The UK Biobank project is a proposed large-scale investigation of the combined effects of genotype and environmental exposures on the risk of common diseases. It is intended to recruit 500,000 subjects aged 40-69, to obtain medical histories and blood samples at outset, and to follow them up for at least 10 years. This will have a major impact on our knowledge of multifactorial genetic disorders, rather than the rare but severe single-gene disorders that have been studied to date. What use may insurance companies make of this knowledge, particularly if genetic tests can identify persons at different risk? We describe here a simulation study of the UK Biobank project. We specify a simple hypothetical model of genetic and environmental influences on the risk of heart attack. A single simulation of UK Biobank consists of 500,000 life histories over 10 years; we suppose that case-control studies are carried out to estimate age-specific odds ratios, and that an actuary uses these odds ratios to parameterise a model of critical illness insurance. From a large number of such simulations we obtain sampling distributions of premium rates in different strata defined by genotype and environmental exposure. We conclude that the ability of such a study reliably to discriminate between different underwriting classes is limited, and depends on large numbers of cases being analysed.

In a simple stationary setting with constant interest rate, we derive pricing formulas for defaultable bonds with stochastic recovery rate using a replication argument. Replication is done by using an insurance contract (i.e. a kind of credit default swap), the price of which is determined by a dynamic premium calculation principle. We consider two cases, a linear one, where pricing amounts to solving an inhomogeneous linear ODE, and a super-linear case where a Riccati ODE has to be solved.

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

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.

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

This paper investigates some models in which non-negative observations from a Poisson or generalised Poisson distribution are possibly damaged according to a binomial or quasi-binomial law. The latter case is appropriate when the observations are over-dispersed. Although the extent of the damage is not known, it is assumed that the event of whether or not damage occurred is discernible. The models are particularly suited for certain applications involving accident counts when evidence of certain accidents may be observed even though the accidents themselves may go unreported. Given the number of observed accidents and knowledge as to whether or not some additional accidents have gone unreported, these models may be used to make inferences concerning the actual number of unreported and total number of accidents in the current period, and the numbers of reported, unreported, and/or total accidents in a future period. The models are applied to a real data set giving reported and unreported patient accidents in a large hospital. Both maximum likelihood and Bayesian estimation methods are presented and discussed.

For a general class of risk models, the dividends-penalty identity is derived by probabilistic reasoning. This identity is the key for understanding and determining the optimal dividend barrier, which maximizes the difference between the expected present value of all dividends until ruin and the expected discounted value of a penalty at ruin (which is typically a function of the deficit at ruin). As an illustration, the optimal barrier is calculated in two classical models, for different penalty functions and a variety of parameter values.

In this paper the dynamic portfolio selection problem is studied for the first time in a dual utility theory framework. The Wang transform is used as distortion function and well diversified optimal portfolios result both with and without short sales allowed.

We revisit the famous Mack formula [2], which gives an estimate for the mean square error of prediction MSEP of the chain ladder claims reserving method: We define a time series model for the chain ladder method. In this time series framework we give an approach for the estimation of the conditional MSEP. It turns out that our approach leads to results that differ from the Mack formula. But we also see that our derivation leads to the same formulas for the MSEP estimate as the ones given in Murphy [4]. We discuss the differences and similarities of these derivations.

We discuss some questionable points of the approach taken in the paper by Buchwalder, Bühlmann, Merz and Wüthrich and come to the conclusion that this approach does not yield an improvement of Mack’s original formula. The main reason is that the new approach disregards the negative correlation of the squares of the development factors. The same applies to the formula by Murphy (PCAS 1994).

We derive the estimation error in a Bayesian framework and discuss the estimates of Mack and of Buchwalder, Bühlmann, Merz and Wüthrich (BBMW) from a Bayesian point of view.

We introduce a class of Bayesian infinite mixture models first introduced by Lo (1984) to determine the credibility premium for a non-homogeneous insurance portfolio. The Bayesian infinite mixture models provide us with much flexibility in the specification of the claim distribution. We employ the sampling scheme based on a weighted Chinese restaurant process introduced in Lo et al. (1996) to estimate a Bayesian infinite mixture model from the claim data. The Bayesian sampling scheme also provides a systematic way to cluster the claim data. This can provide some insights into the risk characteristics of the policyholders. The estimated credibility premium from the Bayesian infinite mixture model can be written as a linear combination of the prior estimate and the sample mean of the claim data. Estimation results for the Bayesian mixture credibility premiums will be presented.

This article discusses risk classification and develops and discusses a framework for estimating the effects of restrictions on risk classification. It is shown that expected losses due to adverse selection depend only on means, variances and covariances of insurance factors and rates of uptake of insurance. Percentage loadings required to avoid losses are displayed. Correlated information, such as family history, is also incorporated and it is seen how such information limits losses and decreases required loadings. Although the evidence suggests that adverse selection is not, at present, a severe problem for insurers, this might change if the authorities impose restrictions on risk classification and/or customers gain an informational advantage (such as better knowledge of their own risk levels). Application is made to unisex annuity pricing in the UK insurance market.

This paper addresses some of the problems a majority of retired individuals face: Why and in what proportion should they invest in a life annuity to maximize the utility of their future consumption or a bequest? The market considered in this work is made up of three assets: a life annuity, a risky asset and a cash account. As this problem doesn’t accept any suitable explicit solution, it is numerically solved by the Markov Chain approximation developed by Kushner and Dupuis. Without a bequest motive, we observe that the optimal planning of consumption is divided into two periods and that optimal asset allocation should include the risky asset. Next, the influence of a bequest on consumption and investment pattern is developed. We demonstrate that even with a bequest motive, pensioners should allocate a part of their wealth to the purchase of life annuities.