Selective Lapse and Anti-Selection Effects

Why Selective Lapse Matters

A lapse assumption affects more than the number of policies expected to remain in force. If lapse rates differ by mortality risk, policy value, or the attractiveness of a guarantee, the remaining portfolio can have a different composition from the original one. Its expected claims and future margins may therefore change even when the total expected number of persisting policies is unchanged.

For universal life business, this is important when policyholders have different account values relative to guaranteed benefits. A policyholder with a valuable guarantee may be less likely to lapse than one whose guarantee is out of the money. The resulting portfolio can retain more policies with costly guarantees. Conversely, if higher-mortality policyholders lapse more often, expected mortality claims among persistors may fall. The direction depends on how lapse varies across risk groups; “anti-selection” describes a mechanism, not a predetermined numerical result.

Definition: Selective lapse occurs when the probability of lapse varies with a characteristic that is associated with future policy cash flows, such as mortality risk, account value, or guarantee moneyness. Anti-selection is the resulting adverse change in the insurer’s remaining risk mix when policies with less costly expected outcomes lapse more readily, or policies with more costly expected outcomes persist more readily.

The calculations below use discrete annual projections. In each example, group-specific lapse probabilities are applied to the group counts to obtain projected expected persistors. Those expected counts can be used in linear expected cash-flow calculations. They do not, by themselves, give the expected realized share of a random portfolio.

Projecting the Remaining Group Mix

Suppose a portfolio has groups indexed by $g$. Let $n_g$ be the number of in-force policies in group $g$, and let $l_g$ be the probability that a policy in that group lapses over the projection period. Assume policies’ lapse outcomes are independent, and that each policy has the same lapse probability within its group. For the expected-count projection, the expected number of persistors from group $g$ is $n_g(1-l_g)$.

E[N_g^{\text{pers}}]=n_g(1-l_g), \qquad \text{projected expected-persistor share}_g =\frac{n_g(1-l_g)}{\sum_h n_h(1-l_h)}.

The second quantity is the group’s share of the projected expected persistors. It is a ratio of expected counts. It is not generally the expected share in the random realized portfolio, because a ratio of random counts does not generally equal a ratio of their expectations.

Worked Example: Expected Counts Versus a Random Portfolio Share

A hypothetical portfolio has 100 policies in group A, with a one-period lapse probability of 0.40, and 200 policies in group B, with a lapse probability of 0.10. Calculate the projected expected persisting counts and shares. Then consider a separate portfolio with one policy in each group, where the persistence probabilities are 0.90 for A and 0.10 for B. For this two-policy portfolio, calculate A’s expected realized share conditional on at least one policy persisting, and explain why it differs from the projected expected-count share.

Method. First apply the group-specific lapse probabilities to the larger portfolio. Then enumerate the possible outcomes for the two-policy portfolio, defining the realized share only when at least one policy persists.

For the first portfolio, group A has expected persistors $100(1-0.40)=60$. Group B has expected persistors $200(1-0.10)=180$. The total projected expected persisting count is $60+180=240$, so the projected expected-persistor shares are $60/240=0.25$ for A and $180/240=0.75$ for B.

For the two-policy portfolio, the ratio of expected counts gives A a projected expected-persistor share of $0.90/(0.90+0.10)=0.90$. To find the expected realized share, list outcomes. A alone persists with probability $0.90(0.90)=0.81$, both persist with probability $0.90(0.10)=0.09$, B alone persists with probability $0.10(0.10)=0.01$, and neither persists with probability $0.10(0.90)=0.09$.

Conditional on at least one persisting, the probability in the denominator is $1-0.09=0.91$. A’s realized share is 1 if only A persists, $1/2$ if both persist, and 0 if only B persists. Thus its conditional expected share is $ (0.81+0.5(0.09))/0.91=0.855/0.91=0.939560\ldots $.

Check. The conditional expected share, approximately 0.9396, is not 0.90. The difference occurs because the realized share is a random ratio, and the calculation conditions on at least one policy persisting. If the share were defined as zero when neither persisted, its unconditional expectation would instead be $0.855$.

Common mistake avoided. The ratio $0.90/(0.90+0.10)$ is a valid projected expected-count share, not the expected realized share. Do not label it as the latter without an argument establishing that the random denominator and the treatment of a zero denominator have been handled.

How Lapse Selection Changes Expected Claims

Once projected expected persisting counts are available, expected future claims can be calculated group by group. Suppose the next period’s death probability for a persisting policy in group $g$ is $q_g$, and the death benefit is $B_g$. Assume lapse selection has already occurred before the period being valued, and conditional mortality probabilities are appropriate for the policies that persist. If death benefits are paid at year-end, the expected claim amount at that time is the sum of each group’s expected persisting count multiplied by its expected claim per policy.

E[C_1]=\sum_g n_g(1-l_g)q_gB_g.

This expression follows from linearity of expectation. For each policy, the expected claim is its probability of persisting to the start of the claim period multiplied by its conditional death probability and benefit amount. Adding those individual expected claims gives the portfolio expectation. Independence between different policies is not required for this expected total, although it matters for the distribution of aggregate claims.

A crucial modeling condition is that $q_g$ must represent mortality among the policies being valued. If lapse and mortality outcomes are modeled jointly over the same period, a simple product of separate lapse and mortality probabilities may be inappropriate unless the assumptions justify it. A competing-decrement model instead specifies mutually exclusive lapse and death probabilities, or suitable transition probabilities. In the examples here, lapse selection occurs first and mortality is applied to the resulting persisting group.

Worked Example: Mortality Selection and a One-Year Reserve

At a valuation date, a hypothetical insurer projects that 90 high-mortality policies and 70 low-mortality policies will remain after selective lapse. For the following year, the death probabilities are 0.04 and 0.01, respectively. Each death benefit is $10,000. Every persisting policy pays a $300 premium at the valuation date, and an expense of $50 is paid at year-end whether or not the policyholder dies. The effective annual interest rate is 4%. Calculate the expected year-end claim amount and the aggregate one-year prospective net liability at the valuation date. Allow negative group liabilities; apply no reserve floor.

Method. Calculate each group’s expected claim and expense outgo, discount the year-end outgo at 4%, and subtract the premium received at the valuation date. The reserve is defined here as the present value of future outgo less future premium income.

For the high-mortality group, expected claims are $90(0.04)(\$10{,}000)=\$36{,}000$. For the low-mortality group, they are $70(0.01)(\$10{,}000)=\$7{,}000$. Total expected claims are therefore $\$36{,}000+\$7{,}000=\$43{,}000$.

The expense applies to all 160 expected persisting policies, so expected expenses are $160(\$50)=\$8{,}000$. Total expected year-end outgo is $\$43{,}000+\$8{,}000=\$51{,}000$. The discount factor is $v=1/1.04$, so the present value of this outgo is $\$51{,}000/1.04=\$49{,}038.461538\ldots$.

Premiums are received at the valuation date, so their present value is $160(\$300)=\$48{,}000$. The aggregate net liability is $\$49{,}038.461538\ldots-\$48{,}000=\$1{,}038.461538\ldots$, or $1,038.46 to the nearest cent.

Check. The high-mortality group contributes more expected claims per policy: $0.04(\$10{,}000)=\$400$, compared with $0.01(\$10{,}000)=\$100$. Their weighted claims total $\$43{,}000$, consistent with the group calculation. The net liability is positive because the discounted outgo exceeds the premiums.

Common mistake avoided. The premium is paid at time zero and must not be discounted. Also, the year-end expense applies to all persistors, not only to those who die.

Guarantee Moneyness, Margins, and Reserve Requirements

Guarantee moneyness can influence lapse behavior because policyholders may respond to the economic value of their contract. For example, a minimum death benefit may be more valuable when the account value is below the guaranteed amount. If policies in that situation lapse less often, the in-force portfolio may retain a larger share of policies with positive guarantee costs. This is a form of anti-selection against the insurer when the retained policies have higher expected net costs than the policies that lapse.

The same selection can affect future margins. If a group generates a positive future margin when it remains in force, a high lapse rate in that group removes some expected margin. If a group has expected guarantee outgo greater than its charges or other income, lower lapse in that group can increase the expected net cost. A sound projection keeps group-specific lapse, mortality, benefits, charges, expenses, and timing aligned; it does not apply a single average lapse rate to unlike groups without justification.

Key result: If group $g$ has $n_g$ policies, lapse probability $l_g$, and expected present value of future net outgo $R_g$ per persisting policy, then the aggregate expected net liability under an expected-count projection is $$ \sum_g n_g(1-l_g)R_g. $$ This linear aggregation is appropriate when $R_g$ is calculated conditional on persistence and the group-specific cash-flow assumptions are consistent with that conditioning. It does not imply that a ratio of expected group counts is the expected realized group share.

Worked Example: A Guarantee That Persists Longer

A hypothetical universal life portfolio has 100 policies with a minimum death benefit $2,000 above account value (group G) and 100 policies whose account value is at least the minimum death benefit (group N). Over the next year, group G has a lapse probability of 0.05 and group N has a lapse probability of 0.30. Conditional on persistence, both groups have a 0.02 death probability. Only group G is expected to incur a guarantee top-up, equal to $2,000 on death. Assume the top-up is paid at year-end, the effective annual interest rate is 4%, and there are no other guarantee cash flows. Calculate expected persisting counts, the expected guarantee claim, and its present value. Compare with a projection that uses the same 0.175 lapse probability for both groups.

Method. Project the two groups separately, then apply conditional mortality and the guarantee top-up only to group G. For the comparison, retain the same mortality and benefit assumptions but use the common lapse probability in both groups.

Under selective lapse, group G has $100(1-0.05)=95$ expected persistors, and group N has $100(1-0.30)=70$. Total projected expected persistors are $95+70=165$. Expected guarantee claims are $95(0.02)(\$2{,}000)=\$3{,}800$. Their present value is $\$3{,}800/1.04=\$3{,}653.846153\ldots$, or $3,653.85.

With a 0.175 lapse probability in both groups, each group has $100(1-0.175)=82.5$ expected persistors. Group G’s expected guarantee claims are $82.5(0.02)(\$2{,}000)=\$3{,}300$, with present value $\$3{,}300/1.04=\$3{,}173.076923\ldots$, or $3,173.08.

The selective-lapse projection therefore increases expected present value of guarantee claims by $\$3{,}653.846153\ldots-\$3{,}173.076923\ldots=\$480.769230\ldots$, or $480.77. Although both projections have 165 expected persistors, the selective projection has more persistors in the group with the guarantee top-up.

Check. Under the common lapse assumption, group G’s expected claim is $82.5/95$ times its selective-lapse claim, matching the lower expected number of guarantee-bearing policies. Group N contributes no guarantee top-up under the stated assumptions.

Common mistake avoided. Applying the total expected persisting count to an average guarantee cost per policy can conceal the changed group mix. The guarantee cost must be calculated for the policies that actually carry the guarantee exposure.

Deriving the Lapse-Adjusted Liability

The lapse adjustment to an expected reserve follows directly from conditioning on persistence. Consider one policy in group $g$. Let $I_g$ equal 1 if it persists to the valuation point and 0 otherwise. Let $Z_g$ be its future net outgo, conditional on persisting, with conditional expected present value $R_g=E[Z_g\mid I_g=1]$. Assume that the conditional cash-flow model used to calculate $R_g$ is appropriate for policies that persist. Then the policy’s present net outgo attributable to post-lapse-period cash flows is $I_gZ_g$, and $E[I_gZ_g]=P(I_g=1)E[Z_g\mid I_g=1]=(1-l_g)R_g$.

Summing over $n_g$ policies in each group and using linearity of expectation gives the aggregate expression below. This justification does not require independent lapse outcomes across policies. Independence is needed for some distributional calculations, such as the probability that no policies persist, but not for the expected aggregate cash flow.

E\!\left[\sum_g\sum_{j=1}^{n_g} I_{gj}Z_{gj}\right] =\sum_g n_g(1-l_g)R_g.

The result does not permit using an arbitrary average lapse rate when lapse propensity is associated with $R_g$. A pooled rate may reproduce the total expected number of persistors but still misstate the expected liability, because the product of persistence and group-specific net cost differs by group.

Common Exam Traps

Confusing expected counts with a random share. $n_g(1-l_g)/\sum_h n_h(1-l_h)$ is the share of projected expected persistors. It is not automatically the expected realized share.
Reversing the direction of selection. A lower lapse rate for high-cost policies increases their weight among expected persistors; a higher lapse rate decreases it. Check which group is costly before describing the effect.
Applying mortality to the wrong population. If mortality rates are conditional on persistence, apply them after projecting persistors. If lapse and death are competing events during one interval, use a coherent joint or competing-decrement model rather than multiplying probabilities without justification.
Discounting cash flows at the wrong time. A premium received at the valuation date is not discounted; a year-end claim is discounted for one year.
Pooling away guarantee exposure. Policies with different account values or guarantee top-ups need separate expected cash-flow calculations when their lapse behavior or benefits differ.

The practical consequence is that lapse assumptions can change both the volume and the quality of the in-force business. A credible projection should identify the characteristics driving lapse, link them to group-specific future claims and margins, and state whether outputs are expected-count projections or statistics of the random realized portfolio. These distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use projected expected counts unless the question explicitly asks for a random realized share. Round money to the nearest cent and probabilities to four decimal places.

A portfolio has 120 policies in group A with lapse probability 0.25 and 80 in group B with lapse probability 0.10. Find each group’s projected expected persisting count and projected expected-persistor share. (Answer: A: 90 persistors and 0.5556 share; B: 72 persistors and 0.4444 share.)
There are 50 high-mortality and 40 low-mortality policies. Their lapse probabilities are 0.10 and 0.30, respectively. Find the projected expected-persistor share of the high-mortality group. (Answer: 45/(45+28)=0.6164.)
Using the expected persisting counts of 45 high-mortality and 28 low-mortality policies, let their next-year death probabilities be 0.03 and 0.01, with a $20,000 benefit for either group. Claims are paid at year-end, and the effective annual interest rate is 5%. Find the present value of expected claims. (Answer: $31,047.62.)
There are 80 policies with a $3,000 guarantee top-up and 120 without a top-up. Their lapse probabilities are 0.05 and 0.25, respectively. Conditional death probability is 0.015 for both groups. The top-up is paid at year-end, and the effective annual interest rate is 3%. Find the present value of expected top-up claims. (Answer: $3,320.39.)
Two policies, A and B, have independent persistence probabilities 0.80 and 0.25. Define A’s realized share to be zero if neither persists; otherwise use A’s fraction of the persisting policies. Find the unconditional expected realized share and, separately, the conditional expected share given that at least one policy persists. (Answer: Unconditional expected share 0.7000; conditional expected share $0.7000/0.8500=0.8235$.)

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