Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Denuit, Michel; Robert, Christian

doi:10.1007/s11009-021-09888-0

Cited by 8 publications

(19 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One obtains convenient results when the multivariate pgf is simple to differentiate, which is sometimes the case for mixture models (which include common shock models). The results from this section supplement the literature on risk allocation or risk sharing for mixture models as studied in Section 3 of [Cossette et al, 2018] or Section 4 of [Denuit and Robert, 2021a].…”

Section: Sum Of Dependent Rvsmentioning

confidence: 62%

Generating function method for the efficient computation of expected allocations

Blier-Wong¹,

Cossette²,

Marceau³

2022

Preprint

View full text Add to dashboard Cite

Consider a risk portfolio with aggregate loss random variable S = X 1 + • • • + X n defined as the sum of the n individual losses X 1 , . . . , X n . The expected allocation, E[X i × 1 {S=k} ], for i = 1, . . . , n and k ∈ N, is a vital quantity for risk allocation and risk-sharing. For example, one uses this value to compute peer-to-peer contributions under the conditional mean risk-sharing rule and capital allocated to a line of business under the Euler risk allocation paradigm. This paper introduces an ordinary generating function for expected allocations, a power series representation of the expected allocation of an individual risk given the total risks in the portfolio when all risks are discrete. First, we provide a simple relationship between the ordinary generating function for expected allocations and the probability generating function. Then, leveraging properties of ordinary generating functions, we reveal new theoretical results on closed-formed solutions to risk allocation problems, especially when dealing with Katz or compound Katz distributions. Then, we present an efficient algorithm to recover the expected allocations using the fast Fourier transform, providing a new practical tool to compute expected allocations quickly. The latter approach is exceptionally efficient for a portfolio of independent risks.

show abstract

Section: Sum Of Dependent Rvsmentioning

confidence: 62%

Generating function method for the efficient computation of expected allocations

Blier-Wong¹,

Cossette²,

Marceau³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Statements (i) and (ii) are the direct results of Definition 2.2 of moment transforms. Statement (iii) is similar to Proposition 3.1 of Denuit and Robert (2021), to which we refer the readers for more details.…”

Section: Letmentioning

confidence: 70%

“…Denuit (2020) derived formulas for the tail conditional expectations (TCE) of some univariate compound distributions. Denuit and Robert (2021) presented some results for the TCE of a compound mixed Poisson model, where both the claim frequencies and sizes depend on several latent variables. Ren (2021) derived the formulas for the TCE and tail variance (TV) of multivariate compound models based on Sundt (1999), where claim frequency is one-dimensional and one claim can yield multiple dependent losses.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, Denuit (2020) applied this method in analyzing the TCE of univariate compound distributions. The concept of multivariate moment transform was studied in Denuit and Robert (2021) and applied in analyzing multivariate risks. Ren (2021) applied this technique to study the tail risk of a multivariate compound sum introduced by Sundt (1999), where the claim frequency is one-dimensional while each claim is a multidimensional random vector with dependent components.…”

Section: Introductionmentioning

confidence: 99%

“…See, for example, Cummins and Wiltbank (1983), Bermúdez (2009), Cossette et al (2012) and the references therein. We also discuss a case where the claim frequencies and sizes are dependent through a common mixing variable, following Denuit and Robert (2021).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Evaluating the Tail Risk of Multivariate Aggregate Losses

Jiang

Ren²

2022

ASTIN Bull.

View full text Add to dashboard Cite

In this paper, we study the tail risk measures for several commonly used multivariate aggregate loss models where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies. We first develop formulas for the moment (or size biased) transforms of the multivariate aggregate losses, showing their relationship with the moment transforms of the claim frequencies and claim sizes. Then, we apply the formulas to compute some popular risk measures such as the tail conditional expectation and tail variance of the multivariate aggregated losses and to perform capital allocation analysis.

show abstract

Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses

Denuit

Robert

2021

Methodol Comput Appl Probab

Self Cite

View full text Add to dashboard Cite

This paper exploits the representation of the conditional mean risk sharing allocations in terms of size-biased transforms to derive effective approximations within insurance pools of limited size. Precisely, the probability density functions involved in this representation are expanded with respect to the Gamma density and its associated Laguerre orthonormal polynomials, or with respect to the Normal density and its associated Hermite polynomials when the size of the pool gets larger. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. The numerical method then consists in truncating the series expansions to a limited number of terms. This results in an approximation in terms of the first moments of the individual loss distributions. Compound Panjer-Katz sums are considered as an application. The proposed method is compared with the well-established Panjer recursive algorithm. It appears to provide the analyst with reliable approximations that can be used to tune system parameters, before performing exact calculations.

show abstract

Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Cited by 8 publications

References 25 publications

Generating function method for the efficient computation of expected allocations

Generating function method for the efficient computation of expected allocations

Evaluating the Tail Risk of Multivariate Aggregate Losses

Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses

Contact Info

Product

Resources

About