Pricing Longevity Bonds Using Affine-Jump Diffusion Models

Historically, actuaries have been calculating premiums and mathematical reserves using a deterministic approach, by considering a deterministic mortality intensity, which is a function of the age only, extracted from available (static) life tables and by setting a flat ("best estimate") interest rate to discount cash flows over time. Since neither the mortality intensity nor interest rates are actually deterministic, life insurance companies and pension funds are exposed to both financial and mortality (systematic and unsystematic) risks when pricing and reserving for any kind of long-term living benefits, particularly on annuities and pensions. In this paper, we assume that an appropriate description of the demographic risks requires the use of stochastic models. In particular, we assume that the random evolution of the stochastic force of mortality of an individual can be modelled by using doubly stochastic processes. The model is then embedded into the well known affine-jump framework, widely used in the term structure literature, in order to derive closed-form solutions for the survival probability. We show that stochastic mortality models provide an adequate framework for the development of longevity risk hedging tools, namely mortality-linked contracts such as longevity bonds or mortality derivatives.Stochastic mortality intensity; Longevity risk; Affine models; Projected lifetables.

Modelling Mortality Using Multiple Stochastic Latent Factors

In this paper we develop a new model for stochastic mortality that considers the possibility of both positive and negative catastrophic mortality shocks. Specifically, we assume that the mortality intensity can be described by an affine function of a finite number of latent factors whose dynamics is represented by affine-jump diffusion processes. The model is then embedded into an affine-jump framework, widely used in the term structure literature, in order to derive closed-form solutions for the survival probability. This framework and model application to the classical Gompertz-Makeham mortality law provides a theoretical foundation for the pricing and hedging of longevity-linked derivatives.Stochastic mortality intensity; Longevity risk; Affine-jump models.

Parametric Immunization in Bond Portfolio Management

In this paper, we evaluate the relative immunization performance of the multifactor parametric interest rate risk model based on the Nelson-Siegel-Svensson specification of the yield curve with that of standard benchmark investment strategies, using European Central Bank yield curve data in the period between January 3, 2005 and December 31, 2011. In addition, we examine the role of portfolio design in the success of immunization strategies, particularly the role of the maturity bond. Considering multiperiod tests, the goal is to assess, in a highly volatile interest rate period, whether the use of the multifactor parametric immunization model contributes to improve immunization performance when compared to traditional single-factor duration strategies and whether durationmatching portfolios constrained to include a bond maturing near the end of the holding period prove to be an appropriate immunization strategy. Empirical results show that: (i) immunization models (single- and multi-factor) remove most of the interest rate risk underlying a naïve or maturity strategy; (ii) duration-matching portfolios constrained to include the maturity bond and formed using a single-factor model outperform the traditional duration-matching portfolio set up using a ladder portfolio and provide appropriate protection against interest rate risk; (iii) the multifactor parametric model outperforms all the other non-duration and duration-matching strategies, behaving almost like a perfect immunization asset; (iv) these results are consistent to changes in the rebalancing frequency of bond portfolios

Modeling Longevity Risk using Extreme Value Theory: An Empirical Investigation using Portuguese and Spanish Population Data

Extreme value theory (EVT) provides a framework to formalize the study of behaviour in the tails of a distribution. In this paper we use EVT to model the statistical behaviour of mortality rates over a given high threshold age and to estimate the significance of rare longevity risk in a given population. We adopt a piecewise approach in estimating the optimal threshold age using an iterative algorithm of maximum likelihood estimation.that statistically determines the cut-off between the central (Gompertz) part of the distribution and the upper tail modelled using the generalized Pareto distribution. The model is empirically tested using the most recent period mortality data for the total, male and female populations of Portugal and Spain. We use some classical results from EVT to estimate the evolution of the theoretical maximum life span over time and to derive confidence intervals for the central estimates. We then use time series methods to forecast the highest attained age. We observe a good fit of the model in all populations and subperiods analysed and on the whole life span considered. We estimate an increase in the theoretical maximum life span over time for all populations, more significant in the male subpopulations

Mortality and Longevity Projections for the Oldest-Old in Portugal

The mortality decline observed in developed countries over the last decades significantly increased the number of those surviving up to older ages. Mortality improvements are naturally viewed as a positive change for individuals and as a substantial social achievement for societies, but create new challenges in a number of different areas, ranging from the planning of all components of social security systems to labour markets. Understanding mortality and survival patterns at older ages is crucial. In this paper, we compare the results provided by a number of different methods designed to project mortality for the oldest-old in the Portuguese population. We identify the merits and limitations of each method and the consequences of their use in constructing complete life tables

Prospective Lifetables: Life Insurance Pricing and Hedging in a Stochastic Mortality Environment

In life insurance, actuaries have traditionally calculated premiums and reserves using a deterministic mortality intensity, which is a function of the age of the insured only. Over the course of the 20th century, the population of the industrialized world underwent a major mortality transition, with a dramatic decline in mortality rates. The mortality decline has been dominated by two major trends: a reduction in mortality due to infectious diseases affecting mainly young ages, and a decrease in mortality at old ages. These mortality improvements have to be taken into account to price long-term life insurance products and to analyse the sustainability of social security systems. In this paper, we argue that pricing and reserving for pension and life insurance products requires dynamic (or prospective) lifetables. We briefly review classic and recent projection methods and adopt a Poisson log-bilinear approach to estimate Portuguese Prospective Lifetables. The advantages of using dynamic lifetables are twofold. Firstly, it provides more realistic premiums and reserves, and secondly, it quantifies the risk of the insurance companies associated with the underlying longevity risks. Finally, we discuss possible ways of transferring the systematic mortality risk to other parties.

Parametric interest rate risk immunization

In this chapter we develop a new immunization model based on a parametric specification of the term structure of interest rates. The model extends traditional duration analysis to account for both parallel and non-parallel term structure shifts that have an economic meaning. Contrary to most interest rate risk models, we formally analyse both first-order and second-order conditions for bond portfolio immunization, emphasizing that the key to successful immunization will be to build up a portfolio such that the gradient of its future value is zero, and such that its Hessian matrix is positive semidefinite. We provide explicit formulae for new parametric interest rate risk measures and present alternative approaches to implement the immunization strategy. Additionally, we develop a more accurate approximation for the price sensitivity of a bond based upon new parametric interest rate risk measures and revise both classic and modern approaches to convexity in order to highlight the risks of convexity when changes other than parallel shifts in the term structure are considered. Furthermore, we provide useful expressions for the sensitivity of interest rate risk measures to changes in term structure shape parameters

PHYSICAL ACTIVITY, FUNCTIONAL CAPACITY AND QUALITY OF LIFE IN OLDER PEOPLE

Gerontological research has shown that there is a marked decline in health with age and an associated demand for expensive medical services (Shepard, 1993). Some previous studies underline sedentary lifestyle as main responsible factor for hypokinetic diseases and reduction in quality of life (Rejeski et al, 1996). By contrast, among the elderly evidences show a positive relationship between regular physical activity and a high physical fitness on reduction of anxiety and depression levels, improving older peopleʼs quality of life (ACSM, 1998, 2000; Blair et al, 1995). Otherwise, the acquisition and maintenance of motor skills are critical to the preservation of an independent lifestyle and quality of life in the elderly (Rikli and Edwards, 1991). The aim of the present study was to evaluate the relationship between physical activity levels, physical fitness and quality of life among institutionalized and non-institutionalized older people. • Older people with institutional community dwelling had lower physical activity levels than non-institutionalized similar individuals. • Physical activity influences older people's functional capacity especially aerobic capacity, and quality of life in several domains. • The above results suggest that caring elderly in their own homes seems to provide higher activity patterns, functional capacities and quality of life