Lapse Effects on Equity-Linked Benefits and Profits

Why Lapse Timing Matters

For an equity-linked policy, a lapse changes more than the number of contracts remaining in force. It may trigger a surrender payment, reduce that payment by a contractual deduction, and end the policy’s exposure to future death or maturity guarantees. The resulting effect on expected benefits or profit depends on when the lapse occurs relative to fund growth and other benefits.

The projections here use an annual model. A year’s fund return is credited first; death is then considered, and lapse applies only to policies that survive death. A policy that lapses receives a surrender value at year-end. If it does not lapse or die, it remains in force to receive a maturity benefit. These are modelling assumptions, not universal contract terms: a contract with a different order of events needs a projection that follows its terms.

Definition: Let $F_0$ be the fund value at the start of a year and $F_1$ its value after the year’s investment return. If the surrender deduction is a proportion $s$ of the fund, the surrender value is $(1-s)F_1$. For a guaranteed death amount $G_D$, the death benefit is $\max(F_1,G_D)$; for a guaranteed maturity amount $G_M$, the maturity benefit is $\max(F_1,G_M)$. In this tutorial, the guarantees apply to death and maturity benefits, not to surrenders.

Write $q$ for the probability of death during the year and $l$ for the probability of lapse during the year conditional on surviving death. Then the probabilities, per policy in force at the start of the year, are

P(\text{death})=q,\qquad P(\text{lapse})=(1-q)l,\qquad P(\text{maturity in force})=(1-q)(1-l).

They sum to one: $q+(1-q)l+(1-q)(1-l)=1$. This convention is useful when the model specifies lapse among survivors. If a lapse probability is instead given as a proportion of the original starting cohort, it should not be multiplied by $1-q$ again.

Expected Benefits and Guarantee Costs

Suppose the fund return can take scenario values $F_1^{(j)}$ with probabilities $w_j$, where $\sum_j w_j=1$. Assume, for this calculation, that the fund scenario is independent of death and lapse. For a given scenario, a death produces $\max(F_1^{(j)},G_D)$, a lapse produces $(1-s)F_1^{(j)}$, and a policy remaining in force to maturity produces $\max(F_1^{(j)},G_M)$.

E[B_1] = \sum_j w_j\left[ q\max(F_1^{(j)},G_D) +(1-q)l(1-s)F_1^{(j)} +(1-q)(1-l)\max(F_1^{(j)},G_M) \right].

Here $E[B_1]$ is the expected year-end benefit per policy in force at the start of the year. To value it at the start of the year using an effective annual discount rate $i$, divide by $1+i$. If the valuation specifies another discounting convention, use that convention consistently.

The insurer’s guarantee cost is not the whole benefit. It is the extra amount paid above the fund value when a death or maturity guarantee binds. A lapse has no guarantee top-up under the contract definition used here.

E[T_1] = \sum_j w_j\left[ q\max(G_D-F_1^{(j)},0) +(1-q)(1-l)\max(G_M-F_1^{(j)},0) \right].

The formula separates three quantities that are easy to conflate: the fund value, the total payment to the policyholder, and the insurer’s guarantee top-up. In particular, the surrender deduction reduces the lapse payment but does not itself change a death or maturity guarantee.

Worked Example: Lapse, Market Scenarios, and a Maturity Guarantee

A hypothetical one-year policy starts with a fund of $10,000. At year-end, the fund is $11,000 or $9,000, each with probability 0.5. The annual death probability is $q=0.02$; among those surviving death, the lapse probability is $l=0.20$. A surrender deduction of 5% applies to the fund. The death guarantee is $10,500 and the maturity guarantee is $10,200. Find the expected year-end benefit, its present value at $i=5\%$, and the present value of guarantee top-ups. Use the timing and independence assumptions stated above.

Method. First find the unconditional outcome probabilities. Death has probability $0.02$, lapse has probability $0.98(0.20)=0.196$, and maturity in force has probability $0.98(0.80)=0.784$. They sum to $0.02+0.196+0.784=1$.

For the $11,000 fund scenario, the death benefit is $\max(11{,}000,10{,}500)=\$11{,}000$, the surrender value is $0.95(11{,}000)=\$10{,}450$, and the maturity benefit is $\max(11{,}000,10{,}200)=\$11{,}000$.

For the $9,000 scenario, the death benefit is $\max(9{,}000,10{,}500)=\$10{,}500$, the surrender value is $0.95(9{,}000)=\$8{,}550$, and the maturity benefit is $\max(9{,}000,10{,}200)=\$10{,}200$.

The scenario-average payments for the three outcomes are therefore

\begin{aligned} \text{death: }&\frac{11{,}000+10{,}500}{2}=\$10{,}750,\\ \text{lapse: }&\frac{10{,}450+8{,}550}{2}=\$9{,}500,\\ \text{maturity: }&\frac{11{,}000+10{,}200}{2}=\$10{,}600. \end{aligned}

Weight each average by its outcome probability:

E[B_1]=0.02(10{,}750)+0.196(9{,}500)+0.784(10{,}600) =215+1{,}862+8{,}310.40 =\$10{,}387.40.

Discounting at 5% gives $10{,}387.40/1.05=\$9{,}892.76$, rounded to the nearest cent.

For the guarantee top-up, the average death top-up is $(0+1{,}500)/2=\$750$. The average maturity top-up is $(0+1{,}200)/2=\$600$. Thus

E[T_1]=0.02(750)+0.784(600)=15+470.40=\$485.40, \qquad \frac{485.40}{1.05}=\$462.29.

The lapse deduction does not appear in the top-up calculation because this contract has no surrender guarantee. Check: the expected benefit exceeds the expected top-up, as it must here because each top-up is only the guaranteed excess over the fund-linked amount.

Why the Probability Weighting Works

The benefit formula follows by conditioning on two mutually exclusive choices: the investment scenario and the policy outcome. For scenario $j$, the expected payment is the sum of each possible payment multiplied by its probability. Independence lets the scenario probability $w_j$ multiply each outcome probability, such as $w_j(1-q)l$ for lapse in scenario $j$. Summing over scenarios gives the expected payment per starting policy. This is the discrete form of the law of total expectation.

The conditioning also clarifies the role of lapse. The lapse payment is weighted by $(1-q)l$, not by $l$, because a policy that dies during the year cannot also lapse under the stated ordering. If death and lapse are not modelled in this order, the probabilities and payment times must be changed to match the specified model.

This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central valuation discipline is to state the basis, event timing, and benefit definition before calculating a value.

Lapse-Sensitive Prospective Reserves

A benefit-only prospective reserve at a valuation time is the expected present value, at that time, of the future benefits included in the stated projection. It is not automatically a net premium reserve: this definition excludes future premiums, expenses, charges, and other cash flows unless they are explicitly added. For an in-force policy, future lapse can reduce the reserve if the surrender payment is less than the benefit expected under continuation. The effect is not necessarily a reduction in every contract: a large surrender guarantee, for example, could change the comparison.

Worked Example: A One-Year Reserve with Conditional Lapse

At a valuation date, a hypothetical in-force policy has a fund of $10,000 and one year remaining. The fund grows deterministically by 4% before any year-end payment, so its year-end value is $10,400. During the year, $q=0.01$, and $l=0.30$ among those surviving death. The surrender deduction is 5%, the death guarantee is $10,500, the maturity guarantee is $10,300, and $i=5\%$. Calculate the benefit-only reserve. No future premiums or other cash flows are included.

Method. The probabilities of death, lapse, and maturity in force are $0.01$, $0.99(0.30)=0.297$, and $0.99(0.70)=0.693$, respectively. They sum to one.

The death payment is $\max(10{,}400,10{,}500)=\$10{,}500$. The lapse payment is $0.95(10{,}400)=\$9{,}880$. The maturity payment is $\max(10{,}400,10{,}300)=\$10{,}400$.

The expected year-end benefit is

0.01(10{,}500)+0.297(9{,}880)+0.693(10{,}400) =105+2{,}934.36+7{,}207.20 =\$10{,}246.56.

Discount one year at 5%:

V=\frac{10{,}246.56}{1.05}=\$9{,}758.63.

As a check on the lapse effect, if lapse were ignored while the death probability remained 1%, the expected payment would be $0.01(10{,}500)+0.99(10{,}400)=\$10{,}401$, giving a reserve of $10{,}401/1.05=\$9{,}905.71$. Lapse replaces a $10,400 maturity payment with a $9,880 surrender payment for 29.7% of the starting in-force group. The reduction in present value is $0.297(10{,}400-9{,}880)/1.05=\$147.09$, matching the difference between the two reserves after rounding.

Profit Emergence When Policies Lapse

Profit testing must follow the contract’s actual cash flows and reserve definition. A surrender deduction may produce an amount retained by the insurer, while lapse may also remove future guarantee exposure. The example below isolates these effects in a simple guarantee-risk account; it does not model the fund principal as insurer profit.

Model definition: In the next example, the guarantee-risk account starts with a reserve of $300 per policy. At year-end, that reserve is released because all policies either die, lapse, or reach maturity. Profit per starting policy is defined as the $300 reserve release, plus any surrender deduction retained by the insurer, less any guarantee top-up paid. Other premiums, expenses, investment margins, and cash flows are excluded.

Worked Example: Surrender Income and Guarantee-Account Profit

A hypothetical one-year policy has a year-end fund of $10,500 after investment growth. The death probability is $q=0.01$; among survivors, the lapse probability is $l=0.20$. A 4% surrender deduction is retained by the insurer. The death guarantee is $11,000 and the maturity guarantee is $10,200. The opening guarantee-risk reserve is $300 per policy. Calculate expected year-end profit under the account definition above.

Method. The death probability is 0.01, the lapse probability is $0.99(0.20)=0.198$, and the maturity-in-force probability is $0.99(0.80)=0.792$. The surrender deduction is $0.04(10{,}500)=\$420$, so surrender pays $10{,}500-420=\$10{,}080$. The death top-up is $11{,}000-10{,}500=\$500$. The maturity guarantee is below the fund, so the maturity top-up is zero.

The profit in each outcome is the reserve release plus any retained deduction, less any top-up:

Death: $300-500=-\$200$.
Lapse: $300+420=\$720$.
Maturity in force: $300+0=\$300$.

Weight these profits by their probabilities:

E[\text{profit}] =0.01(-200)+0.198(720)+0.792(300) =-2+142.56+237.60 =\$378.16.

Without lapse, but with the same death probability and fund, expected profit would be $0.01(-200)+0.99(300)=\$295$. Here lapse increases expected profit by $0.198(420)=\$83.16$: it creates surrender-deduction income, while the maturity guarantee would not have required a top-up anyway. Check: the difference agrees with the expected retained deduction; there is no maturity top-up saving to add in this particular example.

Common Traps and Checks

The main exam trap is to apply a guarantee to an event for which the contract does not provide it. In the first example, one incorrect lapse calculation would be $\max(0.95F_1,G_M)$. For the $9,000 fund scenario, that gives $\max(8{,}550,10{,}200)=\$10{,}200$. But the stated guarantee applies only at maturity, not on surrender. The correct lapse payment is $0.95(9{,}000)=\$8{,}550$. Applying the maturity floor to a surrender overstates the benefit by $1,650 in that scenario.

Other useful checks are to confirm that outcome probabilities sum to one, apply the surrender deduction to the correct fund value, and include each benefit only for the event that triggers it. Also verify whether lapse is specified among starting policies or among survivors. That distinction changes the probability weight, not just the arithmetic.

Key takeaway: Model lapse as a timed exit with its own payment. Weight that payment using the specified lapse probability, exclude exited policies from later guarantees, and apply each guarantee only to the benefit event covered by the contract.

Check Your Understanding

For each problem, state or use the given timing assumptions and keep the surrender payment distinct from guaranteed death or maturity payments.

A policy’s fund is $8,000 at the start of a year and earns 5% before a year-end lapse. The surrender deduction is 3%. What is the surrender payment? (Answer: $8,148.00)
In a one-year model, $q=0.04$ and $l=0.10$ among survivors. The year-end fund is equally likely to be $9,000 or $11,000. Death and maturity benefits are each the greater of the fund and $10,000; lapse receives the fund without a deduction and no guarantee. What is the expected guarantee top-up per starting policy? (Answer: $452.00)
An in-force policy has a $12,000 fund and one year remaining. The fund grows deterministically by 2%; $q=0.02$, $l=0.25$ among survivors, the surrender deduction is 5%, death guarantee is $12,500, maturity guarantee is $12,300, and $i=4\%$. With no other cash flows, calculate the benefit-only reserve. (Answer: $11,672.46)
For a year-end event model, death probability is 0.03 and lapse probability is 0.20 among survivors. The fund is $5,000 before a 4% annual fund return, and the surrender deduction is 6%. What are the unconditional lapse and maturity-in-force probabilities, and what is the surrender payment? (Answer: Lapse probability 0.1940, maturity-in-force probability 0.7760, surrender payment $4,888.00)
A fund of $15,000 is subject to a 2% surrender deduction. The annual death probability is 0.05, and the lapse probability is 0.12 among survivors. What is the expected surrender deduction retained per policy in force at the start of the year? (Answer: $34.20)

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