Academic journal article Journal of Risk and Insurance

Securitization of Longevity Risk: Pricing Survivor Bonds with Wang Transform in the Lee-Carter Framework

Academic journal article Journal of Risk and Insurance

Securitization of Longevity Risk: Pricing Survivor Bonds with Wang Transform in the Lee-Carter Framework

Article excerpt

ABSTRACT

Longevity risk is a major issue for insurers and pension funds, especially in the selling of annuity products. In that respect, securitization of this risk could offer great opportunities for hedging. This article proposes to design survivor bonds which could be issued directly by insurers. In order to guaranty some transparency in the product, the survivor bond is based on a public mortality index. The classical Lee-Carter model for mortality forecasting is used to price a risky coupon survivor bond based on this index.

INTRODUCTION AND MOTIVATION

Longevity Risk

During the 20th century, the human mortality globally declined. Record life expectancy has been rising over the last century at a remarkably regular pace. In most industrialized countries, mortality at adult and old ages reveal decreasing annual death probabilities (see, e.g., McDonald, Cairns, Gwilt, and Miller, 1998). Since 1970, the main factor driving continued gains in life expectancy in industrialized countries is a reduction of death rates among the elderly.

These mortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. When long-term living benefits are concerned, the calculation of expected present values (for pricing or reserving) requires an appropriate mortality projection in order to avoid underestimation of future costs. Actuaries have therefore to resort to lifetables including a forecast of the future trends of mortality (the so-called projected tables). A new risk thus emerges: the risk that the mortality projections turn out to be erroneous, and that the annuitants live longer than predicted by the projected lifetables. This is the so-called longevity risk.

Mortality Projection Methods

Different approaches for building projected lifetables have been developed so far. Elementary aproaches simply extrapolate observed trends in the sequence of annual death probabilities at each age. The exponential model used by the Mortality Investigation Committee of the United Kingdom Institute of Actuaries falls in this category. Since Cramer and Wold (1935), the evolution over time of graduated mortality curves is popular for the purpose of extrapolation. One classical procedure is based on the projection of parameters; see e.g., Felipe, Guillen, and Perez-Marin (2002) for a recent application. This approach of course heavily relies on the appropriateness of the retained parametric models. To avoid this drawback, Lee and Carter (1992) proposed a simple model for describing the secular change in mortality as a function of a single time index. The main statistical tool of Lee and Carter (1992) is least-squares estimation via singular value decomposition of the matrix of the log age-specific observed forces of mortality together with Box-Jenkins modeling for time series. For a review of recent applications of the Lee-Carter methodology, we refer the interested readers to Lee (2000). For a general account of projecting mortality methods, see Pitacco (2004) and Wong-Fupuy and Haberman (2004).

Voluntary Annuity Markets

In a seminal article, Yaari (1965) considered individuals' decisions to purchase annuities. Relying on a series of strong assumption (namely that consumers are Von Neumann-Morgenstern expected-utility maximizers, that their preferences are time independent, that the only risk they face is longevity risk, that complete insurance for this type of risk is available through annuities, that annuities are actuarially fair, that there is only one asset which pays a given interest rate, and that consumers can borrow and lend at the same rate), the prediction of Yaari's model is that full annuitization is optimal in the absence of a bequest motive (see also Davidoff, Brown, and Diamond, 2003).

Empirical studies seem, however, to contradict the prediction that consumers should annuitize all their wealth. …

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