Longevity swaps have been one of the major success stories of pension scheme de-risking in recent years. However, with some few exceptions, all of the transactions to date have been bespoke longevity swaps based upon the mortality experience of a portfolio of named lives. In order for this market to start to meet its true potential, solutions will ultimately be needed that provide protection for all types of members, are cost effective for large and smaller schemes, are tradable, and enable access to the wider capital markets. Index-based solutions have the potential to meet this need; however, concerns remain with these solutions. In particular, the basis risk emerging from the potential mismatch between the underlying forces of mortality for the index reference portfolio and the pension fund/annuity book being hedged is the principal issue that has, to date, prevented many schemes progressing their consideration of index-based solutions. Two-population stochastic mortality models offer an alternative to overcome this obstacle as they allow market participants to compare and project the mortality experience for the reference and target populations and thus assess the amount of demographic basis risk involved in an index-based longevity hedge. In this paper, we systematically assess the suitability of several multi-population stochastic mortality models for assessing basis risks and provide guidelines on how to use these models in practical situations paying particular attention to the data requirements for the appropriate calibration and forecasting of such models.

Insurance companies and pension funds must value liabilities using mortality rates that are appropriate for their portfolio. These can only be estimated in a reliable way from a sufficiently large historical dataset for such portfolios, which is often not available. We overcome this problem by introducing a model to estimate portfolio-specific mortality simultaneously with population mortality. By using a Bayesian framework, we automatically generate the appropriate weighting for the limited statistical information in a given portfolio and the more extensive information that is available for the whole population. This allows us to separate parameter uncertainty from uncertainty due to the randomness in individual deaths for a given realization of mortality rates. When we apply our method to a dataset of assured lives in England and Wales, we find that different prior specifications for the portfolio-specific factors lead to significantly different posterior distributions for hazard rates. However, in short-term predictive distributions for future numbers of deaths, individual mortality risk turns out to be more important than parameter uncertainty in the portfolio-specific factors, both for large and for small portfolios.

Motivated by a recent discovery that the two-step inference for the Lee–Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for a unit root in a Lee–Carter model with an AR(p) process for the mortality index. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed in this paper are unconventional. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the U.S. mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee–Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is conducted to examine the finite sample behavior of the proposed test too.

The new Solvency II Directive and the upcoming IFRS 17 regime bring significant changes to current reporting of insurance entities, and particularly in relation to valuation of insurance liabilities. Insurers will be required to valuate their insurance liabilities on a risk-adjusted basis to allow for uncertainty inherent in cash flows that arise from the liability of insurance contracts. Whilst most European-based insurers are expected to adopt the Cost of Capital approach to calculate reserve risk margin — the risk adjustment method commonly agreed under Solvency II and IFRS 17, there is one additional requirement of IFRS 17 to also disclose confidence level of the risk margin.

Given there is no specific guidance on the calculation of confidence level, the purpose of this paper is to explore and examine practical ways of estimating the risk margin confidence level measured by Probability of Sufficiency (PoS). The paper provides some practical approximation formulae that would allow one to quickly estimate the implied PoS of Solvency II risk margin for a given non-life insurance liability, the risk profile of which is specified by the type and characteristics of the liability (e.g. type/nature of business, liability duration and convexity, etc.), which, in turn, are associated with

• • the level of variability measured by Coefficient of Variation (CoV);

• • the degree of Skewness per unit of CoV; and

• • the degree of Kurtosis per unit of CoV2.

This study considers a pure exchange economy with insurance against ambiguous loss. Ambiguity preferences are represented by the dual theory of the smooth ambiguity model from Iwaki and Osaki (2014). The economic premium principle of Bühlmann (1980, 1984) is extended to ambiguity. We also perform some comparative statics and present sufficient conditions under which an increase in ambiguity aversion increases insurance demand and insurance premiums. Contrary to the result in Tsanakas and Christofides (2006), the optimal demand for insurance is not always comonotonic, because our model permits an economy comprising both ambiguity averse and ambiguity loving agents.

This paper considers the problem of a lifelong health insurance cover where medical inflation is not sufficiently incorporated in the level premium determined at policy issue. We focus on the setting where changes in health benefits, driven by medical inflation, are accounted for by an appropriate update or indexation of the level premium, the policy value, or both premium and policy value, during the term of the contract. Such an updating mechanism is necessary to restore the actuarial equivalence between future health benefits and surrender values on the one hand, and available policy values and future premiums on the other hand. We extend existing literature (Vercruysse et al., 2013; Denuit et al., 2017) by developing updating mechanisms in a discrete-time framework, where medical inflation is only taken into account ex-post as it emerges over time and where surrender values are allowed for. We propose and design two types of surrender values: based on the ageing provision on the one hand and based directly on the premiums paid until surrender on the other hand. We illustrate our updating strategy with numerical examples, using Belgian data, and investigate the sensitivity of our findings with respect to elements from the technical basis (in particular: the lapse rates) used in the actuarial calculations. Our updating mechanism is generic and useful for a wide range of products in life and health insurance, where some elements of the technical basis are guaranteed while others are subject to revision according to policy conditions.

Using data on joint annuities, we conduct an analysis of the inter-couple lifetime dependence. We estimate a mixed proportional hazards model with treatment effects, which disentangles the broken-heart syndrome from the spurious risk dependence induced by unobserved heterogeneities. We use a flexible semi-parametric distribution for the unobserved heterogeneities to allow for either positive or negative spurious risk dependence. We find that the effect of losing one's spouse is asymmetric for the two sexes. Moreover, although the broken-heart syndrome explains a large portion of the dependency, there is evidence of positive spurious risk dependence. These findings have important implications for the pricing of joint insurance products. Failure to take into account either of these two effects leads to pricing error that can be either positive or negative, depending on the characteristics of the couple.

We start by describing how, in some cases, we can use variance-related premium principles in ratemaking, when the claim numbers and individual claim amounts are independent. We use quasi-likelihood generalized linear models, under the assumption that the variance function is a power function of the mean of the underlying random variable. We extend this approach to the cases where the claim numbers are correlated. Some alternatives to deal with dependent risks are presented, taking explicitly into account overdispersion. We present regression models covering the bivariate Poisson, the generalized bivariate negative binomial and the bivariate Poisson–Laguerre polynomial, which nest the bivariate negative binomial. We apply these models to a portfolio of the Portuguese insurance company Tranquilidade and compare the results obtained.

In this paper, we introduce the simplest exponential dispersion model containing the Pareto and exponential distributions. In this way, we obtain distributions with support (0, ∞) that in a long interval are equivalent to the Pareto distribution; however, for very high values, decrease like the exponential. This model is useful for solving relevant problems that arise in the practical use of extreme value theory. The results are applied to two real examples, the first of these on the analysis of aggregate loss distributions associated to the quantitative modelling of operational risk. The second example shows that the new model improves adjustments to the destructive power of hurricanes, which are among the major causes of insurance losses worldwide.

Gini-type correlation coefficients have become increasingly important in a variety of research areas, including economics, insurance and finance, where modelling with heavy-tailed distributions is of pivotal importance. In such situations, naturally, the classical Pearson correlation coefficient is of little use. On the other hand, it has been observed that when light-tailed situations are of interest, and hence when both the Gini-type and Pearson correlation coefficients are well defined and finite, these coefficients are related and sometimes even coincide. In general, understanding how these correlation coefficients are related has been an illusive task. In this paper, we put forward arguments that establish such a connection via certain regression-type equations. This, in turn, allows us to introduce a Gini-type weighted insurance pricing model that works in heavy-tailed situations and thus provides a natural alternative to the classical capital asset pricing model. We illustrate our theoretical considerations using several bivariate distributions, such as elliptical and those with heavy-tailed Pareto margins.

While Bayesian methods have attracted considerable interest in actuarial science, they are yet to be embraced in large-scaled insurance predictive modeling applications, due to inefficiencies of Bayesian estimation procedures. The paper presents an efficient method that parallelizes Bayesian computation using distributed computing on Apache Spark across a cluster of computers. The distributed algorithm dramatically boosts the speed of Bayesian computation and expands the scope of applicability of Bayesian methods in insurance modeling. The empirical analysis applies a Bayesian hierarchical Tweedie model to a big data of 13 million insurance claim records. The distributed algorithm achieves as much as 65 times performance gain over the non-parallel method in this application. The analysis demonstrates that Bayesian methods can be of great value to large-scaled insurance predictive modeling.