Research

16 pages, 459 KiB

Open AccessArticle

Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims

by

Siti Norafidah Mohd Ramli

and

Jiwook Jang

Risks 2014, 2(2), 195-210; https://doi.org/10.3390/risks2020195 - 27 May 2014

Cited by 9 | Viewed by 6276

Abstract

We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, [...] Read more.

We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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25 pages, 460 KiB

Open AccessArticle

Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction

by

Søren Asmussen

Risks 2014, 2(1), 49-73; https://doi.org/10.3390/risks2010049 - 11 Mar 2014

Cited by 8 | Viewed by 7402

Abstract

In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes [...] Read more.

In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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22 pages, 343 KiB

Open AccessArticle

Catastrophe Insurance Modeled by Shot-Noise Processes

by

Thorsten Schmidt

Risks 2014, 2(1), 3-24; https://doi.org/10.3390/risks2010003 - 21 Feb 2014

Cited by 22 | Viewed by 6517

Abstract

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We [...] Read more.

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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21 pages, 340 KiB

Open AccessArticle

Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach

by

Claude Lefèvre

and

Philippe Picard

Risks 2013, 1(3), 192-212; https://doi.org/10.3390/risks1030192 - 13 Dec 2013

Cited by 1 | Viewed by 4326

Abstract

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, [...] Read more.

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

16 pages, 693 KiB

Open AccessArticle

Impact of Climate Change on Heat Wave Risk

by

Romain Biard

,

Christophette Blanchet-Scalliet

,

Anne Eyraud-Loisel

and

Stéphane Loisel

Risks 2013, 1(3), 176-191; https://doi.org/10.3390/risks1030176 - 12 Dec 2013

Cited by 1 | Viewed by 5864

Abstract

We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring [...] Read more.

We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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14 pages, 311 KiB

Open AccessArticle

A Risk Model with an Observer in a Markov Environment

by

Hansjörg Albrecher

and

Jevgenijs Ivanovs

Risks 2013, 1(3), 148-161; https://doi.org/10.3390/risks1030148 - 11 Nov 2013

Cited by 19 | Viewed by 5057

Abstract

We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation [...] Read more.

We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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18 pages, 15023 KiB

Open AccessArticle

Optimal Deterministic Investment Strategies for Insurers

by

Nicole Bäuerle

and

Ulrich Rieder

Risks 2013, 1(3), 101-118; https://doi.org/10.3390/risks1030101 - 07 Nov 2013

Cited by 7 | Viewed by 5353

Abstract

We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility [...] Read more.

We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the Lévy process framework. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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20 pages, 430 KiB

Open AccessArticle

Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates

by

Marcus C. Christiansen

Risks 2013, 1(3), 81-100; https://doi.org/10.3390/risks1030081 - 25 Oct 2013

Cited by 4 | Viewed by 5415

Abstract

In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use [...] Read more.

In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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