Claims Prediction with Dependence using Copula Models Yeo Keng - - PowerPoint PPT Presentation

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 1/28

Claims Prediction with Dependence using Copula Models

Yeo Keng Leong and Emiliano A. Valdez School of Actuarial Studies Faculty of Commerce and Economics University of New South Wales Sydney, AUSTRALIA 11 November 2005

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope

■ Introduction, Motivation

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope

■ Introduction, Motivation ■ Setting and Notations

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope

■ Introduction, Motivation ■ Setting and Notations ■ Modelling Time Dependence with Copulas ◆ Assumptions and Model ◆ Conditional Expectation

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope

■ Introduction, Motivation ■ Setting and Notations ■ Modelling Time Dependence with Copulas ◆ Assumptions and Model ◆ Conditional Expectation ■ Application ◆ Gaussian and Student-t Copulas ◆ Archimedean Copulas - Cook-Johnson Copula ◆ FGM Copulas ◆ Remarks on Choice of Copula ◆ Illustrations of Copula Density Ratio

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope

■ Introduction, Motivation ■ Setting and Notations ■ Modelling Time Dependence with Copulas ◆ Assumptions and Model ◆ Conditional Expectation ■ Application ◆ Gaussian and Student-t Copulas ◆ Archimedean Copulas - Cook-Johnson Copula ◆ FGM Copulas ◆ Remarks on Choice of Copula ◆ Illustrations of Copula Density Ratio ■ Conclusion

• Scope

Introduction, Motivation

• Introduction
• Motivation

Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 3/28

Introduction, Motivation

• Scope

Introduction, Motivation

• Introduction
• Motivation

Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 4/28

Introduction

■ Experience Rating ◆ Rating process that takes into account, at least partially,

the individual risk’s experience

• Scope

Introduction, Motivation

• Introduction
• Motivation

Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 4/28

Introduction

■ Experience Rating ◆ Rating process that takes into account, at least partially,

the individual risk’s experience

■ Credibility Theory

Premium = Z ·Own Experience+(1 − Z)·Group Experience where Z ∈ [0, 1] is known as “credibility factor”

• Scope

Introduction, Motivation

• Introduction
• Motivation

Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 5/28

Motivation

■ Traditional practice in many credibility models to assume

independence of claims

◆ either across time for individual risk or between individuals ◆ Gerber and Jones (1975) and Frees, et al. (1999) are

examples of the former case

• Scope

Introduction, Motivation

• Introduction
• Motivation

Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 5/28

Motivation

■ Traditional practice in many credibility models to assume

independence of claims

◆ either across time for individual risk or between individuals ◆ Gerber and Jones (1975) and Frees, et al. (1999) are

examples of the former case

■ We offer additional insight into modelling dependence of

claims across time periods for a fixed individual by considering use of copulas

• Scope

Introduction, Motivation Setting and Notations

• Setting and Notations

Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 6/28

Setting and Notations

• Scope

Introduction, Motivation Setting and Notations

• Setting and Notations

Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

Setting and Notations

■ T time periods

• Scope

Introduction, Motivation Setting and Notations

• Setting and Notations

Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

Setting and Notations

■ T time periods ■ Claims amount for one individual at time period t, Xt,

t = 1, 2, ..., T

◆ realisation denoted by xt

• Scope

Introduction, Motivation Setting and Notations

• Setting and Notations

Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

Setting and Notations

■ T time periods ■ Claims amount for one individual at time period t, Xt,

t = 1, 2, ..., T

◆ realisation denoted by xt ■ Individual’s claim vector: XT = (X1, X2, ..., XT)′

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 8/28

Modelling Time Dependence with Copulas

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model

■ Multivariate distribution and density functions,

HT (x1, . . . , xT ) and hT (x1, . . . , xT )

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model

■ Multivariate distribution and density functions,

HT (x1, . . . , xT ) and hT (x1, . . . , xT )

■ Marginal distribution and density functions, Ft (xt) and

ft (xt), t = 1, 2, ..., T

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model

■ Multivariate distribution and density functions,

HT (x1, . . . , xT ) and hT (x1, . . . , xT )

■ Marginal distribution and density functions, Ft (xt) and

ft (xt), t = 1, 2, ..., T

■ Copula function, CT (F1 (x1) , . . . , FT (xT )), thus

HT (x1, . . . , xT ) = CT (F1 (x1) , . . . , FT (xT )) = CT (u1, . . . , uT )

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model

■ Multivariate distribution and density functions,

HT (x1, . . . , xT ) and hT (x1, . . . , xT )

■ Marginal distribution and density functions, Ft (xt) and

ft (xt), t = 1, 2, ..., T

■ Copula function, CT (F1 (x1) , . . . , FT (xT )), thus

HT (x1, . . . , xT ) = CT (F1 (x1) , . . . , FT (xT )) = CT (u1, . . . , uT )

■ Copula density function

cT (uT ) = ∂T CT (uT ) ∂u1 . . . ∂uT

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 10/28

Conditional expectation

■ Aim : Compute E (XT +1|XT = xT )

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 10/28

Conditional expectation

■ Aim : Compute E (XT +1|XT = xT ) ■ With assumptions above, we have Proposition 1:

E “ XT +1|XT = xT ” = Z ∞ −∞ xT +1 · cT +1 “ uT +1 ” cT ` uT ´ dFT +1 “ xT +1 ”

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 10/28

Conditional expectation

■ Aim : Compute E (XT +1|XT = xT ) ■ With assumptions above, we have Proposition 1:

E “ XT +1|XT = xT ” = Z ∞ −∞ xT +1 · cT +1 “ uT +1 ” cT ` uT ´ dFT +1 “ xT +1 ”

■ Observe that cT +1(uT +1) cT (uT )

induces change of measure of

• riginal probability measure corresponding to XT +1

E “ XT +1|XT = xT ” = Z ∞ −∞ xT +1 · fQ T +1 “ xT +1 ” dxT +1 = EQ “ XT +1 ”

where

fQ T +1 “ xT +1 ” dxT +1 = dF Q T +1 “ xT +1 ” = cT +1 “ uT +1 ” cT ` uT ´ dFT +1 “ xT +1 ”

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas

• Assumptions and model
• Conditional expectation

Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 10/28

Conditional expectation

■ Aim : Compute E (XT +1|XT = xT ) ■ With assumptions above, we have Proposition 1:

E “ XT +1|XT = xT ” = Z ∞ −∞ xT +1 · cT +1 “ uT +1 ” cT ` uT ´ dFT +1 “ xT +1 ”

■ Observe that cT +1(uT +1) cT (uT )

induces change of measure of

• riginal probability measure corresponding to XT +1

E “ XT +1|XT = xT ” = Z ∞ −∞ xT +1 · fQ T +1 “ xT +1 ” dxT +1 = EQ “ XT +1 ”

where

fQ T +1 “ xT +1 ” dxT +1 = dF Q T +1 “ xT +1 ” = cT +1 “ uT +1 ” cT ` uT ´ dFT +1 “ xT +1 ”

■ Can also interpret as “re-weighting” of density function after

• bserving previous claims

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 11/28

Applications

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 12/28

Gaussian copula

■ Definition:

CT (uT ) = ΦΣT

• Φ−1 (u1) , . . . , Φ−1 (uT )
• where ΦΣT = standardised T-dimensional Normal

distribution function and Φ−1 = quantile function of standard

• ne-dimensional Normal distribution

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 12/28

Gaussian copula

■ Definition:

CT (uT ) = ΦΣT

• Φ−1 (u1) , . . . , Φ−1 (uT )
• where ΦΣT = standardised T-dimensional Normal

distribution function and Φ−1 = quantile function of standard

• ne-dimensional Normal distribution

■ Gaussian copula density:

cT (uT ) = exp

• − 1

2ς′ T Σ−1 T ςT

• (2π)T |ΣT | T

k=1 φ (Φ−1 (uk))

where ς′

T =

• Φ−1 (u1) , . . . , Φ−1 (uT )
• , ΣT = correlation

matrix and φ (z) = (2π)−1/2 exp

• − 1

2z2

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 13/28

Gaussian copula - continued

■ Proposition 2:

cT +1 (uT +1) cT (uT ) = 1 σZ.T +1 · φ

• Φ−1 (˘

uT +1)

• φ (Φ−1 (uT +1))

where Φ−1 (˘ uT +1) =

• Φ−1 (uT +1) − µZ.T +1
• /σZ.T +1,

µZ.T +1 = ρ′

T +1,T Σ−1 T ςT

σZ.T +1 = 1 − ρ′

T +1,T Σ−1 T ρT +1,T and

ρ′

T +1,T = vector of correlations of XT +1 with each element of

XT

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 13/28

Gaussian copula - continued

■ Proposition 2:

cT +1 (uT +1) cT (uT ) = 1 σZ.T +1 · φ

• Φ−1 (˘

uT +1)

• φ (Φ−1 (uT +1))

where Φ−1 (˘ uT +1) =

• Φ−1 (uT +1) − µZ.T +1
• /σZ.T +1,

µZ.T +1 = ρ′

T +1,T Σ−1 T ςT

σZ.T +1 = 1 − ρ′

T +1,T Σ−1 T ρT +1,T and

ρ′

T +1,T = vector of correlations of XT +1 with each element of

XT

■ Corollary 1:

E (XT +1|XT = xT ) = EZ

• F −1

T +1 [Φ (µZ.T +1 + σZ.T +1Z)]

• where unconditional expectation is computed for standard

univariate Normal random variable Z

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 14/28

Student-t copula

■ Definition:

CT (uT ) = TΣT ,r

• T −1

r

(u1) , . . . , T −1

r

(uT )

• where TΣT ,r = standardised T-dimensional Student-t

distribution function and T −1

r

= quantile function of standard one-dimensional Student-t distribution with r degrees of freedom

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 14/28

Student-t copula

■ Definition:

CT (uT ) = TΣT ,r

• T −1

r

(u1) , . . . , T −1

r

(uT )

• where TΣT ,r = standardised T-dimensional Student-t

distribution function and T −1

r

= quantile function of standard one-dimensional Student-t distribution with r degrees of freedom

■ Student-t copula density:

cT (uT ) = Γ r+T

2

1 + 1

r ς′ T Σ−1 T ςT

−(r+T )/2 Γ r

2

(rπ)T |ΣT | T

k=1 tr

• T −1

r

(uk)

• where ς′

T =

• T −1

r

(u1) , . . . , T −1

r

(uT )

• and

Tr (z) =

Γ( r+1

2 )

Γ( r

2)√

(rπ)

• 1 + z2

r

−(r+1)/2

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 15/28

Student-t copula - continued

■ Proposition 3:

cT +1 (uT +1) cT (uT ) = 1 σ∗

Z.T +1

· tr+T

• T −1

r

(˘ uT +1)

• tr
• T −1

r

(uT +1)

• where T −1

r

(˘ uT +1) =

• T −1

r

(uT +1) − µZ.T +1

• /σ∗

Z.T +1,

µZ.T +1 = ρ′

T +1,T Σ−1 T ςT and

σ∗2

Z.T +1 = rσ2 Z.T +1

• 1 + 1

rς′ T Σ−1 T ςT

• / (r + T)

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 15/28

Student-t copula - continued

■ Proposition 3:

cT +1 (uT +1) cT (uT ) = 1 σ∗

Z.T +1

· tr+T

• T −1

r

(˘ uT +1)

• tr
• T −1

r

(uT +1)

• where T −1

r

(˘ uT +1) =

• T −1

r

(uT +1) − µZ.T +1

• /σ∗

Z.T +1,

µZ.T +1 = ρ′

T +1,T Σ−1 T ςT and

σ∗2

Z.T +1 = rσ2 Z.T +1

• 1 + 1

rς′ T Σ−1 T ςT

• / (r + T)

■ Corollary 2:

E (XT +1|XT = xT ) = EZ

• F −1

T +1

• Tr
• µZ.T +1 + σ∗

Z.T +1Z

• ,

where unconditional expectation is computed for standard univariate Student-t random variable Z with (r + T) degrees

• f freedom

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 16/28

Archimedean copulas

■ Definition:

CT (uT ) = ψ−1 (ψ (u1) + · · · + ψ (un)) where ψ = Archimedean generator and ψ−1 = inverse function of ψ

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 16/28

Archimedean copulas

■ Definition:

CT (uT ) = ψ−1 (ψ (u1) + · · · + ψ (un)) where ψ = Archimedean generator and ψ−1 = inverse function of ψ

■ Archimedean copula density:

cT (uT ) = ψ−1(T ) T

• t=1

ψ (ut) T

• t=1

ψ′ (ut) where ψ−1(T ) = T-th derivative of ψ−1 ψ′ = derivative of ψ

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 16/28

Archimedean copulas

■ Definition:

CT (uT ) = ψ−1 (ψ (u1) + · · · + ψ (un)) where ψ = Archimedean generator and ψ−1 = inverse function of ψ

■ Archimedean copula density:

cT (uT ) = ψ−1(T ) T

• t=1

ψ (ut) T

• t=1

ψ′ (ut) where ψ−1(T ) = T-th derivative of ψ−1 ψ′ = derivative of ψ

■ Proposition 4:

cT +1 (uT +1) cT (uT ) = ψ−1(T +1) [ψ (CT +1 (uT +1))] ψ−1(T ) [ψ (CT (uT ))] ψ′ (uT +1)

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 17/28

Archimedean copulas (Cook Johnson)

■ Definition:

CT (uT ) = T

k=1 u−δ k

− T + 1 −1/δ

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 17/28

Archimedean copulas (Cook Johnson)

■ Definition:

CT (uT ) = T

k=1 u−δ k

− T + 1 −1/δ

■ Conditional expectation:

E (XT +1|XT = xT ) = EY

• F −1

T +1

• Y δ

where unconditional expectation is evaluated under translated Pareto with density fY (y) = ((1/δ) + T) (1 + ξ)(1/δ)+T (y + ξ)(1/δ)+T +1 for y > 1

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 18/28

FGM copulas

■ Definition:

CT ` uT ´ = 8 > < > : 1 + T X s=2 X 1≤t1<...<ts≤T αt1,...,ts s Y j=1 » 1 − utj – 9 > = > ; T Y t=1 ut = PT ` uT ´ T Y t=1 ut

where PT (uT ) =

• 1 + T

s=2

• 1≤t1<...<ts≤T αt1,...,ts

s

j=1

• 1 − utj
• and

αt1,...,ts are parameters satisfying certain conditions

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 18/28

FGM copulas

■ Definition:

CT ` uT ´ = 8 > < > : 1 + T X s=2 X 1≤t1<...<ts≤T αt1,...,ts s Y j=1 » 1 − utj – 9 > = > ; T Y t=1 ut = PT ` uT ´ T Y t=1 ut

where PT (uT ) =

• 1 + T

s=2

• 1≤t1<...<ts≤T αt1,...,ts

s

j=1

• 1 − utj
• and

αt1,...,ts are parameters satisfying certain conditions

■ Proposition 5:

E “ XT +1|XT = xT ” = EU 2 4F −1 T +1 (U) · PT +1 ` 2uT , 2U ´ PT +1 ` 2uT , 1 ´ 3 5

where unconditional expectation is evaluated for Uniform(0, 1) random variable U

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 19/28

FGM copulas - continued

■ Further simplification gives:

E “ XT +1|XT = xT ” = E “ XT +1 ” − DT +1 ` uT ´ PT ` uT ´ × Z ∞ −∞ FT +1 “ xT +1 ” h 1 − FT +1 “ xT +1 ”i dxT +1

where DT +1 (uT ) = T

s=1

• 1≤t1<...<ts≤T αt1,...,ts,T +1

s

j=1

• 1 − 2utj

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications

• Gaussian copula
• Gaussian copula - continued
• Student-t copula
• Student-t copula - continued
• Archimedean copulas
• Archimedean copulas (Cook

Johnson)

• FGM copulas
• FGM copulas - continued

Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 19/28

FGM copulas - continued

■ Further simplification gives:

E “ XT +1|XT = xT ” = E “ XT +1 ” − DT +1 ` uT ´ PT ` uT ´ × Z ∞ −∞ FT +1 “ xT +1 ” h 1 − FT +1 “ xT +1 ”i dxT +1

where DT +1 (uT ) = T

s=1

• 1≤t1<...<ts≤T αt1,...,ts,T +1

s

j=1

• 1 − 2utj
• ■ Closed form solutions available for various choices of

marginals, e.g.

◆ exponential ◆ Weibull ◆ Pareto

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 20/28

Choice of Copula

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 21/28

Some remarks - 1

■ Gaussian, Student-t and Archimedean copulas

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 21/28

Some remarks - 1

■ Gaussian, Student-t and Archimedean copulas ◆ modelling of less complex dependence structures

(especially Gaussian and Student-t copulas)

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 21/28

Some remarks - 1

■ Gaussian, Student-t and Archimedean copulas ◆ modelling of less complex dependence structures

(especially Gaussian and Student-t copulas)

◆ has very few parameters therefore simplifying calibration

process

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 21/28

Some remarks - 1

■ Gaussian, Student-t and Archimedean copulas ◆ modelling of less complex dependence structures

(especially Gaussian and Student-t copulas)

◆ has very few parameters therefore simplifying calibration

process

◆ allows for wide range of dependence, e.g. Frank copula

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 22/28

Some remarks - 2

■ FGM copulas

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 22/28

Some remarks - 2

■ FGM copulas ◆ can assign unique dependence parameter for each group

• f risks, allowing for more complicated dependence

structures

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 22/28

Some remarks - 2

■ FGM copulas ◆ can assign unique dependence parameter for each group

• f risks, allowing for more complicated dependence

structures

◆ conditional expectation can be solved analytically

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 22/28

Some remarks - 2

■ FGM copulas ◆ can assign unique dependence parameter for each group

• f risks, allowing for more complicated dependence

structures

◆ conditional expectation can be solved analytically ◆ calibration process more tedious with large number of

parameters involved

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula

• Some remarks - 1
• Some remarks - 2

Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 22/28

Some remarks - 2

■ FGM copulas ◆ can assign unique dependence parameter for each group

• f risks, allowing for more complicated dependence

structures

◆ conditional expectation can be solved analytically ◆ calibration process more tedious with large number of

parameters involved

◆ limitation on its parameter values, therefore allows only

weak dependence

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 23/28

Illustrating the Copula Density Ratio

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 24/28

Case Scenario

■ single period’s claims experience, X1, has been observed

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 24/28

Case Scenario

■ single period’s claims experience, X1, has been observed ■ therefore looking at copula density ratio, c2 (u2) /c1 (u1)

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 24/28

Case Scenario

■ single period’s claims experience, X1, has been observed ■ therefore looking at copula density ratio, c2 (u2) /c1 (u1) ■ X2 assumed to follow Pareto distribution with density

fX2 (x2) = βλβ (λ + x2)β+1 , for x2 > 0

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 24/28

Case Scenario

■ single period’s claims experience, X1, has been observed ■ therefore looking at copula density ratio, c2 (u2) /c1 (u1) ■ X2 assumed to follow Pareto distribution with density

fX2 (x2) = βλβ (λ + x2)β+1 , for x2 > 0

■ assume the Pareto parameters take on values of

λ = 1000 and β = 3 so that E (XT +1) = λ/ (β − 1) = 500

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 24/28

Case Scenario

■ single period’s claims experience, X1, has been observed ■ therefore looking at copula density ratio, c2 (u2) /c1 (u1) ■ X2 assumed to follow Pareto distribution with density

fX2 (x2) = βλβ (λ + x2)β+1 , for x2 > 0

■ assume the Pareto parameters take on values of

λ = 1000 and β = 3 so that E (XT +1) = λ/ (β − 1) = 500

■ Gaussian, Student-t, Cook-Johnson and FGM copulas

considered

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 25/28

Figure 1 - varying dependence

500 1000 1500 2 4 6 8 10 12 14 16 18 20 X2 Copula Density Ratio Gaussian Copula

Prior mean = 500 X1 = 1000 Independence Low Moderate High

500 1000 1500 1 2 3 4 5 6 7 8 X2 Copula Density Ratio Student−t Copula

Prior mean = 500 X1 = 1000 Independence Low Moderate High

500 1000 1500 1 2 3 4 X2 Copula Density Ratio Cook−Johnson Copula

Prior mean = 500 X1 = 1000 Independence Low Moderate High

500 1000 1500 0.5 1 1.5 2 X2 Copula Density Ratio FGM Copula

Prior mean = 500 X1 = 1000 Independence Very Low Low

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio

• Case Scenario
• Figure 1 - varying

dependence

• Figure 2 - varying observed

claim Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 26/28

Figure 2 - varying observed claim

500 1000 1500 1 2 3 4 X2 Copula Density Ratio Gaussian Copula, ρ = 0.707107

Prior mean = 500 Independence X1 = 200 X1 = 500 X1 = 1000

500 1000 1500 1 2 3 X2 Copula Density Ratio Student−t Copula, ρ = 0.707107

Prior mean = 500 Independence X1 = 200 X1 = 500 X1 = 1000

500 1000 1500 1 2 3 X2 Copula Density Ratio Cook−Johnson Copula, δ = 2

Prior mean = 500 Independence X1 = 200 X1 = 500 X1 = 1000

500 1000 1500 0.5 1 1.5 2 X2 Copula Density Ratio FGM Copula, α = 0.9

Prior mean = 500 Independence X1 = 200 X1 = 500 X1 = 1000

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion

• Conclusion

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 27/28

Conclusion

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion

• Conclusion

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 28/28

Conclusion

■ Extension to notion of predicting next period’s claims by

relaxing independence assumptions (across time)

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion

• Conclusion

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 28/28

Conclusion

■ Extension to notion of predicting next period’s claims by

relaxing independence assumptions (across time)

■ Predictive claim can be expressed as expectation under a

new probability measure that reflects ratio of densities of copulas relating to historical claims

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion

• Conclusion

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 28/28

Conclusion

■ Extension to notion of predicting next period’s claims by

relaxing independence assumptions (across time)

■ Predictive claim can be expressed as expectation under a

new probability measure that reflects ratio of densities of copulas relating to historical claims

■ Ratio of densities can also be interpreted as “re-weighting” of

marginal density function after observing one or more claims

• Scope

Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion

• Conclusion

Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 28/28

Conclusion

■ Extension to notion of predicting next period’s claims by

relaxing independence assumptions (across time)

■ Predictive claim can be expressed as expectation under a

new probability measure that reflects ratio of densities of copulas relating to historical claims

■ Ratio of densities can also be interpreted as “re-weighting” of

marginal density function after observing one or more claims

■ Applied to various copulas, i.e. Gaussian, Student-t,

Archimedean and FGM copulas, with illustrations on effect of “re-weighting”