What is longevity risk?
A liability with longevity risk is established whenever cash flows are guaranteed for the lifetime of a recipient; occupational pension benefits and life annuities are therefore the most common sources of longevity risk. In the U.K., there is an estimated $1.4 trillion in pension liabilities and $225 billion in life annuity liabilities whose values could change because of longevity risk. Other developed markets have smaller but ever growing liabilities.

Longevity exposure has a very long tail, in line with human life expectancy. It must be considered a largely systemic risk because factors which influence the odds of survival for one life are expected to have a similar effect on the odds for other lives over the tail period.

Looking ahead at mortality
For mortality pricing and adequate capital allocation, pension and annuity providers must make not only a best-estimate assessment of future mortality, but also an estimate of the risk of deviation from that assessment via stress testing or other methods. While deterministic modeling often provides best-estimate mortality scenarios, stochastic modeling is more appropriate for complex stress testing, although any model imposes a certain level of constraint.


Mathematical models based on mortality data are the starting point of mortality estimations, to these life re/insurers will add medical studies, expert opinion and professional actuarial judgment to arrive at a single deterministic mortality scenario. Capital requirement definitions however increasingly require a full, stress-tested loss distribution, not just a single best-estimate value. Stochastic models (such as the Lee Carter family, the Cairns-Blake-Dowd models or forward mortality models) are better able to provide this information; unfortunately these tend to be complex, to impose a mortality structure which may be different to the company’s best-estimate and to include parameter volatility relating only to model uncertainty rather than to the full potential deviation in mortality.

This paper introduces the stochastic model developed by PartnerRe to turn deterministic scenarios into a full loss distribution while avoiding the aforementioned issues, a model that can potentially be adapted to all company-specific best-estimates of future mortality and which can be extended for example to include pandemic risk and loss corridors or annuity limits. For a full description of the model as well as its validation and calibration, see B.Browne et al. 2009¹.

Modeling the volatility
Mortality rates will vary around a deterministic best-estimate due to fluctuation risks, such as an unpredictable single hot summer or cold winter, trend risks, such as an unexpected new medical treatment, and pandemic risks, shock risks which may then disappear after a period of time. Fluctuation risks become insignificant when studying a large portfolio, pandemic risks can be easily built into models of deviation over time; it is trend risks that are the true drivers of volatility in the result.

The observed mortality in our model is described as the expected mortality (deterministic) multiplied by a stochastic process (C_t) that is time dependent. In simple terms:


As the random noise component (the fluctuation risk) can be left aside for a large portfolio, modeling the systemic risk equates to defining the stochastic process. Our model makes a number of assumptions of value and parameter relationship which ultimately allow us to calibrate the model by defining a single parameter only, in our case the volatility parameter (sigma) of the Log-Normal distribution. All of the assumptions in the model were validated using historical data. With the assumptions validated, sigma can be determined based on past data. Of course, the extent to which future volatility will be solely a reflection of past volatility is subjective; hence user judgment again comes into the final projection.

With volatility (sigma) determined from past data and judgment about the future as to unanticipated extreme events, a number of scenarios of the cumulative deviation (C_t) over time can be projected for e.g. the next 50 years, creating a suite of sample paths (figure 1). The impact of these deviations on annuity cash flows can then be calculated (figure 2). With a sufficient number of C_t paths, one can calculate distributions of net present value of annuity claims outgo, which can be directly used to determine the percentile of loss for capital requirement calculations².





Example results
Under the U.K.’s Individual Capital Assessment (ICA) regime, pre-diversification capital requirement should correspond to the 99.5% percentile, which is also consistent with Solvency II. Using sample mortality assumptions with an assumed volatility (sigma) of 4%, our model shows that for an annuity for a male aged 65 in 2009, the 99.5% VaR measure results in an annuity outgo deviation of 2.7 times the standard deviation, an 8.4% increase in the expected value (mean). This implies that, ignoring other factors, 8.4% of the best-estimate of the annuity outgo should be set aside in order to remain solvent in 199 out of 200 outcomes. With “Tail Value at Risk”, the deviation is 3.1 times the standard deviation and the expected value is 9.6% higher.

Conclusion
This stochastic mortality model can be applied to any deterministic future mortality, is independent of underlying structure, is simple as it depends only on the volatility parameter, is validated by real data, facilitates analysis of extreme scenarios and can be extended to include other risks, such as pandemic. As with all stochastic modeling, calibration and computing time remain challenging. Overall, however, the model provides a very practical and reliable means with which to simulate mortality deviation.

¹ “Longevity: A 'Simple’ Stochastic Modelling of Mortality”, B.Browne, J.Duchassaing, F.Suter; submitted to the special edition of BAJ for papers presented at the “Joining Forces on Longevity and Mortality Conference, Edinburgh 2009”.
² It is important to note that the average of the stochastic present values is not the same as the present value of the deterministic scenario, although close, their ratio deviates significantly over time. This feature requires adjustment of the model by for example using the ratios as a transformation function between the cash flows of a deterministic scenario and the corresponding ones in the stochastic framework.