Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II

Abstract

Recently Cairns et al. introduced a general framework for modeling the dynamics of mortality rates of two related populations simultaneously. Their method ensures that the resulting forecasts do not diverge over the long run by modeling the difference in the stochastic factors between the two populations with a mean-reverting autoregressive process. In this article, we investigate how the modeling of the stochastic factors may be improved by using a vector error correction model. This extension is highly intuitive, allowing us to visualize the cross-correlations and the long-term equilibrium relation between the two populations. Another key benefit is that this extension does not require the user to assume which one of the two populations is dominant. This benefit is important because, as we demonstrate, it is not always easy to identify the dominant population, even if one population is much larger than the other. We illustrate our proposed extension with data from a pair of populations and apply it to the calculation of Solvency II risk capital.

Notes

See http://www.swissre.com/media/news_releases/nr_20121210_longevity_insurance_cover.html.

See Hymans and Robertson, “Buy-outs, Buy-ins and Longevity Hedging” 2012 Q1, available at http://www.hymans.co.uk/media/118860/buy-out_buy-in_report_q12012.pdf.

The Lee-Carter model is referred to as Model M1 in Cairns et al. (2009). This model structure is also used in an earlier (and fuller) version of Cairns et al. (2011) to demonstrate two-population mortality modeling. The earlier version of Cairns et al. (2011) is available at http://www.ma.hw.ac.uk/_andrewc/papers/ajgc54.pdf.

Dowd et al. (2011) used exactly the same sample period and sample age range. Cairn et al. (2011) used the same sample period but a slightly different age range (60–89).

Other parameter constraints may be used, but as Lee and Miller (2001) pointed out, the constraints we use produce more accurate mortality forecasts than other typical parameter constraints do. Over the long run, the difference between the expectations of κ⁽¹⁾ _t and κ⁽²⁾ _t will be a constant that is not necessarily zero, so using the constraint , i=1, 2, does not mean we are assuming that mortality rates in both populations have reached the long-term equilibrium at the forecast origin.

In fitting the Lee-Carter model, we used data for the period 1961–2005. Hence, we have 45 pairs of κ⁽¹⁾ _t and κ⁽²⁾ _t in total. The log-likelihood for RWAR is conditioned on the the first pair of κ⁽¹⁾ _t and κ⁽²⁾ _t , which means for RWAR, n=45−1=44. The log-likelihoods for VAR and VECM are conditioned on the first pair of Δκ⁽¹⁾ _t and Δκ⁽²⁾ _t , which means for VAR and VECM, n=45−2=43. Readers are referred to the Appendix for further information about the log-likelihoods for the models.

European Economic Area, which incluces the member states of the EU as well as some additional countries.

Here apply GBP spot rates based on year-end 2009 (QIS5).

Art. 77 of Directive 2009/138/EC.

According to Solvency II rules, discounting should be carried out based on a current risk-free yield curve. We have simplified this assumption to a flat 1.75% for ease of comparison.

The model used is a single-population Lee-Carter model, with a random walk with drift for the dynamics of the time-varying factor.