Lapse Shocks and Reserve Release

Why Lapses Affect More Than Premium Income

A lapse ends a policy before its scheduled maturity or expiry. In a profit test, a lapse can reduce future premiums, trigger a surrender payment, and eliminate the reserve that would otherwise be held for a continuing policy. Its effect on profit is therefore not determined by lost premiums alone. In particular, the reserve released on termination can offset the surrender payment, or be smaller than it.

The calculations below use a discrete annual model. This makes the event timing explicit: premiums and expenses occur at the start of the policy year; death benefits and surrender payments occur at year-end; and a closing reserve is held only for policies that remain in force after year-end. The results depend on that timing, so it must be stated rather than assumed.

Definition: Let $q_t$ be the probability of death during policy year $t$, conditional on being in force at its start. Let $l_t$ be the probability of lapsing during that year, conditional on surviving it. The probability of remaining in force through year-end is $r_t=(1-q_t)(1-l_t)$. A lapse shock changes $l_t$, with other assumptions held fixed unless the problem states otherwise.

Write $V_{t-1}$ for the opening reserve per policy in force at the start of year $t$, $V_t$ for the closing reserve per policy that continues after year-end, $P_t$ for the start-of-year premium, and $E_t$ for the start-of-year expense. Let $B_t$ be the death benefit, $S_t$ the surrender payment, and $i$ the effective annual investment return. This model assumes no policy can both die and lapse in the same year: lapse is assessed only among those who survive. If a problem instead specifies a competing-decrement model or different payment timing, the probabilities and accumulation factors must be adjusted.

Expected Profit for One Policy-Year

For one policy in force at the beginning of year $t$, the opening reserve, premium, and expense produce an accumulated amount $(V_{t-1}+P_t-E_t)(1+i)$ at year-end. Expected death outgo is $q_tB_t$; expected surrender outgo is $(1-q_t)l_tS_t$; and expected closing reserves are held only for policies continuing in force, with expected amount $r_tV_t$. Thus the expected year-end profit per policy in force at the start of the year is:

\pi_t=(V_{t-1}+P_t-E_t)(1+i)-q_tB_t-(1-q_t)l_tS_t-(1-q_t)(1-l_t)V_t.

This expression includes the reserve release through the fact that a lapsed policy has no closing reserve. To make that effect visible, add and subtract $(1-q_t)V_t$:

\pi_t=(V_{t-1}+P_t-E_t)(1+i)-q_tB_t-(1-q_t)V_t +(1-q_t)l_t(V_t-S_t).

The final term is the lapse-related contribution relative to retaining the reserve for every policy that survives the year. It is the probability of lapse, $(1-q_t)l_t$, multiplied by reserve released less surrender paid, $V_t-S_t$. This is a decomposition, not a claim that a lapse has no other consequences: a lapse also changes the number of policies eligible to pay premiums or generate profits in later years.

Key result: Holding the other year-$t$ assumptions fixed, changing the lapse rate from $l_t^{(0)}$ to $l_t^{(1)}$ changes expected profit per policy in force at the start of the year by $$ \Delta\pi_t=(1-q_t)\bigl(l_t^{(1)}-l_t^{(0)}\bigr)(V_t-S_t). $$ A higher lapse rate increases this year's profit through the reserve-and-surrender component if $V_t>S_t$, reduces it if $V_t<S_t$, and has no direct effect through that component if $V_t=S_t$.

Worked Calculations

Worked Example: Isolating the Reserve Release in a Lapse Shock

A hypothetical annual policy has an opening reserve of $800, a premium of $1,200, a start-of-year expense of $100, a death benefit of $10,000, a closing reserve of $1,500, and a surrender payment of $600. The annual effective investment return is 5%. The probability of death during the year is 0.02. Among survivors, the lapse rate rises from 0.05 to 0.15. Calculate the expected profit under each lapse assumption and the change.

Method. Accumulate the opening reserve, premium, and expense to year-end, then subtract expected death outgo, surrender outgo, and reserves for policies continuing after year-end. The death probability is unchanged by the shock.

The accumulated amount before year-end outgo is:

$$ (800+1{,}200-100)(1.05)=1{,}900(1.05)=1{,}995. $$

Expected death outgo is $0.02(10{,}000)=200$. Under the original lapse rate, the probability of lapse is $0.98(0.05)=0.049$, and the probability of continuing is $0.98(0.95)=0.931$. Therefore:

\pi^{(0)}=1{,}995-200-(0.049)(600)-(0.931)(1{,}500) =1{,}995-200-29.40-1{,}396.50 =369.10.

Under the shocked lapse rate, the lapse probability is $0.98(0.15)=0.147$, and the continuation probability is $0.98(0.85)=0.833$. Hence:

\pi^{(1)}=1{,}995-200-(0.147)(600)-(0.833)(1{,}500) =1{,}995-200-88.20-1{,}249.50 =457.30.

The increase is $457.30-369.10=\$88.20$. Check it using the marginal formula: $(1-0.02)(0.15-0.05)(1{,}500-600)=0.98(0.10)(900)=\$88.20$. The result agrees. The common mistake is to count only the additional surrender payment, which rises by $58.80, and miss that expected closing reserves fall by $147.00. The net change is a profit increase of $88.20.

Worked Example: Building a Two-Year Profit Signature

A hypothetical two-year contract has a 5% annual effective return, no mortality, a $500 premium and $50 expense at the start of each year, and an opening reserve of $0. The year-1 closing reserve is $400 and the year-1 surrender payment is $250. At the end of year 2, a continuing policy receives a maturity benefit of $1,000; a policy lapsing in year 2 instead receives $550. The year-2 closing reserve is zero. Compare a baseline lapse rate of 0.10 in each year with a shock setting the year-1 rate to 0.20 and the year-2 rate to 0.30. Calculate the expected profit signature per original policy and its present value at 5%.

Method. First calculate annual profit per policy in force at the start of each year. Then weight each year's profit by the probability of being in force at that year's start. This distinction is essential: year-2 profit is not earned by every original policy.

For year 1, the accumulated amount is $(0+500-50)(1.05)=\$472.50$. At lapse rate $l_1$, year-1 profit per policy in force at its start is $472.50-l_1(250)-(1-l_1)(400)$. Under the baseline rate:

\pi_1^{(0)}=472.50-0.10(250)-0.90(400)=472.50-25-360=\$87.50.

Under the shock:

\pi_1^{(1)}=472.50-0.20(250)-0.80(400)=472.50-50-320=\$102.50.

The year-2 opening reserve is $400 per policy still in force. The accumulated amount is $(400+500-50)(1.05)=\$892.50$. Since the closing reserve is zero, profit per policy in force at the start of year 2 is $892.50-l_2(550)-(1-l_2)(1{,}000)$. Under the baseline:

\pi_2^{(0)}=892.50-0.10(550)-0.90(1{,}000) =892.50-55-900=-\$62.50.

Under the shock:

\pi_2^{(1)}=892.50-0.30(550)-0.70(1{,}000) =892.50-165-700=\$27.50.

The baseline probability of being in force at the start of year 2 is $1-0.10=0.90$, so the expected profit signature per original policy is $(\$87.50,-\$56.25)$, because $0.90(-62.50)=-56.25$. The shocked year-1 lapse rate makes the year-2 start-of-year persistency $0.80$, giving signature $(\$102.50,\$22.00)$, because $0.80(27.50)=22.00$.

Discount at 5% to time 0. The baseline present value is:

87.50v+(-56.25)v^2 =\frac{87.50}{1.05}-\frac{56.25}{1.05^2} =83.3333-51.0204=\$32.31.

The shocked present value is:

102.50v+22.00v^2 =\frac{102.50}{1.05}+\frac{22.00}{1.05^2} =97.6190+19.9546=\$117.57.

Amounts are rounded to the nearest cent only at the final step. The signatures and present values are per original policy. The common mistake is to add year-2 profit per continuing policy directly to year-1 profit without applying year-2 persistency.

Worked Example: When a Higher Lapse Rate Reduces Profit

A hypothetical policy has an opening reserve of $500, a premium of $300, a start-of-year expense of $20, and a closing reserve of $700. The annual effective return is 4%. The probability of death is 0.01, the death benefit is $5,000, and the surrender payment is $900. The lapse rate among survivors increases from 0.05 to 0.15. Calculate the change in expected profit per policy in force at the start of the year.

Method. Compare lapse-related reserve release and surrender outgo while holding the other terms fixed. First, the accumulated amount is $(500+300-20)(1.04)=780(1.04)=\$811.20$, and expected death outgo is $0.01(5{,}000)=\$50$.

At the original lapse rate, expected surrender outgo is $0.99(0.05)(900)=\$44.55$, while expected closing reserves are $0.99(0.95)(700)=\$658.35$. Thus:

\pi^{(0)}=811.20-50-44.55-658.35=\$58.30.

At the shocked rate, expected surrender outgo is $0.99(0.15)(900)=\$133.65$, and expected closing reserves are $0.99(0.85)(700)=\$589.05$. Therefore:

\pi^{(1)}=811.20-50-133.65-589.05=\$38.50.

The change is $38.50-58.30=-\$19.80$. The marginal formula confirms it: $0.99(0.15-0.05)(700-900)=-\$19.80$. Here the surrender payment exceeds the reserve released, so more lapses worsen current-year profit. This comparison holds all stated reserves and cash-flow assumptions fixed; it does not recalculate them using a new valuation basis.

From Annual Profit to a Profit Signature

Let $w_t$ be the probability that an original policy is in force at the start of year $t$. Under the annual model above, $w_1=1$ and $w_{t+1}=w_t(1-q_t)(1-l_t)$. If $\pi_t$ is the expected profit per policy in force at the start of year $t$, the expected profit for year $t$ per original policy is $\Pi_t=w_t\pi_t$. The sequence $(\Pi_1,\Pi_2,\ldots)$ is the profit signature on this per-original-policy basis.

w_1=1,\qquad w_{t+1}=w_t(1-q_t)(1-l_t),\qquad \Pi_t=w_t\pi_t.

For a discount rate $j$ and $v_j=1/(1+j)$, the present value of the expected profit signature through year $n$ is $\sum_{t=1}^{n}v_j^t\Pi_t$, assuming each year's profit is measured at year-end. The investment return $i$ used to accumulate funds during a year and the discount rate $j$ used to value the profit signature serve different purposes; they need not be equal.

The reason for the survival weighting is the law of total expectation: a policy contributes year-$t$ profit only if it is in force at the start of year $t$. Conditional on being in force, its expected profit is $\pi_t$; the probability of that condition is $w_t$. Thus the unconditional expected contribution per original policy is $w_t\pi_t$. This also explains why a lapse shock can affect the signature twice: it changes the current-year profit per policy through surrender and reserve release, and it changes the number of policies contributing to future years.

Common Traps and Practical Checks

Ignoring the conditional definition of lapse. In this model, the probability of lapse per policy in force at the start is $(1-q_t)l_t$, not $l_t$ alone. The probability of continuing is $(1-q_t)(1-l_t)$.
Counting a reserve release as a cash payment. The surrender payment is cash outgo. The reserve release is reflected by not setting aside a closing reserve for the terminated policy; its profit contribution relative to continuation is $V_t-S_t$.
Using lapse rates to weight the wrong year. A year-$t$ lapse changes future in-force counts, not the start-of-year premium already collected in year $t$, under the timing assumed here.
Assuming higher lapses must reduce profit. For the direct current-year reserve-and-surrender effect, the sign depends on $V_t-S_t$. Future premium and benefit effects still need to be assessed separately.
Mixing per-policy and per-original-policy amounts. Multiply the conditional profit $\pi_t$ by $w_t$ before using it in a profit signature per original policy.
Applying this formula under different event timing. If premiums cease immediately on lapse, lapses occur midyear, or death and lapse are modeled as competing decrements, the cash-flow probabilities and accumulation periods must match that specification.

Key takeaway: To assess a lapse shock, calculate the changed surrender outgo and closing reserve for the current year, then update persistency for later years. A sound profit signature keeps the conditional profit per in-force policy separate from the probability that the policy is still in force.

Check Your Understanding

Use the annual timing and conditional lapse definition stated above. Give monetary answers to the nearest cent where appropriate.

A policy has $V_{t-1}=\$200$, $P_t=\$400$, $E_t=\$20$, $V_t=\$500$, and $S_t=\$350$. There is no mortality, the annual effective return is 3%, and the lapse rate is 0.20. Find expected profit per policy in force at the start of the year. (Answer: $127.40)
A policy has $q_t=0.03$, $V_t=\$800$, and $S_t=\$600$. With all other assumptions fixed, the lapse rate rises from 0.10 to 0.18. Find the change in expected profit per policy in force at the start of the year. (Answer: increase of $15.52)
A year-1 lapse rate rises from 0.12 to 0.20. The year-1 death probability is 0.02, and the year-2 premium is $700 payable at the start of year 2. Find the change in expected year-2 premium per original policy, assuming no other cause of termination and no other change in assumptions. (Answer: decrease of $54.88)
At the start of a year, 600 original policies are in force. The death probability is 0.01 and the lapse rate among survivors is 0.10. Find the expected number still in force at the start of the next year. (Answer: 534.60 policies)
A lapse shock increases the lapse rate by 0.06. Among survivors, the closing reserve is $1,100 and the surrender payment is $1,250; the death probability is 0.04. Find the direct change in expected profit per policy in force at the start of the year from the reserve-and-surrender component. (Answer: decrease of $8.64)

What You'll Learn