## Keeping it simple

Which mortality-improvement basis is tougher — a medium-cohort projection with a 2% minimum value, or a long-cohort projection with a 1% minimum? Unless you are an actuary who works with such things, you have little chance of answering this question. This is unsatisfactory in an age of transparency. There are many non-actuaries — regulators, pension-scheme trustees and equity analysts among them — who need to be able to make this kind of judgment. The advent of the CMI's new projection model means that there will be yet more variety in projection bases.

One solution is to quote specimen life expectancies or annuity values. However, different bases have different impacts at varying ages, as shown in Table 1:

Table 1. Specimen factors for level unit annuities to males, valued at a discount rate of 3% per annum. Mortality is according to S1PA from 2011 onwards, with improvements after that year in line with the stated projection.

Mortality projection | Age 65 | Age 70 | Age 75 |

i) Medium cohort with 2% minimum | 14.606 | 12.042 | 9.527 |

ii) Long cohort with 1% minimum | 14.381 | 11.980 | 9.654 |

(ii) as percentage of (i) | 98.5% | 99.5% | 101.3% |

Table 1 shows that the impact of a projection basis is dependent on age (amongst many other factors), so a quoted life expectancy or annuity factor for age 65 (say) could be quite misleading if the average age in a portfolio is over 70. If we can't quote a specimen annuity value, how else can we eliminate the confounding effects of the portfolio structure and other basis elements?

As it happens, a simple solution is to use the *equivalent-annuity calculation*. This is documented in Richards and Jones (2004), but its use extends back far earlier than this. The idea is to take a portfolio value and solve for a common basis element to allow simple comparison. When dealing with mortality projections, perhaps the simplest approach is to solve for the equivalent constant annual mortality improvement. This is shown in Table 2:

Table 2. Equivalent annual rates of improvement implied by the projection bases in Table 1.

Mortality projection | Age 65 | Age 70 | Age 75 |

i) Medium cohort with 2% minimum | 2.00% | 2.00% | 2.00% |

ii) Long cohort with 1% minimum | 1.66% | 1.89% | 2.32% |

Now things are clearer. We can see that the "medium cohort" bit is a complete misnomer as it has no impact — the medium-cohort improvements are so low that the floor value always applies. We can also see that the "long cohort with 1% minimum" isn't always as tough as it sounds. Using this approach we could compare any two bases we liked by reducing them to the equivalent constant rate of improvement. This would make it much easier for regulators and other users of FSA returns and pension-scheme reports to compare the strengths of mortality-improvement bases.

An equivalent-annuity calculation can be performed over an entire portfolio, thus allowing for the full impact of age distribution and any concentration of liabilities. The end result is a single figure which can be used to compare the strength of a projection basis amongst all portfolios. The method is simple, timeless and does not need to be updated. As such, it can not only be used to compare bases across portfolios, but it can also track the basis strength of a given portfolio over time.

### Comments