A Cautionary Note on Natural Hedging of Longevity Risk

Abstract

In this article, we examine the so-called natural hedging approach for life insurers to internally manage their longevity risk exposure by adjusting their insurance portfolio. In particular, unlike the existing literature, we also consider a nonparametric mortality forecasting model that avoids the assumption that all mortality rates are driven by the same factor(s).

Our primary finding is that higher order variations in mortality rates may considerably affect the performance of natural hedging. More precisely, although results based on a parametric single factor model—in line with the existing literature—imply that almost all longevity risk can be hedged, results are far less encouraging for the nonparametric mortality model. Our finding is supported by robustness tests based on alternative mortality models.

Notes

Similar arguments can be found in other insurance-related studies: e.g., Li and Ng (2010) use a nonparametric framework to price mortality-linked securities.

In particular, in this article, we use a maximal age of 101 though generalizations are possible.

Human Mortality Database. University of California, Berkeley (U.S.), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de.

More precisely, for the estimation of the Lee-Carter parameters, instead of the original approach we use the modified weighted-least-squares algorithm (Wilmoth 1993) and further adjust κ _t by fitting a Poisson regression model to the annual number of deaths at each age (Booth et al. 2002).

Of course, the underlying sample of 45 realizations is rather small for generating a large bootstrap sample, which limits the scope of the approach for certain applications (such as estimating VaR for high confidence levels, which is of practical interest). We come back to this point in our robustness tests (Section 5).

A Principal Component Analysis indicates that 85% of the total variation in the is explained by the leading factor for our dataset. Generally, the percentage of total variation explained is slightly larger for female data in comparison to male data (Zhu and Bauer 2013), suggesting that for female populations a single-factor model is more appropriate.

Yahoo! Finance, http://finance.yahoo.com.

Federal Reserve Economic Data (FRED), http://research.stlouisfed.org/fred2/.

Note that we implicitly assume that the insurer can place arbitrarily many term-life insurance policies in the market place at the same price, which may be unrealistic for large n ^term*. Moreover, we assume that underwriting profits and losses can be transferred between different lines of business and that there are no other technical limitations when pursuing natural hedging. However, such limitations would only cast further doubt on the natural hedging approach, so we refrain from a detailed discussion of these aspects.