Lapse-Level Stresses for Long-Term Policies

Why Lapse-Level Stresses Matter

A lapse assumption determines how many policyholders remain to pay premiums, incur expenses, and receive future benefits. Changing that assumption can therefore affect both sides of a long-term contract’s projected cash flows. A higher lapse rate is not automatically favorable or unfavorable: the result depends on the policy’s margins, the timing and amount of surrender payments, and the future profits lost when policies leave.

A useful stress test changes the lapse assumption while holding other assumptions fixed, then recalculates the projection. The order of events matters. In this tutorial, lapses occur at the start of each year, before that year’s premium, expenses, and mortality exposure. This convention makes the consequences explicit: a policy that lapses at the start of a year does not pay that year’s premium or incur that year’s claim and maintenance expense, but it may generate a surrender payment.

Definition: A lapse-level stress changes the assumed rate at which policies terminate by voluntary withdrawal, while other assumptions—such as mortality, premium rates, expenses, and investment return—are held at their reference values. A uniform lapse stress applies the changed rate to the whole in-force group without changing the characteristics of the policies that lapse.

That last condition is important. The calculations here treat lapse as independent of policy characteristics and future claims. They measure the effect of a change in the overall lapse level, not the effect of particular types of policyholders being more likely to lapse.

Set the Timing Before Projecting

Let $N_t$ be the expected number of policies in force immediately before the lapse decision at the start of year $t$. Let $l_t$ be the lapse probability at that decision, and let $q_t$ be the one-year mortality probability for policies that remain after the lapse decision. If lapse occurs first, then the expected number continuing into the year is $N_t(1-l_t)$, and the expected number of deaths during the year is $N_t(1-l_t)q_t$.

K_t=N_t(1-l_t), \qquad D_t=K_tq_t, \qquad N_{t+1}=K_t(1-q_t).

Here $K_t$ is the expected number exposed to premiums, expenses, and mortality in year $t$, and $D_t$ is the expected number of deaths during that year. The recurrence assumes that the stated annual mortality probability applies to the policies that remain after lapses. It does not apply unchanged if lapse and death are instead modeled with different timing or a joint-event structure.

Suppose each active policy pays annual premium $P_t$, incurs maintenance expense $e_t$, and has death benefit $B_t$. If each lapse at the start of the year receives surrender payment $S_t$, expected profit for the year, before discounting and any fixed expenses, is

\Pi_t =N_tl_t(-S_t) +K_t(P_t-e_t) -D_tB_t =N_t\left[-l_tS_t+(1-l_t)(P_t-e_t-q_tB_t)\right].

This expression treats benefits and expenses as amounts per relevant policy and uses expected claims $D_tB_t$. If there are fixed expenses, commissions, tax, or reserve changes, those must be added using their own timing and cash-flow definitions. A lapse stress must not silently leave such items unchanged if the underlying item varies with the number of policies.

Why the Projection Formulas Work

The formulas follow from conditional expected counts. Of the $N_t$ policies, an expected fraction $l_t$ lapses first, so $N_tl_t$ policies exit and $N_t(1-l_t)$ remain. Applying the mortality probability only to that remaining group gives $D_t=N_t(1-l_t)q_t$. The expected survivors are the remaining policies multiplied by their probability of surviving the year, giving $N_{t+1}=N_t(1-l_t)(1-q_t)$.

Expected profit is the sum of expected cash flows. Each lapse generates an outflow $S_t$, each continuing policy generates premium $P_t$ and expense $e_t$, and each expected death generates benefit $B_t$. By linearity of expectation, these amounts can be multiplied by their corresponding expected counts and added. This justification does not require the expected number of policies to be an integer.

Fix the event order.
State whether lapse is measured before or after premium collection, claim exposure, and any surrender payment.

Recalculate the in-force count.
Apply the stressed lapse rate, then apply mortality or other decrements according to the stated timing.

Recalculate every affected cash flow.
Count premiums, claims, per-policy expenses, and surrender payments using the correct exposed groups.

Compare discounted results consistently.
Use the same valuation date, interest assumption, and projection horizon for the reference and stressed cases.

Worked Example: A Higher Lapse Rate Reduces Profits

A hypothetical term-insurance block has 100 policies at the start of year 1. Lapses occur at the start of each year. Compare annual lapse rates of 10% and 20%. In each year, an active policy pays a premium of $120 and incurs a maintenance expense of $20. The annual mortality probability among active policies is 5%, and the death benefit is $1,000. There is no surrender payment, no fixed expense, and no investment income; use an effective annual interest rate of 0%. Project two years. Calculate expected total profit under each lapse assumption.

Method. Apply the lapse rate first, calculate annual expected profit from the active policies, then carry expected survivors to the next year. The expected claim cost per active policy is $0.05(1{,}000)=\$50$. Thus each active policy contributes $120-20-50=\$50$ expected profit in a year.

10% lapse assumption, year 1. The expected active count is $100(1-0.10)=90$. Expected deaths are $90(0.05)=4.5$, and expected survivors are $90(0.95)=85.5$. Profit is premium income less expenses and claims:

90(120)-90(20)-4.5(1{,}000) =10{,}800-1{,}800-4{,}500 =\$4{,}500.

10% lapse assumption, year 2. The 85.5 expected survivors enter the next lapse decision. The active count is $85.5(0.90)=76.95$, with expected deaths $76.95(0.05)=3.8475$. Profit is $76.95(50)=\$3{,}847.50$. Total two-year expected profit is $\$4{,}500+\$3{,}847.50=\$8{,}347.50$.

20% lapse assumption, year 1. The active count is $100(0.80)=80$, with expected deaths $80(0.05)=4$ and expected survivors $80(0.95)=76$. Profit is $80(50)=\$4{,}000$.

20% lapse assumption, year 2. The active count after the second lapse decision is $76(0.80)=60.8$, and expected deaths are $60.8(0.05)=3.04$. Profit is $60.8(50)=\$3{,}040$. Total expected profit is $\$4{,}000+\$3{,}040=\$7{,}040$.

Check. Both lapse rates use the same $50 expected annual margin per active policy. The higher lapse rate produces fewer active policies in both years, so its total profit is lower by $\$8{,}347.50-\$7{,}040=\$1{,}307.50$. All amounts are shown to the cent; expected policy counts need not be whole numbers.

Separate Reserve Release from Future Profit

When a policy terminates, a reserve held for that policy may be released, while a surrender payment may be due. In a simple comparison at the termination date, if the reserve immediately before lapse is $V$ per policy and the surrender payment is $S$, the immediate change in profit per additional lapse is $V-S$, provided the reserve is actually released in full and there are no other termination cash flows. This is an immediate accounting or valuation effect; it is not, by itself, the total effect of a lapse stress on expected profit.

Key distinction: A lapse can produce an immediate gain when the reserve released exceeds the surrender payment, yet still reduce total expected profit if the policy would otherwise have generated sufficiently large future margins. Conversely, an immediate lapse loss can be outweighed by avoiding future losses.

Worked Example: Reserve Release Versus Lost Future Margin

A hypothetical block has 2,000 policies at a valuation date. A stressed lapse assumption increases the lapse rate at that date from 5% to 15%. Assume the extra lapses are additional to the reference case, each affected policy has a reserve of $80 immediately before lapse and receives a $65 surrender payment, and no other immediate termination cash flows arise. The expected present value of future profit from each policy that would otherwise remain in force is $25. Calculate the immediate reserve-release effect and the total incremental expected-profit effect, ignoring interactions among policies.

The lapse-rate increase is $0.15-0.05=0.10$, so the expected number of additional lapses is $2{,}000(0.10)=200$. For each additional lapse, release of the $80 reserve less the $65 surrender payment gives an immediate gain of $80-65=\$15$. The aggregate immediate gain is $200(15)=\$3{,}000$.

Those 200 policies also no longer generate their expected future profit of $25 each. The lost future profit has present value $200(25)=\$5{,}000$. Therefore, the total incremental expected-profit effect is $\$3{,}000-\$5{,}000=-\$2{,}000$, a reduction of $2,000.

Check. The immediate gain is positive, but it is smaller than the value of the lost future margin. Counting only the reserve release would therefore give the wrong conclusion about the total expected-profit effect.

Higher Lapses Can Also Improve Results

A higher lapse rate can improve expected profit when the policies that terminate would otherwise have negative expected margins, although surrender payments and lost later profits still need to be included. The following one-year example isolates that effect. It assumes no reserve release and deliberately has no later projection years; it should not be read as a general conclusion about long-term contracts.

Worked Example: Higher Lapses Reduce Expected Losses

A hypothetical one-year block begins with 500 policies. At the start of the year, a lapse is paid a surrender amount of $8 and receives no premium for that year. A policy remaining in force pays premium of $90, incurs expense of $80, and has a 5% mortality probability with a $500 death benefit. Assume no other benefits, no reserve effect, and an effective annual interest rate of 0%. Compare lapse rates of 10% and 30%.

For each continuing policy, expected claims are $0.05(500)=\$25$. Its expected operating profit for the year is $90-80-25=-\$15$.

10% lapse rate. Expected lapses are $500(0.10)=50$, costing $50(8)=\$400$. The 450 continuing policies produce expected operating profit $450(-15)=-\$6{,}750$. Total expected profit is $-\$400-\$6{,}750=-\$7{,}150$.

30% lapse rate. Expected lapses are $500(0.30)=150$, costing $150(8)=\$1{,}200$. The 350 continuing policies produce expected operating profit $350(-15)=-\$5{,}250$. Total expected profit is $-\$1{,}200-\$5{,}250=-\$6{,}450$.

Check. The higher lapse rate improves expected profit by $-\$6{,}450-(-\$7{,}150)=\$700$, because it removes more policies with negative expected operating margins. The surrender outgo is included in both results.

Common Exam Trap: Applying the Wrong Cash-Flow Timing

In the preceding example, lapses occur before the year’s premium is collected. A common error is to count the full-block premium even though some policies have already lapsed. At the 30% lapse rate, incorrectly using $500(90)$ rather than $350(90)$ overstates premium income by $150(90)=\$13{,}500$. It also risks counting expenses and claims on policies that are not exposed during the year.

The correction is to identify the policy count for each item: 150 expected lapses receive surrender payments, while 350 expected continuing policies generate premium and are exposed to expenses and mortality. If the problem instead states that premium is paid before lapse, then the cash flows must be recalculated under that different timing. The event order is part of the model, not a detail to infer after doing the arithmetic.

This framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a full long-term projection, apply the stressed lapse assumption at each relevant duration, recalculate the in-force path, and compare results on the same valuation basis. Keep the conclusion conditional on the product’s cash flows: lapse stresses can reduce profitable exposure, remove loss-making exposure, create surrender strain, or release reserves, and several of these effects may occur together.

Check Your Understanding

Use the stated timing and assumptions in each problem. Amounts are hypothetical.

A block has 400 policies. At the start of the year, 8% lapse and each receives $40. The remaining policies pay a $150 premium, incur $30 expense, and face a 2% mortality probability with a $2,000 death benefit. There are no other cash flows and the effective annual interest rate is 0%. What is the expected profit for the year? (Answer: $28,160)
A stress increases the lapse rate from 5% to 9% on 1,000 policies. Each additional lapse releases a reserve of $120 and receives a surrender payment of $95. Assuming no other immediate termination cash flows, what is the immediate aggregate reserve-release gain? (Answer: $1,000)
A block starts with 200 policies. Lapse occurs at the start of each year at 10%, followed by annual mortality of 2% among policies remaining after lapse. What is the expected in-force count immediately after the second year’s lapse decision, before that year’s mortality? (Answer: 158.76 policies)
A one-year block has 100 policies. Lapses occur first; the lapse rate is 20%, and each lapse receives $10. A continuing policy pays $70, incurs $20 expense, and has expected claims of $60. What is expected profit, with no other cash flows? (Answer: $-\$1{,}000$)
In a uniform lapse stress, the lapse rate rises while mortality, premium, and expenses per continuing policy are held fixed. State two distinct reasons the total expected-profit effect cannot be determined from the direction of the lapse change alone. (Answer: It depends on whether continuing policies have positive or negative expected margins, and on termination effects such as surrender payments, reserve release, and lost future profits.)

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