Showing papers in "Scandinavian Actuarial Journal in 2005"
TL;DR: In this paper, a two-parameter smooth continuous composite lognormal-Pareto model is proposed to model the tail behavior of both small and large losses in the actuarial and insurance industries.
Abstract: The actuarial and insurance industries frequently use the lognormal and the Pareto distributions to model their payments data. These types of payment data are typically very highly positively skewed. Pareto model with a longer and thicker upper tail is used to model the larger loss data, while the larger data with lower frequencies as well as smaller data with higher frequencies are usually modeled by the lognormal distribution. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to its monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, we were motivated to look for a new avenue to remedy the situation. Here we introduce a two-parameter smooth continuous composite lognormal-Pareto model that is a two-parameter lognormal density up to an unknown threshold value...
186 citations
TL;DR: In this article, a Poisson log-bilinear projection model is applied to the forecasting of the gender- and age-specific mortality rates for Belgium on the basis of mortality statistics relating to the period 1950-2000.
Abstract: This paper proposes bootstrap procedures for expected ramining lifetimes and life annuity single premiums in a dynamic mortality environment. Assuming a further continuation of the stable pace of mortality decline, a Poisson log-bilinear projection model is applied to the forecasting of the gender- and age-specific mortality rates for Belgium on the basis of mortality statistics relating to the period 1950-2000. Bootstrap procedures are then used to obtain confidence intervals on various actuarial quantities.
166 citations
TL;DR: In this paper, the authors investigated the problem of approximating the tail probability of a weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent.
Abstract: In an insurance context, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approximating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results.
102 citations
TL;DR: In this article, the Laplace transform of the time until ruin for a fairly general risk model is presented, which includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases.
Abstract: This paper presents the Laplace transform of the time until ruin for a fairly general risk model. The model includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases. It also allows for correlated arrival processes, and claim sizes that depend upon environmental factors such as periods of contagion. The paper exploits the relationship between the surplus process and fluid queues, where a number of recent developments have provided the basis for our analysis.
87 citations
TL;DR: Kluppelberg and Stadtmuller as mentioned in this paper obtained a simple asymptotic formula for the ruin probability of the classical model with constant interest force and regularly varying tailed claims.
Abstract: Kluppelberg and Stadtmuller (1998, Scand. Actuar. J., no. 1, 49–58) obtained a simple asymptotic formula for the ruin probability of the classical model with constant interest force and regularly varying tailed claims. This short note extends their result to the renewal model. The proof is based on a result of Resnick & Willekens (1991, Comm. Statist. Stochastic Models 7, no. 4, 511–525).
83 citations
TL;DR: In this paper, the authors employed a systematic outlier detection process to ascertain the timing, magnitude, and persistence of any outliers present in historical mortality trends and then tried to match the identified outliers with imperative events that could possibly justify the vacillations in human mortality levels.
Abstract: The Lee-Carter model and its variants have been extensively employed by actuaries, demographers, and many others to forecast age-specific mortality. In this study, we use mortality data from England and Wales, and four Scandinavian countries to perform time-series outlier analysis of the key component of the Lee-Carter model – the mortality index. We begin by employing a systematic outlier detection process to ascertain the timing, magnitude, and persistence of any outliers present in historical mortality trends. We then try to match the identified outliers with imperative events that could possibly justify the vacillations in human mortality levels. At the same time, we adjust the effect of the outliers for model re-estimation. A new iterative model re-estimation method is proposed to reduce the chance of erroneous model specification. The empirical results indicate that the outlier-adjusted model could achieve more efficient forecasts of variables such as death rates and life expectancies. Finally, we p...
60 citations
TL;DR: In this paper, the authors considered a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution.
Abstract: We consider a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot [10,11] and Gerber and Shiu [21,22]. The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.
57 citations
TL;DR: In this article, the authors considered a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete K m distribution, i.e., the probability generating function (p.f.) of the distribution function is a ratio of two polynomials of order.
Abstract: We consider a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of the distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. A recursive formula is derived for the expected discounted penalty (Gerber-Shiu) function, which can be used to analyze many quantities associated with the time of ruin, e.g., the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.
56 citations
TL;DR: In this article, the authors investigated a model that allows for dividend payments according to a time-dependent linear barrier strategy and derived partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments.
Abstract: In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.
55 citations
TL;DR: In this article, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin.
Abstract: We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.
41 citations
TL;DR: In this article, the authors consider the estimation of the ruin probability in a linear portfolio of insurance risk processes and model the total claim amount of different business activities by compound Poisson processes.
Abstract: We consider the estimation of the ruin probability in a linear portfolio of insurance risk processes. We model the total claim amount of different business activities by compound Poisson processes. We allow for dependence of the components, which we model by a Levy copula. We study in detail a Clayton-Pareto model as representative for a large claims model and a Clayton-exponential model as a small claims model. We compare the ruin probability in the Clayton dependence model with the corresponding independent and completely dependent models.
TL;DR: In this article, the density of the time to ruin given that ruin occurs in a Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are distributed as Erlang(n, β).
Abstract: We derive expressions for the density of the time to ruin given that ruin occurs in a Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are distributed as Erlang(n, β). We provide numerical illustrations of finite time ruin probabilities, as well as illustrating features of the density functions.
TL;DR: In this article, the authors present an explicit characterization for the joint probability density function of the surplus immediately prior to ruin and the deficit at ruin for a general risk process, which includes the Sparre-Andersen risk model with phase-type inter-claim times and claim sizes.
Abstract: This paper presents an explicit characterization for the joint probability density function of the surplus immediately prior to ruin and the deficit at ruin for a general risk process, which includes the Sparre-Andersen risk model with phase-type inter-claim times and claim sizes. The model can also accommodate a Markovian arrival process which enables claim sizes to be correlated with the inter-claim times. The marginal density function of the surplus immediately prior to ruin is specifically considered. Several numerical examples are presented to illustrate the application of this result.
TL;DR: In this paper, the authors consider risk processes with delayed claims in a Markovian environment, and study the asymptotic behaviour of finite and infinite horizon ruin probabilities under the small claim assumption.
Abstract: We consider risk processes with delayed claims in a Markovian environment, and we study the asymptotic behaviour of finite and infinite horizon ruin probabilities under the small claim assumption. We also consider multivariate risk processes of the same kind, and we give upper and lower bounds for the Lundberg parameters of the corresponding total reserve. Our results have strong analogies with those one in the paper by Juri (Super modular order and Lundberg exponents, 2002).
TL;DR: In this paper, the authors compare two different valuation models for assets and liabiliesthat can be considered in the standard approach to solvency assessmentand in particular, in determining the required target capit...
Abstract: We compare two different valuation models for assets and liabilitiesthat can be considered in the standard approach to solvency assessmentand in particular, in determining the required target capit ...
TL;DR: In this article, a real-life insurance portfolio is used to compare four proportional structures: quota share reinsurance, variable quota share, variable and surplus reinsurance with a table of lines.
Abstract: Proportional reinsurance is often thought to be a very simple method of covering the portfolio of an insurer. Theoreticians are not really interested in analysing the optimality properties of these types of reinsurance covers. In this paper, we will use a real-life insurance portfolio in order to compare four proportional structures: quota share reinsurance, variable quota share reinsurance, surplus reinsurance and surplus reinsurance with a table of lines.
TL;DR: In this article, the authors present an inventory of non-exponential bounds for ruin probabilities and stop-loss premiums in the general Sparre-Andersen model (renewal model) of risk theory.
Abstract: We present an inventory of non-exponential bounds for ruin probabilities and stop-loss premiums in the general Sparre-Andersen model (renewal model) of risk theory. Various additional bounds are given if one assumes that the ladder height distribution F associated with the risk process belongs to a certain class of distributions, in particular if it is concave or it exhibits a (positive or negative) aging property. In most cases, these bounds are shown to improve existing ones in the literature and/or possess the correct asymptotic behaviour when the distribution F is subexponential. Since in the classical (compound Poisson) risk model the ladder height distribution is always concave, all the bounds given in the paper are also valid for this model. Finally, in many cases the results of the paper are also valid for any compound geometric distribution.
TL;DR: In this paper, the authors consider the infinite-time ruin probability of an insurance company investing in the stock market and give asymptotic expressions for the minimal ruin probability, as well as for the optimal investment strategy.
Abstract: We consider the infinite-time ruin probability of an insurance company investing in the stock market. This is done under the following assumptions: For the claim surplus process we use the classical Cramer-Lundberg model, where the claims have a distribution function, belonging to certain subclasses of the class of subexponential distributions. The stock price movement is modeled by geometric Brownian motion, and we allow positive interest rates for the riskless bond. In this setting we analyze the Hamilton-Jacobi-Bellman equation for the minimal ruin probability. We give asymptotic expressions for the minimal ruin probability, as well as for the optimal investment strategy. It turns out that the asymptotic order of the minimal ruin probability is different from the previous considered case of zero interest on the bond.
TL;DR: In this article, the authors adapted the results of Salminen and Yor [7] in several ways, such as using spectral theory to obtain a series expansion for the distribution of I + that renders this quantity applicable to actuarial purposes.
Abstract: In a recent paper, Salminen and Yor [7] relate the distribution of the Dufresne's reflected perpetuity We thank Moshe Milevsky for fruitful discussions on this subject. where B μ is Brownian motion with drift μ>0, to the hitting time of a reflected Bessel process. In this contribution, we adapt the results of Salminen and Yor [7] in several ways. First, we use spectral theory to obtain a series expansion for the distribution of I + that renders this quantity applicable to actuarial purposes. We also study the exponential functionals where X μ α is a skew Brownian motion with drift μ>0.
TL;DR: In this article, the problem of optimal redistribution of surplus in life and pension insurance when the interest rate is modelled as a continuous time Markov chain with a finite state space was studied.
Abstract: We study the problem of optimal redistribution of surplus in life and pension insurance when the interest rate is modelled as a continuous time Markov chain with a finite state space. We work with traditional participating life insurance policies with payments consisting of a specified contractual payment stream and an unspecified additional bonus payment stream. Our model allows for interest rates below the technical interest rate. We apply stochastic control techniques in our search for optimal strategies, and we prove the dynamic programming principle for our particular type of problem. Furthermore, we state and prove a verification theorem and obtain an explicit solution that leads to a characterization of optimal strategies, indicating that some widely used redistribution schemes are suboptimal.
TL;DR: In this paper, a general framework for a bonus-malus system based on the number and the size of the claims is presented, the set of the bonus classes being an interval [a, b], say 0
Abstract: A general framework for a Bonus–Malus system (BMS) based on the number and the size of the claims is presented, the set of the bonus classes being an interval [a, b], say 0
TL;DR: In this paper, the authors derived a partial differential equation, the solution of which provided the ruin probability for fractional Brownian motion with Hurst exponent in the range (0.5, 1).
Abstract: In this paper we study the ruin probability at a given time for liabilities of diffusion type, driven by fractional Brownian motion with Hurst exponent in the range (0.5, 1). Using fractional Ito calculus we derive a partial differential equation the solution of which provides the ruin probability. An analytical solution is found for this equation and the results obtained by this approach are compared with the results obtained by Monte-Carlo simulation.
TL;DR: In this paper, the authors consider the problem of maximizing a utility function of the form U(y) for a fixed terminal time, where y is the risk neutral utility function.
Abstract: Consider an investor, e.g. an insurance company whose assets evolve according to where the are i.i.d. Thus if is the time of ruin, for , i.e. the process is put on hold after ruin. In an insurance context, the can be absolute net returns of its underwriting business (minus expenses), and the the proportion of the business the company chooses to cover itself. Alternatively, the can be seen as the combined profit ratio, i.e. one minus the combined ratio, and the will then be net premiums. In this context, using the reparametrization , the are the net premium/assets ratio, and in practical insurance this quantity enters frequently in solvency considerations. We seek to choose the , or equivalently the , in order to maximize , with T a fixed terminal time. The maximization is done for “utility” functions of the form with , the case c=1 gives the risk neutral utility function U(y)=y. The optimal solution turns out to be of the form , where is constant for each t, finite or infinite. Given a tolerance level , i...
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