In this paper, the optimal safety loading that the reinsurer should set in the reinsurance pricing is studied, which is novel in the literature. It is first assumed that the insurer will choose the form of the reinsurance contract by following the results derived in Cai et al. Different optimality criteria from the reinsurer’s perspective are then studied, such as maximising the expectation of the profit, maximising the utility of the profit and minimising the value-at-risk of the reinsurer’s total loss. By applying the concept of comonotonicity, the problem in which the reinsurer is facing two risks with unknown dependency structure is also solved. Closed-form solutions are obtained when the underlying losses are zero-modified exponentially distributed. Finally, numerical examples are provided to illustrate the results derived.

This paper presents a new method of graduation which uses parametric formulae together with Bayesian reversible jump Markov chain Monte Carlo methods. The aim is to provide a method which can be applied to a wide range of data, and which does not require a lot of adjustment or modification. The method also does not require one particular parametric formula to be selected: instead, the graduated values are a weighted average of the values from a range of formulae. In this way, the new method can be seen as an automatic graduation method which we believe can be applied in many cases without any adjustments and provide satisfactory graduated values. An advantage of a Bayesian approach is that it allows for model uncertainty unlike standard methods of graduation.

This paper investigates the benefits of incorporating diversification effects into the pricing process of insurance policies from two different business lines. The paper shows that, for the same risk reduction, insurers pricing policies jointly can have a competitive advantage over those pricing them separately. However, the choice of competitiveness constrains the underwriting flexibility of joint pricers. The paper goes a step further by modelling explicitly the relationship between premiums and the number of customers in each line. Using the total collected premiums as a criterion to compare the competing strategies, the paper provides conditions for the optimal pricing decision based on policyholders’ sensitivity to price discounts. The results are illustrated for a portfolio of annuities and assurances. Further, using non-life data from the Brazilian insurance market, an empirical exploration shows that most pairs satisfy the condition for being priced jointly, even when pairwise correlations are high.

In Finance and Actuarial Science, the multivariate elliptical family of distributions is a famous and well-used model for continuous risks. However, it has an essential shortcoming: all its univariate marginal distributions are the same, up to location and scale transformations. For example, all marginals of the multivariate Student’s t-distribution, an important member of the elliptical family, have the same number of degrees of freedom. We introduce a new approach to generate a multivariate distribution whose marginals are elliptical random variables, while in general, each of the risks has different elliptical distribution, which is important when dealing with insurance and financial data. The proposal is an alternative to the elliptical copula distribution where, in many cases, it is very difficult to calculate its risk measures and risk capital allocation. We study the main characteristics of the proposed model: characteristic and density functions, expectations, covariance matrices and expectation of the linear regression vector. We calculate important risk measures for the introduced distributions, such as the value at risk and tail value at risk, and the risk capital allocation of the aggregated risks.

This paper aims to evaluate the aggregate claims distribution under the collective risk model when the number of claims follows a so-called generalised (a, b, 1) family distribution. The definition of the generalised (a, b, 1) family of distributions is given first, then a simple matrix-form recursion for the compound generalised (a, b, 1) distributions is derived to calculate the aggregate claims distribution with discrete non-negative individual claims. Continuous individual claims are discussed as well and an integral equation of the aggregate claims distribution is developed. Moreover, a recursive formula for calculating the moments of aggregate claims is also obtained in this paper. With the recursive calculation framework being established, members that belong to the generalised (a, b, 1) family are discussed. As an illustration of potential applications of the proposed generalised (a, b, 1) distribution family on modelling insurance claim numbers, two numerical examples are given. The first example illustrates the calculation of the aggregate claims distribution using a matrix-form Poisson for claim frequency with logarithmic claim sizes. The second example is based on real data and illustrates maximum likelihood estimation for a set of distributions in the generalised (a, b, 1) family.

The funding position of a defined benefit pension plan is often closely linked to the performance of the sponsoring company’s business. For example, a plan sponsor whose financial health is dependent on high oil prices may struggle during periods of oil price weakness. If the pension plan’s assets perform poorly at this time, the ability of the sponsor to address any funding requirement could be restricted precisely when the need for funding is heightened. In this paper, we propose an approach to dealing with joint plan and sponsor risk that can provide protection against extreme adverse events for the sponsor. In particular, adopt a strategy of minimising a portfolio’s expected losses in the event of an assumed drop of x% in the oil price. Our methodology relies on an asset allocation framework that takes into account the impact of serial correlation in asset returns, as well as the negative skewness and leptokurtosis resulting from the non-normal shape of marginal distributions of historical asset returns. We also make use of copulas to measure the dependence between asset class returns.

It is well known that the presence of outliers can mis-estimate (underestimate or overestimate) the overall reserve in the chain-ladder method, when we consider a linear regression model, based on the assumption that the coefficients are fixed and identical from one observation to another. By relaxing the usual regression assumptions and applying a regression with randomly varying coefficients, we have a similar phenomenon, i.e., mis-estimation of the overall reserves. The lack of robustness of loss reserving regression with random coefficients on incremental payment estimators leads to the development of this paper, aiming to apply robust statistical procedures to the loss reserving estimation when regression coefficients are random. Numerical results of the proposed method are illustrated and compared with the results that were obtained by linear regression with fixed coefficients.

Actuaries often encounter censored and masked survival data when constructing multiple-decrement tables. In this paper, we propose estimators for the cause-specific failure time density using LOESS smoothing techniques that are employed in the presence of left-censored data, while still allowing for right-censored and exact observations, as well as masked causes of failure. The smoothing mechanism is incorporated as part of an expectation-maximisation algorithm. The proposed models are applied to a bivariate African sleeping sickness data set.

This paper argues that most of the problems that actuaries have to deal with in the context of non-life insurance can be usefully cast in the framework of computational intelligence (a.k.a. artificial intelligence), the discipline that studies the design of agents which exhibit intelligent behaviour. Finding an adequate framework for actuarial problems has more than a simply theoretical interest: it also allows a technological transfer from the computational intelligence discipline to general insurance, wherever techniques have been developed for problems which are common to both contexts. This has already happened in the past (neural networks, clustering, data mining have all found applications to general insurance) but not in a systematic way. One of the objectives of this paper will therefore be to introduce some useful techniques such as sparsity-based regularisation and dynamic decision networks that are not yet known to the wider actuarial community.

Whilst in the first part of this paper we dealt mainly with data-driven loss modelling under the assumption that all the data were accurate and fully relevant to the exercise, in this second part of the paper we explore how to deal with uncertain knowledge, whether this uncertainty comes from the fact that the data are not fully reliable (e.g. they are estimates) or from the fact that the knowledge is “soft” (e.g. expert beliefs) or not fully relevant (e.g. market information on a given risk). Most importantly, we will deal with the problem of making pricing, reserving and capital decisions under uncertainty. It will be concluded that a Bayesian framework is the most adequate for dealing with uncertainty, and we will present a number of computational intelligence techniques to do this in practice.

This paper provides a user-friendly approach to explain how variation in fundamental price-determining variables ‘translates into’ variation in the fundamental value of equities, based on the standard dividend-growth model. The analysis is illustrated with UK data using estimates of real interest rate forecasts and real dividend growth rate forecasts in the past. An important application of this approach is that stock market volatility can be analysed in terms of its component parts. Actual market volatility does not appear to be excessive when compared with the notional volatility implied by changes over time in our estimates of forecast real interest rates and forecast real dividend growth rates.

This paper starts with the Beta transform and discusses the stochastic ordering properties of this transform under different parameter settings. Later, the distribution of discounted aggregate claims in a compound renewal risk model with dependence between inter-claim times and claim sizes is studied. Recursive formulas for moments and joint moments are expressed in terms of the Beta transform of the inter-claim times and claim severities. Particularly, our moments formula is more explicit and computation-friendly than earlier ones in the references. Lastly, numerical examples are provided to illustrate our results.

The International Accounting Standards Board (IASB) is introducing new International Financial Reporting Standards (IFRS) which aim to make financial statements more useful. The process has generated considerable debate. This paper is a contribution to the debate, in the particular context of insurance accounting, and attempts to provide a coherent framework for accounting theory which makes a clear distinction between retrospective statements required for administrative accountability, fair value for current market transactions and to measure value creation, and a prospective prudence required to protect policyholders, depositors and other creditors. It is argued that the IASB's founding purpose to provide a single set of accounts is therefore incoherent; different purposes require different numbers. This also implies that fair value accounts should attempt to value intangible assets. In this context, actuarial analyses of surplus would greatly assist in measuring whether model assumptions are appropriate.

The aim of the paper is to derive a simple, implementable machine learning method for general insurance losses. An algorithm for learning a general insurance loss triangle is developed and justified. An argument is made for applying support vector regression (SVR) to this learning task (in order to facilitate transparency of the learning method as compared to more “black-box” methods such as deep neural networks), and SVR methodology derived is specifically applied to this learning task. A further argument for preserving the statistical features of the loss data in the SVR machine is made. A bespoke kernel function that preserves the statistical features of the loss data is derived from first principles and called the exponential dispersion family (EDF) kernel. Features of the EDF kernel are explored, and the kernel is applied to an insurance loss estimation exercise for homogeneous risk of three different insurers. Results of the cumulative losses and ultimate losses predicted by the EDF kernel are compared to losses predicted by the radial basis function kernel and the chain-ladder method. A backtest of the developed method is performed. A discussion of the results and their implications follows.