Parametric mortality indexes: From index construction to hedging strategies

In this paper, we investigate the construction of mortality indexes using the time-varying parameters in common stochastic mortality models. We first study how existing models can be adapted to satisfy the new-data-invariant property, a property that is required to ensure the resulting mortality indexes are tractable by market participants. Among the collection of adapted models, we find that the adapted Model M7 (the Cairns–Blake–Dowd model with cohort and quadratic age effects) is the most suitable model for constructing mortality indexes. One basis of this conclusion is that the adapted model M7 gives the best fitting and forecasting performance when applied to data over the age range of 40–90 for various populations. Another basis is that the three time-varying parameters in it are highly interpretable and rich in information content. Based on the three indexes created from this model, one can write a standardized mortality derivative called K-forward, which can be used to hedge longevity risk exposures. Another contribution of this paper is a method called key K-duration that permits one to calibrate a longevity hedge formed by K-forward contracts. Our numerical illustrations indicate that a K-forward hedge has a potential to outperform a q-forward hedge in terms of the number of hedging instruments required.     

On age-period-cohort parametric mortality rate projections

An enhanced version of the Lee–Carter modelling approach to mortality forecasting, which has been extended to include an age modulated cohort index in addition to the standard age modulated period index, is described and tested for prediction robustness. Life expectancy and annuity value predictions, at pensioner ages and for various periods are compared, both with and without the age modulated cohort index, for the England & Wales male mortality experience. The simulation of prediction intervals for these indices of interest is discussed in detail.     

A multivariate evolutionary credibility model for mortality improvement rates

The present paper proposes an evolutionary credibility model that describes the joint dynamics of mortality through time in several populations. Instead of modeling the mortality rate levels, the time series of population-specific mortality rate changes, or mortality improvement rates are considered and expressed in terms of correlated time factors, up to an error term. Dynamic random effects ensure the necessary smoothing across time, as well as the learning effect. They also serve to stabilize successive mortality projection outputs, avoiding dramatic changes from one year to the next. Statistical inference is based on maximum likelihood, properly recognizing the random, hidden nature of underlying time factors. Empirical illustrations demonstrate the practical interest of the approach proposed in the present paper.     

Estimation and sensitivity of Gompertz parameters with mortality deceleration rate

Studies in the evolutionary biology of aging require good estimates of theage-dependent mortality rate coefficient (one of the Gompertz parameters). In this paper we introduce an alternative algorithm for estimating this parameter. And we discuss the sensitivity of the estimates to changes in the other model parameters.     

Forecasting mortality rate by multivariate singular spectrum analysis

In this paper, we investigate the possibility of using multivariate singular spectrum analysis (SSA), a nonparametric technique in the field of time series analysis, for mortality forecasting. We consider a real data application with 9 European countries: Belgium, Denmark, Finland, France, Italy, Netherlands, Norway, Sweden, and Switzerland, over a period 1900 to 2009, and a simulation study based on the data set. The results show the superiority of multivariate SSA in comparison with the univariate SSA, in terms of forecasting accuracy.     

Valuation of equity-indexed annuity under stochastic mortality and interest rate

An equity-indexed annuity (EIA) contract offers a proportional participation in the return on a specified equity index, in addition to a guaranteed return on the single premium. In this paper, we discuss the valuation of equity-indexed annuities under stochastic mortality and interest rate which are assumed to be dependent on each other. Employing the method of change of measure, we present the pricing formulas in closed form for the most common product designs: the point-to-point and the annual reset. Finally, we conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.     

Valuation of contingent claims with mortality and interest rate risks

We consider the pricing of life insurance contracts under stochastic mortality and interest rates assumed not independent of each other. Employing the method of change of measure together with the Bayes’ rule for conditional expectations, solution expressions for the value of common contracts are obtained. A demonstration of how to apply our proposed stochastic modelling approach to value survival and death benefits is provided. Using the Human Mortality Database and UK interest rates, we illustrate that the dependence between interest rate and mortality dynamics has considerable impact in the value of even a simple survival benefit.     

Forecasting mortality rate improvements with a high-dimensional VAR

Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial–temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the differences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main factor-based mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyper-parameters. The variance–covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort and the period effects which are implicitly captured by our model. Third, our approach can be extended to model the several populations in long run perspectives, without raising issue in the estimation process. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort and period effects for these populations.     

Predicting the false alarm rate in multi-institution mortality monitoring

Statistical process control is increasingly used by single hospitals or centres to monitor their performance, but national monitoring across multiple centres, measures and groups incurs higher false alarm rates unless the method is modified. We consider setting the threshold for cumulative sum charts to produce the desired false alarm rate, taking into account the centre volume and expected outcome rate. We used simulation to estimate the false alarm and successful detection rates for a variety of chart thresholds. We thereby calculated the ‘cost’ of a higher threshold compared with one set to give a false alarm rate of 5% for three clinical groups of common interest. The false alarm rate often showed non-linear relations with the threshold, volume and expected mortality rate but an equation was found with good approximation to the simulated values. The relation between these factors and the ‘cost’ of a higher threshold was not straightforward. The ‘cost’ (difference in number of deaths) incurred by raising the chart threshold provides an intuitive measure and is applicable to other settings.     

Parametric modelling and numerical simulation of natural-convective transport of radon-222 from a phosphogypsum stack into open air

An existing finite-volume computational simulator for heat and mass transfer in media fully or partially filled with porous material has been adapted to predict radon-222 exhalation rates. For validation purposes, this paper numerically examines the extent of natural-convective effects on radon-222 steady-state transfer from a phosphogypsum stack into the surrounding atmosphere. The stack is approximated by a dry rectangular porous matrix having uniform porosity and isotropic permeability whereas the supposedly laminar buoyancy-driven air flow is modelled following Darcy–Brinkman–Boussinesq approach. Differential governing equations are cast in dimensionless form in order to encompass simultaneous effects from physical factors involved. Dimensionless groups related to decay and emanation processes are put forward apart from usual controlling parameters such as Prandtl, Schmidt, Darcy and Grashof numbers. Results are reported for 10⁶  Gr  10⁸ and 10⁻¹³  Da  10⁻⁷. Natural-convective effects on typical low-permeability phosphogypsum stacks proved to be of minor importance, as radon-222 transfer becomes diffusive dominant as expected.     

OR and the challenge to improve the NHS: modelling for insight and improvement in in-patient flows

This paper considers efforts to improve in-patient flows, a particularly urgent issue in the National Health Service (NHS). The context is described and related to reasons why OR has been making relatively little contribution. The paper argues that large complex models may often be unnecessary and even get in the way of providing clear insight and guidance for problem owners. The importance of understanding the generic working of systems to lead to improvement, and the limitations of simply describing them, is stressed. It is demonstrated that some very simple models can be of significant practical value in understanding and managing complex systems, changing mindsets and driving collection and use of operationally valuable data. Recommendations for more effective engagement with the NHS are offered.     

Parametric Statistical Uncertainty Relations and Parametric Statistical Fundamental Equations

Multivariate parametric statistical uncertainty relations are proved to specify multivariate basic parametric statistical models. The relations are expressed by inequalities. They generally show that we cannot exactly determine simultaneously both a function of observation objects and a parametric statistical model in a compound parametric statistical system composed of observations and a model. As special cases of the relations, statistical fundamental equations are presented which are obtained as the conditions of attainment of the equality sign in the relations. Making use of the result, a generalized multivariate exponential family is derived as a family of minimum uncertainty distributions. In the final section, several multivariate distributions are derived as basic multivariate parametric statistical models.     

Dualizing,projecting, and restricting GKZ systems

Let A be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N}A]$ is normal. Fix a face F of the cone of A. We show that the projection and restriction of an A-hypergeometric system to the coordinate subspace corresponding to F are essentially F-hypergeometric; moreover, at most one of them is nonzero.We also show that, if A is in addition homogeneous, the holonomic dual of an A-hypergeometric system is itself A-hypergeometric. This extends a result from [16], proving a conjecture of Nobuki Takayama in the normal homogeneous case.     

Parametric and non-parametric minima

We prove that if uBV() is a local minimum for the non-parametric functional F(u) associated to a parametric integrand g××ⁿ⁺¹[0,+), then the hypograph of u is locally minimizing for the parametric functionalG associated to g, among the sets of finite perimeter in ×.