We consider a Tweedie's compound Poisson regression model with fixed and random effects, to describe the payment numbers and the incremental payments, jointly, in claims reserving. The parameter estimates are obtained within the framework of hierarchical generalized linear models, by applying the h-likelihood approach. Regression structures are allowed for the means and also for the dispersions. Predictions and prediction errors of the claims reserves are evaluated. Through the parameters of the distributions of the random effects, some external information (e.g. a development pattern of industry wide-data) can be incorporated into the model. A numerical example shows the impact of external data on the reserve and prediction error evaluations.

Fuelled by the big data explosion, a new methodology to estimate sub-annual death probabilities has recently been proposed, opening new insurance business opportunities. This new approach exploits all the detailed information available from millions of microdata records to develop seasonal-ageing indexes (SAIs) from which sub-annual (quarterly) life tables can be derived from annual tables. In this paper, we explore whether a shortcut could be taken in the estimation of SAIs and (life insurance) sub-annual death rates. We propose three different approximations, in which estimates are attained by using just a small bunch of thousands of data records and assess their impact on several competitive markets defined from an actual portfolio of life insurance policies. Our analyses clearly point to the shortcuts as good practical alternatives that can be used in real-life insurance markets. Noticeably, we see that embracing the new quarterly based approach, even using only an approximation (shortcut), is economically preferable to using the associated annual table, offering a significant competitive advantage to the company adopting this innovation.

Dans trois études récentes (1977a, 1977b, 1979), J. Lemaire a appliqué à un ensemble d'observations du risque automobile quelques méthodes de sélection fréquemment utilisées en analyse de la régression. Les variables explicatives (traitées comme variables indépendantes) sont les variables décrivant le risque. Les variables étudiées (traitées comme variables dépendantes d'un modèle de régression) sont les deux variables généralement prises en considération dans ce contexte: le nombre des sinistres et leur montant cumulé.

Nous avons déjà souligné (Hallin, 1977) combien les hypothèses qui se trouvent à la base de l'analyse de la régression — normalité, homoscédasticité et linéarité de la régression — sont loin d'être remplies dans le contexte de l'assurance automobile. Le montant cumulé annuel prend la valeur zéro avec une probabilité proche de 0.9! Le nombre annuel de sinistres vaut 0, 1, 2, rarement plus! Même très approximativement, de telles variables peuvent difficilement être considérées comme normales. En outre, la plupart des variables explicatives sont de type nominal ou ordinal, ce qui rend délicate l'utilisation de modèles linéaires, les interactions de tous ordres étant très importantes, ainsi qu'on pourra le constater. Ces réserves sont d'ailleurs prévues par Jean Lemaire lui-même, qui ne propose ses conclusions qu'à titre de première approximation. Les mêmes données ont encore été soumises (Masure, 1978) aux méthodes de l'analyse discriminante, et les mêmes réserves peuvent être faites en ce qui concerne l‘utilisation des méthodes et l'interprétation des résultats (combinaisons linéaires, etc.).

Nous avons proposé dans Hallin (1977a, b) un ensemble de méthodes qui, selon les hypothèses distributionnelles pouvant être faites (et qui vont des hypothèses classiques de l'analyse de la variance à celles, beaucoup moins restrictives, des méthodes de rangs), constituent des généralisations de celles qui sont utilisées par Jean Lemaire. En particulier, celle que nous appliquons ici est entièrement “distribution free”. Ces méthodes sont également une extension de celle qu'a proposée Pitkänen (1975, 1976).

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equation

which is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense that

for an arbitrarily chosen point y. Alternatively, one can consider the risk aversion

which is the same for all affine transformations of a utility function.

Given the risk aversion r(x), the standardized utility function can be retrieved from the formula

It is easily verified that this expression satisfies (2) and (3).

The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

Soit x1 ≤ x2, ≤ … ≤ xn—m + 1 ≤… ≤xn—p+1≤…≤ xn un échantillon ordonné de taille n d'une distribution dont F(x) et T(x) sont respectivement la fonction de répartition et la fonction de densité.

Nous appelons xn—m+1 la valeur de rang m de l'échantillon (x1, x2,…xn).

Pour simplifier l'écriture nous remplacerons partout l'indice n — m + I par m.

Avec cette convention nous notons la valeur caractéristique de rang m et l'intensité de rang m respectivement par um et αm [I].

Par définition on a:

Notons enfin par Fm (x) et Tm (x) la fonction de répartition et la fonction de densité de xm pour n → ∞ et m fixe.

2.1. Avant de résumer les différents points traités dans cette note, rappelons les particularités qui caractérisent la distribution asymptotique de la plus grande valeur x1 d'un échantillon. On sait que cette distribution existe si et seulement si la fonction de répartition F(x) appartient à un des trois types suivants [2]:

a) Type I: F(x) est du type exponentiel.

Le domaine de x est illimité vers la droite et F(x) tend vers I pour x → + ∞ au moins aussi rapidement qu'une fonction exponentielle.

Tous les moments de F(x) existent.

b) Type II: F(x) est du type de Cauchy.

Le domaine de x est illiminté vers la droite.

Target benefit (TB) plans that incorporate intergenerational risk sharing have been demonstrated to be welfare improving over the long term. However, there has been little discussion of the short-term benefits for members in a defined benefit (DB) plan that is transitioning to TB. In this paper, we adopt a two-step approach that is designed to ensure the long-term sustainability of the new plan, without unduly sacrificing the benefit security of current retirees. We propose a cohort-based transition plan for reducing intergenerational inequity. Our study is based on simulations using an economic scenario generator with some theoretical results under simplified settings.

We study the time-consistent investment and contribution policies in a defined benefit stochastic pension fund where the manager discounts the instantaneous utility over a finite planning horizon and the final function at constant but different instantaneous rates of time preference. This difference, which can be motivated for some uncertainties affecting payoffs at the end of the planning horizon, will induce a variable bias between the relative valuation of the final function and the previous payoffs and will lead the manager to show time-inconsistent preferences. Both the benefits and the contribution rate are proportional to the total wage of the workers that we suppose is stochastic. The aim is to maximize a CRRA utility function of the net benefit relative to salary in a bounded horizon and to maximize a CRRA final utility of the fund level relative to the salary. The problem is solved by means of dynamic programming techniques, and main results are illustrated numerically.

Variable rate savings accounts have two main features. The interest rate paid on the account is variable and deposits can be invested and withdrawn at any time. However, customer behaviour is not fully rational and withdrawals of balances are often performed with a delay. This paper focuses on measuring the interest rate risk of variable rate savings accounts on a value basis (duration) and analyzes the problem how to hedge these accounts. In order to model the embedded options and the customer behaviour we implement a partial adjustment specification. The interest rate policy of the bank is described in an error-correction model.

We study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.

Due to the presence of reporting and settlement delay, claim data sets collected by non-life insurance companies are typically incomplete, facing right censored claim count and claim severity observations. Current practice in non-life insurance pricing tackles these right censored data via a two-step procedure. First, best estimates are computed for the number of claims that occurred in past exposure periods and the ultimate claim severities, using the incomplete, historical claim data. Second, pricing actuaries build predictive models to estimate technical, pure premiums for new contracts by treating these best estimates as actual observed outcomes, hereby neglecting their inherent uncertainty. We propose an alternative approach that brings valuable insights for both non-life pricing and reserving. As such, we effectively bridge these two key actuarial tasks that have traditionally been discussed in silos. Hereto, we develop a granular occurrence and development model for non-life claims that tackles reserving and at the same time resolves the inconsistency in traditional pricing techniques between actual observations and imputed best estimates. We illustrate our proposed model on an insurance as well as a reinsurance portfolio. The advantages of our proposed strategy are most compelling in the reinsurance illustration where large uncertainties in the best estimates originate from long reporting and settlement delays, low claim frequencies and heavy (even extreme) claim sizes.

A diminishing number of Reinsurers still seem to work with percentage loadings which are applied with more or less sophistication to the pure risk premium rate.

A more sensible approach is to study the distribution of the total claims amount of the layer reinsured (standard deviation, variance, skewness).

In practice this distribution is not known and we have to work with estimates E* and σ*2 of the first moment which is the pure risk premium (E) and of the second central moment (σ2). Because of the positive skewness of the distribution, E* will fall below E in more than half of the cases and sometimes considerably below.

One security loading principle could be to add a proportion k of σ* to E* in such a way that

If ε is put equal to 0.25 this reflects our wish to ensure that our security loaded rates are too low only in 25% of the cases. A σ- loading here emerges as a result of the statistical uncertainty.

If we analyse the result fluctuations we are led to apply a variance loading (1), which can be interpreted as a price for capacity (2). (See also discussion contributions by B. Ajne, G. Benktander, G. Berger, H. Bühlmann; Transactions of the Congress, 5, Oslo 1972, p. 169 ff.)

An underwriter who is confronted with two portfolios having the same first and second moments should, if conditions are the same, prefer the one with the lower third moment. A special loading for skewness is thus indicated.

This paper deals with some requirements put forward by the Finnish Supervisory Service on technical reserves mainly for solvency reasons. Consequently only one but nevertheless an important aspect of solvency problem is touched upon. The solvency question in its entirety is discussed in a paper of Dr. Pentikainen at this same colloquium.

According to the instructions of the Supervisory Service, given by virtue of the Insurance Companies Act, insurance companies are obliged to calculate a so-called equalisation reserve in a specific way. Simplifying slightly, this equalisation reserve is that part of the technical reserves which exceeds the premium reserve and the claims reserve calculated according to classical actuarial principles; that is to say, the present value of the expectation value of the net insurance liabilities. The Insurance Companies Act calls for this reserve because conventional technical reserves do not take into account the random feature of claims amount.

Much of the risk theory literature deals with the total claims distribution where pk = the probability of k claims and S(x) is the distribution function of severity. Both pk and S(x) are univariate probability distributions. Thus, F(x) can be interpreted as a model of claims from one class of policies or as an aggregate model where pk and S(x) represent mixed probability distributions from a heterogeneous portfolio of policies. An alternative approach to modelling total claims in the latter case would be to recognize explicitly that total claims are the result of the interaction of multivariate processes. In the most general case, total claims arise from a multivariate accident process where each accident triggers multivariate claims frequency and severity processes.

The purpose of this article is to present a multivariate model of total claims and to develop the cumulant generating function of this distribution. Such a model is superior to the traditional model in two respects: (1) It permits explicit recognition of shifts in the overall portfolio composition. Applications of the traditional model, in contrast, rely on the assumption that the portfolio composition is relatively constant over time. (2) It facilitates the evaluation of the effects of reinsurance on the total claims distribution when the reinsurance arrangements are not the same in different segments of the portfolio.

Interest rate is one of the main risks for the liability of the variable annuity (VA) due to its long maturity. However, most existing studies on the risk measures of the VA assume a constant interest rate. In this paper, we propose an efficient two-dimensional willow tree method to compute the liability distribution of the VA with the joint dynamics of the mutual fund and interest rate. The risk measures can then be computed by the backward induction on the tree structure. We also analyze the sensitivity and impact on the risk measures with regard to the market model parameters, contract attributes, and monetary policy changes. It illustrates that the liability of the VA is determined by the long-term interest rate whose increment leads to a decrease in the liability. The positive correlation between the interest rate and mutual fund generates a fat-tailed liability distribution. Moreover, the monetary policy change has a bigger impact on the long-term VAs than the short-term contracts.

The present paper considers the present value, Z(t), of a series of cashflows up to some time t. More specifically, the cashflows and the interest rate process will often be stochastic and not necessarily independent of one another or through time. We discuss under what circumstances Z(t) will converge almost surely to some finite value as t→∞. This problem has previously been considered by Dufresne (1990) who provided a sufficient condition for almost sure convergence of Z(t) (the Root Test) and then proceeded to consider some specific examples of such processes. Here, we develop Dufresne's work and show that the sufficient condition for convergence can be proved to hold for quite a general class of model which includes the growing number of Office Models with stochastic cashflows.

This paper provides a detailed quantitative assessment of the impact of capital and default probability on product pricing and shareholder value for a life insurer providing life annuities. A multi-period cash flow model, allowing for stochastic mortality and asset returns, imperfectly elastic product demand, as well as frictional costs, is used to derive value-maximizing capital and pricing strategies for a range of one-year default probability levels reflecting differences in regulatory regimes including Solvency II. The model is calibrated using realistic assumptions. The sensitivity of results is assessed. The results show that value-maximizing life insurers should target higher solvency levels than the Solvency II regulatory one-year 99.5% probability under assumptions of reasonable levels of policyholder's aversion to insolvency risk. Even in the case of less restrictive solvency probabilities, policyholder price elasticity and solvency preferences are shown to be important factors for a life insurer's value-maximizing strategy.

In Bohman the following model is considered. Our notation follows Bohman.

Let Z1, Z2, … be a sequence of independent random variables with distribution function F and put

Put

and define X by

X = inf{n; Sn > U, Sk ≤ U for k = 1, …, n − 1}.

Bohman shows that if U → ∞ in such a way that U/σ → ∞ and then

where G(α, x) is the distribution function for the time when a Wiener process X(t) with EX(t) = αt and Var X(t) = t first crosses the level 1.

Let N be an integer, which in a certain sense corresponds to “time”, and consider P(X ≤ N). This is thus the probability of ruin before the N:th claim.

The marginal approach to risk and return analysis compares the marginal return from a business decision to the marginal risk imposed. Allocation distributes the total company risk to business units and compares the profit/risk ratio of the units. These approaches coincide when the allocation actually assigns the marginal risk to each business unit, i.e., when the marginal impacts add up to the total risk measure. This is possible for one class of risk measures (scalable measures) under the assumption of homogeneous growth and by a subclass (transformed probability measures) otherwise. For homogeneous growth, the allocation of scalable measures can be accomplished by the directional derivative. The first well known additive marginal allocations were the Myers-Read method from Myers and Read (2001) and co-Tail Value at Risk, discussed in Tasche (2000). Now we see that there are many others, which allows the choice of risk measure to be based on economic meaning rather than the availability of an allocation method. We prefer the term “decomposition” to “allocation” here because of the use of the method of co-measures, which quantifies the component composition of a risk measure rather than allocating it proportionally to something.

Risk adjusted profitability calculations that do not rely on capital allocation still may involve decomposition of risk measures. Such a case is discussed. Calculation issues for directional derivatives are also explored.

Policyholders often decide to buy, renew, or cancel insurance based on the premium charged by the insurer compared with what they expect their claims will be. It is important for actuaries to consider the persistency of policyholders because the financial well-being of the insurer depends on spreading its risk over a large book of business. We use statistical decision theory to develop premium formulas that account for the past experience of a given policyholder, the experience of the entire collection of policyholders, and the likelihood of the policyholder renewing with or buying from a given insurer, that is, persistency.

We assume that the persistency of policyholders depends on the arithmetic difference between the premium charged and their anticipated claims. We extend the work of Taylor (1975) in which he obtains linear credibility formulas by minimizing loss functions that incorporate the persistency of policyholders. We consider Taylor's loss functions and other objective functions, including those that account for the amount of business the insurer writes or renews.