Bayesian modelling of the time delay between diagnosis and settlement for Critical Illness Insurance using a Burr generalised-linear-type model

We discuss Bayesian modelling of the delay between dates of diagnosis and settlement of claims in Critical Illness Insurance using a Burr distribution. The data are supplied by the UK Continuous Mortality Investigation and relate to claims settled in the years 1999-2005. There are non-recorded dates of diagnosis and settlement and these are included in the analysis as missing values using their posterior predictive distribution and MCMC methodology. The possible factors affecting the delay (age, sex, smoker status, policy type, benefit amount, etc.) are investigated under a Bayesian approach. A 3-parameter Burr generalised-linear-type model is fitted, where the covariates are linked to the mean of the distribution. Variable selection using Bayesian methodology to obtain the best model with different prior distribution setups for the parameters is also applied. In particular, Gibbs variable selection methods are considered, and results are confirmed using exact marginal likelihood findings and related Laplace approximations. For comparison purposes, a lognormal model is also considered.     

Premium rates based on genetic studies: How reliable are they?

Underwriting the risk of rare disorders in long-term insurance often relies on rates of onset estimated from quite small epidemiological studies. These estimates can have considerable sampling uncertainty and any function based upon them, such as a premium rate, is also an estimate subject to uncertainty. This is particularly relevant in the case of genetic disorders, because the acceptable use of genetic information may depend on establishing its reliability as a measure of risk. The sampling distribution of a premium rate is hard to estimate without access to the original data, which is rarely possible. From two studies of adult polycystic kidney disease (APKD) we obtain, not the original data, but the cases and exposures used for Kaplan-Meier estimates of the survival probability. We use three resampling methods with these data, namely: (a) the standard bootstrap; (b) the weird bootstrap; and (c) simulation of censored random lifetimes. Rates of onset were obtained from each simulated sample using kernel-smoothed Nelson-Aalen estimates, hence critical illness insurance premium rates for a mutation carrier or a member of an affected family. From 10,000 such samples we estimate the sampling distributions of the premium rates, finding considerable uncertainty. Very careful consideration should be given before using small-sample epidemiological data to deal with insurance problems.     

Forward mortality and other vital rates — Are they the way forward?

This paper presents a comparative study of stochastic interest and stochastic mortality showing that, despite a virtual similarity, the two concepts are fundamentally different. The notion of forward mortality rate, fetched from finance and now the latest thing in actuarial science, is predicted to soon go out of fashion. Trying it on, it does not fill the measurements of a well-made theoretical concept: there is an element of arbitrariness in its very definition, it disobeys certain self-evident parity requirements, and it fails to generalize to more complex models. It is concluded that forward rate modeling, while passable in the context of interest, is not the way forward in the context of mortality and more general life history analysis.     

Finite‐dimensional Realizations of Regime‐switching HJM Models

This paper studies Heath–Jarrow–Morton‐type models with regime‐switching stochastic volatility. In this setting the forward rate volatility is allowed to depend on the current forward rate curve as well as on a continuous time Markov chain y with finitely many states. Employing the framework developed by Björk and Svensson we find necessary and sufficient conditions on the volatility guaranteeing the representation of the forward rate process by a finite‐dimensional Markovian state space model. These conditions allow us to investigate regime‐switching generalizations of some well‐known models such as those by Ho–Lee, Hull–White, and Cox–Ingersoll–Ross.     

Dynamic hybrid products in life insurance: Assessing the policyholders’ viewpoint

Dynamic hybrid life insurance products are intended to meet new consumer needs regarding stability in terms of guarantees as well as sufficient upside potential. In contrast to traditional participating or classical unit-linked life insurance products, the guarantee offered to the policyholders is achieved by a periodical rebalancing process between three funds: the policy reserves (i.e. the premium reserve stock, thus causing interaction effects with traditional participating life insurance contracts), a guarantee fund, and an equity fund. In this paper, we consider an insurer offering both, dynamic hybrid and traditional participating life insurance contracts and focus on the policyholders’ perspective. The results show that higher guarantees do not necessarily imply a higher willingness-to-pay, but that in case of dynamic hybrid contracts, a minimum guarantee level should be offered in order to ensure that the willingness-to-pay exceeds the minimum premium the insurer has to charge when selling the contract. In addition, strong interaction effects can be found between the two products, which particularly impact the willingness-to-pay of the dynamic hybrids.     

Delayed stability switches in singularly perturbed predator–prey models

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed planar systems in which there occurs a transcritical bifurcation of the quasi steady states. The proof uses the one-dimensional result proved by V.F. Butuzov, N.N. Nefedov and K.R. Schneider, and an appropriate monotonicity assumption on the vector field. The result is applied to identify all possible predator–prey models with quadratic vector fields allowing for the existence of canard solutions.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).