In this paper, the optimal insurance design is studied from the perspective of an insured, who faces an insurable risk and a background risk. For the reduction of ex post moral hazard, alternative insurance contracts are asked to satisfy the principle of indemnity and the incentive-compatible condition. As in the literature, it is assumed that the insurer calculates the insurance premium solely on the basis of the expected indemnity. When the insured has a general mean-variance preference, an explicit form of optimal insurance is derived explicitly. It is found that the stochastic dependence between the background risk and the insurable risk plays a critical role in the insured’s risk transfer decision. In addition, the optimal insurance policy can often change significantly once the incentive-compatible constraint is removed.

The Erlang mixture model has been widely used in modeling insurance losses due to its desirable distributional properties. In this paper, we consider the problem of efficient estimation of the Erlang mixture model. We present a new thresholding penalty function and a corresponding EM algorithm to estimate model parameters and to determine the order of the mixture. Using simulation studies and a real data application, we demonstrate the efficiency of the EM algorithm.

In this paper we estimate operational risk by using the convex risk measure Expected Shortfall (ES) and provide an approximation as the confidence level converges to 100% in the univariate case. Then we extend this approach to the multivariate case, where we represent the dependence structure by using a Lévy copula as in Böcker and Klüppelberg (2006) and Böcker and Klüppelberg, C. (2008). We compare our results to the ones obtained in Böcker and Klüppelberg (2006) and (2008) for Operational VaR and discuss their practical relevance.

In this paper, a new methodology based on the use of the inverse of the circular tangent function that allows us to add a scale parameter (say α) to an initial survival function is presented. The latter survival function is determined as limiting case when α tends to zero. By choosing as parent the classical Pareto survival function, the Pareto ArcTan (PAT) distribution is obtained. After providing a comprehensive analysis of its statistical properties, theoretical results with reference to insurance are illustrated. Its performance is compared, by means of the well-known Norwegian fire insurance data, with other existing heavy-tailed distributions in the literature such as Pareto, Stoppa, Shifted Lognormal, Inverse Gamma and Fréchet distributions.

In this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

The normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various “good” competitive models including the log-normal, the Burr, and the symmetric α-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hürlimann (1995), the normal mixture by Praetz (1972), the symmetric α-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed.

Stimulated by a recent contribution by G. Venter in this journal the adequate-ness of (re-)insurance premium calculation based on the hypothesis of arbitrage free (re-)insurance markets is questioned. It is argued that—in contrast to the theory of financial markets—it is not reasonable to demand that insurance markets are arbitrage free. In addition the adjusted distribution principles put forward by Venter are claimed to be invalid.

In this paper we consider compound distributions where the counting distribution is a bivariate distribution with the probability function (Pn1,n2)n1,n2≥0 that satisfies a recursion in the form

We present an algorithm for recursive evaluation of the corresponding compound distributions and some examples of distributions in this class.

The very title of this paper may cause some surprise, since economic theory so far has found virtually no application in insurance. Insurance is obviously an economic activity, and it is indeed strange that general economic theory should seem inapplicable to insurance.

This apparent paradox may to some extent be explained by the historic development. Actuarial mathematics and the essential scientific basis of insurance were developed into a self-contained and fairly complete theory long before economists could claim the name of science for their subject. Actuaries and other insurance people may from time to time have turned to economic theory for help on their problems. In most cases they must have turned away in disappointment, being convinced that actuarial mathematics was well ahead of general economic theory.

The last point is brought out clearly in a paper on the safety loading of insurance premiums which Tauber (19) presented at the Sixth International Congress of Actuaries in Vienna in 1909. In the introduction to this paper Tauber seems to consider risk bearing as a service and appears to assume that it like all other goods and services must have a price, determined by supply and demand in the market. Tauber did not develop this idea, apparently because the economic theory of his time was utterly unable to analyse the problem. That his approach was sound, and that the problem can be formulated and solved by modern economic theory, has been indicated in a previous paper (6).

During the last thirty years there has been an extremely rapid development in economic theory. The “General Theory” of Keynes (12) which appeared in 1936, is usually considered to have caused a revolution in economic theory. However, this “revolution” has not lead to any developments which seem to have an immediate application in insurance. Post- and pre-Keynesian economics are equally powerless when confronted with the problem which Tauber tried to formulate. It is therefore premature to conclude from the title of this paper that economic theory has caught up with actuarial science, and that the theory of insurance has now become a part of a general theory comprising all economic activities.

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

Using telematics technology, insurers are able to capture a wide range of data to better decode driver behavior, such as distance traveled and how drivers brake, accelerate, or make turns. Such additional information also helps insurers improve risk assessments for usage-based insurance, a recent industry innovation. In this article, we explore the integration of telematics information into a classification model to determine driver heterogeneity. For motor insurance during a policy year, we typically observe a large proportion of drivers with zero accidents, a lower proportion with exactly one accident, and a far lower proportion with two or more accidents. We here introduce a cost-sensitive multi-class adaptive boosting (AdaBoost) algorithm we call SAMME.C2 to handle such class imbalances. We calibrate the algorithm using empirical data collected from a telematics program in Canada and demonstrate an improved assessment of driving behavior using telematics compared with traditional risk variables. Using suitable performance metrics, we show that our algorithm outperforms other learning models designed to handle class imbalances.

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant, and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability.

This paper is concerned with two methods to estimate the parameters of the Poisson-Goncharov distribution introduced recently by Lefevre and Picard (1996). These methods are applied to fit, inter alia, the six observed claims distributions, from automobile insurance third party liability portfolios, studied by Gossiaux and Lemaire (1981) and analysed afterwards by several authors.

In an arbitrage-free economy, it is well-known that financial risks can be priced using equivalent martingale measures. We establish in this paper that, for general stochastic processes, the Wang Transform does not lead to a price which is consistent with the arbitrage-free price. Based on these results we must conclude that the Wang Transform cannot be a universal framework for pricing financial and insurance risks.

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

This paper deals with the Bonus-Malus system obtained when the claims frequency is submitted to trend. This system is specified in the two particular cases of Poisson-Gamma and Poisson-Inverse Gaussian distributions. The theoretical results are checked on data issued from automobile insurance policies observed during three years.

The Cox regression model is a standard tool in survival analysis for studying the dependence of a hazard rate on covariates (parametrically) and time (nonparametrically). This paper is a case study intended to indicate possible applications to non-life insurance, particularly occurrence of claims and rating.

We studied individuals from one Danish county holding policies in auto, property and household insurance simultaneously at some point during the four year period 1988-1991 in one company. The hazard of occurrence of claims of each type was studied as function of calendar time, time since the last claim of each type, age of policy holder, urbanization and detailed type of insurance. Particular emphasis was given to the technical advantages and disadvantages (particularly the complicated censoring patterns) of considering the nonparametrically underlying time as either calendar time or time since last claim. In the former case the theory is settled, but the results are somewhat complicated. The latter choice leads to several issues still under active methodological development. We develop a goodness-of-fit criterion which shows the lack of fit of some models, for which the practical conclusions might otherwise have been useful.

This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion.

From a theoretical point of view, we also provide below a generalization of Thorin’s formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times.

In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or “exponentially killed probability’’), in terms of the roots with positive real part of the Lundberg’s equations, which complements Asmussen’s representation (1992) in terms of the roots with negative real part.

In this paper, we develop a theoretical basis for drawing up a “Swedish” type actuarial balance sheet for a defined benefit pay-as-you-go (DB PAYG) scheme with retirement and disability benefits. Our model enables us to obtain the system's expected average turnover duration, measure the scheme's solvency and explore the phenomenon identified as “pension reclassification”, a widespread practice that masks the system's real status unless further pension information becomes available. The model is clearly linked to actuarial practice in social security and gives partial support to the practical adaptation of Swedish methodology carried out by OSFI (2012) in applying the concept of the contribution asset to the Canadian Pension Plan (CPP) balance sheet, which includes disability and survivor benefits.

Excess claims lead to an unsatisfactory behavior of standard linear credibility estimators. We suggest in this paper to use robust methods in order to obtain better estimators. Our first proposal is the linear credibility estimator with the claims replaced by a robust M-estimator of scale calculed from the claims. This corresponds to a truncation of the claims with a truncation point depending on the data and different for each contract. We discuss the properties of the robust M-estimator and present several examples. In order to improve the performance for a very small number of years, we propose a second estimator, which incorporates information from other claims into the M-estimator.