Susanna Levantesi

Pricing Critical Illness Insurance from Prevalence Rates: Gompertz versus Weibull

ABSTRACT The pricing of critical illness insurance requires specific and detailed insurance data on healthy and ill lives. However, where the critical illness insurance market is small or national commercial insurance data needed for premium estimates are unavailable, national health statistics can be a viable starting point for insurance ratemaking purposes, even if such statistics cover the general population, are aggregate, and are reported at irregular intervals. To develop a critical illness insurance pricing model structured on a multiple state continuous and time-inhomogeneous Markov chain and based on national statistics, we do three things: First, assuming that the mortality intensity of healthy and ill lives is modeled by two parametrically different Weibull hazard functions, we provide closed formulas for transition probabilities involved in the multiple state model we propose. Second, we use a dataset that allows us to assess the accuracy of our multiple state model as a good estimator of incidence rates under the Weibull assumption applied to mortality rates. Third, the Weibull results are compared to corresponding results obtained by substituting two parametrically different Gompertz models for the Weibull models of mortality rates, as proposed previously. This enables us to assess which of the two parametric models is the superior tool for accurately calculating the multiple state model transition probabilities and assessing the comparative efficiency of Weibull and Gompertz as methods for pricing critical illness insurance.

A longevity basis risk analysis in a joint FDM framework

Purpose - The improvements of longevity are intensifying the need for capital markets to be used to manage and transfer the risk through longevity-linked securities. Nevertheless, the difference between the reference population of the hedging instrument and the population of members of a pension plan, or the beneficiaries of an annuity portfolio, determines a significant heterogeneity causing the so-called basis risk. In particular, it is shown that if insurers use financial instruments based on national indices to hedge longevity risk, this hedge can become imperfect. For this reason, it is fundamental to arrange a model allowing to quantify the basis risk for minimising it through a correct calibration of the hedging instrument. Design/methodology/approach - The paper provides a framework for measuring the basis risk impact on the. To this aim, we propose a model that measures the population basis risk involved in a longevity hedge, in the functional data model setting. hedging strategies. Findings - The innovative contribution of the paper occurs in two key points: the modelling of mortality and the hedging strategy. Regarding the first point, the paper proposes a functional demographic model framework (FDMF) for capturing the basis risk. The FDMF model generally designed for single population combines functional data analysis, nonparametric smoothing and robust statistics. It allows to capture the variability of the mortality trend, by separating out the effects of several orthogonal components. The novelty is to set the FDMF for modelling the mortality of the two populations, the hedging and the exposed one. Regarding the second point, the basic idea is to calibrate the hedging strategy determining a suitable mixture of q-forwards linked to mortality rates to maximise the degree of longevity risk reduction. This calibration is based on the key q-duration intended as a measure allowing to estimate the price sensitivity of the annuity portfolio to the changes in the underlying mortality curve. Originality/value - The novelty lies in linking the shift in the mortality curve to the standard deviation of the historical mortality rates of the exposed population. This choice has been determined by the observation that the shock in a mortality rate is age dependent. The main advantage of the presented framework is its strong versatility, being the functional demographic setting a generalisation of the Lee-Carter model commonly used in mortality forecasting, it allows to adapt to different demographic scenarios. In the next developments, we set out to compare other common factor models to assess the most effective longevity hedge. Moreover, the parsimony for considering together two trajectories of the populations under consideration and the convergence of long-term forecast are important aspects of our approach.

On longevity risk securitization and Solvency capital requirements in life annuities

In the current work we analyze two longevity-linked securities and try to price them coherently in the Solvency II framework. We consider a vanilla survivor swap and a survivor option. The mortality index underlying these derivatives is built on the survivors of a specific cohort of individuals. Although extensively discussed, it does not exist yet a satisfactory methodology for pricing these products. At the root of the problem lies the incompleteness of the market of longevity-linked securities. Innovative solutions continue to be presented. Moving from the consideration that the market price of longevity risk is intrinsic in the risk margin computed for the same risk, some authors suggest using the risk margin to price longevity risk. We follow their suggestion to price the vanilla survivor swap and the survivor option.

Managing demographic risk in enhanced pensions

This paper deals with demographic risk analysis in Enhanced Pensions, i.e., long-term care (LTC) insurance cover for the retired. Both disability and longevity risks affect such cover. Specifically, we concentrate on the risk of systematic deviations between projected and realised mortality and disability, adopting a multiple scenario approach. To this purpose we study the behaviour of the random risk reserve. Moreover, we analyse the effect of demographic risk on risk-based capital requirements, explaining how they can be reduced through either safety loading or capital allocation strategies. A profit analysis is also considered.

A biometric risks analysis in Long Term Care insurance

This paper deals with problem of analyzing uncertainty arising from both mortality and disability risks in Long Term Care (LTC) covers. To this purpose some projected demographical scenarios are developed. A biometric risks analysis is performed at portfolio level according to both loss function and risk reserve. The probabilistic structure adopted is consistent with multiple state models, based on a time-continuous Markov chain.