A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.

where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.

If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.

Even though the trend in mortality improvements has experienced several permanent changes in the past, the uncertainty regarding future mortality trends is often left unmodeled when pricing longevity-linked securities. In this paper, we present a stochastic modeling framework for the valuation of longevity-linked securities which explicitly considers the risk of random future changes in the long-term mortality trend. We construct a set of meaningful probability distortions which imply equivalent risk-adjusted pricing measures under which the basic model structure is preserved. Inspired by risk-based capital requirements for (re)insurers, we also establish a cost-of-capital pricing approach which then serves as the appropriate reference framework for finding a reasonable range for the market price of longevity risk. In a numerical application, we demonstrate that our model produces plausible risk loadings and show that a greater proportion of the risk loading is allocated to longer maturities when the risk of random future mortality trend changes is adequately modeled.

The present paper is concerned with optimal premium plans for a reinsurance contract with reinstatements. In the reinsurance contract considered here, constant reinstatement premiums are due when the reinsurer’s loss exceeds certain bounds. For this reinsurance contract we examine the existence of a premium plan which minimizes the expected squared difference between the loss and the total premium income of the reinsurer. We show that an optimal premium plan exists, that it is unique, and that it satisfies the net premium principle.

Les colloques successifs d'Astin ent eu le mérite de clarifier certaines idées théoriques importantes. En invitant au colloque d'Arnhem les actuaires á penser aux méthodes de Monte Carlo, les organisateurs ont forcément orienté les recherches vers le domaine des applications. Car simuler, c'est á. dire établir des résultats conformes á un processus aléatoire bien déterminé, suppose á priori que les données numériques d'introduction (ou de définition) ont été fixées.

Nous ne reprendrons pas les principes de la méthode de simulation, ils sont donnés de maniére magistrale dans “Notes on operations research 1959” du centre de recherche opérationnelle du Massachusetts Institute of Technology. Le chapitre II, portant titre „Simulation of Random processes” rédigé par Herbert P. Galliher donne un exposé complet du probléme. Nous renvoyons á cette référence.

Parmi les idées que l'on doit retenir, car elle est d'application dans tous les domaines, nous citons „Never simulate a single numerical problem;always simulate several of a class of problems”.

Dans le domaine non life, cet aspect est fondamental. Ce qui y fait défaut, ce sont des algorithmes généraux de résolution numériques. Une des difficultés majeures rencontrées par les actuaires a été et est encore le calcul effectif des convolutions. Ceux qui ont établi cette note se sont posé la question de savoir si la méthode de Monte Carlo ne pouvait fournir un procédé:

— évitant les convolutions.

— suffisamment général pour être applicable à une classe de risques arbitraire.

— donnant comme sous-produit des éléments intéressant aussi bien les théoriciens que les praticiens.

By the term actual run-off triangle we shall mean the two-way tabulation—according to year of origin and year of payment—of claims paid to date, which has the following form:

where Cij is the amount paid during development year j in respect of claims whose year of origin is i.

The information relating to the area below and/or to the right of this triangle is unknown since it represents the future development of various cohorts of claims.

Now in seeking to use this triangle as a basis for projection of claims in future development years for each of the years of origin 0, 1, 2, etc., we must recognise that the entries Cij in the above triangle, being random variables, contain random deviations from their expected values uij. It is the corresponding triangle of these expected values in which we are interested, and which shall be called the expected run-off triangle.

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.

As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expression

where c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.

In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we have

where

Cramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

This paper studies a problem of optimal reinsurance design under asymmetric information. The insurer adopts distortion risk measures to quantify his/her risk position, and the reinsurer does not know the functional form of this distortion risk measure. The risk-neutral reinsurer maximizes his/her net profit subject to individual rationality and incentive compatibility constraints. The optimal reinsurance menu is succinctly derived under the assumption that one type of insurer has a larger willingness to pay than the other type of insurer for every risk. Some comparative analyses are given as illustrations when the insurer adopts the value at risk or the tail value at risk as preferences.

Risk Theory for Life Insurance is simplified by the fact that the distribution Ψ (x) of claim amounts x approximately coincides with the distribution of “Risk sums” (not exactly, owing to differaences in the claim frequency with age and actual state of health), ond this distribution is comparatively stable.—The dependence on the claim frequency is eliminated by the introduction of a new time variable, and the system reduced to a (stationary) Poisson Process, which should be valid at least for large risk systems and for the total Life branch for a moderate sequence of years.

In almost all non-life branches, partial claims will dominate and Ψ (x) can only be determined by risk statistics, leaving a certain space of indetermination, in particular for large claims and for mediumsized statistical risk groups.

In my previous analyses, in particular New York 1957, interest has been concentrated on traffic and motor car insurance, where the risk depends on cars insured and on the meeting traffic (including road conditions). In one year the same car can be involved in many accidents and double claims (=collisions) are rather frequent.—According to my experience, this system is best represented by a sequence of single and double risk situations in time (for individual cars or for risk groups).

Analysis is simplified for Fire Insurance (and many other non-life branches), because the risk system is composed of mostly independent insurances (or risk objects), which are best described by the ordinary Individual Risk Theory.

The Capital Asset Pricing Model arises in an economy where agents have exponential utility functions and aggregate consumption is normally distributed, and gives the prices of assets with payoffs which are jointly normal with consumption. Such assets have normal marginal distributions and have dependence with consumption characterised by a normal copula. Wang has derived a transform which extends the CAPM by allowing pricing of assets in such an economy which have non-normal marginal distributions but still are normal-copula with consumption.

Here we set out the stochastic discount factors corresponding to this version of the CAPM and to Wang’s transform, and show how to calculate stochastic discount factors and hence asset prices for assets whose dependence with consumption is non-normal. We show that the impact of dependency structure on asset prices is significant.

It has been known for generations that the fire risk rate increases with the size of the insured object in a similar way as the death rate increases with the age.

Professor d'Addario and other Italian mathematicians have shown that statistical data often can be graduated by the formulas where S denotes the sum insured:

In 1940, d'Addario) in a practical case found the values α = 0.78, β = 0.44. In 1956, Blandin and Depoid have used the same formulas. Although these formulas in practical cases often lead to good approximations, one can hardly say that a proper mathematical or physical model describing the behaviour of fires lies behind.

About 1950, I worked with the statistics of a group of Danish fire insurance companies, in particular covering farm buildings. This investigation was organized by Gunnar Benktander. Our data confirmed the increase of the fire risk rate with the size of the buildings.

In one special group: Farm houses with thatched roofs, this increase was so important that the risk premium was approximately proportionate to the square of the sum insured. The statistics fully justified the tariffing and we tried to construct a model describing and explaining the observed facts.

The risk group in question was characterized by the overwhelming importance of total or practically total losses. Only a few per cent of the damages went to minor fires. When a fire breaks out in such a building, and reaches a certain slight extent, then it is not possible to save the building from total destruction. With sufficient approximation, we may say that only total losses occur (as in life assurance).

A great attention has been devoted, in the actuarial literature, to premium calculation principles and it has been often emphasized that these principles should not only be defined in strictly actuarial terms, but should also take into account the market conditions (Bühlmann (1980), de Jong (1981)).

In this paper we propose a decision model to define the pricing policy of an insurance company that operates in a market which is stratified in k risk classes .

It is assumed that any class constitutes a homogeneous collective containing independent risks Sj(t) of compound Poisson type, with the same intensity λj. The number nj of risks of that are held in the insurance portfolio depends on the premium charged to the class by means of a demand function which captures the concept of risk aversion and represents the fraction of individuals of , that insure themselves at the annual premium xj.

With these assumptions, the return Y on the portfolio is a function of the vector x = (x1, x2, …, xk) of the prices charged to the single classes (and of the time) and x is therefore the decision policy instrument adopted by the company for the selection of the portfolio, whose optimal composition is evaluated according to a risk-return type performance criterion.

As a measure of risk we adopt the ultimate ruin probability q(w) that, in the assumptions of our model, can be related to a safety index τ, by means of Lundberg-de Finetti inequality. Even though it has been widely debated in the actuarial field, the use of q(w) offers undeniable operational advantages. In our case the safety index τ can be expressed as a function of x and therefore, in the phase of selecting an efficient portfolio, it becomes the function to be maximized, for a given level M of the expected return.

Modeling taxation of Variable Annuities has been frequently neglected, but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity. In particular, they consider the simple Black–Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long-term products, and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policyholder behavior, as observed by Goudenège et al. (2018). In this paper, we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement.

The tail index is an important parameter that measures how extreme events occur. In many practical cases, this tail index depends on covariates. In this paper,we assume that it takes a finite number of values over a partition of the covariate space. This article proposes a tail index partition-based rules extraction method that is able to construct estimates of the partition subsets and estimates of the tail index values. The method combines two steps: first an additive tree ensemble based on the Gamma deviance is fitted, and second a hierarchical clustering with spatial constraints is used to estimate the subsets of the partition. We also propose a global tree surrogate model to approximate the partition-based rules while providing an explainable model from the initial covariates. Our procedure is illustrated on simulated data. A real case study on wind property damages caused by tornadoes is finally presented.

In practical statistical work one frequently meets certain problems. For instance, we may have the following data about loss ratios in certain insurance companies and corresponding, numbers of insurance in force:

I assume further that we have no reason to believe that the companies, their loss ratios and their structure of insurances in force differ in any other way than by the size of companies. The problem is how to get quick estimates of mean losses and their variances in different companies?

A straitforward way to estimate the mean loss ratio would be to compute the usual mean of numbers pi, (Σpi/6) = 14; its standard deviation is 6,5. As this procedure of the “first statistician” seems to be too simple and naive, a “more cautious” statistician would compute the weighted mean loss ratio

The “more cautious” statistician would arque that his result is much better than the other result 14. But what would be the variance of the estimate 7,1, and what is the variance in the different companies?

Motivated by a recent discovery that the two-step inference for the Lee–Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for a unit root in a Lee–Carter model with an AR(p) process for the mortality index. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed in this paper are unconventional. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the U.S. mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee–Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is conducted to examine the finite sample behavior of the proposed test too.

This paper investigates an operation mechanism for mutual aid platforms to develop more sustainably and profitably. A mutual aid platform is an online risk-sharing platform for risk-heterogeneous participants, and the platform extracts revenues by charging participants commission and subscription fees. A modeling framework is proposed to identify the optimal commissions and subscriptions for mutual aid platforms. Participants are divided into different types based on their loss probabilities and values derived from the platform. We present how these commissions and subscriptions should be set in a mutual aid plan to maximize the platform’s revenues. Our analysis emphasized the importance of accounting for risk heterogeneity in mutual aid platforms. Specifically, different types of participants should be charged different commissions/subscriptions depending on their loss probabilities and values on the platform. Participants’ shared costs should be determined based on their loss probabilities. Adverse selection occurs on the platform if participants with different risks pay the same shared costs. Our results also show that the platform’s maximum revenue will be lower if the platform charges the same fee to all participants. The numerical results of a practical example illustrate that the optimal commission/subscription scheme and risk-sharing rule result in considerable improvements in platform revenue over the current scheme implemented by the platform.

We study the relation between one-year premium risk and ultimate premium risk. In practice, the one-year risk is sometimes related to the ultimate risk by using a so-called emergence pattern formula which postulates a linear relation between both risks. We define the true emergence pattern of the ultimate loss for the one-year premium risk based on a conditional distribution of the ultimate loss derived from a multivariate distribution of the claims development process. We investigate three models commonly used in claims reserving and prove that the true emergence pattern formulas are different from the linear emergence pattern formula used in practice. We show that the one-year risk, when measured by VaR, can be under and overestimated if the linear emergence pattern formula is applied. We present two modifications of the linear emergence pattern formula. These modifications allow us to go beyond the claims development models investigated in the first part and work with an arbitrary distribution of the ultimate loss.

The transition from defined benefit to defined contribution (DC) pension schemes has increased the interest in target annuitization funds that aim to fund a minimum level of retirement income. Prior literature has studied the optimal investment strategies for DC funds that provide minimum guarantees, but far less attention has been given to portfolio insurance strategies for DC pension funds focusing on retirement income targets. We evaluate the performance of option-based and constant proportion portfolio insurance strategies for a DC fund that targets a minimum level of inflation-protected annuity income at retirement. We show how the portfolio allocation to an equity fund varies depending on the member’s age upon joining the fund, displaying a downward trend through time for members joining the fund before ages in the mid-30s. We demonstrate how both portfolio insurance strategies provide strong protection against downside equity risk in financing a minimum level of retirement income. The option-based strategy generally leads to higher accumulated savings at retirement, whereas the constant proportion strategy provides better downside risk protection robust to equity market jumps/volatilities.

Our purpose is to introduce some models of inference for risk processes. The bayesian viewpoint is adopted and for our treatment the concepts of exchangeability and partial exchangeability (due to B. de Finetti, [6], [7]) are essential.

We recall the definitions:

The random variables of a sequence (X1, X2 …) are exchangeable if, for every n, the joint distribution of n r.v. of the sequence is always the same, whatever the n r.v. are and however they are permuted.

From a structural point of view an exchangeable process X1, X2 … can be intended as a sequence of r.v. equally distributed among which a “stochastic dependence due to uncertainty” exists. More precisely the Xi are independent conditionally on any of a given set (finite or not) of exhaustive and exclusive hypothesis. These hypotheses may concern, for instance, the values of a parameter (number or vector) on which the common distribution, of known functional form, of Xi depends. We shall restrict ourselves to this case. Therefore, we shall assume that, conditionally on each possible value θ of a parameter Θ, the Xi are independent with F(x/θ) as known distribution function. According to the bayesian approach, a probability distribution on Θ must be assigned.