Non-life insurance payouts consist of two factors: claimsizes and claim frequency. When calculating e.g. next years premium, it is vital to correctly model these factors and to estimate the unknown parameters. A standard way is to separately estimate in the claimsize and the claim frequency models.

Often there is a deductible with each single claim, and this deductible can be quite large, particularly in inhomogeneous cases such as industrial fire insurance or marine insurance. Not taking the deductibles into account can lead to serious bias in the estimates and consequent implications when applying the model.

When the deductibles are nonidentical, in a full maximum likelihood estimation all unknown parameters have to be estimated simultaneously. An alternative is to use pseudo-maximum likelihood, i.e. first estimate the claimsize model, taking the deductibles into account, and then use the estimated probability that a claim exceeds the deductible as an offset in the claim frequency estimation. This latter method is less efficient, but due to complexity or time considerations, it may be the preferred option.

In this paper we will provide rather general formulas for the relative efficiency of the pseudo maximum likelihood estimators in the i.i.d. case. Two special cases will be studied in detail, and we conclude the paper by comparing the methods on some marine insurance data.

We investigate the question how the development pattern in the Bornhuetter-Ferguson method should be estimated and derive the corresponding conditional mean square error of prediction (MSEP) of the ultimate claim prediction. An estimator of this conditional MSEP in a distribution-free model was given by Mack [9], whereas in Alai et al. [2] this conditional MSEP was studied in an over-dispersed Poisson model using the chain ladder development pattern. First we consider distributional models and derive estimators (maximum likelihood) for the development pattern taking all relevant information into account. Moreover, we suggest new estimators of the correlation matrix of these estimators and new estimators of the conditional MSEP. Our findings supplement some of Mack's results. The methodology is illustrated at two numerical examples.

This paper investigates market-consistent valuation of insurance liabilities in the context of Solvency II among others and to some extent IFRS 4. We propose an explicit and consistent framework for the valuation of insurance liabilities which incorporates the Solvency II approach as a special case.

The proposed framework is based on replication over multiple (one-year) time periods by a periodically updated portfolio of assets with reliable market prices, allowing for “limited liability” in the sense that the replication can in general not always be continued. The asset portfolio consists of two parts: (1) assets whose market price defines the value of the insurance liabilities, and (2) capital funds used to cover risk which cannot be replicated. The capital funds give rise to capital costs; the main exogenous input in the framework is the condition on when the investment of the capital funds is acceptable.

We investigate existence of the value and show that the exact calculation of the value has to be done recursively backwards in time, starting at the end of the lifetime of the insurance liabilities. We derive upper bounds on the value and, for the special case of replication by risk-free one-year zero-coupon bonds, explicit recursive formulas for calculating the value. In the paper, we only partially consider the question of the uniqueness of the value.

Valuation in Solvency II and IFRS 4 is based on representing the value as a sum of a “best estimate” and a “risk margin”. In our framework, it turns out that this split is not natural. Nonetheless, we show that a split can be constructed as a simplification, and that it provides an upper bound on the value under suitable conditions. We illustrate the general results by explicitly calculating the value for a simple example.

This paper introduces a new framework for modelling the joint development over time of mortality rates in a pair of related populations with the primary aim of producing consistent mortality forecasts for the two populations. The primary aim is achieved by combining a number of recent and novel developments in stochastic mortality modelling, but these, additionally, provide us with a number of side benefits and insights for stochastic mortality modelling. By way of example, we propose an Age-Period-Cohort model which incorporates a mean-reverting stochastic spread that allows for different trends in mortality improvement rates in the short-run, but parallel improvements in the long run. Second, we fit the model using a Bayesian framework that allows us to combine estimation of the unobservable state variables and the parameters of the stochastic processes driving them into a single procedure. Key benefits of this include dampening down of the impact of Poisson variation in death counts, full allowance for paramater uncertainty, and the flexibility to deal with missing data. The framework is designed for large populations coupled with a small sub-population and is applied to the England & Wales national and Continuous Mortality Investigation assured lives males populations. We compare and contrast results based on the two-population approach with single-population results.

We quantify the overall impact of genetic information on the insurance industry using the ‘bottom-up’ approach, in which detailed models are constructed of representative major genetic disorders. We consider six such disorders, namely adult polycystic kidney disease, early-onset Alzheimer's disease, Huntington's disease, myotonic dystrophy (MD), hereditary non-polyposis colorectal cancer; and breast/ovarian cancer. Actuarial models based on the epidemiological literature exist for all these except MD. We parameterise a suitable model of MD, then synthesize the results from all six models to estimate the adverse selection costs arising from restrictions on insurers' use of genetic information. These are all very small, only in the most extreme cases rising above 1% of premiums. In the worst case — females displaying ‘extreme’ adverse selection in a ‘small’ critical illness insurance market, with the use of family history banned — the cost is about 3% of premiums. Our model includes the most common single-gene disorders relevant to insurance, and includes representatives of most important classes of these disorders. While the ‘bottom-up’ approach could be continued by modelling more and more diseases, we suggest that our model is adequate to draw robust conclusions.

Conditional indexation offers a middle way between defined benefit and defined contribution pension schemes. In this paper, we consider a fully-funded pension scheme with conditional indexation. We show how the pension fund can be managed to reduce the risks associated with promised pension benefits when declared benefits are adjusted regularly during the working life. In particular, we derive an investment strategy that provides protection against underfunding at retirement and which is self-financing on average. Our results are illustrated in an extensive simulation study.

The mortality evolution of small populations often exhibits substantial variability and irregular improvement patterns making it hard to identify underlying trends and produce plausible projections. We propose a methodology for robust forecasting based on the existence of a larger reference population sharing the same long-term trend as the population of interest. The reference population is used to estimate the parameters in a frailty model for the underlying intensity surface. A multivariate time series model describing the deviations of the small population mortality from the underlying mortality is then fitted and forecasted. Coherent long-term forecasts are ensured by the underlying frailty model while the size and variability of short- to medium-term deviations are quantified by the time series model. The frailty model is particularly well suited to describe the changing improvement patterns in old age mortality. We apply the method to Danish mortality data with a pooled international data set as reference population.

In this paper we propose an additive mixture model, where one component is the Generalized Pareto distribution (GPD) that allows us to estimate extreme quantiles. GPD plays an important role in modeling extreme quantiles for the wide class of distributions belonging to the maximum domain of attraction of an extreme value model. One of the main difficulty with this modeling approach is the choice of the threshold u, such that all observations greater than u enter into the likelihood function of the GPD model. Difficulties are due to the fact that GPD parameter estimators are sensible to the choice of u. In this work we estimate u, and other parameters, using suitable priors in a Bayesian approach. In particular, we propose to model all data, extremes and non-extremes, using a semiparametric model for data below u, and the GPD for the exceedances over u. In contrast to the usual estimation techniques for u, in this setup we account for uncertainty on all GPD parameters, including u, via their posterior distributions. A Monte Carlo study shows that posterior credible intervals also have frequentist coverages. We further illustrate the advantages of our approach on two applications from insurance.

The forecasting of the future mortality of the very old presents additional challenges since data quality can be poor at such ages. We consider a two-factor model for stochastic mortality, proposed by Cairns, Blake and Dowd, which is particularly well suited to forecasting at very high ages. We consider an extension to their model which improves fit and also allows forecasting at these high ages. We illustrate our methods with data from the Continuous Mortality Investigation.

In this paper we develop a full stochastic cash flow model of outstanding liabilities for the model developed in Verrall, Nielsen and Jessen (2010). This model is based on the simple triangular data available in most non-life insurance companies. By using more data, it is expected that the method will have less volatility than the celebrated chain ladder method. Eventually, our method will lead to lower solvency requirements for those insurance companies that decide to collect counts data and replace their conventional chain ladder method.

In this paper, we study the fair valuation of participating life insurance contract, which is one of the most common life insurance products, under the jump diffusion model with the consideration of default risk. The participating life insurance contracts considered here can be expressed as portfolios of options as shown by Grosen and Jørgensen (1997). We use the Laplace transforms methods to price these options.

The chain ladder is considered in relation to certain recursive and non-recursive models of claim observations. The recursive models resemble the (distribution free) Mack model but are augmented with distributional assumptions. The non-recursive models are generalisations of Poisson cross-classified structure for which the chain ladder is known to be maximum likelihood. The error distributions considered are drawn from the exponential dispersion family.

Each of these models is examined with respect to sufficient statistics and completeness (Section 5), minimum variance estimators (Section 6) and maximum likelihood (Section 7). The chain ladder is found to provide maximum likelihood and minimum variance unbiased estimates of loss reserves under a wide range of recursive models. Similar results are obtained for a much more restricted range of non-recursive models.

These results lead to a full classification of this paper's chain ladder models with respect to the estimation properties (bias, minimum variance) of the chain ladder algorithm (Section 8).

This paper intends to develop a feasible framework which incorporates ambiguity aversion into the pricing of insurance products and investigate the implications of ambiguity aversion on the pricing by comparing it with risk aversion. As applications of the framework, we present the closed-form pricing formulae for some insurance products appearing in life insurance and property insurance. Our model confirms that the effects of ambiguity aversion on the pricing of insurance do differ from those of risk aversion. Implications of our model are consistent with some empirical evidences documented in the literature. Our results suggest that taking advantage of natural hedge mechanism can help us control the effects of model uncertainty.

Modeling dependencies among multiple loss triangles has important implications for the determination of loss reserves, a critical element of risk management and capital allocation practices of property-casualty insurers. In this article, we propose a copula regression model for dependent lines of business that can be used to predict unpaid losses and hence determine loss reserves. The proposed method, relating the payments in different run-off triangles through a copula function, allows the analyst to use flexible parametric families for the loss distribution and to understand the associations among lines of business. Based on the copula model, a parametric bootstrap procedure is developed to incorporate the uncertainty in parameter estimates.

To illustrate this method, we consider an insurance portfolio consisting of personal and commercial automobile lines. When applied to the data of a major US property-casualty insurer, our method provides comparable point prediction of unpaid losses with the industry's standard practice, chain-ladder estimates. Moreover, our flexible structure allows us to easily compute the entire predictive distribution of unpaid losses. This procedure also readily yields accident year reserves, calendar year reserves, as well as the aggregate reserves. One important implication of the dependence modeling is that it allows analysts to quantify the diversification effects in risk capital analysis. We demonstrate these effects by calculating commonly used risk measures, including value at risk and conditional tail expectation, for the insurer's combined portfolio of personal and commercial automobile lines.

In this paper, we study two classes of optimal reinsurance models by minimizing the total risk exposure of an insurer under the criteria of value at risk (VaR) and conditional value at risk (CVaR). We assume that the reinsurance premium is calculated according to the expected value principle. Explicit solutions for the optimal reinsurance policies are derived over ceded loss functions with increasing degrees of generality. More precisely, we establish formally that under the VaR minimization model, (i) the stop-loss reinsurance is optimal among the class of increasing convex ceded loss functions; (ii) when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the stop-loss reinsurance with an upper limit is shown to be optimal; (iii) and finally under the set of general increasing and left-continuous retained loss functions, the truncated stop-loss reinsurance is shown to be optimal. In contrast, under CVaR risk measure, the stop-loss reinsurance is shown to be always optimal. These results suggest that the VaR-based reinsurance models are sensitive with respect to the constraints imposed on both ceded and retained loss functions while the corresponding CVaR-based reinsurance models are quite robust.

In this contribution, a new measure of comonotonicity for m-dimensional vectors is introduced, with values between zero, representing the independent situation, and one, reflecting a completely comonotonic situation. The main characteristics of this coefficient are examined, and the relations with common dependence measures are analysed. A sample-based version of the comonotonicity coefficient is also derived. Special attention is paid to the explanation of the accuracy of the convex order bound method of Goovaerts, Dhaene et al. in the case of cash flows with Gaussian discounting processes.

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation and approximations based on moment matching methods.

We analyze different types of guaranteed withdrawal benefits for life, the latest guarantee feature within variable annuities. Besides an analysis of the impact of different product features on the clients' payoff profile, we focus on pricing and hedging of the guarantees. In particular, we investigate the impact of stochastic equity volatility on pricing and hedging. We consider different dynamic hedging strategies for delta and vega risks and compare their performance. We also examine the effects if the hedging model (with deterministic volatility) differs from the data-generating model (with stochastic volatility). This is an indication for the model risk an insurer takes by assuming constant equity volatilities for risk management purposes, whereas in the real world volatilities are stochastic.

It is known that the partial stop-loss contract is an optimal reinsurance form under the VaR risk measure. Assuming that market premiums are set according to the expected value principle with varying loading factors, the optimal reinsurance parameters of this contract are obtained under three alternative single and joint party reinsurance criteria: (i) strong minimum of the total retained loss VaR measure; (ii) weak minimum of the total retained loss VaR measure and maximum of the reinsurer's expected profit; (iii) weak minimum of the total retained loss VaR measure and minimum of the total variance risk measure. New conditions for financing in the mean simultaneously the cedent's and the reinsurer's required VaR economic capital are revealed for situations of pure risk transfer (classical reinsurance) or risk and profit transfer (design of internal reinsurance or reinsurance captive owned by the captive of a corporate firm).

Many papers in the literature have adopted the expected utility paradigm to analyze insurance decisions. Insurance companies manage policies by growing, by adding independent risks. Even if adding risks generally ultimately decreases the probability of insolvency, the impact on the insurer's expected utility is less clear. Indeed, it is not true that the risk aversion toward the additional loss generated by a new policy included in an insurance portfolio always decreases with the number of contracts already underwritten. The present paper derives conditions under which zero-utility premium principles are subadditive for independent risks. It is shown that subadditivity is the exception rather than the rule: the zero-utility premium principle generates a superadditive risk premium for most common utility functions. For instance, all completely monotonic utility functions generate superadditive zero-utility premiums. The main message of the present paper is thus that the zero-utility premium for a marginal policy is generally not sufficient to guarantee the formation of insurance portfolios without additional capital.