Lapse And Surrender Cash Flows

Introduction

When a policy lapses, two different things happen to the insurer’s finances. The insurer may pay a surrender benefit, which is an actual cash outgo. It also ceases to hold a reserve for a policy that is no longer in force. That reserve release affects profit through the closing reserve, but it is not another cash payment to the policyholder.

Premium timing matters as well. A premium already collected is not automatically removed when a policy lapses; a premium that becomes due only while the policy remains in force may instead be lost on lapse. This tutorial sets out a timed model for including these effects in a policy-year profit calculation. Its treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

A Policy-Year Model for Lapse

Consider one policy that is in force at time 0, the start of a policy year. Assume for now that there are only two possible outcomes: lapse or persistence to the end of the year. Let $q_l$ be the probability of lapse during the year and $1-q_l$ the probability of remaining in force. Suppose lapse occurs at time $\tau$, measured in years, if it occurs. For the examples below, $\tau=0.5$.

The policy has an opening reserve $V_0$ and an end-of-year reserve $V_1$ if it remains in force. An initial premium $P_0$ is collected at time 0. A further premium $R$ is due at time $\tau$ only if the policy is still in force immediately before that premium is due. A surrender payment $S$ is made at time $\tau$ if the policy lapses. This ordering means a lapse at $\tau$ happens before the premium due there.

Definition: In this model, expected profit per policy in force at time 0 is the accumulated value at time 1 of opening funds and premiums, less accumulated benefits, less the reserve required at time 1 for policies still in force. The surrender payment is a benefit cash flow; the reduction in the closing reserve caused by lapse is accounted for through the number of policies for which $V_1$ is held.

Assume an effective annual interest rate $i$, with accumulation over $t$ years of $(1+i)^t$. Ignore expenses, taxes, death claims, and any other benefits in the model. The opening reserve and initial premium are accumulated for a full year. A premium or surrender payment at time $\tau$ is accumulated for the remaining $1-\tau$ years. Only policies that persist require the closing reserve.

\Pi = (V_0+P_0)(1+i) +(1-q_l)R(1+i)^{1-\tau} -q_lS(1+i)^{1-\tau} -(1-q_l)V_1

Here $\Pi$ is expected profit at time 1 per policy in force at time 0. This is a model-specific expression, not a universal formula for every timing convention. If a premium is due before lapse can occur, it should not be multiplied by the probability of persistence. If lapse occurs at a different time, the accumulation period for the surrender payment and any conditional premium must be changed. If there are other decrements or cash flows, they must also be included consistently.

Why the Formula Has These Terms

The formula follows by tracing the value at time 1 of each item. Every opening policy contributes the opening reserve and the time-0 premium, so these amounts are accumulated by one year. A policy that persists contributes the time-$\tau$ premium, if one is due. A lapse instead produces a surrender payment. Finally, a reserve is needed at time 1 only for a policy that is still in force.

Accumulate opening funds.
At time 0, $V_0+P_0$ is available for every policy in force, giving $(V_0+P_0)(1+i)$ at time 1.

Add conditional premium income.
The probability of persistence is $1-q_l$. If premium $R$ is due only to a policy still in force at $\tau$, its expected accumulated value is $(1-q_l)R(1+i)^{1-\tau}$.

Subtract surrender outgo.
The lapse probability is $q_l$, so the expected accumulated surrender payment is $q_lS(1+i)^{1-\tau}$.

Subtract required closing reserves.
Only the fraction $1-q_l$ still in force needs reserve $V_1$, giving an expected closing reserve of $(1-q_l)V_1$.

This derivation also explains the central distinction. Lapse reduces the reserve held at time 1 because fewer policies remain in force. The surrender amount is a separate cash outgo, paid only on lapses. Adding a further “reserve released” amount to this formula would count the effect of fewer closing reserves a second time.

Worked Example: Lapse, Surrender, and a Conditional Premium

A hypothetical policy has opening reserve $V_0=\$500$, a premium $P_0=\$100$ at time 0, and a further premium $R=\$60$ at time 0.5 if the policy has not lapsed. Lapse occurs at time 0.5 with probability $q_l=0.20$, before the further premium is due, and the surrender payment is $S=\$300$. A policy persisting to year-end has reserve $V_1=\$550$. The effective annual interest rate is $i=4\%$. Calculate expected profit at time 1 per policy initially in force.

Method. Apply the timed expected-profit formula. The time-0 amounts accumulate for a full year; the time-0.5 amounts accumulate for half a year. The further premium is collected only on persistence.

The half-year accumulation factor is $(1.04)^{0.5}=1.0198039$. The accumulated opening reserve and initial premium are \[ (\$500+\$100)(1.04)=\$624.00. \] The probability of persistence is $1-0.20=0.80$, so the accumulated expected further premium is \[ 0.80(\$60)(1.0198039)=\$48.9506. \] The accumulated expected surrender outgo is \[ 0.20(\$300)(1.0198039)=\$61.1882. \] The expected closing reserve is \[ 0.80(\$550)=\$440.00. \] Therefore, \[ \Pi=\$624.00+\$48.9506-\$61.1882-\$440.00 =\$171.7624. \] Rounded to the nearest cent, expected profit is \$171.76 per policy initially in force.

Check. The premium at time 0 is included for every policy because it was collected before lapse. The time-0.5 premium is included only on persistence, while the surrender payment is included only on lapse. The closing reserve is held only on persistence. The calculation avoids treating the surrender payment and reserve reduction as the same item.

Reserve Release Is Not a Second Surrender Payment

The closing-reserve term is an expected amount per policy initially in force, not the reserve for a hypothetical policy that certainly persists. If a fraction $q_l$ lapses, then the expected reserve held at time 1 is $(1-q_l)V_1$. Relative to a calculation that reserved $V_1$ for every policy, the reduction is $q_lV_1$. That reduction is already present when the closing reserve is calculated using only policies that persist.

A surrender payment can be less than, equal to, or greater than the reserve that would otherwise have been held. The difference affects profit, but the surrender payment itself is determined by the contract’s surrender terms; it is not automatically equal to the reserve. The insurer’s cash payment and its reserve liability are distinct quantities.

Worked Example: Detecting Double Counting

A hypothetical contract has $V_0=\$400$, $P_0=\$80$, and no conditional premium. Lapse occurs at time 0.5 with probability $q_l=0.10$; the surrender payment is $S=\$250$, and the reserve for a policy persisting to year-end is $V_1=\$450$. The effective annual interest rate is $i=5\%$. There are no other cash flows. Calculate expected profit and identify the overstatement from adding a separate reserve-release item.

The half-year factor is $(1.05)^{0.5}=1.0246951$. The accumulated opening funds are \[ (\$400+\$80)(1.05)=\$504.00. \] The expected accumulated surrender outgo is \[ 0.10(\$250)(1.0246951)=\$25.6174. \] The expected closing reserve is \[ 0.90(\$450)=\$405.00. \] Thus, \[ \Pi=\$504.00-\$25.6174-\$405.00 =\$73.3826, \] or \$73.38 to the nearest cent.

The closing reserve is $\$45$ less than the $\$450$ that would be required if every policy persisted: $\$450-\$405=\$45$. That lower closing reserve is already reflected in the calculation. Adding a further $\$45$ “reserve release” would produce $\$118.38$, overstating profit by $\$45.00$. The correction is to include the surrender payment as cash outgo and use the expected closing reserve once.

Premium Cessation and the Profit Effect of Lapse

Premium cessation depends on the timing and terms of the contract. A premium collected at the start of the year remains part of the year’s cash flows even if the policy lapses later, unless a separate contractual payment reverses it. In contrast, a premium due after the lapse time is not collected on a lapsed policy under the model above. Do not deduct an already collected premium merely because the policy terminates.

One useful check is to compare the lapse outcome with persistence, holding the opening funds and initial premium fixed. At time 1, a lapse has no closing reserve, pays the surrender amount, and loses any conditional premium. Persistence receives the conditional premium and requires a closing reserve. At the lapse probability level, the difference in expected profit attributable to replacing persistence with lapse is therefore

q_l\left[V_1-(R+S)(1+i)^{1-\tau}\right].

This comparison assumes the same timing as the main model: both $R$ and $S$ occur at $\tau$, with the premium due only on persistence. The expression is a diagnostic for the incremental effect of lapse, not a substitute for checking all cash flows. It shows why lapse does not necessarily increase profit: the reserve no longer required may be outweighed by surrender outgo and the lost premium.

Worked Example: Lapse Can Reduce Profit

A hypothetical policy has $V_0=\$300$, a time-0 premium $P_0=\$50$, and a conditional premium $R=\$80$ due at time 0.5 only if the policy persists. Lapse occurs at time 0.5 with probability $q_l=0.25$, and the surrender payment is $S=\$260$. The year-end reserve on persistence is $V_1=\$280$. Assume an effective annual interest rate of $i=0\%$. Find expected profit and compare it with the profit if every policy persisted.

With zero interest, the accumulation factor is 1 at every time. Opening funds are \[ V_0+P_0=\$300+\$50=\$350. \] The expected conditional premium is $0.75(\$80)=\$60$. The expected surrender outgo is $0.25(\$260)=\$65$, and the expected closing reserve is $0.75(\$280)=\$210$. Hence \[ \Pi=\$350+\$60-\$65-\$210=\$135.00. \]

If every policy persisted, profit would be $\$350+\$80-\$280=\$150.00$. The lapse effect is therefore $\$135-\$150=-\$15.00$. The diagnostic gives the same result: \[ 0.25[\$280-(\$80+\$260)]=0.25(-\$60)=-\$15.00. \] Although lapse removes the need for a closing reserve, it also removes the conditional premium and triggers a surrender payment. Here those two effects more than offset the reserve release.

Common Mistakes and a Practical Check

A reliable calculation starts with a timeline. Mark when the initial premium is paid, when the lapse decision occurs, when any further premium is due, when the surrender amount is paid, and when the closing reserve is measured. Then assign each cash flow to the policies that actually generate it and accumulate all amounts to the same valuation date.

Counting a surrender payment as reserve release. The surrender payment is a benefit cash flow. The reduced closing reserve is separately reflected by reserving only for policies still in force.
Subtracting the closing reserve for every initial policy. If only a fraction persists, the expected closing reserve is that fraction multiplied by the conditional reserve.
Stopping a premium too early. A premium already received remains in the calculation unless the contract requires a refund or other payment. Premium refunds and return-of-premium features require their own cash-flow terms.
Including a premium after lapse. A premium due only while active must be weighted by the probability of being in force at its due time.
Mixing valuation dates. A payment at time 0.5 and a reserve at time 1 cannot be added or subtracted directly without accumulating or discounting them to a common date.

For a more complex policy-year model, use the same logic for each outcome: list its probability, cash flows, and terminal reserve, then combine the outcome-specific amounts at a common valuation time. The two-outcome lapse model here isolates the key distinction needed in profit testing: surrender is paid in cash, while reserve release is represented through the reserve no longer held.

Key takeaway: Include the surrender payment as an outgo on lapse, include a premium only when it is due under the stated in-force condition, and calculate the closing reserve only for policies remaining in force. Do not add a separate reserve-release amount when that effect is already captured by the reduced closing reserve.

Check Your Understanding

Use the timing convention stated in each question. Unless specified otherwise, ignore expenses and other decrements.

A policy has $V_0=\$200$ and $P_0=\$40$. At time 0.5, it lapses with probability $0.10$ before a premium of $R=\$20$ is due; the surrender payment is $S=\$100$. The year-end reserve on persistence is $V_1=\$210$, and $i=0\%$. Find expected profit at time 1 per policy initially in force. (Answer: $\$59.00$)
A policy has $V_0=\$300$, $P_0=\$100$, $R=\$50$, $S=\$180$, $V_1=\$320$, and $q_l=0.20$. Lapse occurs at time 0.5 before $R$ is due. If $i=4\%$, find expected profit at time 1, rounded to the nearest cent. (Answer: $\$164.08$)
A policy has $V_0=\$500$, $P_0=\$60$, $R=\$40$, $S=\$300$, $V_1=\$520$, and $q_l=0.25$. Lapse at time 0.5 occurs before $R$ is due, and $i=0\%$. Find expected profit. By how much would profit be overstated if $q_lV_1$ were added again as a separate release? (Answer: Profit $\$125.00$; overstatement $\$130.00$)
A policy has $V_0=\$250$, $P_0=\$30$, $R=\$25$, $V_1=\$240$, and $q_l=0.20$. Lapse at time 0.5 precedes the conditional premium, and $i=0\%$. What surrender payment $S$ makes expected profit equal to zero? (Answer: $\$540.00$)

What You'll Learn