Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Goffard, Pierre-Olivier; Loisel, Stéphane; Pommeret, Denys

doi:10.1007/s11009-015-9470-7

Cited by 13 publications

(15 citation statements)

References 23 publications

(29 reference statements)

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“…Note also that the damping procedure employed when integrability problems arise is quite similar to considering the exponentially tilted distribution instead of the real one. The use of the gamma distribution as reference is applied to actuarial science in [29,30]. 3…”

Section: Lognormal Sums Via a Gamma Reference Distributionmentioning

confidence: 99%

Orthonormal Polynomial Expansions and Lognormal Sum Densities

Asmussen

Goffard

Laub

2019

Risk and Stochastics

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Approximations for an unknown density g in terms of a reference density fν and its associated orthonormal polynomials are discussed. The main application is the approximation of the density f of a sum S of lognormals which may have different variances or be dependent. In this setting, g may be f itself or a transformed density, in particular that of log S or an exponentially tilted density. Choices of reference densities fν that are considered include normal, gamma and lognormal densities. For the lognormal case, the orthonormal polynomials are found in closed form and it is shown that they are not dense in L2(fν ), a result that is closely related to the lognormal distribution not being determined by its moments and provides a warning to the most obvious choice of taking fν as lognormal. Numerical examples are presented and comparison are made to established approaches such as the Fenton-Wilkinson method and skew-normal approximations. Also extension to density estimation for statistical data sets and non-Gaussian copulas are outlined.

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Section: Lognormal Sums Via a Gamma Reference Distributionmentioning

confidence: 99%

Orthonormal Polynomial Expansions and Lognormal Sum Densities

Asmussen

Goffard

Laub

2019

Risk and Stochastics

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“…Proposition 2 represents a practical refinement in comparison to the works Goffard et al [13,14] as the formulas derived may be readily evaluated without using numerical integration.…”

Section: Approximating Densities Of Positive Random Variablesmentioning

confidence: 95%

“…Thus it is possible to evaluate the infinite-time ruin probability via Panjer's algorithm. If we are able to determine the Laplace transform of S N then we can also apply the polynomial method of Goffard et al [13], the fractional moment based method of Gzyl et al [15], and the exponential moments based method of Mnatsakanov et al [27].…”

Section: Risk Theorymentioning

confidence: 99%

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Orthogonal polynomial expansions to evaluate stop-loss premiums

Goffard

Laub

2020

Journal of Computational and Applied Mathematics

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A numerical method is proposed to evaluate the survival function of a compound distribution and the stop-loss premiums associated with a non-proportional global reinsurance treaty. The method relies on a representation of the probability density function in terms of Laguerre polynomials and the gamma density. We compare the method against a well established Laplace transform inversion technique at the end of the paper.MSC 2010: 60G55, 60G40, 12E10. This paper concerns approximations of f S N and F S N , though we begin with a discussion of how S N is used in actuarial science.Frequently, S N models the aggregated losses of a non-life insurance portfolio over a given period of time-here N represents the number of claims and U k the claim sizes-yet other applications also exist. Actuaries and risk managers typically want to quantify the risk of large losses by a single comprehensible number, a risk measure.One popular risk measure is the Value-at-Risk (VaR). In actuarial contexts, the VaR at level α ∈ (0, 1) is defined such that the probability of (aggregated) losses exceeding the level VaR is at most 1 − α. We denote this α-quantile asFollowing the European recommendation of the Solvency II directive, the standard value for α is 0.995, see [18]. It is used by risk managers in banks, insurance companies, and other financial institutions to allocate risk reserves and to determine solvency margins. Also, we have stop-loss premiums (slp's) which are risk measures that are commonly used in reinsurance agreements.A reinsurance agreement is a common risk management contract between insurance companies, one called the "cedant" and the other the "reinsurer". Its aim is to keep the cedant's long-term earnings stable by protecting the cedant against large losses. The reinsurer absorbs part of the cedant's loss, say f (S N ) where 0 ≤ f (S N ) ≤ S N , leaving the cedant with I f (S N ) = S N − f (S N ). In return, the cedant pays a premium linked tounder the expected value premium principle.

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“…Overall, the results using the orthogonal polynomial approach can be impressive, cf. [84,85] for some applications in insurance. As f S is in such a simple form, integrals involving it can often be solved analytically -for example, Dufresne and Li [68] give an explicit form for the price of a (discrete) Asian option using the orthogonal polynomial pdf approximation.…”

Section: Beyond the Central Limit Theoremmentioning

confidence: 99%

Computational methods for sums of random variables

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A typical problem in the field of rare-event estimation is to find the probability P(S > γ) where S := X 1 + · · · + X d for a fixed d ∈ N + and where the γ ∈ R is large or increasing. In applications we often wish to understand the behaviour of a combination of random factors. Hence the random variable S is ubiquitous in real-world modeling problems. It can model, for example, aggregate risk or portfolio value for holding d risky assets [124,152], the aggregate losses for d insurance policy claims [14,107], and the combined signal interference from d wireless transmission sources [73]. Probabilities of this form are used to understand how a system would behave under extreme scenarios such as a market crash, a power surge, or a natural disaster. One is typically interested in, not just the quantity P(S > γ), but the behaviour of the summands when the extreme event {S > γ} occurs.

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Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Cited by 13 publications

References 23 publications

Orthonormal Polynomial Expansions and Lognormal Sum Densities

Orthonormal Polynomial Expansions and Lognormal Sum Densities

Orthogonal polynomial expansions to evaluate stop-loss premiums

Computational methods for sums of random variables

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