A Trend-Change Extension of the Cairns-Blake-Dowd Model

This paper builds on the two-factor mortality model known as the Cairns-Blake-Dowd (CBD) model, which is used to project future mortality. It is shown that these two factors do not follow a random walk, as proposed in the original model, but that each should instead be modelled as a random fluctuation around a trend, the trend changing periodically. The paper uses statistical techniques to determine the points at which there are statistically significant changes in each trend. The frequency of change in each trend is then used to project the frequency of future changes, and the sizes of historical changes are used to project the sizes of future changes. The results are then presented as fan charts, and used to estimate the range of possible future outcomes for period life expectancies. These projections show that modelling mortality rates in this way leaves much greater uncertainty over future life expectancy in the long term

Still living with mortality: The longevity risk transfer market after one decade

This paper updates Living with Mortality published in 2006. It describes how the longevity risk transfer market has developed over the intervening period, and, in particular, how insurance-based solutions – buy-outs, buy-ins and longevity insurance – have triumphed over capital markets solutions that were expected to dominate at the time. Some capital markets solutions – longevity-spread bonds, longevity swaps, q-forwards, and tail-risk protection – have come to market, but the volume of business has been disappointingly low. The reason for this is that when market participants compare the index-based solutions of the capital markets with the customized solutions of insurance companies in terms of basis risk, credit risk, regulatory capital, collateral, and liquidity, the former perform on balance less favourably despite a lower potential cost.We discuss the importance of stochastic mortality models for forecasting future longevity and examine some applications of these models, e.g., determining the longevity risk premiumand estimating regulatory capital relief. The longevity risk transfer market is now beginning to recognize that there is insufficient capacity in the insurance and reinsurance industries to deal fully with demand and new solutions for attracting capital markets investors are now being examined – such as longevity-linked securities and reinsurance sidecars

Modelling and forecasting mortality in Spain

[EN] Experience shows that static life tables overestimate death probabilities. As a consequence of this overestimation the premiums for annuities, pensions and life insurance are not what they actually should be, with negative effects for insurance companies or policy-holders. The reason for this overestimation is that static life tables, through being computed for a specific period of time, cannot take into account the decreasing mortality trend over time. Dynamic life tables overcome this problem by incorporating the influence of the calendar when graduating mortality. Recent papers on the topic look for the development of new methods to deal with this dynamism. Most methods used in dynamic tables are parametric, apply traditional mortality laws and then analyse the evolution of estimated parameters with time series techniques. Our contribution consists in extending and applying Lee–Carter methods to Spanish mortality data, exploring residuals and future trends.This work was partially supported by a grant from MEyC (Ministerio de Educacio´n y Ciencia, Spain, project MTM-2004-06231). The research of Francisco Montes has also been partially supported by a grant from DGITT (Direccio´ General d’Investigacio´ i Transfere`ncia Tecnolo`gica de la Generalitat Valenciana, project GRUPOS03/189).Debón Aucejo, AM.; Montes, F.; Puig, F. (2008). Modelling and forecasting mortality in Spain. European Journal of Operational Research. 189(3):624-637. https://doi.org/10.1016/j.ejor.2006.07.050S624637189

Comonotonic approximations to quantiles of life annuity conditional expected present values: extensions to general arima models and comparison with the bootstrap

This paper aims to provide accurate approximations for the quantiles of the conditional expected present value of the payments made by the annuity provider, given the future path of the Lee-Carter time index. Conditional cohort and period life expectancies are also considered. The paper also addresses some associated simulation issues, which, hitherto, have been unresolved

Modelling and projecting mortality improvement rates using a cohort perspective

We investigate the feasibility of defining, modelling and projecting of (scaled) mortality improvement rates along cohort years-of-birth, that is, using a cohort perspective. This is in contrast to the approach in the literature which has considered mortality improvement rates that are defined by reference to changes in mortality rates over successive calendar years, that is, using a period perspective. In this paper, we offer a comparison of the 2 parallel approaches to modelling and forecasting using mortality improvement rates. Comparisons of simulated life expectancy and annuity value predictions (mainly by the cohort method) using the England & Wales population mortality experiences for males and females under a variety of controlled data trimming exercises are presented and comparisons are also made between the parallel cohort and period based approaches

Forecasting mortality in subpopulations using Lee-Carter type models: A comparison

The relative performance of multipopulation stochastic mortality models is investigated. When targeting mortality rates, we consider five extensions of the well known Lee–Carter single population extrapolative approach. As an alternative, we consider similar structures when mortality improvement rates are targeted. We use a dataset of deaths and exposures of Italian regions for the years 1974–2008 to conduct a comparison of the models, running a battery of tests to assess the relative goodness of fit and forecasting capability of different approaches. Results show that the preferable models are those striking a balance between complexity and flexibility

Multiple mortality modeling in Poisson Lee-Carter framework

The academic literature in longevity field has recently focused on models for detecting multiple population trends (D'Amato et al., 2012b; Njenga and Sherris, 2011; Russolillo et al., 2011, etc.). In particular, increasing interest has been shown about "related" population dynamics or "parent" populations characterized by similar socioeconomic conditions and eventually also by geographical proximity. These studies suggest dependence across multiple populations and common long-run relationships between countries (for instance, see Lazar et al., 2009). In order to investigate cross-country longevity common trends, we adopt a multiple population approach. The algorithm we propose retains the parametric structure of the Lee-Carter model, extending the basic framework to include some cross-dependence in the error term. As far as time dependence is concerned, we allow for all idiosyncratic components (both in the common stochastic trend and in the error term) to follow a linear process, thus considering a highly flexible specification for the serial dependence structure of our data. We also relax the assumption of normality, which is typical of early studies on mortality (Lee and Carter, 1992) and on factor models (see e.g., the textbook by Anderson, 1984). The empirical results show that the multiple Lee-Carter approach works well in the presence of dependence

Pricing reverse mortgages in Spain

[EN] In Spain, as in other European countries, the continuous ageing of the population creates a need for long-term care services and their financing. However, in Spain the development of this kind of services is still embryonic. The aim of this article is to obtain a calculation method for reverse mortgages in Spain based on the fit and projection of dynamic tables for Spanish mortality, using the Lee and Carter model. Mortality and life expectancy for the next 20 years are predicted using the fitted model, and confidence intervals are obtained from the prediction errors of parameters for the mortality index of the model. The last part of the article illustrates an application of the results to calculate the reverse mortgage model promoted by the Spanish Instituto de Crédito Oficial (Spanish State Financial Agency), for which the authors have developed a computer application.The authors are indebted to Jose Garrido, whose suggestions improved the original manuscript, and to the anonymous referee for his/her valuable comments. This work was partially supported by grants from the MEyC (Ministerio de Educacio´n y Ciencia, Spain), projects MTM2010- 14961 and MTM2008-05152.Debón Aucejo, AM.; Montes, F.; Sala, R. (2013). Pricing reverse mortgages in Spain. European Actuarial Journal. 3:23-43. https://doi.org/10.1007/s13385-013-0071-yS23433Blay-Berrueta D (2007) Sistemas de cofinaciaciación de la dependencia: seguro privado frente a hipoteca inversa. Cuadernos de la Fundación, Fundación Mapfre Estudios, Madrid.Booth H (2006) Demographic forecasting: 1980 to 2005 in review. Int J Forecast 22(3):547–582Booth H, Hyndman R, Tickle L, de Jong P (2006) Lee–Carter mortality forecasting: a multi-country comparison of variants and extensions. Demogr Res 15(9):289–310Booth H, Maindonald J, Smith L (2002) Applying Lee–Carter under conditions of variable mortality decline. Popul Stud 56(3):325–336Booth H, Tickle L (2003) The future aged: new projections of Australia’s ederly population. Popul Stud 22(4):38–44Brouhns N, Denuit M, Keilegom IV (2005) Bootstrapping Poisson log-bilinear model for mortality forecasting. Scand Actuar J 2005(3):212–224Brouhns N, Denuit M, Vermunt J (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur Math Econ 31(3):373–393Carter L, Lee R (1992) Modeling and forecasting US sex differentials in mortality. Int J Forecast 8(3):393–411Carter L, Prkawetz A (2001) Examining structural shifs in mortality using the Lee–Carter method. Mpidr wp 2001–2007, Center for Demography and Ecology Information, University of Wisconsin-Madison.Chinloy P, Megbolugbe I (1994) Reverse mortgages: contracting and crossover. J Am Real Estate Urban Econ Assoc 22(2):367–386Coale A, Guo G (1989) Revisited regional model life tables at very low levels of mortality. Popul Index 55:613–643Coale A, Kisker E (1990) Defects in data old age mortality in the United States: New procedures for calculating approximately accurate mortality schedules and lifes tables at the highest ages. Asian Pac Popul Forum 4:1–31Cossette H, Delwarde A, Denuit M, Guillot F, Étienne M (2007) Pension plan valuation and mortality projection: a case study with mortality data. N Am Actuar J 11(2):1–34.Costa-Font J (2009) Ageing in place? exploring elderly people’s housing preferences in Spain. Urban Stud 46(2):295–316Costa-Font J (2013) Housing-related well-being in older people: the impact of environmental and financial influences. Urban Stud 50(4):657–673Currie I, Kirkby J, Durban M, Eilers P (2004) Smooth Lee–Carter models and beyond. In: Workshop on Lee–Carter Methods, http://www.ma.hw.ac.uk/~iain/workshop/workshop.html . Accessed 4 Mar 2005Czado C, Delwarde A, Denuit M (2005) Bayesian Poisson log-bilinear mortality projections. Insur Math Econ 36(3):260–284D’Amato V, Haberman S, Piscopo G, Russolillo M (2012) Modelling dependent data for longevity projections. Insur Math Econ 51(3):694–701Davidoff T (2012) Can ‘high costs’ justify weak demand for the home equity conversion mortgage? Technical report, available at SSRNDavidoff T, Welke G (2007) Selection and moral hazard in the reverse mortgage market. Technical report, Haas School of Business, UC BerkeleyDebón A, Montes F, Mateu J, Porcu E, Bevilacqua M (2008) Modelling residuals dependence in dymanic life tables. Comput Stat Data Anal 52(3):3128–3147Debón A, Montes F, Puig F (2008) Modelling and forecasting mortality in Spain. Eur J Oper Res 189(3):624–637Debón A, Montes F, Sala R (2009) Tablas de mortalidad dinámicas. Una aplicación a la hipoteca inversa en España. Fundación ICO. Publicaciones de la Universitat de Valéncia, ValenciaDebón A, Montes F, Martínez-Ruiz F (2011) Statistical methods to compare mortality for a group with non-divergent populations: an application to Spanish regions. Eur Actuar J 1:291–308Delwarde A, Denuit M, Eilers P (2007) Smoothing the Lee–Carter and poisson log-bilinear models for mortality forecasting: a penalized log-likelihood approach. Stat Modell 7(1):29–48Denuit M (2007) Distribution of the random future life expectancies in log-bilinear mortality projections models. Lifetime Data Anal 13(3):381–397Denuit M, Goderniaux A (2004) Closing and projecting lifetables using log-linear models. Mitteilungen. der Schweizerischen Aktuarvereingung 1:29–49Felipe A, Guillén M, Pérez-Marín A (2002) Recent mortality trends in the Spanish population. Br Actuar J 8(4):757–786.Forfar D, McCutcheon J, Wilkie A (1988) On graduation by mathematical formula. J Inst Actuar 115(459):1–149Guillen M, Vidiella-i-Anguera A (2005) Forecasting Spanish natural life expectancy. Risk Anal 25(5):1161–1170Heligman L, Pollard J (1980) The age pattern of mortality. J Inst Actuar 107:49–80Herranz-Gonzalez R (2006) Hipoteca inversa y figuras afines. Informes Portal Mayores 49, IMSERSO, Madrid, http://www.imsersomayores.csic.es/documentos/documentos/herranz-hipoteca-01.pdfHoriuchi S, Wilmoth J (1998) Decelaration in the age pattern of mortality at older ages. Demography 35:391–412Hyndman RJ (2008) Forecast: forecasting functions for time series. R package version 1.11Koissi M, Shapiro A, Hgns G (2006) Evaluating and extending the Lee–Carter model for mortality forecasting confidence interval. Insur Math Econ 38(1):1–20Kutty N (1998) The scope for poverty alleviation among elderly home-owners in the United States through reverse mortgages. Urban Stud 35(1):113–129Lee R (2000) The Lee–Carter method for forecasting mortality, with various extensions and applications. N Am Actuar J 4(1):80–91Lee R, Carter L (1992) Modelling and forecasting US mortality. J Am Stat Assoc 87(419):659–671Lee R, Nault F (1993) Modeling and forecasting provincial mortality in Canada. Montreal world congress of the International Union for Scientific Study of PopulationLee R, Rofman R (1994) Modelación y Proyección de la mortalidad en Chile. Notas Poblacin 22(59):182–213Li N, Lee R (2005) Coherent mortality forecast for a group of populations: an extension of the Lee–Carter method. Demography 42(3):575–593Li S-H, Hardy M, Tan K (2009) Uncertainty in mortality forecasting: an extensin to the classical Lee–Carter approach. Astin Bull 31:137–164Lindbergson M (2001) Mortality among the elderly in Sweden. Scan Actuar J 1:79–94Liu X, Braun WJ (2010) Investigating mortality uncertainty using the block bootstrap. J Probab Stat 2010:385–399McNown R, Rogers A (1989) Forecasting mortality: a parametrized time series aproach. Demography 26(4):645–660McNown R, Rogers A (1992) Forecasting cause-specific mortality using time series methods. Int J Forecast 8(3):413–432Miceli T, Sirmans C (1994) Reverse mortgages and borrower maintenance risk. J Am Real Estate Urban Econ Assoc 22(2):433–450Pedroza C (2006) A bayesian forecasting model: predicting US male mortality. Biostatistics 7(4):530–550Renshaw A, Haberman S (2003) Lee–Carter mortality forecasting: a parallel generalized linear modelling aproach for England and Wales mortality projections. J R Stat Soc C 52(1):119–137Renshaw A, Haberman S (2003) Lee–Carter mortality forecasting with age specific enhancement. Insur Math Econ 33(2):255–272Renshaw A, Haberman S (2003) On the forecasting of mortality reduction factors. Insur Math Econ 32(3):379–401Renshaw A, Haberman S (2006) A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insur Math Econ 38(3):556–570Renshaw A, Haberman S (2008) On simulation-based approaches to risk measurement in mortality with specific reference to poisson Lee–Carter modelling. Insur Math Econ 42(2):797–816Shiller R, Weiss A (2000) Moral hazard in home equity conversion. Real Estate Econ 28(1):1–31Skarr D (2008) Financial planner’s guide to the FHA insured home equity conversion mortgage. J Financ Plan 21(5):68–75Sánchez-Álvarez I, Lpez-Ares S, Quiroga-García R (2007) Diseño de hipotecas inversas en el mercado español. Proyecto 205/05 3, Instituto de Mayores y Servicios SocialesTaffin C (2006) La hipoteca inversa o vitalicia. Informes externos, Asociación Hipotecaria EspañolaThatcher A, Kannisto V, Andreev K (2002) The survivor ratio method for estimating numbers at high ages. Demogr Res 6(1):1–18Thatcher A, Kannisto V, Vaupel J (1998) The force of mortality at ages 80 to 120. Odense University Press, OdenseTuljapurkar S, Li N, Boe C (2000) A universal pattern of mortality decline in the G7 countries. Nature 405(6788):789–792Wang L, Valdez E, Piggott J (2008) Securization of longevity risk in reverse mortgages. N Am Actuar J 12(4):345–370Wilmoth J (1993) Computational methods for fitting and extrapolating the Lee–Carter model of mortality change. Technical report, Departament of Demography, University of California, BerkeleyWilmoth J (1996) Health and mortality among elderly populations, chapter mortality projections for Japan: a comparison of four methods. Oxford University Press, Oxford, pp 266–28

A parameterized approach to modeling and forecasting mortality

A new method is proposed of constructing mortality forecasts. This parameterized approach utilizes Generalized Linear Models (GLMs), based on heteroscedastic Poisson (non-additive) error structures, and using an orthonormal polynomial design matrix. Principal Component (PC) analysis is then applied to the cross-sectional fitted parameters. The produced model can be viewed either as a one-factor parameterized model where the time series are the fitted parameters, or as a principal component model, namely a log-bilinear hierarchical statistical association model of Goodman [Goodman, L.A., 1991. Measures, models, and graphical displays in the analysis of cross-classified data. J. Amer. Statist. Assoc. 86(416), 1085–1111] or equivalently as a generalized Lee–Carter model with p interaction terms. Mortality forecasts are obtained by applying dynamic linear regression models to the PCs. Two applications are presented: Sweden (1751–2006) and Greece (1957–2006)