Polynomial approximation of mutivariate aggregate claim amounts distribution Applications to reinsurance P.O. Goffard Axa France - Mathematics Institute of Marseille I2M Aix-Marseille University 19 th International Congress on Insurance Mathematics and Economics.
Sommaire Introduction to multivariate aggregate claim amounts model Study of a bivariate model with applications to reinsurance A numerical method based on orthogonal polynomials June 24 th of 2015 Liverpool 2/25
Multivariate risk models: Capturing dependency Modelization of aggregate claim amounts associated to individuals in a portfolio Different lines of business that belong to a given insurance company A given line of business of that belong to several insurance companies. Claims are caused by an event and the severities are correlated to the magnitude of it. Third-party liability motor: Bodily injured claims and property damage claims Workers Compensation: Medical claims and income replacement claims June 24 th of 2015 Liverpool 3/25
Univariate risk model: How to include dependency? A univariate total claim amount random variable is defined by X = N U i i=1 N is a counting random variable {U i } i N is a sequence of i.i.d. random variables. Dependency occurs Between the claim severities Between the claim frequencies June 24 th of 2015 Liverpool 4/25
Multivariate collective model Model #1 Risk are modeled jointly via X 1. X n j=1 M = V 1j. V nj {V i } i N = {(V 1i... V ni )} i N is a sequence of i.i.d. random vectors M is a counting random variable {V i } i N and M are mutually independent. June 24 th of 2015 Liverpool 5/25
Multivariate collective model Model #2 Risk are modeled jointly via X 1. X n = N1 j=1 U 1j. Nn j=1 U nj N = (N 1... N n ) is a counting random vectors {U i } i N = ({U 1i } i N... {U ni } i N ) are independant sequences of i.i.d. random variables N and {U i } i N are mutually independent. June 24 th of 2015 Liverpool 6/25
Multivariate collective model Model #3 Risk are modeled jointly via X 1. X n = N1 j=1 U 1j. Nn j=1 U nj M + j=1 V 1j. V nj N = (N 1... N n ) is a counting random vectors {U i } i N = ({U 1i } i N... {U ni } i N ) are independant sequences of i.i.d. random variables {V i } i N = {(V 1i... V ni )} i N is a sequence of i.i.d. random vectors M is a counting random variable {V i } i N N M and {U i } i N are mutually independent. June 24 th of 2015 Liverpool 7/25
Two insurers and one reinsurer are in a pub. Let 2 insurance portfolios associated to the same line of business that belong to two insurance companies The aggregate claim amounts are given by ( X 1 X 2 ) = ( N1 N 1 and N 2 are independent j=1 U 1j N2 j=1 U 2j ) + ( M j=1 V 1j V 2j Polynomial approximation of the joint PDF of (X 1 X 2 ) ) June 24 th of 2015 Liverpool 8/25
A non-proportional global reinsurance treaty Reinsurer offers insurer i a usual stop-loss reinsurance treaty with priority b i and limit c i i = 1 2. The joint distribution of (X 1 X 2 ) is useful Pricing purposes: Determination of the stop loss premium. Risk Management purposes: The risk exposure of the reinsurer is modeled through Z = min [ (X 1 b 1 ) + c 1 ] + min [ (X2 b 2 ) + c 2 ]. Value-at-Risk computation and solvency capital determination. June 24 th of 2015 Liverpool 9/25
A 2-dimensional Polynomial approximation Let (X 1 X 2 ) be a random vector with probability measure P X1 X 2 and PDF f X1 X 2. ν is a reference probability measure constructed via the product of two univariate probability measure ν(x 1 x 2 ) = ν 1 (x 1 ) ν 2 (x 2 ) f ν (x 1 x 2 ) = f ν1 (x 1 ) f ν2 (x 2 ) {Q ν i k } k N is an orthonormal polynomial system ν i i = 1 2. {Q kl } kl N is an orthonormal polynomial system ν where Q kl (x 1 x 2 ) = Q ν 1 k (x 1)Q ν 2 l (x 2 ) k l N. June 24 th of 2015 Liverpool 10/25
A 2-dimensional Polynomial approximation If dp X 1 X 2 dν L 2 (ν) then where The Parseval identity is checked dp X1 X 2 (x 1 x 2 ) = dν + kl=0 a kl Q kl (x 1 x 2 ) a kl = E [Q kl (X 1 X 2 )] = E [ Q ν 1 k (X 1)Q ν 2 l (X 2 ) ] dp X1 X 2 dν 2 = + kl=0 a 2 kl < + June 24 th of 2015 Liverpool 11/25
Méthode d approximation polynomiale en dimension 2 The PDF of (X 1 X 2 ) admits a polynomial representation f X1 X 2 (x 1 x 2 ) = + kl=0 PDF Approximations follow from truncations f K L X 1 X 2 (x 1 x 2 ) = K k=0 l=0 a kl Q kl (x 1 x 2 )f ν (x 1 x 2 ). L a kl Q kl (x 1 x 2 )f ν (x 1 x 2 ) where K and L denote the orders of truncation. Survival function approximations follow from integration F K L X 1 X 2 (u 1 u 2 ) = + + u 1 u 2 K k=0 l=0 L a kl Q kl (x 1 x 2 )f ν (x 1 x 2 )dx 1 dx 2. June 24 th of 2015 Liverpool 12/25
Polynomial approximation for a bivariate aggregate claim model The probability measure associated to ( ) X 1 X 2 = = ( N1 ( j=1 U 1j N2 W 1 W 2 j=1 U 2j ) + ( ) + Y 1 Y 2 ( M j=1 ) V 1j V 2j ) has a lot of singularities. June 24 th of 2015 Liverpool 13/25
On the choice of the reference probability measure The probability measure of ( Y 1 Y 2 ) = ( M j=1 V 1j V 2j ) is dp Y1 Y 2 (y 1 y 2 ) = f M (0)δ 00 (y 1 y 2 ) + dg Y1 Y 2 (y 1 y 2 ). dg Y1 Y 2 is a defective probability measure having support on R 2 +. ν is defined as the product of gamma measures. ν i is a gamma measure Γ(m i r i ) i = 1 2. f νi (x) = e x/m i x r i Γ(r i )m r i i x 0. {Q ν i k } k N is sequence of generalized Laguerre polynomials i = 1 2. June 24 th of 2015 Liverpool 14/25
On the choice of the reference probability measure The probability measure of ( W 1 W 2 ) = ( N1 j=1 U 1j N2 j=1 U 2j ) is dp W1 W 2 (w 1 w 2 ) = f N1 (0)f N2 (0)δ 00 (w 1 w 2 ) + dg W1 (w 1 ) dg W2 (w 2 ) + f N1 (0)dG W2 (w 2 ) δ 0 (w 1 ) + f N2 (0)dG W1 (w 1 ) δ 0 (w 2 ). Univariate polynomial approximation for G Wi i = 1 2. Gamma probability measure and generalized Laguerre polynomials. June 24 th of 2015 Liverpool 15/25
On the integrability condition Theorem Let Y = (Y 1 Y 2 ) be a random vector. If H1 The set is not empty. Γ Y = inf{(s 1 s 2 ) R 2 + L Y (s 1 s 2 ) = + } H2 There exists a = (a 1 a 2 ) R 2 + such that y i f Y (y 1 y 2 ) are strictly decreasing for y i a i and i = 1 2. Then for y a where γ Y Γ Y. f Y (y 1 y 2 ) A Y (s 1 s 2 )e (s 1y 1 +s 2 y 2 ) s γ Y June 24 th of 2015 Liverpool 16/25
On the integrability condition June 24 th of 2015 Liverpool 17/25
On the integrability condition 1. Find (γ Y1 γ Y2 ) such that L Y1 Y 2 (γ Y1 γ Y2 ) < + and γ Y1 γ Y2 > 0 2. Choose r 1 r 2 m 1 and m 2 such that 0 < r 1 r 2 1 and 0 < 1 m 1 < 2γ Y1 0 < 1 m 2 < 2γ Y2. June 24 th of 2015 Liverpool 18/25
Decay of {a kl } kl N : Generating function study The defective PDF of (Y 1 Y 2 ) admits the polynomial representation g Y1 Y 2 (y 1 y 2 ) = + kl=0 a kl Q kl (y 1 y 2 )f ν (y 1 y 2 ). Taking the Laplace transform leads to [ ] C(z 1 z 2 ) = (1 + z 1 ) r 1 (1 + z 2 ) r 2 z 1 L gy1 Y 2 m 1 (1 + z 1 ) z 2 m 2 (1 + z 2 ) where C(z 1 z 2 ) = + kl=0 a klc ν 1 c ν i k = k cν 2 l (k + ri 1 k z k 1 zl 2 and ) i = 1 2. June 24 th of 2015 Liverpool 19/25
Numerical illustrations: Survival function of (Y 1 Y 2 ) M is governed by a geometric distribution N B(1 3/4) (V 1 V 2 ) is governed by a bivariate exponential distribution DBVE(ρ µ 1 µ 2 ) ρ = 1 4 µ 1 = µ 2 = 1 Polynomial approximations are compared to Monte-Carlo based approximations. The parametrization m 1 = 1 1 m 2 = r 1 = r 2 = 1. (1 p)µ 1 (1 p)µ 2 leads to a generating function of the form and C(z 1 z 2 ) = 1 1 + z 1 z 2 (p 2 ρ(1 p) 2 p). a kl = [ p 2 ρ(1 p) 2 p ] k δkl k l N June 24 th of 2015 Liverpool 20/25
Numerical illustrations: Survival function of (Y 1 Y 2 ) June 24 th of 2015 Liverpool 21/25
Numerical illustrations: Distribution of (X 1 X 2 ) N 1 and N 2 are geometrically distributed N B(1 3/4) {U 1j } j N are {U 2j } j N are sequences of i.i.d. Γ(1 1)-distributed The PDF is available in a closed form. Polynomial approximation of the survival function of (X 1 X 2 ) with increasing truncation order. Priorities: c 1 = c 2 = 1. Limits: b 1 = b 2 = 4. Polynomial approximations with orders of truncation equal to 10 of the survival function of Z = min [ ] [ ] (X 1 b 1 ) + c 1 + min (X2 b 2 ) + c 2. The polynomial approximations are compared to Monte Carlo approximations. June 24 th of 2015 Liverpool 22/25
Numerical illustrations: Survival function of (X 1 X 2 ) June 24 th of 2015 Liverpool 23/25
Numerical illustrations: Reinsurance cost z Monte Carlo approximation Polynomial approximation 0 0.90385 0.898808 2 0.73193 0.724774 4 0.44237 0.435013 6 0.24296 0.237576 June 24 th of 2015 Liverpool 24/25
Conclusion and Perspectives The polynomial approximation is an efficient numerical method: A good approximation of a bivariate aggregate claim amounts going along with motivation in reinsurance. Perspectives Applications of the method to the approximation of other function in actuarial science and other fields of applied probabilities. Statistical extensions when data are available The approximation can turn into a semi-parametric estimator of the PDF. June 24 th of 2015 Liverpool 25/25