Deriving Lapse-Contingent Profit

Why Lapse-Contingent Profit Needs Its Own Calculation

A universal life policy that lapses produces a different set of cash flows from one that remains in force. The insurer may release a liability reserve, pay a surrender value, incur a termination expense, and write off acquisition costs that have not yet been recovered. The policy account also determines the amount available for surrender and may already have generated charges or interest-crediting cash flows during the period. These items need to be kept distinct to calculate the profit attributable to lapse.

This tutorial uses an end-of-period profit-test convention. It calculates the profit at the lapse event, then weights that amount by the probability of reaching the lapse date and lapsing there. Cash flows earned or paid before the event, such as premium loads and cost-of-insurance charges, belong in the period’s regular profit calculation; they are not added again to the event profit unless the example explicitly includes them.

Definition: Let $V_t^-$ be the insurer’s liability reserve immediately before a policy lapses at time $t$, $S_t$ the surrender payment, $E_t^L$ the expense paid because of the lapse, and $D_t$ the unamortized acquisition expense carried as an asset just before lapse. Under the stated accounting convention, lapse-event profit is $$ \Pi_t^L=V_t^- - S_t - E_t^L - D_t. $$ The reserve released is a source of profit; the surrender payment, lapse expense, and write-off of the acquisition-cost asset reduce profit. This formula assumes that $D_t$ is recorded separately as an asset and is not already deducted in $V_t^-$.

A reserve may include the value of future benefits and expenses. Releasing it at lapse therefore captures the liability that no longer needs to be held for a continuing policy. The acquisition-cost asset has the opposite balance-sheet effect: if the policy terminates before the cost is recovered, the asset is written off and the remaining balance is a loss. If an expense item is already included in the reserve or in another cash-flow term, it must not be counted a second time.

Account Value and the Lapse Settlement

The account-value calculation is often needed before the lapse profit can be determined. For a simple annual model, suppose a premium $P_t$ is received at the start of the year, a premium load $L_t$ and other account deductions $C_t$ are taken before interest crediting, and the account receives an effective annual crediting rate $j_t$. Then the account value at the end of the year is

A_t=(A_{t-1}+P_t-L_t-C_t)(1+j_t).

This expression depends on the specified timing: the premium is added at the start, the listed deductions occur before crediting, and interest is credited at year-end. A different timing convention requires a different account-value recursion. The surrender payment may equal the account value less a surrender charge, but that is a product-specific assumption, not a universal rule.

When the liability reserve includes the account-value liability, it is helpful to write $V_t^-=A_t+G_t$, where $G_t$ is the reserve in excess of the account value on the chosen valuation basis. If the surrender payment is $A_t-K_t$, where $K_t$ is a surrender charge, then

\Pi_t^L=G_t+K_t-E_t^L-D_t.

This decomposition is an algebraic check on the full calculation. It shows that the account-value portion of the liability is largely settled by the surrender payment; the surrender charge and any reserve above account value contribute to the lapse margin, while the expense and acquisition-cost write-off reduce it.

Worked Example: Building Lapse Profit From Account Value

A hypothetical policy starts the year with account value $A_0=\$8{,}000$. At the start of the year, the policy receives a premium of $\$1{,}200$. A 5% premium load is deducted, followed by $\$100$ of expense charges and $\$140$ of cost-of-insurance charges. The remaining account earns an effective annual crediting rate of 3%. At year-end, the reserve is $\$9{,}500$, the surrender charge is 2% of account value, the lapse expense is $\$40$, and unamortized acquisition expense is $\$300$. Find the account value, surrender payment, and lapse-event profit.

Method. Apply the stated account timing first, then calculate the surrender payment and use the lapse-profit balance-sheet formula.

The premium load is $0.05(1{,}200)=\$60$. The account amount before interest crediting is $8{,}000+1{,}200-60-100-140=\$8{,}900$. Thus $A_1=8{,}900(1.03)=\$9{,}167$.

The surrender charge is $0.02(9{,}167)=\$183.34$, so the surrender payment is $S_1=9{,}167-183.34=\$8{,}983.66$.

The lapse-event profit is $9{,}500-8{,}983.66-40-300=\$176.34$. As a check, the reserve above the account value is $9{,}500-9{,}167=\$333$, so the decomposition gives $333+183.34-40-300=\$176.34$, the same result.

Common mistake avoided: Subtracting the account value and the surrender payment as separate outflows would count the account-related settlement twice. The reserve is the liability released; the surrender payment is the cash paid.

From Conditional Lapse Rates to Expected Profit

For a policy in force at the start of year $t$, let $l_t$ be the conditional probability of lapse during that year and $m_t$ the conditional probability of death. Assume these are mutually exclusive decrements over the period, with no other way to leave the in-force population. The conditional probability of remaining in force to the next year is then $p_t=1-l_t-m_t$. These probabilities are conditional on being in force at the beginning of the year.

Write ${}_{t-1}P_{\mathrm{in}}$ for the probability, per policy issued, of still being in force at the beginning of year $t$. With ${}_0P_{\mathrm{in}}=1$, the recursion is

{}_{t-1}P_{\mathrm{in}}=\prod_{r=1}^{t-1}p_r.

Therefore, the probability per policy issued of lapsing during year $t$ is ${}_{t-1}P_{\mathrm{in}}l_t$, not simply $l_t$. If $\Pi_t^L$ is the profit conditional on lapse in year $t$, and the effective annual interest rate is $i$, the issue-date expected present value through year $n$, with profit paid at each year-end, is

\operatorname{EPV}_0 =\sum_{t=1}^{n}v^t\,{}_{t-1}P_{\mathrm{in}}\,l_t\,\Pi_t^L, \qquad v=\frac{1}{1+i}.

This expression treats death as a competing decrement: a policy that dies before year $t$ cannot lapse in that year. The formula also assumes the lapse-event profit is paid at year-end and that the stated interest rate is appropriate for discounting those profits. A different payment time changes the discount factor.

Why the Expected Present Value Formula Holds

Let $I_t^L$ equal 1 if a policy issued lapses in year $t$, and 0 otherwise. The present value of lapse profits for one issued policy is the random quantity $\sum_{t=1}^{n}v^t I_t^L\Pi_t^L$, where each $\Pi_t^L$ is the conditional event profit for its year. Taking expectations and using linearity of expectation gives

E\left[\sum_{t=1}^{n}v^t I_t^L\Pi_t^L\right] =\sum_{t=1}^{n}v^t\Pi_t^L E[I_t^L] =\sum_{t=1}^{n}v^t\Pi_t^L P(I_t^L=1).

To lapse in year $t$, the policy must first remain in force through years $1$ to $t-1$, and then lapse in year $t$. Under the conditional decrement assumptions, $P(I_t^L=1)={}_{t-1}P_{\mathrm{in}}l_t$. Substituting this probability yields the EPV formula. This derivation explains why multiplying each year’s profit by the conditional lapse rate alone is insufficient when valuing per policy issued.

Worked Example: Two Years With Lapse and Death

A hypothetical block earns an effective annual interest rate of 4%. In year 1, the conditional lapse and death probabilities are 0.10 and 0.02. In year 2, they are 0.08 and 0.02. Conditional lapse-event profits are $\$500$ in year 1 and $\$700$ in year 2. Find the issue-date expected present value of lapse profit through year 2.

Method. Find each year’s lapse probability per policy issued by multiplying the conditional lapse rate by the probability of reaching that year in force. Then discount each expected profit to issue.

The year-1 lapse probability per issued policy is $0.10$, so its expected year-end profit is $0.10(500)=\$50$. Its present value is $50/1.04=\$48.0769$.

The probability of remaining in force after year 1 is $1-0.10-0.02=0.88$. Thus the year-2 lapse probability per issued policy is $0.88(0.08)=0.0704$. Its expected year-end profit is $0.0704(700)=\$49.28$, and its issue-date present value is $49.28/1.04^2=49.28/1.0816=\$45.5621$.

The total is $48.0769+45.5621=\$93.6390$, or $\$93.64$ per policy issued, rounded to the nearest cent. The result uses the stated death probabilities to determine how many policies can reach year 2.

Worked Example: Three-Year Lapse Profit Signature

A hypothetical product has an effective annual interest rate of 5%. Conditional on being in force at the start of each year, the lapse probabilities in years 1, 2, and 3 are 0.15, 0.10, and 0.08, respectively; the corresponding death probabilities are 0.02, 0.03, and 0.02. Conditional lapse-event profits are $\$300$, $\$500$, and $\$200$. Find the issue-date expected present value through year 3.

Method. Construct the probability of reaching each year in force, then apply the conditional lapse probability, event profit, and year-end discount factor.

Year 1 lapse probability per issue is $0.15$. The present value contribution is $0.15(300)/1.05=\$42.8571$.

The year-1 persistence probability is $1-0.15-0.02=0.83$. The year-2 lapse probability per issue is $0.83(0.10)=0.083$. Its present value contribution is $0.083(500)/1.05^2=41.5/1.1025=\$37.6417$.

The probability of reaching year 3 in force is $0.83(1-0.10-0.03)=0.83(0.87)=0.7221$. The year-3 lapse probability per issue is $0.7221(0.08)=0.057768$. Its present value contribution is $0.057768(200)/1.05^3=11.5536/1.157625=\$9.9804$.

Adding the contributions gives $42.8571+37.6417+9.9804=\$90.4792$, or $\$90.48$ per policy issued, rounded to the nearest cent. The conditional lapse probabilities are used only after applying the probability of reaching the relevant year.

Common Exam Traps

A frequent error is to apply a conditional lapse probability as though it were already the probability per policy issued. For example, if year-1 lapse and death probabilities are 0.12 and 0.03, then the probability of reaching year 2 in force is $1-0.12-0.03=0.85$. With a year-2 conditional lapse probability of 0.10, the year-2 lapse probability per issued policy is $0.85(0.10)=0.085$, not 0.10. The year-2 contribution must use 0.085 before discounting.

Another trap is a timing mismatch. A profit at the end of year $t$ is discounted by $v^t$ to issue; a value at the end of year 1 is discounted only for cash flows after that date. For example, at an effective annual rate of 10%, future profits of $\$420$ at the end of year 2 and $\$220$ at the end of year 3 have value at the end of year 1 of $420/1.10+220/1.10^2=\$563.64$, rounded to the nearest cent. Comparing a lapse profit of $\$500$ at that same date with the persistence value gives $500-563.64=-\$63.64$. The calculation must use a common valuation date.

Check the decrement basis. If death or another decrement competes with lapse, subtract all mutually exclusive decrement probabilities when determining persistence.
Check the cost sign. A separately carried unamortized acquisition-cost asset is written off on lapse, reducing profit.
Check for double counting. Do not add account value as a separate payment if the surrender payment already reflects the account value.
Check the accounting definition. Do not subtract an expense again if it is already included in the reserve or another cash-flow item.

A disciplined lapse-profit calculation therefore has two stages: first derive the event profit on a consistent balance-sheet and cash-flow basis; then weight and discount it using the probability of lapse per policy issued. This framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the stated timing and assumptions in each problem. Round monetary answers to the nearest cent where needed.

At a hypothetical lapse date, a policy has reserve $\$7{,}200$, surrender payment $\$6{,}800$, lapse expense $\$25$, and separately carried unamortized acquisition expense $\$200$. Find lapse-event profit. (Answer: $\$175.00$)
At an effective annual interest rate of 4%, year-1 lapse and death probabilities are 0.12 and 0.03. The year-2 conditional lapse probability is 0.10, and conditional lapse-event profits are $\$250$ in year 1 and $\$400$ in year 2. Find the issue-date expected present value through year 2. (Answer: $\$60.28$)
A hypothetical account starts at $\$5{,}000$, receives a start-of-year premium of $\$800$, has a 4% premium load, and then has $\$60$ and $\$90$ of deductions before receiving 2% annual interest. At year-end the surrender charge is 3% of account value, the reserve is $\$5{,}900$, lapse expense is $\$20$, and unamortized acquisition expense is $\$250$. Find the lapse-event profit. (Answer: $\$71.55$)
At an effective annual interest rate of 5%, conditional lapse and death probabilities are 0.08 and 0.02 in year 1, and 0.12 and 0.03 in year 2. Conditional lapse-event profits are $\$600$ and $\$700$. Find the issue-date expected present value through year 2. (Answer: $\$114.29$)
At the end of year 1, lapse-event profit is $\$520$. Conditional on persistence, expected profits are $\$300$ at the end of year 2 and $\$180$ at the end of year 3. At an effective annual rate of 6%, find the end-of-year-1 difference between lapsing immediately and the stated future profits. (Answer: $\$76.78$)

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