A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

By Cairns, Andrew J. G.; Blake, David et al. | Journal of Risk and Insurance, December 2006 | Go to article overview

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration


Cairns, Andrew J. G., Blake, David, Dowd, Kevin, Journal of Risk and Insurance


ABSTRACT

In this article, we consider the evolution of the post-age-60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so-called longevity risk. We introduce a two-factor stochastic model for the development of this curve through time. The first factor affects mortality-rate dynamics at all ages in the same way, whereas the second factor affects mortality-rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly.

A key component of the article is the proposal and development of a method for calculating the market risk-adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.

INTRODUCTION

Recently, it has become clear that mortality is a stochastic process: longevity has not only been improving, but it has been improving, to some extent, in an unpredictable way. These unanticipated improvements have proved to be of greatest significance at higher ages, and have caused life offices (and pension plan sponsors in the case where the plan provides the pension) to incur losses on their life annuity business. The problem is that pensioners are living much longer than was anticipated, say, twenty years ago. As a result, life offices are paying out for much longer than was anticipated, and their profit margins are being eroded in the process. The insurance industry is therefore bearing the costs of unexpectedly greater longevity. Looking forward, possible changes in lifestyle, medical advances, and new discoveries in genetics are likely to make future improvements to life expectancy highly unpredictable as well. This, in turn, will lead to smaller books of life annuity business, smaller profit margins, or both.

There are a number of possible types of systematic, mortality-related risks that annuity providers and life insurers are exposed to. For the sake of clarity, in this article we will use the following conventions.

* The term mortality risk should be taken to encompass all forms of uncertainty in future mortality rates, including increases and decreases in mortality rates.

* Longevity risk should be interpreted as uncertainty in the long-term trend in mortality rates and its impact on the long-term probability of survival of an individual. Longevity risk is normally taken to mean the risk that survival rates are higher than anticipated, although we strictly take it to mean uncertainty in either direction.

* Short-term, catastrophic mortality risk should be interpreted as the risk that, over short periods of time, mortality rates are much higher (or lower) than would normally be experienced. Examples of such "catastrophes" include the influenza pandemic in 1918 and the tsunami in December 2004. Once the catastrophe has past, we expect mortality rates to revert to their previous levels and to continue along previous trends. (1)

The idea of using the capital markets to securitize and trade specific insurance risks is relatively new, and picked up momentum in the 1990s with a number of securitizations of non-life insurance risks (see, for example, Lane, 2000). December 2003 saw the issue by Swiss Re of the first bond to link payments to mortality risk: specifically short-term, catastrophic mortality risk. …

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