We propose a new neighbouring prediction model for mortality forecasting. For each mortality rate at age x in year t, mx,t, we construct an image of neighbourhood mortality data around mx,t, that is, Ꜫmx,t (x1, x2, s), which includes mortality information for ages in [x-x1, x+x2], lagging k years (1 ≤ k ≤ s). Combined with the deep learning model – convolutional neural network, this framework is able to capture the intricate nonlinear structure in the mortality data: the neighbourhood effect, which can go beyond the directions of period, age, and cohort as in classic mortality models. By performing an extensive empirical analysis on all the 41 countries and regions in the Human Mortality Database, we find that the proposed models achieve superior forecasting performance. This framework can be further enhanced to capture the patterns and interactions between multiple populations.

Suppose that an automobile insurance plan is characterized by a double classification. The risks are thus divided into classes, i = 1, 2, …, p (e.g. defined by use of car and age of operator), and groups j = 1, 2, …, q (e.g. defined by licence and by accidents during the last three years). The experience of the company is described by the observed “relative loss ratios” rij and some measure of exposure nij. A general model, often used, is that the rij: s are observations of random variables with the expected values gij = g(αi, βj), where the relativities αi are parameters representing the classes i and the relativities βj represent the influence of the groups j. One of the ratemaker's problems is to find a realistic function g(α, β) and to obtain estimates ai of αi and bj of βj.

In their paper “Two Studies in Automobile Insurance Rate-making” (ASTIN Bulletin Vol. I, Part IV, page 192-217) Robert Bailey and LeRoy Simon have thoroughly analyzed this problem for private passenger automobiles in Canada. They have principally studied three different types of the function g(α β), namely g(α β) = αβ (Method 2), g(α β) = α + β (Method 3) and g(α β) = 3αβ— 2 (Method 4). The authors show in an appendix, that the variance of rij is approximately g(αi βj)/Knij where K ≅ 0.005 for the Canadian data. They estimate the relativities αi and βj; by making χ2 = K. Σnij(rij — gij)2/gij a minimum. For the Canadian material, the “method 4” agrees best with the observations. This method gives an observed χ2 value of about 8 for 11 degrees of freedom.

The standard methods for the calculation of total claim size distributions and ruin probabilities, Panjer recursion and algorithms based on transforms, both apply to lattice-type distributions only and therefore require an initial discretization step if continuous distribution functions are of interest. We discuss the associated discretization error and show that it can often be reduced substantially by an extrapolation technique.

The generalized Poisson distribution with parameters θ and λ was introduced by Consul and Jain (1973) and has recently found several instances of application in the actuarial literature. The most frequently used version of the distribution assumes that θ > 0 and 0 ≤ λ < 1, in which case the mean and variance are θ/(1 − λ) and θ/(1 − λ)3, respectively. These simple moment expressions, along with nearly all of the other theoretical results available for this distribution, fail when λ < 0 or λ > 1 (e.g., Johnson, Kotz, and Kemp, 1992, page 397). In these cases, even the definition of the probability mass function usually given in the literature is not properly normalized so that its values do not sum to unity. For this reason, it is common to truncate the support of the distribution and explicitly normalize the probability mass function. In this paper we discuss the estimation of the parameters of this truncated generalized Poisson distribution using a Bayesian method.

This paper contains little which can be considered as new. It gives a survey of results which have been presented over the last 10-15 years. At one time these results seemed very promising, but in retrospect it is doubtful if they have fulfilled the expectations they raised. In this situation it may be useful to retrace one's steps and see if problems can be reformulated or if new approaches can be found.

Mathematical models have been used in insurance for a long time. One of the first was the Gompertz mortality law; a more recent model, which has been intensively studied is the Compound Poisson Distribution in Lundberg's risk theory.

When a model is introduced, one usually proceeds by stages. The first step is to see if the model appears acceptable on a priori reasons. If it does, the second step is to examine the implications of the model, to see if any of these are in obvious contradiction with observations. If the result of this examination is satisfactory, the third step is usually a statistical analysis to find out how well the model approximates the situation in real life, which one wants to analyse. If the model passes this second examination, the next and final step may be to estimate the parameters of the model, and use it in practice, i.e. to make decisions in the real world.

The advantage of working with a model is that it gives an overall purpose to the collection and analysis of data. A good model should tell us which data we need, and why.

In this paper motor liability insurance is considered from the viewpoint of an excess of loss reinsurer.

The reinsurer pays portions of claim amounts in excess of a contractually agreed limit, further referred to as excess point.

Aspects of practical interest are the rating of new contracts and cost-projections on claims occurred in past years which have not yet been reported.

What causes a major difficulty to a rating approach is the timelag between the occurrence of a claim and its settlement.

During this timelag, which may extend to ten years or even longer, the size of a claim will usually increase considerably. This is due to such factors as inflation and more victim oriented legal procedures.

As a result of this phenomenon, a claim, initially not involving the reinsurer, may confront him many years later, when its size overtakes the excess point. Hence, each risk year produces a generation of claims stretching far into the future.

The problem facing the reinsurer trying to compute premiums is that he must look, say ten years into the future, on the basis of incomplete data.

The table below is a schematic presentation of the time history of a contract that started k years ago.

The experience is assumed to be expressed in numbers of claims exceeding a constant excess point.

Successive rows show the generations of excess claims down to the last completed risk year k.

In the pages that follow we will attempt to project expected numbers of excess claims into the future, using information as schematized in this table.

In this paper we discuss the concept of excess of loss reinsurance with reinstatements. The main objective is to provide a methodology to calculate the distribution of total aggregate losses for two or more consecutive layers when there is a limited number of reinstatements. We also compare different premium principles and their properties to price these treaties for any number of free or paid reinstatements.

The Lee–Carter model has become a benchmark in stochastic mortality modeling. However, its forecasting performance can be significantly improved upon by modern machine learning techniques. We propose a convolutional neural network (NN) architecture for mortality rate forecasting, empirically compare this model as well as other NN models to the Lee–Carter model and find that lower forecast errors are achievable for many countries in the Human Mortality Database. We provide details on the errors and forecasts of our model to make it more understandable and, thus, more trustworthy. As NN by default only yield point estimates, previous works applying them to mortality modeling have not investigated prediction uncertainty. We address this gap in the literature by implementing a bootstrapping-based technique and demonstrate that it yields highly reliable prediction intervals for our NN model.

A method to hedge variable annuities in the presence of basis risk is developed. A regime-switching model is considered for the dynamics of market assets. The approach is based on a local optimization of risk and is therefore very tractable and flexible. The local optimization criterion is itself optimized to minimize capital requirements associated with the variable annuity policy, the latter being quantified by the Conditional Value-at-Risk (CVaR) risk metric. In comparison to benchmarks, our method is successful in simultaneously reducing capital requirements and increasing profitability. Indeed the proposed local hedging scheme benefits from a higher exposure to equity risk and from time diversification of risk to earn excess return and facilitate the accumulation of capital. A robust version of the hedging strategies addressing model risk and parameter uncertainty is also provided.

It is shown how the distribution-free method of Mack (1993) can be extended in order to estimate the prediction error of the Chain Ladder method for a portfolio of several correlated run-off triangles.

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation:

For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

An approximation of the distribution of the present value of the benefits of a portfolio of temporary insurance contracts is suggested for the case where the size of the portfolio tends to infinity. The model used is the one presented in Parker (1922b) and involves random interest rates and future lifetimes. Some justifications of the approximation are given. Illustrations for limiting portfolios of temporary insurance contracts are presented for an assumed Ornstein-Uhlenbeck process for the force of interest.

A machine learning approach to zero-inflated Poisson (ZIP) regression is introduced to address common difficulty arising from imbalanced financial data. The suggested ZIP can be interpreted as an adaptive weight adjustment procedure that removes the need for post-modeling re-calibration and results in a substantial enhancement of predictive accuracy. Notwithstanding the increased complexity due to the expanded parameter set, we utilize a cyclic coordinate descent optimization to implement the ZIP regression, with adjustments made to address saddle points. We also study how various approaches alleviate the potential drawbacks of incomplete exposures in insurance applications. The procedure is tested on real-life data. We demonstrate a significant improvement in performance relative to other popular alternatives, which justifies our modeling techniques.

The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.

The result is a generalization of Lundberg (1940).

Consider the distribution of

where X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution function

and N is an integer valued random variable with probability function

Then the distribution function of S is given by

If MX(t), MN(t) and MS(t) are the associated moment generating functions, then

Several recent papers treated robust and efficient estimation of tail index parameters for (equivalent) Pareto and truncated exponential models, for large and small samples. New robust estimators of “generalized median” (GM) and “trimmed mean” (T) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, quantiles, and percentile matching. Here we investigate performance of the above mentioned estimators on real data and establish — via the use of goodness-of-fit measures — that favorable theoretical properties of the GM and T type estimators translate into an excellent practical performance. Further, we arrive at guidelines for Pareto model diagnostics, testing, and selection of particular robust estimators in practice. Model fits provided by the estimators are ranked and compared on the basis of Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling statistics.

We consider a general credibility model for the prediction of IBNR-claims which allows for random fluctuations in the underlying delay distribution. Such fluctuations always bring about decreasing credibility. It is shown that even negative credibility is achieved for more substantial fluctuations in the delay distribution. Special attention is paid to the mixed Poisson case for claim numbers including the discussion of parameter estimation.

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

This article, based on a result of Borch and an extension of Bühlmann, gives a complete characterization of Pareto optimal risk exchanges by a system of differential equations linking the derivate of agents contributions to their risk aversion coefficients.

Consider the classical compound Poisson model of risk theory, in which dividends are paid to the shareholders according to a barrier strategy. Let b* be the level of the barrier that maximizes the expectation of the discounted dividends until ruin. This paper is inspired by Dickson and Waters (2004). They point out that the shareholders should be liable to cover the deficit at ruin. Thus, they consider b0 , the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. In this paper, b* and b0 are compared, when the claim amount distribution is exponential or a combination of exponentials.