# Term Life: Understanding Post-Level Experience April  2010

The life insurance industry is beginning to experience the effects of the so-called “shock lapse” as level-premium term products transition into their post-level periods. To understand the mortality implications of these extreme lapse scenarios, it seems appropriate to review the selective lapsation theory presented in the ground-breaking 1980 article “Pricing a Select and Ultimate Renewable Term Product,” by Jeffery Dukes and Andrew MacDonald. This will provide good context for analyzing emerging post-level lapse and mortality experience.

Selective Lapsation Theory

The Dukes-MacDonald selective lapsation model assumes that policy lapsation in excess of a baseline set of rates is anti-selective. The total of the expected deaths from the cohort in the excess lapse group (the “reverters”) and the expected deaths from the cohort continuing their insurance (the “persisters”) must equal the expected deaths arising from the original cohort using the baseline lapses. This is the conservation of death principle.

Dukes-MacDonald assumes that the mortality for the reverters follows the select mortality of a newly underwritten attained-issue-age group. Then, conservation of deaths is used to mathematically solve for the mortality of the persisters. As long as there are excess lapse rates, this process is repeated year after year as the persisting cohort continues to be divided into new reverters and new persisters.

The theory allows for only some portion (“effectiveness rate”) of the reverters to follow attained-issue-age mortality. The remainder, while still lapsing, follow the point-in-scale mortality of the original cohort.

For example, if the effectiveness rate is 50 percent, then only half of the reverters exhibit attained-issue-age mortality. The other half continue to exhibit the original point-in-scale mortality. The sum of the attained-issue-age deaths plus the point-in-scale deaths plus the persister anti-selective deaths must equal the original cohort deaths.

Testing Dukes-MacDonald

To test the Dukes-MacDonald theory, we use the proprietary Transamerica Experience Database (TED) on a closed block of 10-year term policies. Since we are only interested in measuring the shock experience for policies entering their post-level period, we limit the issue years to 1993 through 1997. Exposures run from January 1, 2004 through June 30, 2008.

As seen in Table 1, results of the lapse study indicate that the rates (by amount) in durations 10 and 11 are approximately 64 percent and 52 percent respectively, compared to a base rate of around 10 percent for earlier durations.

 Table 1 Duration Lapse Count Lapse Rate by Amt 7 3,173 12% 8 6,091 9% 9 7,091 9% 10 59,666 64% 11 18,251 52% 12 2,721 22% 13 875 17% 14 356 14% 15 132 12% 16 12 7%

Lapse study results by count and amount for 10-year level-premium term policies.

A closer inspection shows that nearly all of the excess lapses in duration 10 occur at the end of the policy year, while the excess lapses in duration 11 occur in the first few policy months after renewal.

To simplify the Dukes-MacDonald analysis, we will use a heaped lapse rate of 83 percent [1-(1-0.64)*(1-0.52)] at the end of policy year 10. With the Society of Actuaries 2001 Valuation Basic Table (VBT) as our expected basis, mortality results (by amount) for the same block of 10-year term policies indicate that mortality immediately prior to the shock is running at around 55 percent of the VBT, while mortality in the durations following the shock are at 126 percent of the VBT. Using Dukes-MacDonald terminology, this means that mortality for the persisters is approximately 230 percent of the mortality for the original cohort.

Dukes-MacDonald Predictions

Using the 2001 VBT, we can derive the Dukes-MacDonald predicted increase in mortality that should result from a 10th-duration lapse rate of 83 percent with an underlying base rate of 10 percent. To simplify the calculations, we will use the mortality for a male nonsmoker cohort with an original issue age of 40 (male nonsmokers represent nearly 75 percent of the exposures in our mortality study). At the end of policy year 10, this group is now at attained age 50. We use the VBT mortality rates for issue ages 40 and 50 in our calculations.

In determining the theoretical mortality of the persisters, the initial increase immediately following the shock lapse at the end of duration 10 begins to decline in subsequent durations. This is due to the very different pattern of mortality followed by the persisters as compared to the reverters.

However, for our analysis we will focus on the post-level mortality averaged over durations 11-16. This is consistent with the methodology used to calculate the 230 percent from our mortality study. The Dukes-MacDonald mortality prediction is somewhat sensitive to the effectiveness rate assumption. Table 2 shows the predicted results for our group of male nonsmoker 10-year term policyholders.

Table 2

 Effectiveness Rate Average Mortality Multiple 45% 190% 55% 210% 65% 230% 75% 250% 85% 270%

Mortality results are sensitive to the "effectiveness rate" or how selective
policyholders behave given an understanding of their own mortality (and ability to replace coverage).

Scanning the mortality column reveals that our study result of 230 percent corresponds to an effectiveness rate of 65 percent. This means that although there is selective behavior in the reverters, only 65 percent of the attained age 50 lapsing policies have the mortality of a freshly underwritten issue age 50 cohort.

If policyholders were 100 percent efficient in assessing their health status to determine whether to lapse, mortality for the persisters would be over 300 percent. That is, if every policyholder had perfect information about their health and behaved rationally, mortality associated with the persisting business would more than triple.

The Dukes-MacDonald selective lapsation model provides useful insights into the effects of post-level period policyholder behavior. While there are undoubtedly additional factors (e.g., the size of the impending premium increase) that determine lapse and mortality patterns during this period, the theory presents reasonable boundaries for prudent pricing assumptions.

What should be clear, however, is that shock lapse and expected mortality are closely related. As a result, it is unadvisable to price shock lapse and expected mortality increases independently – setting the assumption for one will lead to the predicted outcome of the other.