0 ≤ k ≤ n, ∀x, (−1)k ∂k

∂xk f(x) ≥ 0}.• Cn exists for all n ≥ 2 if and only if ϕ is completely monotonic

(i.e. ϕ ∈ L∞).• Cn exists for 2 ≤ n ≤ m if and only if ϕ ∈ Lm (Kimberling,

1974).

• Bernstein’s theorem and link with frailty models : ϕ−1

completely monotone if and only if it ϕ−1 = LΘ for somenon-negative random variable Θ. In that case U1, . . . , Um canbe seen as being conditionally independent w.r.t. Θ.

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Drawbacks of Archimedean copulas

• Exchangeable distributions : pairwise dependence is the samefor every pair of risks

• Basic example in insurance with three lines of business : zoomon QIS4 parameters for some life insurance risks.

LoB’s Mortality Longevity Disability

Mortality 1

Longevity 0 1

Disability 0.5 0 1

These values are just QIS4 parameters and do not correspond toany real-world linear correlation coefficient or concordance measure.

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Drawbacks of Archimedean copulas

• Exchangeable distributions : pairwise dependence is the samefor every pair of risks

• Basic example in insurance with three lines of business : zoomon QIS4 parameters for some life insurance risks.

LoB’s Mortality Longevity Disability

Mortality 1

Longevity 0 1

Disability 0.5 0 1

Negative dependence and positive dependence, with(probably) positive tail dependence index.

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Outline




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Nested Archimedean copulas• A four-dimensional fully nested Archimedean copula :

C(u1, . . . , u4) = ϕ[−1]3

(ϕ3

[ϕ

[−1]2

(ϕ2

[ϕ

[−1]1 (ϕ1(u1) + ϕ1(u2))

]+ ϕ2(u3)

)]+ ϕ3(u4)

)

Uniform variates (U1, . . . , U4) with copula C are such that

• (U1, U2) has copula C1 with generator ϕ1,• (U1, U3) and (U2, U3) both have copula C2 with generator ϕ2,• and the couples (U1, U4), (U2, U4) and (U3, U4) all have

copula C3 with generator ϕ3.• A four-dimensional partially nested Archimedean copula :

C(u1, . . . , u4) = ϕ[−1]0

(ϕ0

[ϕ

[−1]12 (ϕ12(u1) + ϕ12(u2))

]+ ϕ0

[ϕ

[−1]34 (ϕ34(u3) + ϕ34(u4))

])Uniform variates (U1, . . . , U4) with copula C are such that

• (U1, U2) has copula C12 with generator ϕ12,• (U3, U4) has copula C34 with generator ϕ34,• and the couples (U1, U3), (U2, U3), (U1, U4) and (U2, U4) all

have copula C0 with generator ϕ0.

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Sufficient conditions

• Sufficient conditions to have a copula in the fully nested case :

C(u1, . . . , u4) = ϕ[−1]3

(ϕ3

[ϕ

[−1]2

(ϕ2

[ϕ

[−1]1 (ϕ1(u1) + ϕ1(u2))

]+ ϕ2(u3)

)]+ ϕ3(u4)

)

• ϕ−11 , ϕ−1

2 and ϕ−13 completely monotonic

• (ϕ−12 ◦ ϕ1)′ and (ϕ−1

3 ◦ ϕ2)′ completely monotonic

• Sufficient conditions to have a copula in the partially nestedcase :

C(u1, . . . , u4) = ϕ[−1]0

(ϕ0

[ϕ

[−1]12 (ϕ12(u1) + ϕ12(u2))

]+ ϕ0

[ϕ

[−1]34 (ϕ34(u3) + ϕ34(u4))

])

• ϕ−10 , ϕ−1

12 and ϕ−134 completely monotonic

• (ϕ−10 ◦ ϕ12)′ and (ϕ−1

0 ◦ ϕ34)′ completely monotonic

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Nesting copulas of the same family

• Many ordered families can be nested (Clayton, AMH,Cook-Johnson, Gumbel, ...)

• Useful transitivity theorem (see McNeil (2007) e.g.) : if(ϕ−1

2 ◦ ϕ1)′ and (ϕ−13 ◦ ϕ2)′ are completely monotonic, so

does (ϕ−13 ◦ ϕ1)′.

• Simulation methods using the Laplace transform approach.

• Some classical references (among others) : Joe (1997),McNeil (2007), Savu and Trede (2006), Embrechts et al.(2003), Hofert (2007), ...

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Nesting copulas of different families

Hofert (2007) characterizes the set of pairs of copulas of Nelsen’slist (1999) that can be nested, in the case where generator inversesare completely monotonic, and where each involved (ϕ−1

r ◦ ϕs)′ iscompletely monotonic.In particular, these 7 combinations include :

• AMH-Clayton (with some parameter restrictions)

• Clayton-Families 12, 14, 19, 20 (with some parameterrestrictions)

Limitation : only positive dependence is possible with the same ordifferent families if one considers the completely monotonic case.Can we get nested Archimedean copulas involving some negativedependence in low and reasonable dimensions (3-15) ?

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Nesting copulas and negative dependence• It is possible to build models for the three risks (mortality,

longevity, disability) ?

LoB’s Mortality Longevity DisabilityMortality (1) ?Longevity (2) - ?Disability (3) ++ - ?

? : comonotonic, - : Negative dependence and ++ : positive

dependence, with (probably) positive tail dependence index.

• One possible solution (among many others) :

C(u1, u2, u3) = ϕ[−1]2

(ϕ2

[ϕ

[−1]1 (ϕ1(u1) + ϕ1(u3))

]+ ϕ2(u2)

),

with

• ϕ1 generator of a Gumbel copula with parameter θ1 = 3,

• and ϕ2 generator of a Frank copula with parameter θ2 = −2.

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Nested Archimedean copulas with negativedependence

• This kind of models can be useful for partial internal modelsin Solvency II.

• Sufficient conditions can be adapted from completelymonotonic conditions depending on the dimension.

• Existence of such nested Archimedean copulas is of courselimited to low dimensions.

• Simulation methods are more complicated as we do not haveanymore the frailty model representation.

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