Showing papers in "Scandinavian Actuarial Journal in 2004"
TL;DR: In this article, a simple Poisson process N with constant intensity λ>0 is considered, and the process Y such that Y 0∈ℝ and where a, σ are predictable bounded stochastic processes, and γ is a predictable process which is bounded away from zero.
Abstract: We consider a filtered probability space with a standard Brownian motion W, a simple Poisson process N with constant intensity λ>0, and we consider the process Y such that Y 0∈ℝ and where a, σ are predictable bounded stochastic processes, and γ is a predictable process which is bounded away from zero. A discrete record of n+1 observations {Y 0, Y t 1 , …, Y t n−1 , Y t n } is available, with t i =ih. Using such observations, we construct estimators of N t i , i=1, …, n, λ and γ τ j , where τ j are the instants of jump within [0, nh]. They are consistent and asymptotically controlled when the number of observations increases and the step h tends to zero.
95 citations
TL;DR: In this article, the authors consider a risk process with the possibility of investment into a risky asset and obtain the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claims case.
Abstract: We consider a risk process with the possibility of investment into a risky asset The aim of the paper is to obtain the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claims case In addition we prove convergence of the optimal investment level as the initial capital tends to infinity
69 citations
TL;DR: The actuarial profession is increasingly teaming up with financial economists for a fruitful cooperation on the proper valuation of life insurance and pension (L&P) liabilities as mentioned in this paper, which has been a natural consequence of a recent sharply increased focus on market values in financial reports of L&P companies from regulators, standard setters, the financial press, stakeholders, and others with an interest in the business.
Abstract: The actuarial profession is increasingly teaming up with financial economists for a fruitful cooperation on the proper valuation of life insurance and pension (L&P) liabilities. This has been a natural consequence of a recent sharply increased focus on market values in financial reports of L&P companies from regulators, standard setters, the financial press, stakeholders, and others with an interest in the L&P business. This article provides a financial economist's point of view on recent developments in relation to the fair valuation of L&P liabilities. The role of accounting standards and the background for the international harmonization in this field are first discussed. We then review and explain the concept of fair value and provide a general view on appropriate techniques for estimating fair values of L&P liabilities in accordance with the definition of the concept. The paper also contains a section which briefly reviews recent and quite innovative regulatory initiatives in relation to market value...
64 citations
TL;DR: In this article, an explicit expression for the latter probability, under an excess of loss (XL) treaty is derived, using the improved version of the Ignatov and Kaishev's ruin probability formula.
Abstract: A certain volume of risks is insured and there is a reinsurance contract, according to which claims and total premium income are shared between a direct insurer and a reinsurer in such a way, that the finite horizon probability of their joint survival is maximized. An explicit expression for the latter probability, under an excess of loss (XL) treaty is derived, using the improved version of the Ignatov and Kaishev's ruin probability formula (see Ignatov, Kaishev & Krachunov. 2001a) and assuming, Poisson claim arrivals, any discrete joint distribution of the claims, and any increasing real premium income function. An explicit expression for the probability of survival of the cedent only, under an XL contract is also derived and used to determine the probability of survival of the reinsurer, given survival of the cedent. The absolute value of the difference between the probability of survival of the cedent and the probability of survival of the reinsurer, given survival of the cedent is used for the choice...
58 citations
TL;DR: In this article, the authors investigated the ultimate ruin probability of a discrete time risk model with a positive constant interest rate under the assumption that the gross loss of the company within one year is subexponentially distributed, and derived a simple asymptotic relation for the ruin probability.
Abstract: This paper investigates the ultimate ruin probability of a discrete time risk model with a positive constant interest rate. Under the assumption that the gross loss of the company within one year is subexponentially distributed, a simple asymptotic relation for the ruin probability is derived and compared to existing results.
56 citations
TL;DR: In this paper, the authors consider the distribution of the deficit at ruin in the Sparre Andersen renewal risk model given that ruin occurs and show that the individual claim amounts have a phase-type distribution.
Abstract: We consider the distribution of the deficit at ruin in the Sparre Andersen renewal risk model given that ruin occurs. We show that if the individual claim amounts have a phase-type distribution, then there is a simple phase-type representation for the distribution of the deficit. We illustrate the application of this result with several examples.
54 citations
TL;DR: In this paper, the authors derive optimal, from the cedent's point of view, reinsurance arrangements balancing the risk measured by variance and expected profits under various mean-variance premium principles of the reinsurer.
Abstract: Reinsurance reduces the risk but it also reduces the potential profit. The aim of the paper is to derive optimal, from the cedent's point of view, reinsurance arrangements balancing the risk measured by variance and expected profits under various mean-variance premium principles of the reinsurer. We find that quota share, excess of loss or combinations of excess of loss with quota share are the optimal rules according to a fixed expected gain of the cedent
51 citations
TL;DR: In this article, the current state of affairs in the EU Solvency II project is described and the background and international context of the project is discussed, as well as the general outline of a future EU solvency system.
Abstract: This article describes the current state of affairs in the EU Solvency II project. The background and international context of the project is discussed, as well as the general outline of a future EU solvency system. In particular, several areas where further technical work is needed are outlined. These topics could provide interesting objects of study for professionals of actuarial sciences as well as to those of other related sciences.
44 citations
TL;DR: An insurance model where the risk process can be controlled by reinsurance and by investment in a financial market is studied, and a weak contraction property is derived in order to treat models without discount factor.
Abstract: This paper studies an insurance model where the risk process can be controlled by reinsurance and by investment in a financial market. The performance criterion is either the expected exponential utility of the terminal surplus or the ruin probability. It is shown that the problems can be imbedded in the framework of discrete-time stochastic dynamic programming but with some special features. A short introduction to control theory with infinite state space is provided which avoids the measure-theoretic apparatus by use of the so-called structure assumption. Moreover, in order to treat models without discount factor, a weak contraction property is derived. Explicit conditions are obtained for the optimality of employing no reinsurance.
39 citations
TL;DR: In this article, the authors derived the asymptotic behavior of the optimal ruin probability of an insurer who invests optimally in a stock in the presence of positive interest force and claims with tails of regular variation.
Abstract: This paper consists of three parts. In the first part we derive the asymptotic behavior of the optimal ruin probability of an insurer who invests optimally in a stock in the presence of positive interest force and claims with tails of regular variation. Our results extend previously obtained results by Gaier & Grandits (2002) with zero interest, and by Kluppelberg & Stadtmuller (1998) without investment possibility. In the second part we prove an existence theorem for the integro-differential equation for the survival probability of an insurer, who invests a constant fraction of his wealth in a risky stock, and his remaining wealth in a bond with nonnegative interest. Our result extends a previously known result by Wang & Wu (2001). Finally, in the third part we derive the asymptotic behavior of the ruin probability of the insurer, introduced in the second part, in the presence of claims with tails of regular variation.
34 citations
TL;DR: A model of Huntington's disease, a highly penetrant, dominantly inherited, fatal neurological disorder, is described and the possible costs of adverse selection, in terms of increased premiums, under various possible moratoria on the use of genetic information, including family history are considered.
Abstract: We describe briefly a model of Huntington's disease (HD), a highly penetrant, dominantly inherited, fatal neurological disorder. Although it is a single-gene disorder, mutations are variable in their effects, depending on the number of times that the CAG trinucleotide is repeated in a certain region of the HD gene. The model covers: (a) rates of onset, depending on CAG repeat length as well as age; (b) post-onset rates of mortality; and (c) the distribution of CAG repeat lengths in the population. Using these, we study the critical illness and life insurance markets. We calculate premiums based on genetic test results that disclose the CAG repeat length, or more simply on a family history of HD. These vary widely with age and policy term; some are exceptionally high, but in a large number of cases cover could be offered within normal underwriting limits. We then consider the possible costs of adverse selection, in terms of increased premiums, under various possible moratoria on the use of genetic informat...
TL;DR: This article derives credibility estimators and mean square errors for normal hierarchical linear models and establishes the link between these estimator and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.
Abstract: Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods. Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.
TL;DR: This work investigates developments in Danish mortality based on data from 1974–1998 working in a two-dimensional model with chronological time and age as the two dimensions and suggests that life insurance companies use the estimation technique and the cross-validation for bandwidth selection when analyzing their portfolio mortality.
Abstract: We investigate developments in Danish mortality based on data from 1974–1998 working in a two-dimensional model with chronological time and age as the two dimensions. The analyses are done with non-parametric kernel hazard estimation techniques. The only assumption is that the mortality surface is smooth. Cross-validation is applied for optimal bandwidth selection to ensure the proper amount of smoothing to help distinguishing between random and systematic variation in data. A bootstrap technique is used for construction of pointwise confidence bounds. We study the mortality profiles by slicing up the two-dimensional mortality surface. Furthermore we look at aggregated synthetic population metrics as ‘population life expectancy’ and ‘population survival probability’. For Danish women these metrics indicate decreasing mortality with respect to chronological time. The metrics can not directly be used for prediction purposes. However, we suggest that life insurance companies use the estimation technique and ...
TL;DR: In this article, recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t -th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials.
Abstract: Recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t-th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. Also, some general results are derived for the evaluation of the t-th order moments of stop-loss transforms. The recursions can be applied for the exact evaluation of the probability function, distribution function, tail probability and stop-loss premium of compound mixed Poisson distributions and the corresponding mixed Poisson distributions. Several examples are also presented.
TL;DR: A unification and integration of the level premium and paid-up valuation principles is obtained and it is demonstrated how to avoid capital being required in the future to cover valuation strains.
Abstract: Surrender and paid-up states are incorporated in the valuation of guaranteed benefits and payments of a level premium paying life insurance policy. We present different valuation methods and examine to what extent they avoid capitalizing and releasing future loadings which are associated with the payment of future premiums. We demonstrate how to avoid capital being required in the future to cover valuation strains. The paid-up benefit valuation method is being extended so that it does not require the premium basis to be on the safe-side of the valuation basis. We obtain a unification and integration of the level premium and paid-up valuation principles.
TL;DR: In this paper, a mixture representation for the conditional distribution of the deficit at ruin (given that ruin occurs) is derived, as well as a stochastic decomposition involving the residual lifetime associated with the maximal aggregate loss.
Abstract: Properties of the distribution of the deficit at ruin in the stationary renewal risk model are studied. A mixture representation for the conditional distribution of the deficit at ruin (given that ruin occurs) is derived, as well as a stochastic decomposition involving the residual lifetime associated with the maximal aggregate loss. When the individual claims have a phase-type distribution, the deficit at ruin is also of phase-type.
TL;DR: In this paper, the authors consider a general problem of modeling a mortality law of a population of failing units with some parametric function, and define a mortality table of crude rates as a statistical estimator with multinomial distribution and show its consistency as well as asymptotic normality.
Abstract: We consider a general problem of modeling a mortality law of a population of failing units with some parametric function. In this setting we define a mortality table of crude rates as a statistical estimator with multinomial distribution and show its consistency as well as asymptotic normality. We further derive the statistical properties of parameter estimators in a parametric mortality model based on a weighted square loss function. We use the obtained results to study consistency and appropriateness of the parametric bootstrap method in our setting. We derive the conditions on the assumed parametric mortality law and the loss function, under which the bootstrap is consistent for estimating the model parameters, their standard errors and corresponding confidence intervals. We apply our results to a model of Aggregate US Mortality Table based on a so called mixture of extreme value distributions suggested by Carriere (1992).
TL;DR: In this paper, the probability of ruin is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model.
Abstract: In this paper the probability of ruin is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model. The idea is originally enlightened by Jasiulewicz (2001). We make a slight modification on the model and a generalization on the intensity process. By a “backward differential argument” and the Markov property of the intensity process we strictly derive the integral equation satisfied by the probability of ruin. Further, we solve the equation when the intensity process is a homogeneous n-state Markov process by Laplace transforms. At the end of the paper, an example is given.
TL;DR: Using the language of copulas, the famous Fisher-Tippett Theorem of extreme value theory is generalized to the case with sequences of dependent random variables and the dependence structure is modelled using archimedean copulas.
Abstract: Using the language of copulas, we generalize the famous Fisher-Tippett Theorem of extreme value theory to the case with sequences of dependent random variables. The dependence structure is modelled...
TL;DR: In this paper, the marginal distributions of a discrete-time claim process with correlated claims are derived based on stochastic comparison in convex order and in Laplace transform order of the process S with two corresponding processes and having, respectively, uncorrelated and weakly correlated claims.
Abstract: The purpose of this paper is to derive bounds on the marginal distributions of a discrete-time claim process S with correlated claims. These bounds are based on stochastic comparison in convex order and in Laplace transform order of the process S with two corresponding processes and having, respectively, uncorrelated and weakly correlated claims. The relevance of these comparisons is due to the simple structure of the processes and , which are nothing else than a random walk and a mixed random walk. The paper also contains the proof of the closure under mixture property of some dependence orders, like supermodular and PQD, and some applications of the main results.
TL;DR: In this paper, a stochastic risk reserve process whose risk exposure can be controlled dynamically by applying proportional reinsurance and by issuing CAT bonds is considered, where the CAT Bond payments are only partly correlated with the insurers losses.
Abstract: We consider a stochastic risk reserve process whose risk exposure can be controlled dynamically by applying proportional reinsurance and by issuing CAT Bonds. The CAT Bond payments are only partly correlated with the insurers losses. The aim is to minimize the probability of ruin. Using a two-dimensional diffusion approximation we obtain a controlled diffusion problem which can be solved explicitly with the help of the HJB equation. We present some numerical results and discuss to which extend the proportional reinsurance can be replaced by issuing CAT Bonds.
TL;DR: In this article, the authors consider two portfolios: one of endowment insurance contracts and one of whole life insurance contracts, and study the effect of stochastic heterogeneity on the premiums and death benefits of the insurance contracts.
Abstract: In this paper we consider two portfolios: one of m endowment insurance contracts and one of m whole life insurance contracts. We introduce the majorization order, Schur functions, and parametric families of distribution functions. We assume that the owners of the portfolios are exposed to different members of a known parametric family of distributions and study the effect of this stochastic heterogeneity on the premiums and death benefits of the insurance contracts. We show that the premiums paid in both contracts are Schur concave and that the death benefit awarded in the whole life contract is Schur convex. We provide upper and lower bounds for the premiums and for the death benefit, and compute the bounds for four parametric families of distribution functions used frequently in the Actuarial Sciences.
TL;DR: In this paper, the authors study a bonus malus system with deductibles, where an insured has to find an optimal strategy of submitting claims, where optimal is defined as minimizing the total expected present value (epv) costs.
Abstract: In this paper we study a bonus malus system (bms) with deductibles. A bms is characterized by its premium levels and the transition rules among them. An insured is being moved among premium levels according to his/her claim record. Thus, an insured has to find an optimal strategy of submitting claims. Here optimal is in the sense of minimizing the total expected present value (epv) costs. Such strategies are found both for finite and infinite horizons. Furthermore, premium levels balancing the cost to the insured and the payoff of the insurer are given. The methods used to analyze the problem are from dynamic programming and Markov chains.
TL;DR: In this paper, the authors extended Reinhard and Snoussi's work to cover the case where the premium is an integer value and no restriction on the annual result is imposed.
Abstract: This paper deals with the severity of ruin in a discrete semi-Markov risk model. It is shown that the work of Reinhard and Snoussi (Stochastic Models, 18) can be extended to cover the case where the premium is an integer value and no restriction on the annual result is imposed. In particular, it is shown that the severity of ruin without initial surplus is solution of a system of equations. It can be obtained by a monotonically converging algorithm when the claims are bounded.