Asset-Share Sensitivity to Lapse Experience

Why Lapse Experience Changes Asset Shares

When a policyholder lapses, the insurer may pay a surrender benefit and the policy leaves the in-force cohort. Both changes matter to asset shares. The surrender payment reduces the cohort’s accumulated funds, while the smaller number of remaining policies must support later expenses that are shared across the cohort. The direction of the change in the per-persisting-policy asset share is not determined by the lapse rate alone: it also depends on surrender amounts, cash flows and the expenses allocated to the remaining policies.

The calculations below isolate lapse experience. They use an annual model with no deaths or other exits, so that lapses are the only reason the cohort shrinks. This is a useful sensitivity analysis, not a claim that actual lapse experience affects no other cash flows. In practice, lapse behavior may be associated with selection, changes in future claims, or other experience differences; those effects need separate assumptions.

Definition: Let $N_t$ be the number of policies in force at the beginning of year $t$, and let $\lambda_t$ be the proportion of those policies that lapse at the end of that year. The number remaining is $N_{t+1}=N_t(1-\lambda_t)$. Let $A_t$ be the aggregate asset share for the cohort at the beginning of the year. The per-policy asset share is $a_t=A_t/N_t$. A surrender amount $S_t$ is paid at year-end for each policy that lapses.

Assume premiums and expenses are paid at the beginning of the year, net cash flows earn the effective annual investment return $i_t$, and surrender payments are made at year-end after investment has been earned. Let $P_t$ be the premium per in-force policy and $E_t$ its allocated expense for that year. The expense may include a variable per-policy amount and a share of fixed cohort expenses.

A_{t+1} = \left[A_t+N_t(P_t-E_t)\right](1+i_t)-N_t\lambda_tS_t, \qquad N_{t+1}=N_t(1-\lambda_t).

The first equation is an aggregate balance: accumulate the opening assets and net beginning-of-year cash flows, then deduct surrender outgo. It does not by itself give the asset share of a policy that remains in force. That requires dividing by the number of policies remaining.

a_{t+1} = \frac{\left[a_t+P_t-E_t\right](1+i_t)-\lambda_tS_t} {1-\lambda_t}, \qquad 0\leq\lambda_t<1.

Why the Per-Policy Formula Has a Denominator

The denominator $1-\lambda_t$ is essential. Before surrender payments, the accumulated amount in the numerator is expressed per policy in force at the beginning of the year. After lapses, that remaining amount is supported by fewer policies. Dividing by the proportion that persists converts the result to an amount per continuing policy.

To justify the formula, write $A_t=N_ta_t$. The aggregate balance becomes

A_{t+1} = N_t\left(\left[a_t+P_t-E_t\right](1+i_t)-\lambda_tS_t\right).

There are $N_t(1-\lambda_t)$ policies remaining. Dividing the aggregate balance by this number gives

a_{t+1} = \frac{A_{t+1}}{N_t(1-\lambda_t)} = \frac{\left[a_t+P_t-E_t\right](1+i_t)-\lambda_tS_t} {1-\lambda_t}.

This derivation also identifies the model’s boundary: if $\lambda_t=1$, no policy persists, so a per-persisting-policy asset share is undefined. The aggregate balance after surrender payments can still be calculated, but it cannot be divided among a zero-sized continuing cohort.

Worked Example: One-Year Lapse Sensitivity

A hypothetical cohort begins the year with 100 policies and an asset share of $500 per policy. Each policy pays a beginning-of-year premium of $120 and incurs a beginning-of-year expense of $20. The effective annual investment return is 4%. A policy that lapses at year-end receives $550. Compare lapse rates of 10% and 30%, assuming no other exits.

Method. First calculate the amount accumulated before surrender outgo, per opening policy:

$$ G=(500+120-20)(1.04)=600(1.04)=624. $$

At a 10% lapse rate, the surrender outgo per opening policy is $0.10(550)=55$. The remaining proportion is $0.90$, so

a_{t+1} = \frac{624-55}{0.90} = \frac{569}{0.90} = 632.2222\ldots

Thus the asset share is $632.22 per persisting policy, rounded to the nearest cent. The cohort has $100(0.90)=90$ policies and aggregate assets of $100(569)=$56{,}900$. Dividing $56,900 by 90 gives $632.22 per policy, confirming the result.

At a 30% lapse rate, surrender outgo per opening policy is $0.30(550)=165$, and the remaining proportion is $0.70$:

a_{t+1} = \frac{624-165}{0.70} = \frac{459}{0.70} = 655.7142\ldots

The result is $655.71 per persisting policy, rounded to the nearest cent. The cohort has 70 policies and aggregate assets of $100(459)=$45{,}900$; $45,900 divided by 70 is $655.71. The higher lapse rate leaves fewer aggregate assets, but it also leaves fewer policies among which those assets are allocated. In this example, the surrender amount is below the $624 accumulated per opening policy, so the reduction in cohort size more than offsets the surrender outgo for the per-policy measure.

Common mistake avoided: $459 is the post-surrender amount per opening policy at the 30% lapse rate, not the asset share per persisting policy. Reporting $459 as the answer skips division by 0.70 and confuses the two measures.

Shared Expenses and a Shrinking Cohort

If a cohort incurs a fixed expense, fewer in-force policies mean a larger fixed-expense allocation per policy. Suppose year $t$ has a variable expense $e_t$ per policy and a fixed shared expense $K_t$ for the cohort. Under an equal allocation across policies in force at the start of the year,

E_t=e_t+\frac{K_t}{N_t}.

A lapse at the end of a year affects the following year’s allocation, not the allocation already made at the beginning of the current year. Holding future fixed expenses constant, a smaller $N_{t+1}$ increases $K_{t+1}/N_{t+1}$. The resulting reduction in future per-policy asset shares can offset, or exceed, any per-policy increase caused by dividing current assets among fewer survivors.

Worked Example: Two Years With Shared Expenses

A hypothetical cohort begins with 100 policies and zero asset shares. In each year, each in-force policy pays a beginning-of-year premium of $100 and incurs a variable expense of $10. The cohort also incurs a fixed expense of $1,000 at the beginning of each year, allocated equally across policies then in force. The effective annual investment return is 5%. Surrender payments are $50 at the end of year 1 and $60 at the end of year 2. Compare constant annual lapse rates of 10% and 30%. There are no other exits.

Method. For each year, calculate the fixed expense per opening policy, accumulate the opening asset share and net cash flow, deduct surrender outgo per opening policy, then divide by the proportion persisting.

In year 1, the fixed expense is $1{,}000/100=$10$ per policy, so total expenses are $20 per policy. The pre-surrender amount per opening policy is

$$ G_1=(0+100-20)(1.05)=84. $$

At a 10% lapse rate, the year-1 asset share is $(84-0.10(50))/0.90=79/0.90=$87.7778\ldots$, or $87.78. There are 90 policies remaining. At a 30% lapse rate, it is $(84-0.30(50))/0.70=69/0.70=$98.5714\ldots$, or $98.57, with 70 policies remaining.

For the 10% lapse scenario in year 2, the fixed expense per opening policy is $1{,}000/90=$11.1111\ldots$. The pre-surrender amount is

G_2 = \left(87.7778\ldots+100-10-11.1111\ldots\right)(1.05) = 166.6666\ldots(1.05) = 175.

Year-2 surrender outgo per opening policy is $0.10(60)=$6$. Therefore,

a_2 = \frac{175-6}{0.90} = 187.7777\ldots

The two-year asset share is $187.78 per persisting policy. There are $90(0.90)=81$ policies, so aggregate assets are $81(187.7777\ldots)=$15{,}210$.

For the 30% lapse scenario in year 2, there are 70 opening policies and the fixed expense per policy is $1{,}000/70=$14.2857\ldots$. Thus,

G_2 = \left(98.5714\ldots+100-10-14.2857\ldots\right)(1.05) = 174.2857\ldots(1.05) = 183.

Surrender outgo per opening policy is $0.30(60)=$18$, giving

a_2 = \frac{183-18}{0.70} = 235.7142\ldots

The result is $235.71 per persisting policy. There are $70(0.70)=49$ policies, so aggregate assets are $49(235.7142\ldots)=$11{,}550$. The high-lapse scenario has a higher asset share per survivor but lower aggregate assets. The result reflects both the fewer policies sharing assets and the larger fixed expense per policy in year 2. Each displayed per-policy result is rounded to the nearest cent; intermediate calculations retain unrounded values.

Common mistake avoided: Allocating the year-2 fixed expense over the original 100 policies would use $10 rather than the correct $11.1111... or $14.2857... per opening policy. The expense allocation must use the cohort size at the start of the year in which the expense is incurred.

When Does a Higher Lapse Rate Raise the Survivor Asset Share?

For a single year, let $G$ be the amount per opening policy after premiums, expenses and investment return, but before surrender outgo. With surrender amount $S$ and lapse rate $\lambda$, the per-persisting-policy asset share is $(G-\lambda S)/(1-\lambda)$. Compare it with $G$, the per-policy amount if no policies lapse:

\frac{G-\lambda S}{1-\lambda}-G = \frac{\lambda(G-S)}{1-\lambda}, \qquad 0\leq\lambda<1.

Since $1-\lambda$ is positive, the sign of the difference is determined by $G-S$. If $G>S$, lapses raise the asset share per persisting policy relative to the no-lapse amount. If $G<S$, they lower it. If $G=S$, the per-persisting-policy amount is unchanged by the lapse rate in this one-year comparison. These statements hold with $G$ and $S$ fixed; in a multi-year projection, lapse rates can also change future expense allocations and other assumptions.

Worked Example: Finding a Lapse Rate for a Target Asset Share

A one-year projection has a pre-surrender amount of $700 per opening policy and a year-end surrender amount of $760. There are no other exits. Find the lapse rate that produces an asset share of $690 per persisting policy.

Method. Substitute the target into the per-persisting-policy formula and solve for the lapse rate $\lambda$:

\frac{700-760\lambda}{1-\lambda}=690.

Multiplying by $1-\lambda$ gives $700-760\lambda=690-690\lambda$. Rearranging,

10=70\lambda, \qquad \lambda=\frac{10}{70}=\frac{1}{7}=0.142857\ldots

The required lapse rate is 14.29%, rounded to two decimal places. Check by substitution: the numerator is $700-760/7=591.4286\ldots$, and the remaining proportion is $6/7$. Their ratio is $690. Here $G<S$, so lapses reduce the survivor asset share below the no-lapse amount of $700, consistent with the comparison above.

Common mistake avoided: Solving $700-\lambda(760)=690$ would treat the post-surrender amount per opening policy as the target per persisting policy. The target applies after the remaining funds have been allocated among the policies that persist, so the equation must include the divisor $1-\lambda$.

Interpreting a Lapse Sensitivity

A reliable sensitivity analysis keeps three quantities distinct: the cohort’s aggregate assets, the number of policies remaining, and the asset share per persisting policy. A higher lapse rate can lower aggregate assets because more surrender payments are made. At the same time, it can raise the per-policy asset share if fewer policies share the remaining assets and surrender amounts are relatively small. Higher fixed expenses per remaining policy can push in the opposite direction in later years.

The comparison is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. For any projection, state the timing of premiums, expenses, investment return and surrender payments; specify which policies receive each payment; and identify the cohort size used to allocate expenses. If mortality or another decrement is also present, the number of policies remaining must reflect the relevant decrement assumptions and the payment rules for each exit. The lapse-only equations here should not be used as though they already include those additional decrements.

Key takeaway: Calculate aggregate assets after surrender outgo, update the cohort size, and only then divide to obtain the per-persisting-policy asset share. Recalculate shared-expense allocations using the in-force cohort for the year in which those expenses are incurred.

Check Your Understanding

Use the annual timing assumptions stated in each question. Round dollar amounts to the nearest cent where needed.

A cohort has a pre-surrender amount of $500 per opening policy, a $540 surrender payment per lapse, and a 25% lapse rate. What is the asset share per persisting policy? (Answer: $486.67)
A cohort has 80 opening policies and a 12.5% lapse rate. Each policy that lapses receives $200 at year-end. How many policies remain, and what is aggregate surrender outgo? (Answer: 70 policies remain; aggregate surrender outgo is $2,000.)
A policy cohort has an opening asset share of $300, beginning-of-year premium of $80, beginning-of-year expense of $20, an effective annual investment return of 5%, a 20% year-end lapse rate and a $250 surrender payment. Find the asset share per persisting policy. (Answer: $410.00)
A cohort begins with 50 policies and zero asset share. At the beginning of the year, each pays a $100 premium and incurs a $10 variable expense plus an equal share of a $500 fixed expense. The effective annual investment return is 0%, the lapse rate is 20%, and the year-end surrender payment is $40. Find the next-year asset share per persisting policy. (Answer: $90.00)
A one-year projection has a pre-surrender amount of $640 per opening policy and a surrender payment of $700 per lapse. What lapse rate produces an asset share of $620 per persisting policy? (Answer: 25.00%)

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