Showing papers in "Scandinavian Actuarial Journal in 2012"
TL;DR: For the uniform distance ρ(F n, Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3, the best known upper estimate of the absolute constant in the classical Berry-Esseen inequality is 0.3041 which is strictly less than the least possible value 0.4097 as discussed by the authors.
Abstract: By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequalit...
124 citations
TL;DR: In this article, the authors analyzed both with and without a surplus distribution mechanism, and the analysis was undertaken both with a bonus account that serves as a buffer in the following sense: in (bad) years when the investment returns are lower than the minimum rate of return guarantee, funds are transferred from the bonus account to the customer's account.
Abstract: Annual minimum rate of return guarantees are analyzed together with rules for distribution of positive excess return, i.e. investment returns in excess of the guaranteed minimum return. Together with the level of the annual minimum rate of return guarantee both the customer's and the insurer's fractions of the positive excess return are determined so that the market value of the insurer's capital inflow (determined by the fraction of the positive excess return) equals the market value of the insurer's capital outflow (determined by the minimum rate of return guarantee) at the inception of the contract. The analysis is undertaken both with and without a surplus distribution mechanism. The surplus distribution mechanism works through a bonus account that serves as a buffer in the following sense: in (‘bad’) years when the investment returns are lower than the minimum rate of return guarantee, funds are transferred from the bonus account to the customer's account. In (‘good’) years when the investment return...
119 citations
TL;DR: In this paper, a mixed copula approach is proposed to allow for dependency between the number of claims and its corresponding average claim size using a Gaussian copula, and the performance of the estimation procedure is validated in an extensive simulation study.
Abstract: A crucial assumption of the classical compound Poisson model of Lundberg for assessing the total loss incurred in an insurance portfolio is the independence between the occurrence of a claim and its claims size. In this paper we present a mixed copula approach suggested by Song et al. to allow for dependency between the number of claims and its corresponding average claim size using a Gaussian copula. Marginally we permit for regression effects both on the number of incurred claims as well as its average claim size using generalized linear models. Parameters are estimated using adaptive versions of maximization by parts (MBP). The performance of the estimation procedure is validated in an extensive simulation study. Finally the method is applied to a portfolio of car insurance policies, indicating its superiority over the classical compound Poisson model.
113 citations
TL;DR: In this paper, the authors investigated the latest developments in longevity risk modelling, and explored the key risk management challenges for both the financial and insurance industries, and provided a global view of the practical issues for longevity-linked insurance and pension products that have evolved concurrently with the steady increase in life expectancy since 1960s.
Abstract: This article investigates the latest developments in longevity risk modelling, and explores the key risk management challenges for both the financial and insurance industries. The article discusses key definitions that are crucial for the enhancement of the way longevity risk is understood; providing a global view of the practical issues for longevity-linked insurance and pension products that have evolved concurrently with the steady increase in life expectancy since 1960s. In addition, the article frames the recent and forthcoming developments that are expected to action industry-wide changes as more effective regulation, designed to better assess and efficiently manage inherited risks, is adopted. Simultaneously, the evolution of longevity is intensifying the need for capital markets to be used to manage and transfer the risk through what are known as Insurance-Linked Securities (ILS). Thus, the article will examine the emerging scenarios, and will finally highlight some important potential developments for longevity risk management from a financial perspective with reference to the most relevant modelling and pricing practices in the banking industry.
111 citations
TL;DR: In this paper, a compound Markov binomial model based on the Markov Bernoulli process is presented, which introduces dependency between claim occurrences and a Lundberg exponential bound is derived for the ruin probability.
Abstract: In this paper, we present a compound Markov binomial model which is an extension of the compound binomial model proposed by Gerber (1988a, b) and further examined by Shiu (1989) and Willmot (1993). The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences. Recursive formulas are provided for the computation of the ruin probabilities over finite- and infinite-time horizons. A Lundberg exponential bound is derived for the ruin probability and numerical examples are also provided.
69 citations
TL;DR: In this paper, an overview of the survival models commonly found in statistical packages and compares their suitability for actuarial work with the mortality "laws" proposed by actuaries over the past two centuries.
Abstract: Traditional actuarial techniques for mortality analysis are being supplanted by statistical models. Chief amongst these are survival models, which model mortality continuously at the level of the individual. An assumption of a mathematical form for the hazard function or, equivalently, the assumption of a continuous distribution for an individual's lifetime, leads automatically to smooth fitted mortality rates. This note gives an overview of the survival models commonly found in statistical packages and compares their suitability for actuarial work with the mortality ‘laws’ proposed by actuaries over the past two centuries. We find that the actuarial laws provide substantially better fits at post-retirement ages. We also give a common structure of parameterisation which gives consistent behaviour and interpretation of risk factors across all 16 survival models listed here. Finally, we consider the benefits of working directly with the log-likelihood function, including making allowance for the left trunca...
50 citations
TL;DR: In this article, a new criterion for the evaluation of automobile bonus systems is proposed, which states that a bonus system should be constructed such as to minimize a weighted average of the expected squared rating errors in various insurance periods.
Abstract: A new criterion for the evaluation of automobile bonus systems is proposed. It states that a bonus system should be constructed such as to minimize a weighted average of the expected squared rating errors in various insurance periods. The criterion generalizes an asymptotic criterion given earlier by Norberg in 1976. In addition, the new nonasymptotic criterion makes it possible to discuss various short term aspects such as the optimal choice of starting class and the time heterogeneity of risks. Our treatment is illustrated by examples with numerical results.
42 citations
TL;DR: In this paper, a nonparametric estimator of the Gerber-Shiu function is proposed for a risk process with a compound Poisson claim process plus a diffusion perturbation; the Wiener-Poisson risk model.
Abstract: A non-parametric estimator of the Gerber–Shiu function is proposed for a risk process with a compound Poisson claim process plus a diffusion perturbation; the Wiener–Poisson risk model. The estimator is based on a regularized inversion of an empirical-type estimator of the Laplace transform of the Gerber–Shiu function. We show the weak consistency of the estimator in the sense of an integrated squared error with the rate of convergence.
41 citations
TL;DR: In this paper, the authors considered a stochastic model in which a business enterprise is subject to constant rate of expenses over time and gains which are random in both time and amount.
Abstract: In this paper, we consider a stochastic model in which a business enterprise is subject to constant rate of expenses over time and gains which are random in both time and amount. Inspired by Albrecher & Boxma (2004), it is assumed in general that the size of a given gain has an impact on the time until the next gain. Under such a model, we are interested in various quantities related to the survival of the business after default, which include: (i) the fair price of a perpetual insurance which pays the expenses whenever the available capital reaches zero; (ii) the probability of recovery by the first gain after default if money is borrowed at the time of default; and (iii) the Laplace transforms of the time of recovery and the first duration of negative capital. To this end, a function resembling the so-called Gerber–Shiu function (Gerber & Shiu (1998)) commonly used in insurance analysis is proposed. The function's general structure is studied via the use of defective renewal equations, and its applicati...
34 citations
TL;DR: In this article, the least square estimatcs for a common factor structure were derived for a single factor structure and the asymptotic variances and covariances of the loading estimates were evaluated.
Abstract: Summary In § 2 the least square estimatcs are derived for a common factor structure §§ 3–4 are devoted to further discussion of the estimation and test problems, while the asymptotic variances and covariances of the loading estimates are evaluated in § 5
33 citations
TL;DR: A pair-copula construction will be used for the fit of the continuous copula allowing to choose appropriate copulas for each pair of margins, and how to express the joint pf by copulas with discrete and continuous margins is illustrated.
Abstract: In insurance applications yearly claim totals of different coverage fields are often dependent. In many cases there are numerous claim totals which are zero. A marginal claim distribution will have an additional point mass at zero, hence this probability function (pf) will not be continuous at zero and the cumulative distribution functions will not be uniform. Therefore using a copula approach to model dependency is not straightforward. We will illustrate how to express the joint pf by copulas with discrete and continuous margins. A pair-copula construction will be used for the fit of the continuous copula allowing to choose appropriate copulas for each pair of margins.
TL;DR: In this paper, the authors developed a simple first version of such a method and investigated life-cycle products recommended on the web of the four biggest commercial Danish pension companies on one day in February 2007.
Abstract: The Danish pension market of life-cycle products have expanded considerably since its introduction in the beginning of the millennium. The market is maturing and pensioners have the choice between a wide area of different products. It is therefore about time that financial insurance technology is developed to guide the performance measurement of available products. In this paper we develop a simple first version of such a method and we investigate life-cycle products recommended on the web of the four biggest commercial Danish pension companies on one day in February 2007. All considered products are outperformed by trivial benchmark products with constant stock proportion over time. Our approach is the following: for each life-cycle product we first find a trivial benchmark product with the same longterm risk and then we compare the long-term return of the two equivalent products. We primarily consider value at risk (VaR) and tail VaR as risk measures, but we also include a study where the fair value of ...
TL;DR: The first two moments and covariance of the aggregate discounted claims have been found for a stochastic interest rate, from which the inflation rate has been subtracted, and for a claims number process that is an ordinary or a delayed renewal process.
Abstract: The first two moments and the covariance of the aggregate discounted claims have been found for a stochastic interest rate, from which the inflation rate has been subtracted, and for a claims number process that is an ordinary or a delayed renewal process. Hereafter we extend the preceding results by presenting recursive formulas for the joint moments of this risk process, for a constant interest rate, and non-recursive formulas for higher joint moments when the interest rate is stochastic. Examples are given for exponential claims inter-arrival times and for the Ho-Lee-Merton interest rate model.
TL;DR: In this paper, the joint density of the time of ruin and the deficit at ruin in the Erlang(n) risk model is considered. But the authors focus on the special case when n = 2.
Abstract: We consider the joint density of the time of ruin and deficit at ruin in the Erlang(n) risk model. We give a general formula for this joint density and illustrate how the components of this formula can be found in the special case when n=2. We then show how the formula can be implemented numerically for a general value of n. We also discuss how the ideas extend to the generalised Erlang(n) risk model.
TL;DR: In this article, the authors investigate how to estimate the ultimate value of large losses based on the development of individual losses and therefore allow to compute the netting impact of excess of loss reinsurance.
Abstract: In this paper we investigate how to estimate ultimate values of large losses. The method is based on the development of individual losses and therefore allows to compute the netting impact of excess of loss reinsurance. In particular the index clause is properly accounted for. A numerical example based on real-life data is provided. © 2012 Taylor and Francis Group, LLC.
TL;DR: In this paper, the authors present an outline and historical summary of the Whittaker-Henderson method of data smoothing, together with an extension of the method in which the graduated values are obtained by minimising a WHDH criterion subject to constraints.
Abstract: We present an outline and historical summary of the Whittaker–Henderson method of graduation (or data smoothing), together with an extension of the method in which the graduated values are obtained by minimising a Whittaker–Henderson criterion subject to constraints. Examples are given, using data for the global average temperature anomaly and for a set of share prices, in which the proposed method appears to give good results.
TL;DR: In this article, the analysis of a generalized Gerber-Shiu function is considered in a discrete-time (ordinary) Sparre Andersen renewal risk process with time-dependent claim sizes.
Abstract: Analysis of a generalized Gerber–Shiu function is considered in a discrete-time (ordinary) Sparre Andersen renewal risk process with time-dependent claim sizes. The results are then applied to obtain ruin-related quantities under some renewal risk processes assuming specific interclaim distributions such as a discrete K n distribution and a truncated geometric distribution (i.e. compound binomial process). Furthermore, the discrete delayed renewal risk process is considered and results related to the ordinary process are derived as well.
Journal Article•
TL;DR: In this article, the authors considered the problem of finding the distribution of interruptions among the first m trials, given that the probability that the event S occurs in the experiments Ej, is equal to p for all j...
Abstract: I. Introduction. The following problem is considered. A sequence of experiments E 1, E 2, ... En , En is given and in each of them the occurrence or non-occurrence of a certain event S is observed. A trial consists of performing these experiments independently of each other one at a time until the event S eventually occurs. If the event S does not occur in any of the experiments all n experiments are performed and we shall in this case say that the trial is not interrupted. A second trial is performed in the same way but this time the experiment at which the first trial was eventually interrupted is excluded. This procedeure is repeated so that in the mth trial those experiments at which the first m - 1 trials were eventually interrupted are excluded from the sequence of experiments. The fundamental question asked is to find the distribution of the number of interruptions X among the first m trials, given that the probability that the event S occurs in the experiments Ej , is equal to p for all j...
Journal Article•
TL;DR: In this article, the authors study the problem of what an insurance company will do in the absence of any regulation, and find that if the management of the company takes a long-term view, no regulation should be necessary.
Abstract: The answer to the question posed must obviously depend on what an insurance company will do in the absence of any regulation. There should be no need for governmental intervention unless the company, when left to its own devices, is expected to engage in unfair or socially harmful practices. In the following we shall study this problem with the help of some simple models, based on assumptions which seem reasonable. We shall find, not surprisingly, that if the company primarily is interested in making a quick profit, some regulation may be necessary. On the other hand we shall find that if the management of the company takes a long-term view, no regulation should be necessary. We shall also see that there are limits to what a government can achieve by regulation of private insurance companies which operate in a free economy.
Journal Article•
TL;DR: In this article, the authors present a Bayesian approach to estimate the underlying death rates for the observation period (graduation), estimation of the under-approximation for C a future calendar periods (extrapolation), and prediction of the observed death rate for the c 2 fut...
Abstract: The estimation of survival functions is fundamental to the disciplines of reliability engineering, biostatistics, demography, and actuarial science. ln actuarial applications we deal with populations of insureds, annuitants, and pensioners. We need to estimate probabilities of individuals remaining in the populations and moving from the populations for reasons of death, change in health status, voluntary withdrawal, etc. Estimates of these probabilities aid actuaries in premium and reserve determination and, as a consequence, in developing investment strategies and cash flow projections. Let there be K age groups in a life table. Suppose that for each age group a death rate has been observed for each of c 1 calendar periods. We present a Bayesian approach to (1) estimation of the underlying death rates for the observation period (graduation), (2) estimation of the underlying death rates for C a future calendar periods (extrapolation), and (3) prediction of the observed death rates for the c 2 fut...
TL;DR: The renewal theorem for one-dimensional, positive, lattice random variables was proved by Kolmogorov in 1936 as the ergodic theorem for denumerable Markov chains.
Abstract: Renewal theory has been treated by many pure and applied mathematicians. Among the former we may mention Feller, Tack-lind and Doob. The principal limit theorem (for one-dimensional, positive, lattice random variables) was however proved earlier by Kolmogorov in 1936 as the ergodic theorem for denumerable Markov chains. A partial result for the non-lattice case was first proved by Doob using the theory of Markov processes, and the complete result by Blackwell. The extension of the renewal theorem to random variables taking both positive and negative values was first given by Wolfowitz and the author [1], for the lattice case. A partial result for the non-lattice case, using a purely analytical approach, was obtained by Pollard and the author [3].* For the literature see [1].
TL;DR: In this paper, the authors considered a chance variable which is constant with probability one and whose distribution is unknown, except that it is known that with probability 1, where 0 and p are unknown constants, the expected values of u and v are both zero.
Abstract: 1. Summary of results. Let E and Eo be chance variables at least one of which is not normally distributed (throughout the present paper a chance variable which is constant with probability one will be considered to be normally distributed with variance zero), and whose distribution is otherwise unknown, except that it is known that with probability one, where 0 and p are unknown constants, . Let (u; v) be jointly normally distributed chance variables with unknown covariance matrix, distributed independently of (e, e0). Without loss of generality we assume that the expected values E u and E v, of u and v respectively, are both zero. Define
TL;DR: In this paper, the basic concepts of the life insurance mathematics are discussed and the importance of the probability calculus as a hasis for the actuarial science has been disclaimed in recent literature.
Abstract: 1. Introduction. In this paper the basic concepts of the life insurance mathematics will be discussed. Due to the fact that the importance of the probability calculus as a hasis for the actuarial science has repeatedly been disclaimed in recent literature (See e.g. Ernst ZWinggi (1]), the present author feels that there is a justification for reconsidering the fundamental ideas of the actuarial science.
TL;DR: In this article, the observed random variable (x) is the number of individual trials that have produced a certain event A. The observed variable is defined as the total number of trials that produced A.
Abstract: Suppose that N experiments have been carried out, each experiment consisting of y individual random trials, and that the observed random variable (x) is the number of individual trials that have produced a certain event A. Suppose further, that these N -experiments are grouped in k classes.
TL;DR: In this article, Feller's logistic population model and Bartlett's infection moqeI as being representative of the nonlinear type of scheme were considered. And the main results are the different series solutions for the moments given by equations (28), (39) and (312).
Abstract: Summary Certain properties are considered of those models describing the distribution with time of persons or things among different states (e g in an epidemio· model we have the states of susceptibility, infection, death and immunity) for which the transition probabilities depend only upon the numbers in the different states Special attention is paid to Feller's logistic population model and Bartlett's infection moqeI as being representative of the nonlinear type of scheme The main results are the different series solutions for the moments given by equations (28), (39) and (312)