Academic journal article Journal of Risk and Insurance

Modeling Mortality with Jumps: Applications to Mortality Securitization

Academic journal article Journal of Risk and Insurance

Modeling Mortality with Jumps: Applications to Mortality Securitization

Article excerpt

ABSTRACT

In this article, we incorporate a jump process into the original Lee-Carter model, and use it to forecast mortality rates and analyze mortality securitization. We explore alternative models with transitory versus permanent jump effects and find that modeling mortality via transitory jump effects may be more appropriate in mortality securitization. We use the Swiss Re mortality bond in 2003 as an example to show how to apply our model together with the distortion measure approach to value mortality-linked securities. Pricing the Swiss Re mortality bond is challenging because the mortality index is correlated across countries and over time. Cox, Lin, and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries, but the problem of correlation over time remains unsolved. We show in this article how to account for the correlations of the mortality index over time by simulating the mortality index and changing the measure on paths.

INTRODUCTION

Mortality risk management is fundamental to life insurance and pension industries. Mortality models make it available to quantify the risks and provide the basis of pricing and reserving. Traditionally, reinsurance, and more recently, securitization, renders a mean of transferring or hedging mortality risks. Naturally, mortality models are fundamental to these transactions.

Mortality securitizations differ from reinsurance transactions in several ways. Perhaps the most important is that investors in a securitization typically do not have the mortality expertise that a reinsurer has. Also, in order to avoid moral hazard problems, the basis of a securitization may be a public index rather than the actual lives insured by the ceding party. Therefore, it is important that a mortality model clearly conveys the nature of risk transfer to investors, reflects the characteristics of available data, and provides for scenario analysis.

A wide variety of stochastic models have been proposed for modeling the dynamics of mortality over time. Cairns, Blake, and Dowd (2006a) provide a detailed overview and categorization. Most of the literature in this field is in the framework of short-rate models, among which continuous time models focus on the spot force of mortality and discrete time models concentrate on the spot mortality rates. Continuous time models (e.g., Milevsky and Promislow, 2001; Dahl, 2004; Biffis, 2005; Dahl and Moller, 2006; Miltersen and Persson, 2005; Schrager, 2006) help us understand the evolution of mortality rates over time, but they are relatively intractable. We prefer discrete time models because mortality rates are measured at most once a year and discrete time models are relatively easy to implement in practice.

The Lee-Carter model is among the earliest discrete time models. Lee and Carter (1992) model the central mortality rates to be log-linearly correlated with a time-dependent mortality factor and adjust for age-specific effects using two sets of age-dependent coefficients. In this way, the model captures both the mortality trend overall and the age-specific changes on different age groups. Thus, it describes the development of the mortality curve over time quite well. The age adjustment is necessary because mortality improvement varies across age groups. Moreover, the adverse mortality shocks, such as the 1918 influenza pandemic, attack different groups with varied intensities. We will discuss these two points in detail in the data section. The Lee-Carter approach has been extended by Brouhns, Denuit, and Vermunt (2002), Renshaw and Harberman (2003), Denuit, Devolder, and Goderniaux (2007), and further revisited by Li and Chan (2007). Cairns, Blake, and Dowd (2006b) propose a two-factor model that is similar in the spirit of that of the Lee-Carter approach. The first factor indicates the mortality trend over time and equally affects mortality at all ages, whereas the second factor's effect on mortality is proportional to age. …

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