# Product Development News

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1 Article from: Product Development News August 2005 Issue 62

4 Term Mortality and Lapses from page 5 Lapses in excess of some set of baseline rates are assumed to be anti-selective. Total deaths for the excess lapse group ( reverters ) and those who do not lapse ( persisters ) must equal expected deaths with baseline lapses (conservation of deaths). You need to make an assumption for reverter mortality. Then you can use conservation of deaths to solve for expected persister mortality. One consequence of conservation of deaths is that the anti-selection will wear off m years after the last excess lapse, where m equals the select period for the base mortality table. It seems to me that this internal consistency with the base mortality table also represents one advantage of B-K and D-M over a pure (S)WAG approach. For purposes of illustration, let s suppose that: D-M calculations assume that n% of excess lapses at attained age x+s (duration s for issue age x) are fully select and that the remaining (100-n)% percent have mortality equal to what persister mortality would be if there were no excess lapses at ages x+s+t, t = 1, 2, 3,... I believe this is what Doug Doll calls Method 2 in his July 2003 article. The formulas become somewhat involved when excess lapses occur at more than one duration. Formulas for n% = 100% were presented in the original D-M paper in the 1980 TSA. B-K calculations assume that reverter mortality for excess lapses at age x+s equals: q r [x+s]+t-1 = F(t)* q [x+s]+t-1 F(t) = 1 + G(t)*R*[(q [x] +s / q [x+s] ) 1] G(t) grades from 1.0 for t=1 to 0 for t=16, the first ultimate duration. For purposes In the original B-K article, they recommended that R be between 0.2 and 0.4. I believe this is the original formulation of B-K, except that I have omitted an accidental death refinement. Male, issue age 40 Base lapse rates = 10 percent per year, annual mode Base mortality = S&U, ALB Total lapse rates = base lapse rates, except for policy years Total lapse rates (QW) equal 85% or 90% in year 10 and 30%, 20% and 15% for years 11, 12 and 13, respectively. Resulting persister mortality as a multiple of base mortality is shown in the table on page 7 for selected policy years of a 10-year level premium term product and a few choices for n%, QW 10 and R. The table gives some indication of the sensitivity of post-level premium period mortality to the choice of parameters and to the level of excess lapses. Not surprisingly, decreasing expected reverter mortality increases expected persister mortality. To estimate reverter or persister mortality, it strikes me that it would be very useful to know: The fraction of the in-force at the end of the level premium period that would fall in each underwriting category, including various levels of substandard, if subjected to underwriting at that time. The answer would almost certainly vary by gender, issue age, underwriting class at issue and length of the level term period. The relationships between (a) the premiums payable by persisters after the level premium period and (b) corresponding new issue ART or level premium term premiums. of the sample calculations, I have assumed that this occurs linearly. R is a parameter that controls the level of reverter mortality smaller values of R translate into lower levels of reverter and higher levels of persister mortality. 6 August 2005

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