In the present paper we deal with the problem of calculating a premium for the largest claims and ECOMOR reinsurance treaties. Ammeter derived already in 1964 formulas for calculating the premiums of the largest claims and ECOMOR reinsurance treaties (compare also Seal (1969), Thépaut (1950)), which we will restate in the following Section 2. Lately Benktander (1978) has established an interesting connection between the premiums of the largest claims and excess of loss reinsurance treaties. He proved that the net risk premium of the largest claims treaty covering the p largest claims is bounded by the risk premium of an excess of loss treaty plus p times its priority, which has to be determined such that the mean number of excess claims equals p. Furthermore Benktander showed in examples that the upper bound is quite good in case of the Poisson-Pareto risk process. Nevertheless he did not give a formal proof for the quality of the bound in the Poisson-Pareto case nor for other risk processes.

In the following note we take up this last point and prove that for general risk process Benktander's upper bound is equivalent to the premium of the largest claims reinsurance cover when the size of the collective approaches infinity. Consequently, for large portfolios the risk premium of the largest claims cover may be replaced by the upper bound, e.i., calculated from the premium of the corresponding excess of loss treaty. Moreover we state a similar result for the ECOMOR treaty.

We consider the conditional mean risk allocation for an insurance pool, as defined by Denuit and Dhaene (2012). Precisely, we study the asymptotic behavior of the respective relative contributions of the participants as the total loss of the pool tends to infinity. The numerical illustration in Denuit (2019) suggests that the application of the conditional mean risk sharing rule may produce a linear sharing in the tail of the total loss distribution. This paper studies the validity of this empirical finding in the class of compound Panjer–Katz sums consisting of compound Binomial, compound Poisson, and compound Negative Binomial sums with either Gamma or Pareto severities. It is demonstrated that such a behavior does not hold in general since one term may dominate the other ones conditional of large total loss.

Bühlmann-Straub credibility is used to find an estimate of the mortality loss ratio for a company, relative to a standard table, for use in the statutory valuation of life insurance business. A method for calculating the margin for adverse deviation to be added to the mortality rate (in accordance with the general principle of Canadian statutory valuation) is derived. Applying credibility further to the variance of the mortality loss ratio gives a methodology for calculating the amount of the surplus (i.e. capital) required to cover annual fluctuations in mortality experience. The necessary structural parameters are calculated from industry statistics; the methodology is illustrated using Canadian life insurance data.

In this paper, we consider the dual of the classical Cramér-Lundberg model when gains follow a phase-type distribution. By using the property of phase-type distribution, two pairs of upcrossing and downcrossing barrier probabilities are derived. Explicit formulas for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin under a variety of dividend strategies can then be obtained without the use of Laplace transforms.

In nonlife insurance, actuaries usually resort to random effects to take unexplained heterogeneity into account (in the spirit of the Bühlmann-Straub model). This paper aims to study the kind of dependence induced by the introduction of correlated latent variables in the annual numbers of claims reported by policyholders. The effect of reporting claims on the a posteriori distribution of the random effects will be made precise. This will be done by establishing some stochastic monotonicity property of the a posteriori distribution with respect to the claims history.

We discuss some questionable points of the approach taken in the paper by Buchwalder, Bühlmann, Merz and Wüthrich and come to the conclusion that this approach does not yield an improvement of Mack’s original formula. The main reason is that the new approach disregards the negative correlation of the squares of the development factors. The same applies to the formula by Murphy (PCAS 1994).

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.

Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

In this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.

Individual automobile insurance claims are characterized by over-dispersion relative to the Poisson model. In addition, claim propensities vary among individuals in any insurance portfolio. This paper presents a model which takes account of both characteristics. The model employs the negative-binomial distribution as the distribution for individual-level claims and a Pareto distribution as the distribution for claim propensities within the portfolio. The paper shows that the resulting model is tractable and has a number of attractive properties which make it suitable for this application. The fit of the model to actual claim numbers for automobile third party liability insurance is examined and found acceptable. Bayes theorem is then applied to this model to calculate illustrative optimal premiums under the Bonus-Malus System (BMS).

Let us denote by E(x) the pure risk premium of an unlimited excess cover with the retention x and by H(x) and m(x) the corresponding expected frequency and severity.

We thus have E(x) = H(x) · m(x).

H(x) is a non-increasing function of x and for practical purposes we can assume that it is decreasing; H′(x) < o. The equation H(x) = n has then only one solution xn, where n is a fixed integer.

Let En denote the risk premium for a reinsurance covering the n largest claims from the bottom.

Let us define Intuitively we feel that is a good approximation for En.

We shall first show that when the claims size distribution is Pareto and the number of claims is Poisson distributed, is a good approximation for En, being slightly on the safe side. We further include a proof given by G. Ottaviani that the inequality always holds.

In the Pareto case we have

where the Poisson parameter t stands for the expected number of claims in excess of 1 (equal to a suitably chosen monetary unit) and

.

In a typical equity-linked life insurance contract, the insurance company is entitled to a share of return surpluses as compensation for the return guarantee granted to the policyholders. The set of possible contract terms might, however, be restricted by a regulatory default constraint — a fact that can force the two parties to initiate sub-optimal insurance contracts. We show that this effect can be mitigated if regulatory policy is more flexible. We suggest that the regulator implement a traffic light system where companies are forced to reduce the riskiness of their asset allocation in distress. In a utility-based framework, we show that the introduction of such a system can increase the benefits of the policyholder without deteriorating the benefits of the insurance company. At the same time, default probabilities (and thus solvency capital requirements) can be reduced.

The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).

Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.

It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

Regression modelling involving heavy-tailed response distributions, which have heavier tails than the exponential distribution, has become increasingly popular in many insurance settings including non-life insurance. Mixed Exponential models can be considered as a natural choice for the distribution of heavy-tailed claim sizes since their tails are not exponentially bounded. This paper is concerned with introducing a general family of mixed Exponential regression models with varying dispersion which can efficiently capture the tail behaviour of losses. Our main achievement is that we present an Expectation-Maximization (EM)-type algorithm which can facilitate maximum likelihood (ML) estimation for our class of mixed Exponential models which allows for regression specifications for both the mean and dispersion parameters. Finally, a real data application based on motor insurance data is given to illustrate the versatility of the proposed EM-type algorithm.

La sélection des risques est un des premiers principes fondamentaux que la théorie de l'assurance a dégagés. Il se traduit essentiellement par la répartition des risques en classes de tarif homogènes et il prémunit l'assureur contre le danger de voir „le mauvais risque chasser le bon”.

Pour créer des classes de tarif homogènes, il faut théoriquement repérer tous les facteurs influençant le risque et en chiffrer l'effet. Si cela est fait, la fluctuation des résultats individuels autour de la moyenne n'est que l'effet accidentel du hasard et ne peut donner lieu a posteriori à rectification de la prime: il n'y a rien d'inéquitable à ce que les titulaires des contrats non sinistrés paient pour les autres puisque tous sont égaux devant le risque.

Comme dans toute science appliquée, la mise en oeuvre des principes se heurte parfois à de grandes difficultés d'exécution. Dans la pratique il n'est généralement pas question de repérer tous les facteurs du risque, mais seulement les principaux, ne fût-ce qu'en raison de l'impossibilité de conduire des études statistiques valables sur une multitude de groupes peu étoffés. On ne peut donc faire grief au praticien d'estimer qu'il a créé des classes homogènes lorsque celles-ci tiennent compte des principaux facteurs de risque.

Mais l'optique change si, pour l'une ou l'autre raison qu'il importe peu de préciser, le tarif néglige un ou des facteurs dont le bon sens et l'expérience — à défaut du calcul — prouvent l'influence appréciable.

This paper deals with the bivariate generalized Poisson distribution. The distribution is fitted to the aggregate amount of claims for a compound class of policies submitted to claims of two kinds whose yearly frequencies are a priori dependent. A comparative study with the bivariate Poisson distribution and with two bivariate mixed Poisson distributions has been carried out, based on data concerning natural events insurance in the USA and third party liability automobile insurance in France.

Natural hazards—floods, hurricanes, tornadoes, earthquakes, windstorms and hailstorms—cause considerable property damage in various parts of the world. In the United States, average annual damage resulting from these hazards is increasing rapidly. A large percentage of the damages occur as a result of infrequent, but severe, geophysical events (individual storms or earthquakes). If aggregate damage resulting from the event is exceptionally large, the event is called a natural disaster. The number of natural disasters in the United States is increasing each year. Resultant property losses are increasing even more rapidly. Increased density of properties susceptible to damage, increased value of these properties, and increased cost of repair have raised the probability of natural disaster occurrence in recent years even though the magnitude and character of the natural hazards have not changed.

Insurance is one means of protection against the natural hazards for fixed property. In this report, one-to-four family dwelling structures represent fixed property. To provide protection, two components of risk must be evaluated: (1) risk per individual structure and (2) risk of a large number of simultaneous losses—catastrophe potential. The latter component has attained added importance recently with the increased number and magnitude of natural disasters.

Information available for risk evaluation is (1) past damage experience; (2) data on the damage susceptibility of structures to be insured and the cost of repair; and (3) knowledge of physical characteristics of the natural hazards from the natural sciences. Damage experience results from the interaction of a natural hazard (frequency, location and severity of a geophysical event—storm, earthquake or flood) with characteristics and geographical distribution of exposed properties. Occasionally, interaction between the geophysical event and the distribution of properties leads to the creation of catastrophic losses resulting in a natural disaster. Evaluation of risk must be based either upon a retrospective or prospective measure of this damage experience.

Past damage experience, a retrospective measure of loss potential, is a poor measure of future risk because of (1) non-stationarity of property characteristics; (2) bias introduced by chance interactions of hazard and property array; and (3) by random occurrence (or non-occurrence) of a severe geophysical event during the short sampling period of years that is usually available for study. A pure extrapolation into the future of past loss experience, including the chance combinations of events that led to past natural disasters, does not provide a great amount of insight into the character of future risk. What is needed is not actual damage that occurred as a result of past geophysical events, but damage resulting to the present distribution of properties from a recurrence of these past events. For example, to estimate future earthquake risk in California, emphasis should not be on what the 1906 San Francisco earthquake cost, but what it would cost if a comparable earthquake occurred today and affected the present type, distribution, and value of properties.

A supplementary approach to the use of loss experience (called Natural Hazard Simulation) is presented which provides a prospective measure of risk. A mathematical approximation of the natural hazard mechanism is constructed which artificially produces geophysical events that mathematically interact with a given geographical array of properties. Natural Hazard Simulation utilizes and ties together available pertinent information. Use of an electronic computer permits calculation of a large number of, say, 25-year sequences of synthetic loss experience which can be used to estimate the two measures of natural hazard risk. “Natural disasters” occur at irregular intervals in the simulation analysis when a severe geophysical event occurs near a center of population.

Using this approach, effects of a recurrence of past geophysical events or simulated future events upon present or hypothetical future distributions and types of properties can be estimated. It produces, in effect, a weighted measure of the many possible interactions between natural hazard and property array which, because of his short life span, man cannot afford to wait for nature to produce. Characteristics of an insurance operation needed to cover the hazard (rating, underwriting, claim settlement, loss reserving, and reinsurance) also are simulated.

Examples of the application of Natural Hazard Simulation to flood, earthquake, hurricane wind and tides, winter windstorm, and thunderstormspawned hazards (such as tornadoes, wind and hail hazard) are presented. Purpose of the application to flood hazard was to (1) estimate magnitude of the hazard to more than fifty-million dwelling structures in the United States for which very little damage experience was available and (2) determine characteristics of a joint Insurance Industry/Federal Government flood insurance program need to cover the hazard. Characteristics of a joint program were needed to establish relationships and financial arrangements between the Federal Government and the Insurance Industry. This work was done as consultants to the U.S. Department of Housing and Urban Development during development of a National Flood Insurance Program which is now operational. In this plan, the Federal Government assumes a portion of risk by acting as a reinsurer against excessive losses on industry's share of the Program.

An application of Natural Hazard Simulation to the earthquake hazard on the West Coast of the United States has been made. Examples of mathematically produced earthshock patterns are given. Correspondence between calculated and observed patterns is good. Measures of both components of risk are discussed for the present array of 625,000 dwellings in the San Francisco Metropolitan area when a recurrence of all earthquakes in the historical past (170 years) is used as a measure of earthquake hazard. A similar type of analysis also has been made for the Los Angeles Metropolitan area.

An application of the approach to the hurricane wind hazard is illustrated using computer printouts of the geographical pattern of highest wind expected during a hurricane's passage as obtained from the computerized mathematical model. Calculated patterns of wind speed severity provide realistic approximations of observed patterns. An example of the interaction between natural hazard and property array in producing a “natural disaster” is illustrated by calculating “loss experience” to dwelling properties in Louisiana from an intense hurricane whose path is successively changed relative to centers of population. The effect of changing the intensity of a hurricane upon resulting damage when the path is held constant is also shown. Both measures of hurricane wind risk—expected loss per exposure and catastrophe potential—are being estimated by developing “loss experience” to the present array of dwelling properties in the Gulf and Atlantic States based upon two measures of the magnitude of the hurricane wind hazard; namely, (1) a recurrence of hurricanes of various intensities and paths which have been recorded in the historical past and (2) a number of series of 25-year sequences of “synthetic loss experience” based upon computer simulation techniques.

An application of Natural Hazard Simulation to winter windstorm and thunderstorm-spawned tornadoes, wind, and hail in the Middle Western United States has been carried out for the first measure of risk. The mathematical model for obtaining a measure of catastrophe potential for these hazards is currently being developed. Future applications will include development of an integrated procedure for simulating “loss experience” from all of the natural hazards to a given array of structures in various geographical areas.

Natural hazard simulation offers a supplementary approach to the sole use of past loss experience for (1) estimating the two components of natural hazard risk and (2) developing characteristics of an insurance program needed to cover the natural hazards at a time when average annual property damages caused by natural hazards are increasing rapidly because of the increased number and magnitude of natural disasters.

This paper examines the possibility of embedding public long-term care (LTC) insurance within the retirement pension system, i.e. introducing life care annuities into a notional defined contribution framework. To do this, we develop a multistate overlapping generations model that includes the so-called survivor dividend and give special attention to the assumptions made about mortality rates for dependent persons and LTC incidence rates, which largely determine the contribution rate assigned to LTC. The proposed model could be of interest to policymakers because it could be implemented without too much difficulty, it would universalize LTC coverage with a “fixed” cost, and it would discourage politicians from making promises about future LTC benefits without the necessary funding support.

An iteration scheme is derived for calculating the aggregate claims distribution in the individual life model. The (exact) procedure is an efficient reformulation of De Pril's (1986) algorithm, considerably reducing both the number of arithmetic operations to be carried out and the number of data to be kept at each step of iteration. Scaling functions are used to stabilize the algorithm in case of a portfolio with a large number of policies. Some numerical results are displayed to demonstrate the efficiency of the method.

In motorcar insurance is widely used a merit rating system characterized by the fact that only the number of claims occurred (and not their amount) modifies the premium. This system induces the insured drivers to support themselves the cost of the cheap claims. We analyze this “hunger for bonus” and solve this decision problem by means of an algorithm related to dynamic programming. The method is then applied to the Belgian bonus system.

This paper examines an insurance or risk premium calculation method called the mean-value-distortion pricing principle in the general framework of anticipated utility theory. Then the relationship between comonotonicity and independence is explored. Two types of risk aversion and optimal reinsurance contracts are also discussed in the context of the pricing principle.