## Model risk

Investors in longevity risk are particularly interested in extremes — they want to know the maximum loss they are likely to bear for a given probability. Reinsurers can be even more strongly interested in extremes, especially if they have written stop-loss reinsurance. Regulators have the same interest — they want to know how well reserved the insurers and reinsurers are.

A major uncertainty in longevity risk is the path taken by future mortality improvements. A common question under Solvency II in the EU is to ask what annuity reserves are required to be 99.5% sure that they are adequate. To answer this, it is common to use a stochastic projection model as this gives a range of outcomes, each with an attaching probability. But which model should you use? Different models produce different answers, as shown in Table 1.

Table 1. Best-estimate and stressed annuity values for a male aged 65 following population mortality in England & Wales. Continuous temporary annuities to age 105, valued at 3% per annum. DDE model is that of Delwarde, Denuit and Eilers (2007), while CBD5 model is that of Cairns, Blake and Dowd (2006), as modified by Currie (2010) to allow for a non-linear effect by age. The DDE model projects a single constant drift term, while the CBD5 model projects a bivariate drift term.

Model | Best-estimate reserve | 99.5% reserve | Capital required |

DDE | 13.92 | 14.42 | 3.6% |

CBD5 | 13.96 | 15.04 | 8.0% |

The two models illustrated in Table 1 broadly agree on the best-estimate reserve, but they disagree sharply on the 99.5th percentile. According to the DDE model, a reserve of 14.42 has only a 0.5% probability of being insufficient to cover trend risk. This sounds safe. However, according to the CBD5 model, the same reserve has a 12.7% probability of being insufficient. This doesn't sound quite so safe.

Unfortunately, there is no way of knowing if any one particular model is correct, or even close. This is called *model risk*, and Iain Currie and I illustrated the impact of this on annuities in a paper presented to a meeting of the Faculty of Actuaries. In this paper we found that the confidence interval for one model was about twice as wide as the interval for a second model. Simply put, the uncertainty *around* the model was as financially important as the uncertainty produced *within* a model.

As an investor, the most thorough way of considering longevity risk is not just to use a stochastic projection model, but to use several different models. Relying on a single projection model is dangerous, as you could be much more exposed than you think.

**References**

Cairns, A. J. G., Blake, D., Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, *Journal of Risk and Insurance*, **73**, *687–718*.

Currie, I. D. (2010) On a model of Cairns, Blake and Dowd, *submitted paper*.

Delwarde, A., Denuit, M. and Eilers, P.H.C. (2007). Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: a penalized likelihood approach, *Statistical Modelling*, **7**, *29–48*.

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