Mortality improvement (generational mortality)

A mortality table with an improvement scale, aka a generational mortality table, has several inputs required to calculate adjusted mortality rates:

1) = the base year for projecting mortality rates, e.g., 2000

2) = mortality base rate for a person aged x in the base year y

and

3) improvement scale for the mortality rate. This factor is a function of age and, optionally, year of improvement:

Either = annual rate of mortality improvement for age x in year of improvement z, or

= cumulative factor of mortality improvement for age x in year of improvement z

If mortality improvement is based on age alone, all rates are assumed to be annual (and there are no inputs for year of improvement). Details about the methodology used for projecting mortality improvement are provided in separate sections of this article, below, for scales that vary by age and scales that vary by both age and year of improvement.

Mortality projection may be fully generational, i.e., projection extends indefinitely into the future, or only to a fixed end year, which produces a static table. Note the following, with respect to calculating the projected (adjusted) mortality rates:

In the case of fully generational projection, the variable z (i.e., the year of improvement) represents each decrement or payment year for the member or spouse in question. For static tables, z is replaced with a constant z₀ equal to the specified fixed end year, used for all decrement years and payment years for all individuals.
This improvement leads to adjusted mortality rates that are always rounded to six decimal places in a static table; adjusted mortality rates in fully generational tables, however, are not rounded and will retain their full precision in subsequent calculations.

Age by Year Improvement

If the mortality improvement scale varies by both age and year of improvement, an adjusted mortality rate in year z is calculated as the product of the mortality base rate and an annual rate f or cumulative factor F of mortality improvement (described below):

= mortality rate for a life aged x in year z

The improvement factors f or F may be input directly. If annual improvement rates f are entered, ProVal will calculate the cumulative factors F from them, as follows for the standard case z > y:

Note that the index z represents the year of the resulting adjusted mortality rate. That is, the mortality improvement from year 2000 to year 2001 is given by the year of improvement 2001 and not 2000. More information regarding this convention is available in the Society of Actuaries RPEC Response to Comments on Mortality Improvement Scale BB Exposure Draft, Section 7.8.

For the special case where z < y (e.g., for entry age normal liability methods), the adjustment is instead performed as follows:

Note that if the Interpolate rates from January to valuation month option is selected the interpolation adjustment will be performed on the cumulative improvement factors F. Also, if the cumulative improvement factors F are input directly, the factors in year y are expected to be 1. If they are not, the rates for each age will be normalized through division by the factors in year y:

For example, consider the case with these mortality base rates and improvement scale rates:

Age (x)	q_x²⁰⁰⁰	f_x,2001	f_x,2002	f_x,2003
65	0.012737	0.0261	0.0242	0.0230
66	0.014409	0.0275	0.0269	0.0255
67	0.016075	0.0274	0.0281	0.0278

With a base year of y = 2000, the adjusted mortality rates in years 2001-2003 would be calculated as follows:

Age (x)	q_x	q_x,2001	q_x,2002	q_x,2003
65	0.012737	0.012737 (1 - 0.0261) = 0.012405	0.012737 (1 - 0.0261) (1- 0.0242) = 0.012104	0.012737 (1 - 0.0261)(1 - 0.0242) (1- 0.0230) = 0.011826
66	0.014409	0.014409 (1 - 0.0275) = 0.014013	0.014409 (1 - 0.0275) (1 - 0.0269) = 0.013636	0.014409 (1- 0.0275)(1 - 0.0269) (1 - 0.0255) = 0.013288
67	0.016075	0.016075 (1- 0.0274) = 0.015635	0.016075 (1- 0.0274) (1- 0.0281) = 0.015195	0.016075 (1- 0.0274)(1 - 0.0281) (1- 0.0278) = 0.014773

Age-based Improvement

In the simplified case that the improvement scale does not depend on year of improvement , i.e., the rates f are the same for all years, the relation above further simplifies (for any z) to:

Therefore,

Thus an improvement scale without the dimension for year of improvement produces a simpler adjustment to mortality rates.

Note that this example assumes different base rates and a different scale from those in the example above:

Age (x)	q_x²⁰⁰⁰	f_x	q_x,2001	q_x,2002	q_x,2003
65	0.015629	0.014	0.015629 (1 - 0.014) = 0.015410	0.015629 (1 - 0.014)² = 0.015194	0.015629 (1 - 0.014)³ = 0.014982
66	0.017462	0.013	0.017462 (1 - 0.013) = 0.017235	0.017462 (1 - 0.013)² = 0.017011	0.017462 (1 - 0.013)³ = 0.016790
67	0.019391	0.013	0.019391 (1 - 0.013) = 0.019139	0.019391 (1 - 0.013)² = 0.018890	0.019391 (1 - 0.013)³ = 0.018645