Model Of Risk Theory Research Articles

Relations between integrated tails and moments based on the deficit at ruin in the renewal risk model

In this paper, a renewal model of risk theory is considered, and relations between tails of the equilibrium distribution of the deficit at the time of ruin and a generalization of the equilibrium distribution of non-ruin are obtained. Extensions to higher order equilibrium distribution and Gerber-Shiu functions are also provided. Distributional properties for Pareto and exponential claims are studied. The results are illustrated by numerical examples.

Ensuring the operational reliability of roads on the horizontal curves using TST-Risk software

Due to the urgency of the task to ensure the operational reliability of roads and compliance with the requirements of Technical Regulations of the Customs Union 014/2011 on road safety, we developed software to determine the danger of moving along the horizontal curve. The developed software is based on the office processing of experimental studies on the horizontal curve and mathematical risk theory models.

Size-Biased Risk Measures of Compound Sums

The size-biased, or length-biased transform is known to be particularly useful in insurance risk measurement. The case of continuous losses has been extensively considered in the actuarial literature. Given their importance in insurance studies, this article concentrates on compound sums. The zero-augmented distributions that naturally appear in the individual model of risk theory are obtained as particular cases when the claim frequency distribution is concentrated on {0, 1}. The general results derived in this article help actuaries to understand how risk measurement proceeds because the formulas make explicit the loadings corresponding to each source of randomness. Some simple and explicit expressions are obtained when losses are modeled by independent compound Poisson sums and compound mixed Poisson sums, including the compound negative binomial sums. Extensions to correlated risks are briefly discussed in the concluding section.

Collective loss reserving with two types of claims in motor third party liability insurance

This paper adopts the new loss reserving approach proposed by Denuit and Trufin (2016), inspired from the collective model of risk theory. But instead of considering the whole set of claims as a collective, two types of claims are distinguished, those claims with relatively short development patterns and claims requiring longer developments. In each case, the total payment per cell is modelled by means of a Compound Poisson distribution with appropriate assumptions about the severities. A case study based on a motor third party liability insurance portfolio observed over 2004–2014 is used to illustrate the approach proposed in this paper. Comparisons with Chain-Ladder are performed and reveal significant differences in best estimates as well as in Value-at-Risk at high probability levels.

Beyond the Tweedie Reserving Model: The Collective Approach to Loss Development

This article proposes a new loss reserving approach, inspired from the collective model of risk theory. According to the collective paradigm, we do not relate payments to specific claims or policies, but we work within a frequency-severity setting, with a number of payments in every cell of the run-off triangle, together with the corresponding paid amounts. Compared to the Tweedie reserving model, which can be seen as a compound sum with Poisson-distributed number of terms and Gamma-distributed summands, we allow here for more general severity distributions, typically mixture models combining a light-tailed component with a heavier-tailed one, including inflation effects. The severity model is fitted to individual observations and not to aggregated data displayed in run-off triangles with a single value in every cell. In that respect, the modeling approach appears to be a powerful alternative to both the crude traditional aggregated approach based on triangles and the extremely detailed individual reserving approach developing each and every claim separately. A case study based on a motor third-party liability insurance portfolio observed over 2004–2014 is used to illustrate the relevance of the proposed approach.

Optimal Dynamic Proportional and Excess of Loss Reinsurance under Dependent Risks

In this paper, we study an optimal reinsurance strategy combining a proportional and an excess of loss reinsurance. We refer to a collective risk theory model with two classes of dependent risks; particularly, the claim number of the two classes of insurance business has a bivariate Poisson distribution. In this contest, our aim is to maximize the expected utility of the terminal wealth. Using the control technique, we write the Hamilton-Jacobi-Bellman equation and, in the special case of the only excess of loss reinsurance, we obtain the optimal strategy in a closed form, and the corresponding value function.

Optimal Proportional Reinsurance in a Bivariate Risk Model

The paper deals with the optimal proportional reinsurance in a collective risk theory model involving two classes of insurance business. These classes are dependent through the number of claims. The objective of the insurer is to choose an optimal reinsurance strategy that maximizes the expected exponential utility of terminal wealth. We are able to derive the evolution of the insurer surplus process under the assumption that the number of claims of the two classes of the insurance business has a Poisson bivariate distribution. We face the problem of finding the optimal strategy using the dynamic programming approach. Therefore, we determine the infinitesimal generator for the surplus process and for the value function, and we give the Hamilton Jacobi Bellmann (HJB) equation. Under particular assumptions, we obtain explicit form of the optimal reinsurance strategy on correspondent value function.

Analysis of Effects of Race on Insurance Companies’ Operations in Chicago, December 1977 ~ November 1978

Dataset Chicago is about the residential fire insurance policies issued in Chicago over December 1977 through February 1978; categorized as either voluntary, or involuntary, which includes state-offered Fair Access to Insurance Requirement (FAIR) 1 plans (Wriggins 2010) for people who would be denied insurance because of high risk classification, after Fair Housing Act (FHA) has been in act since 1968 2 (Calmore 1997). Main purpose of this article is to explore the relationship between insurance activity and the variable race by S-Plus, rather than numerical methods such as risk theory models. The final result proves insurance companies were using race as a determinative factor of underwriting insurance contacts, but not the only determinative factor. Data size is relatively small as n = 47 Zip codes. Theft, fire and the age of the house are also provided, along with the median income on the size of the expected loss and insolvency.

Some Optimization and Decision Problems in Proportional Reinsurance

Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function (Gerber & Shiu, 1998) is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs.We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phase-type 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and the Tail Value at Risk).

A Risk-Theory Model to Assess the Capital Requirement for Mortality and Longevity Risk

There is considerable uncertainty regarding the future development of life expectancy that leads to significant change in many fields of the insurance market. Pricing annuity products and mortality-linked securities seem primary goals of actuarial literature. At the same time, the valuation of non-hedgeable liabilities (as technical provisions for contracts where risk is not entirely borne by the policyholders) and the estimation of capital requirement appear very important issues in Solvency II framework. In this context, we propose a model based on Risk Theory in order to evaluate the capital requirement for mortality and longevity risk. We assume a life portfolio characterized by traditional and with-profit products divided in several homogeneous generations of contracts. Each cohort includes equal contracts that differ only by the insured sum with the aim to consider the effect of variability coefficient. Some assumptions allow to obtain closed formulae for the exact characteristics of demographic profit distribution regardless of contract types (i.e either with survival or death benefits). Furthermore Monte-Carlo methods provide the simulated distribution of mortality and longevity profit for each generation. Some case studies show the moments and the capital requirements for different life portfolios. Finally, further research will regard both the aggregation effect between several generations and a valuation of liabilities consistent to Solvency II context.

Special issue on Actuarial Statistics

This special issue of AStA aims to offer an overview of some relevant problems in actuarial science, involving statistical aspects. For centuries, the function of (re)insurers has been to sell risk coverage. Insurance companies charge premiums for covering these risks. The premiums are collected in advance while claim amounts are known only later. In most cases, the amount of premium is fixed so that the insurer cannot counteract adverse experience by charging additional premiums. The liabilities of insurance companies have, thus, to be carefully assessed in order to ensure solvency and to achieve profitability. The increasing complexity of insurance policies (sometimes covering several perils or catastrophic events, or running over decades in a changing environment) requires sophisticated risk theory models and advanced statistical methods, such as those presented in this special issue.

Maximum-likelihood and marginal-sum estimation in some particular collective models

In the present paper we consider collective models in risk theory and their thinning and decomposition. We give three applications to reinsurance, multiplicative tariffs, and loss reserving. For each of these applications we show how maximum-likelihood and marginal-sum estimation can be used to estimate the parameters.

RATIO MONOTONICITY FOR TAIL PROBABILITIES IN THE RENEWAL RISK MODEL

A renewal model in risk theory is considered, where $\overline{H}(u,y)$ is the tail of the distribution of the deficit at ruin with initial surplus u and $\overline{F}(y)$ is the tail of the ladder height distribution. Conditions are derived under which the ratio $\overline{H}(u,y)/\overline{F}(u+y)$ is nondecreasing in u for any y≥0. In particular, it is proven that if the ladder height distribution is stable and DFR or phase type, then the above ratio is nondecreasing in u. As a byproduct of this monotonicity, an upper bound and an asymptotic result for $\overline{H}(u,y)$ are derived. Examples are given to illustrate the monotonicity results.

Obtaining the dividends–penalty identities by interpretation

The dividends–penalty identity is a relation between three functions: the discounted penalty function without dividends, the discounted penalty function if a barrier dividend strategy is applied, and the expected discounted dividends until ruin. The classical model of risk theory is modified in that the deterministic premiums are replaced by a compound Poisson process with exponential jumps. In this model, the dividends–penalty identity is new and can be derived by interpretation. Then the dividends–penalty identity in the classical model is obtained as a limit.

Monotonicity properties and the deficit at ruin in the Sparre Andersen model

Let H u (y) be the (proper) distribution function of the deficit at ruin, given that ruin occurs with initial surplus u, in the Sparre Andersen model of risk theory. Dickson & dos Reis (1996) discussed the monotonicity of H u (y) as a function of u. In this paper, we obtain various monotonicity results for H u (y) and other related quantities for the decreasing/increasing failure rate (DFR/IFR) and the increasing/decreasing mean residual lifetime (IMRL/DMRL) classes of distributions. These results in particular extend and make more concrete the results of Dickson & dos Reis (1996) and Willmot & Lin (1998). A new class of distributions (increasing convolution ratio; ICR) is introduced. This class extends the well-known class of distributions with IFR. Specifically, we show that if the ladder height distribution F in the model is ICR, the ratio is a non-decreasing function of u, where ψ(u) denotes the ruin probability and . Further, we obtain generalizations (expressed in terms of the distribution of the deficit) of the well-known new worse than used (NWU) property of the probability of non-ruin.

A Generalization of the Lundberg Condition in the Sparre Andersen Model and Some Applications

We generalize the well-known Lundberg condition for the existence of the adjustment coefficient, R, in the Sparre Andersen model of risk theory. The new condition is given in terms of the distribution of the deficit at ruin in that model. As an application of this condition, we show that the function e Ru ψ(u) is nonincreasing when the claim size distribution in the model has a decreasing failure rate. Further, we obtain some new bounds for ruin probabilities and stop-loss premiums that are easy to compute and we give some examples to compare these bounds against existing ones in the literature.

Stochastic comparisons of multivariate mixture models

In this paper we consider sufficient conditions in order to stochastically compare random vectors of multivariate mixture models. In particular we consider stochastic and convex orders, the likelihood ratio order, and the hazard rate and mean residual life dynamic orders. Applications to proportional hazard models and mixture models in risk theory are also given.

Multivariate risk processes with interacting intensities

The classical models in risk theory consider a single type of claim. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous-time Markov chain of pure-birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions, we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.

Multivariate risk processes with interacting intensities

The classical models in risk theory consider a single type of claim. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous-time Markov chain of pure-birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions, we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.

Optimal dividend strategies for a risk process under force of interest

In the classical Cramér–Lundberg model in risk theory the problem of maximizing the expected cumulated discounted dividend payments until ruin is a widely discussed topic. In the most general case within that framework it is proved [Gerber, H.U., 1968. Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1, 185–227; Azcue, P., Muler, N., 2005. Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 (2) 261–308; Schmidli, H., 2008. Stochastic Control in Insurance. Springer] that the optimal dividend strategy is of band type. In the present paper we discuss this maximization problem in a generalized setting including a constant force of interest in the risk model. The value function is identified in the set of viscosity solutions of the associated Hamilton–Jacobi–Bellman equation and the optimal dividend strategy in this risk model with interest is derived, which in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. Finally, an example is constructed for Erlang(2)-claim sizes, in which the bands for the optimal strategy are explicitly calculated.