We analyze optimal consumption in the life cycle model by introducing life and pension insurance contracts. The model contains a credit market with biometric risk, and market risk via risky securities. This idealized framework enables us to clarify important aspects of life insurance and pension contracts. We find optimal pension plans and life insurance contracts where the benefits are state dependent. We compare these solutions both to the ones of standard actuarial theory, and to policies offered in practice. Implications of this include what role the insurance industry may play to improve welfare. The relationship between substitution of consumption and risk aversion is highlighted in the presence of a consumption puzzle. One problem related portfolio choice is discussed the horizon problem. Finally, we present some comments on longevity risk and cohort risk.

Various types of structures that enable a group of individuals to pool their mortality risk have been proposed in the literature. Collectively, the structures are called pooled annuity funds. Since the pooled annuity funds propose different methods of pooling mortality risk, we investigate the connections between them and find that they are genuinely different for a finite heterogeneous membership profile. We discuss the importance of actuarial fairness, defined as the expected benefits equalling the contributions for each member, in the context of pooling mortality risk and comment on whether actuarial unfairness can be seen as solidarity between members. We show that, with a finite number of members in the fund, the group self-annuitization scheme is not actuarially fair: some members subsidize the other members. The implication is that the members who are subsidizing the others may obtain a higher expected benefit by joining a fund with a more favorable membership profile. However, we find that the subsidies are financially significant only for very small or highly heterogeneous membership profiles.

The literature on Bayesian chain ladder models is surveyed. Both Mack and cross-classified forms of the chain ladder are considered. Both cases are examined in the context of error terms distributed according to a member of the exponential dispersion family. Tweedie and over-dispersed Poisson errors follow as special cases. Bayesian cross-classified chain ladder models may randomise row, column or diagonal parameters. Column and diagonal randomisation has been largely absent from the literature until recently. The present paper allows randomisation of row and column parameters. The Bayes estimator, the linear Bayes (credibility) estimator, and the MAP estimator are shown to be identical in the Mack case, and in the cross-classified case provided that the error terms are Tweedie distributed. In the Mack case the variance structure is generalised considerably from the existing literature. In the cross-classified case the model structure differs somewhat from the existing literature, and a comparison is made between the two. MAP estimators for the cross-classified case are often given by implicit equations that require numerical solution. Recursive formulas are given for these in the general case of error terms from the exponential dispersion family. The connection between the cross-classified case and Bornhuetter-Ferguson prediction is explored.

Providers of motor extended warranties with limited usage often face difficulty evaluating the impact of usage limits on warranty price because of incomplete usage data. To address this problem, this paper employs a non-parametric interval-censored survival model of time to accumulate a specific usage. This is used to develop an estimator of the probability that a provider is on risk at a specific time in service. The resulting pricing model is applied to a truck extended warranty case study. The case study demonstrates that interval-censored survival models are ideal for use in pricing motor extended warranties with limited usage cover. The results also suggest that employing a usage rate distribution to forecast the number of vehicles on risk can be misleading, especially on an extended warranty with a relatively high usage limit.

The dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

We discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

In this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

This paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.