## Benchmarking VaR for longevity trend risk

I recently wrote about an objective approach to setting the value-at-risk capital for longevity trend risk. This approach is documented in Richards, Currie & Ritchie (2012), which was recently presented to a meeting of actuaries in Edinburgh. One of the topics which came up during the discussion was how the answers from the value-at-risk (VaR) method squared with how life offices actually change their projection bases in practice. In particular, commentators were interested in what might be regarded as a "real world" example of a sudden change in projection basis, and how this might compare with the results from the VaR framework.

As it happens, we do have an historical example to call upon, and furthermore it is recent enough to be directly relevant. At the end of the last millennium, Willets (1999) brought the cohort effect to the attention of the UK actuarial profession. It was such an important piece of work that CMI (2002) was released as a stop-gap replacement for CMI (1999). The change in projections was large enough for one UK life office to make a stock-exchange announcement because of the increase in its annuity reserves. This is exactly the sort of "real world" example of a change in mortality projection we are looking for, and one unconnected with the VaR framework or any stochastic model. We can therefore use this change from CMIR17 to the cohort projections as a benchmark example of a sudden change in trend expectation.

Under the outgoing CMIR17 projection the annuity factor at age 70 was 10.89(*). Using the replacement short-cohort projection the same annuity factor became 11.28, a 3.60% increase. Using the more commonly used medium-cohort projection the annuity factor became 11.35, a 4.20% increase. Life offices switching from the CMIR17 projection to one of the new cohort projections would therefore have experienced a relatively sudden increase in capital requirement in line with the sort of figures produced by the VaR framework. In practice a life office would often stage the basis transition over a few years, but it would know at outset the size of the change it was aiming for.

Of course, modern economic conditions are different from a decade ago, and we know that the trend-risk capital depends on the yield curve. Indeed, with lower discount rates the sensitivity of annuity reserves to mortality changes has increased. Nevertheless, the historical example of the switch from CMIR17 to the cohort projections provides a broad corroboration of the one-year VaR capital figures cited in Richards, Currie & Ritchie (2012).

(*) Using 100% of PMA92 for a single-life annuity paid continuously to a male aged 70, starting in 1992. Annuity payments are discounted at 3% interest per annum.

**References**

CMI (1999) Projection factors for mortality improvements. C.M.I.R. 17.

CMI (2002) An interim basis for adjusting the “92” Series mortality projections for cohort effects, London.

Willets, R. C. (1999) Mortality in the next Millennium, *Staple Inn Actuarial Society*, London.

Richards, S. J., Currie, I. D. & Ritchie, G. P. (2012) A value-at-risk framework for longevity trend risk, *British Actuarial Journal* (to appear).

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