## Age rating

Back in the days before personal computers, actuaries relied solely on published tables for their calculations. These were not just the mortality tables, but monetary functions of these tables known as *commutation factors*. My old student tables from 1980 list commutation and other factors at discount rates of 4%, 6% and 8% (the latter rate seems almost comically high by current standards). If you wanted to use a different discount rate, you would have to perform the same calculation twice and interpolate.

However, actuaries couldn't interpolate if they wanted to use higher or lower mortality rates — all monetary functions were calculated using a fixed (and implied) 100% of the rates in the stated mortality table. Instead, a common way to allow for mortality differentials was to add or subtract a number of years to the chronological age. This *rated age* would make the life older (to reflect higher mortality) or younger (to reflect lower mortality) in the actuarial calculations. This habit of adding a rating to the policyholder's age extended even into this millennium: at least one large UK life office was still using ratings to age in published reserves in 2005.

Of course, nowadays modern actuaries use statistical models for mortality, not just comparisons with standard tables. However, just as Iain Currie showed with smoothing, there is a connection between modern statistical methods and the traditional approach of applying an age rating. To show this we start by supposing that the mortality hazard at age \(x\), \(\mu_x\), is the same as the mortality hazard under a published table, \(\mu^*\), with an age rating of \(r\) years, i.e.:

\[\mu_x=\mu_{x+r}^*\qquad(1)\]

where \(r\) can be positive or negative, depending on whether mortality is to be raised or lowered. To take an example, assume that we only had a mortality table for males; to get approximate mortality rates for females we might use a negative age rating, say \(r=-3\) years (this example is not as ridiculous as it might seem to modern readers, as even some post-war assurance tables were still only published for males).

We now switch to a more modern approach, where the mortality hazard follows a functional form that can be fitted to portfolio experience data. There is a wide variety of choices, but we find from experience that the Makeham-Beard model is usually one of the best for pensioner and annuitant mortality (see Richards, 2012):

\[\mu_x=\frac{e^\epsilon+e^{\alpha+\beta x}}{1+e^{\alpha+\rho+\beta x}}\qquad(2)\]

where \(\alpha\), \(\beta\), \(\epsilon\) and \(\rho\) are real-valued parameters to be estimated. In a modern context we would estimate the effect of gender by defining \(\alpha_i\) for life \(i\) as follows:

\[\alpha_i=\alpha_0+\alpha_Fz_{i,F}\qquad(3)\]

where \(\alpha_0\) is the baseline shared by all lives, \(\alpha_F\) is the effect of being female (as measured against males, who form the baseline) and \(z_{i,F}\) is an indicator factor taking the value 1 if life \(i\) is female and 0 otherwise. We fit this model to the experience data of a Dutch pension scheme and get the parameter estimates shown in Table 1:

Table 1. Parameter estimates for Age+Gender model. Source: Own calculations using mortality-experience data of Dutch pensioners aged 60 and over in the period 2006–2012.

Name | Parameter | Estimate |
---|---|---|

Age | \(\beta\) | 0.154752 |

Beard | \(\rho\) | 0.482312 |

Gender.F | \(\alpha_F\) | -0.417574 |

Intercept | \(\alpha_0\) | -15.3273 |

Makeham | \(\epsilon\) | -5.81069 |

To see the relationship with age rating, assume that females follow male mortality in equation (2) with an age rating of \(r\). The female mortality rate, expressed as an age-rated male mortality rate, would then be:

\[\mu_x^{\rm females} = \frac{e^\epsilon+e^{\alpha_0+\beta(x+r)}}{1+e^{\alpha_0+\rho+\beta(x+r)}}\qquad(4)\]

Using equation (3) for females the right-hand exponent on the numerator in equation (2) is \(\alpha_0+\alpha_F+\beta x\). From equation (4) the equivalent right-hand exponent on the numerator is \(\alpha_0+\beta (x+r)=\alpha_0+\beta r+\beta x\). By matching the \(\alpha_0\) and \(\beta x\) terms we see that \(\alpha_F=\beta r\), and thus that \(r=\alpha_F/\beta\) (the same can be done for the exponents on the denominators, and will reach the same answer). Thus, the rating-to-age approach is mathematically identical to fitting a simple risk factor. The same equivalence with the age-rating approach applies to the Gompertz, Makeham and other mortality laws.

In the case of the Dutch pensioners in Table 1, this would mean the age rating for females would be \(r=-0.4176/0.1548=-2.70\) years. A similar exercise can be carried out for any risk factor, or even combinations of risk factors: a rating to age could be a sum of (partially offsetting) ratings to age for each risk factor, just as \(\alpha_i\) can be built up as a sum of effects for each risk factor. Straight away we see the greater flexibility of the statistical model: dealing with fractional values for \(r\) requires further approximation with the published table, which is an unnecessary step with the statistical model. Furthermore, the statistical model can also vary the effect of a risk factor with age, something which would be very hard to do using a published table.

**References**

Richards, S. J. (2012) A handbook of parametric survival models for actuarial use, *Scandinavian Actuarial Journal*, 2012 (4), 233–257.