Lapses, Withdrawals, and Persistency-Related Cash Flows

Why Lapses Change an Asset-Share Projection

A lapse changes both sides of an asset-share calculation. The insurer may pay a surrender benefit when a policy ends, reducing the fund. At the same time, that policy is no longer part of the continuing cohort. The asset share per continuing policy must therefore be calculated from the fund remaining after exit payments divided by the number of policies still in force.

The calculation depends on cash-flow timing. A premium paid at the start of the year earns investment return during the year; a surrender payment made at year-end does not. The model below uses that convention unless a problem states otherwise. It is a deterministic cohort projection: counts and cash flows are calculated from the stated lapse assumptions, rather than from a realized random pattern of individual exits.

Definition: Let $N_0$ be the number of policies in force at the start of a year, and let $A_0$ be the opening asset share per policy. The opening aggregate fund is $N_0A_0$. If a proportion $q_l$ lapses during the year, the expected number of lapses is $N_0q_l$. When lapse is the only exit, the continuing count is $N_0(1-q_l)$. A surrender payment is cash outgo paid when a policy lapses; it is distinct from the asset share of a continuing policy.

Annual Projection With Lapses

Suppose each policy pays a premium $P$ and incurs an expense $e$ at the start of the year. The opening fund and these start-of-year cash flows are invested at an effective annual return $i$. At year-end, $N_0q_l$ policies lapse and each receives a surrender payment $S$. Assume there are no other exits or year-end payments.

Immediately before investment, the aggregate fund is $N_0(A_0+P-e)$. After investment, it is $N_0(A_0+P-e)(1+i)$. Surrender payments then reduce that fund by $N_0q_lS$. Dividing the remainder by the continuing count gives the asset share per continuing policy:

A_1= \frac{N_0(A_0+P-e)(1+i)-N_0q_lS} {N_0(1-q_l)} = \frac{(A_0+P-e)(1+i)-q_lS} {1-q_l}.

Here $A_1$ is the projected year-end asset share for a continuing policy. The simplification by $N_0$ is valid because the cohort is homogeneous and all policies have the same stated cash flows and decrement assumptions. If continuing policies also receive a year-end withdrawal $W$ each, subtract $N_0(1-q_l)W$ from the aggregate fund before dividing by the continuing count. If there are other exits, the continuing count and exit payments must also reflect those decrements.

Timing condition: The displayed formula assumes premiums and expenses occur at the start of the year, investment is credited over the full year, and surrender payments occur at year-end. If a lapse or payment occurs earlier, the cash flow may earn investment return only for part of the year, or not at all. Use the stated timing rather than applying the formula unchanged.

Why the Formula Works

The formula follows from a fund identity, not from an assumption that a lapsed policy’s opening asset share is paid out. The aggregate fund is built up with the cash flows received and paid while the cohort is invested. Surrenders are then deducted as actual exit outgo, and the remainder belongs to the continuing cohort under this allocation convention.

Form the invested fund.
At the start of the year, the cohort has $N_0A_0$. Add $N_0P$ of premiums and subtract $N_0e$ of expenses, giving $N_0(A_0+P-e)$.

Accumulate to year-end.
Because those amounts are invested for the full year, multiply by $1+i$, giving $N_0(A_0+P-e)(1+i)$.

Pay exiting policies.
There are $N_0q_l$ lapses, so total surrender outgo is $N_0q_lS$. Subtract this from the accumulated fund.

Allocate to continuing policies.
The remaining count is $N_0(1-q_l)$. Divide the post-surrender fund by this count to obtain the asset share per continuing policy.

This derivation also shows why the denominator changes. The projected asset share is not the remaining fund divided by the original cohort size when the question asks for the value per continuing policy.

Worked Projections

Worked Example: Year-End Lapses and Surrender Benefits

A hypothetical cohort has 100 policies with an opening asset share of $500 each. Each policy pays a $200 premium and incurs a $20 expense at the start of the year. The annual effective investment return is 4%. At year-end, 10% lapse and each receives a $350 surrender payment. There is no mortality or other outgo. Calculate the year-end asset share per continuing policy.

Method. Project the aggregate fund through investment, deduct the year-end surrender payments, and divide by the number continuing.

1. Opening aggregate fund. $100\times \$500=\$50{,}000$.

2. Start-of-year net cash flow. Each policy contributes $\$200-\$20=\$180$. For 100 policies this is $100\times \$180=\$18{,}000$. The fund invested is $\$50{,}000+\$18{,}000=\$68{,}000$.

3. Accumulate the fund. $\$68{,}000\times 1.04=\$70{,}720$.

4. Count lapses and continuing policies. $100\times 0.10=10$ policies lapse, leaving $100-10=90$ continuing policies.

5. Deduct surrender outgo. Total surrender payments are $10\times \$350=\$3{,}500$. The remaining fund is $\$70{,}720-\$3{,}500=\$67{,}220$.

6. Calculate the continuing asset share. $\$67{,}220/90=\$746.888\ldots$, or $746.89 per continuing policy, rounded to the nearest cent.

Check. $90\times \$746.888\ldots=\$67{,}220$, the fund remaining after surrender payments. The common mistake is to divide $\$70{,}720$ by 90 and ignore the $\$3{,}500$ paid to lapsers; that would give $\$785.78$, which overstates the continuing fund per policy.

Worked Example: Lapses and Withdrawals From Continuing Policies

A hypothetical cohort has 80 policies, each with an opening asset share of $900. Each pays a $250 premium and incurs a $30 expense at the start of the year. The annual effective return is 5%. At year-end, 10% lapse and receive $500 each. Every continuing policy receives a $40 partial withdrawal at year-end. Find the remaining fund and the asset share per continuing policy.

Method. Accumulate the cohort fund first, then deduct both surrender payments to lapsers and withdrawals to continuing policies.

1. Opening aggregate fund. $80\times \$900=\$72{,}000$.

2. Start-of-year net cash flow. The net amount per policy is $\$250-\$30=\$220$, so the cohort receives $80\times \$220=\$17{,}600$. The amount invested is $\$72{,}000+\$17{,}600=\$89{,}600$.

3. Accumulate to year-end. $\$89{,}600\times 1.05=\$94{,}080$.

4. Determine the exit and continuing counts. $80\times 0.10=8$ policies lapse, so $80-8=72$ continue.

5. Deduct year-end payments. Surrenders total $8\times \$500=\$4{,}000$. Withdrawals total $72\times \$40=\$2{,}880$. The remaining fund is $\$94{,}080-\$4{,}000-\$2{,}880=\$87{,}200$.

6. Calculate the continuing asset share. $\$87{,}200/72=\$1{,}211.111\ldots$, or $1,211.11 per continuing policy, rounded to the nearest cent.

Check. The post-withdrawal continuing asset shares total $72\times \$1{,}211.111\ldots=\$87{,}200$. Do not subtract the 72 withdrawals and then divide by 80: those withdrawals are paid to continuing policies, and the requested denominator is 72.

Worked Example: Mutually Exclusive Death and Lapse Exits

A hypothetical cohort has 100 policies with an opening asset share of $600 per policy. At the start of the year, each pays a $100 premium and incurs a $20 expense. The annual effective return is 3%. During the year, 2% die and receive a $3,000 death benefit, while 8% lapse and receive a $250 surrender payment. Death and lapse are mutually exclusive year-end exits. Calculate the asset share per policy that remains in force.

Method. Accumulate the aggregate fund, deduct each type of exit payment separately, and use the count remaining after both mutually exclusive exits.

1. Opening aggregate fund and start-of-year cash flows. The opening fund is $100\times \$600=\$60{,}000$. The net start-of-year cash flow per policy is $\$100-\$20=\$80$, or $\$8{,}000$ for the cohort. The invested fund is $\$68{,}000$.

2. Accumulate. $\$68{,}000\times 1.03=\$70{,}040$.

3. Count the exits. There are $100\times 0.02=2$ deaths and $100\times 0.08=8$ lapses. Because the exits are mutually exclusive, the continuing count is $100-2-8=90$, not a count that subtracts an overlap twice.

4. Deduct exit benefits. Death benefits total $2\times \$3{,}000=\$6{,}000$. Surrender payments total $8\times \$250=\$2{,}000$. The fund for continuing policies is $\$70{,}040-\$6{,}000-\$2{,}000=\$62{,}040$.

5. Calculate the asset share. $\$62{,}040/90=\$689.333\ldots$, or $689.33 per continuing policy, rounded to the nearest cent.

Check. $90\times \$689.333\ldots=\$62{,}040$. The decrement percentages are both measured against the original 100 policies for this one-year model, so 2% and 8% correspond to 2 and 8 distinct exits.

Model Assumptions and Common Traps

A lapse percentage must be interpreted consistently. In these examples it is the proportion of the start-of-year cohort that lapses during the year. Where death is also included, the stated decrement probabilities are mutually exclusive. If a problem instead gives a lapse rate conditional on survival, or gives sequential decrement rates, counts must be calculated according to that definition before payments and remaining policies are determined.

The aggregate approach also assumes that the policies in the cohort can be represented by common cash flows, investment return, and decrement assumptions. If policies have different surrender values or different asset shares, the relevant exit outgo must be calculated policy by policy or by appropriate subgroups before aggregating.

Common exam trap: Deducting a surrender payment is not enough; also update the number of policies used in the per-policy denominator. Conversely, changing the denominator does not itself pay the surrender benefit. Both steps are required: subtract exit cash flows from the fund, then divide by the continuing count.

A useful final check is to multiply the calculated continuing asset share by the continuing policy count. The result should equal the projected fund after all year-end surrender payments, benefits, and withdrawals. A mismatch often identifies an omitted payment or an incorrect denominator. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, follow the stated cash-flow timing and round monetary answers to the nearest cent.

A hypothetical cohort has 50 policies with an opening asset share of $400 each. Each pays a $100 premium and incurs a $10 expense at the start of the year. The annual effective return is 2%. At year-end, 10% lapse and receive $250 each. Find the asset share per continuing policy. (Answer: $527.56 per policy)
A cohort of 120 hypothetical policies has an opening asset share of $300 each. Each pays a $150 start-of-year premium and incurs a $20 start-of-year expense. The annual effective return is 4%. At year-end, 5% lapse and receive $200 each. Find the continuing asset share. (Answer: $460.21 per policy)
A cohort of 40 policies has an opening asset share of $1,000 each. Each pays a $100 premium and incurs a $20 expense at the start of the year. The annual effective return is 5%. At year-end, 15% lapse and receive $600 each; each continuing policy receives a $25 withdrawal. Find the remaining fund and continuing asset share. (Answer: $40,910 remaining fund; $1,203.24 per continuing policy)
A cohort of 200 policies has an opening asset share of $500 each. Each pays a $80 premium and incurs a $10 expense at the start of the year. The annual effective return is 3%. At year-end, 2% die and receive $2,000, while 8% lapse and receive $300. The exit types are mutually exclusive. Find the number continuing and their asset share. (Answer: 180 policies; $581.22 per continuing policy)

What You'll Learn