## The cost of uncertainty

In an earlier blog I wrote about how stochastic volatility in run-off increases with age. This applies when you exactly know (or think you know) the current and future mortality rates.

Of course, in practice we are not certain about current or future rates. What impact does this have? A good way of exploring this is to use a stochastic projection model. Table 1 shows the annuity factors at key ages using a best-estimate projection and a typical ICA or Solvency II stress test (50^{th} and 99.5^{th} percentiles, respectively).

Table 1. Annuity factors and relative capital increase for level annuity when moving from 50^{th} to 99.5^{th} percentile. ONS data for mortality of males in England & Wales, P-spline age-period projection. Annuity cashflows discounted at 2.5% per annum.

Age | 50^{th} percentile | 99.5^{th} percentile | Capital cost as % |

60 | 18.26 | 19.08 | 4.5% |

70 | 12.99 | 13.49 | 3.8% |

80 | 7.85 | 8.09 | 3.1% |

90 | 4.08 | 4.19 | 2.7% |

Other models will give different percentages, but Table 1 shows that the relative uncertainty due to trend risk decreases with age, largely because there is a progressively shorter period of time over which the adverse trend can make itself felt. Anything which increases the duration of an annuity will increase the relative capital cost in Table 1. It could be a younger average age in the portfolio, or a higher proportion of spouse's benefits. The same effect can also be caused by a higher aggregate rate of escalation on the benefits in payment, or a lower discount rate.

There is an interesting corollary here for investment policy, too. Most insurers attempt to match their assets to the liabilities, for example by cashflow or duration matching. Table 1 shows that this is tricky, since it is uncertain what the cashflows or duration will be. Similarly, the mismatch consequences of an adverse trend emerging are greater for business with longer duration.

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