Lapse Transitions in Profit Tests and Reserves

Why Lapse Belongs in the State Model

A policy that lapses is no longer an active contract: future premiums and insured benefits stop, and a surrender payment may become due. Treating lapse only as a reduction in the number of policies can therefore omit a cash flow and misstate both the reserve for active policies and projected profit. In a profit test, lapse is a transition out of the active state, with its own probability and any associated surrender benefit.

The model below uses constant forces during each year to determine the probabilities of death, lapse, and continued activity. For the annual profit calculations, all transition benefits are treated as paid at the year-end. This timing convention is an assumption, not a consequence of the constant-force model. If surrender benefits are paid immediately when policies lapse, their payment times must instead be reflected in the cash-flow valuation.

Definition: Let an active policy be subject to a constant force of mortality $\mu$ and a constant force of lapse $\lambda$ during a year. Assume the two causes of exit act as competing transitions, so at most one occurs first. The probability of remaining active for one year is $p=e^{-(\mu+\lambda)}$. The probability of exit by death is $q_d=\frac{\mu}{\mu+\lambda}(1-p)$, and the probability of exit by lapse is $q_l=\frac{\lambda}{\mu+\lambda}(1-p)$, when $\mu+\lambda>0$. These probabilities satisfy $p+q_d+q_l=1$.

The probabilities are conditional on being active at the start of the year. In particular, $q_l$ is not simply $1-e^{-\lambda}$ when mortality also operates: some policies die before they can lapse. Under this model the same one-year probabilities apply in each year for which the forces are unchanged.

Worked Example: Converting Competing Forces to Transition Probabilities

A hypothetical policy has a constant annual mortality force $\mu=0.02$ and lapse force $\lambda=0.08$. Calculate the probabilities of remaining active, dying, and lapsing over one year. The year-end surrender payment is $60 if the policy lapses.

Method. Add the competing forces to obtain $\mu+\lambda=0.10$. Thus the probability of any exit is $1-e^{-0.10}=0.0951626$, and the exit probabilities are divided in proportion to their respective forces.

p=e^{-0.10}=0.9048374,\qquad q_d=\frac{0.02}{0.10}(0.0951626)=0.0190325,\qquad q_l=\frac{0.08}{0.10}(0.0951626)=0.0761301.

To four decimal places, the probabilities are 0.9048, 0.0190, and 0.0761, respectively. The expected year-end surrender payment per policy active at the start of the year is $60q_l=60(0.0761301)=\$4.57$, rounded to the nearest cent. The unrounded transition probabilities sum to $0.9048374+0.0190325+0.0761301=1.0000000$, as required.

Common mistake avoided: using $1-e^{-0.08}$ as the lapse probability ignores the competing mortality transition. The competing-risk calculation gives $q_l=0.0761301$, not $1-e^{-0.08}=0.0768837$.

Prospective Reserves with a Surrender Benefit

A prospective reserve for an active policy is the present value of future benefits and expenses less the present value of future premiums, conditional on the policy being active at the valuation time. If lapse terminates coverage and triggers surrender, the surrender payment is one of the future benefits in this calculation. A reserve for a terminating state is zero after all payments due at termination have been made.

For a one-year period with premiums and expenses at the start, and death, lapse, or maturity payments at the end, let $P$ be the premium, $E$ the expense, $B_d$ the death benefit, $B_l$ the surrender benefit, and $B_m$ the maturity benefit payable only if the policy remains active. At effective annual interest rate $i$, the reserve for an active policy at the start of this period is:

V=\frac{q_dB_d+q_lB_l+pB_m}{1+i}-(P-E).

This formula uses $P-E$ as the net cash inflow at the start of the period. It discounts the expected end-of-period payments one year, then subtracts that start-of-period net inflow. It applies only under the stated timing and benefit assumptions; different payment timing requires different discounting.

Worked Example: Valuing a Reserve Including Surrender

Consider a hypothetical two-year policy with effective annual interest rate $i=0.05$. At the start of each year, an active policy receives a premium of $100 and incurs an expense of $10. During each year, the forces are $\mu=0.02$ and $\lambda=0.08$. A death payment of $500 or surrender payment of $60 is made at the year-end on the corresponding transition. A policy still active at the end of year 2 receives $250. Calculate the reserve at time 1 for a policy active then, and the expected profit for the first year if the issue reserve is zero.

Method. First calculate the prospective reserve for the one remaining year. Then use the reserve change in the first-year profit calculation. The transition probabilities are those already calculated: $p=0.9048374$, $q_d=0.0190325$, and $q_l=0.0761301$.

The expected end-of-year-2 payments, conditional on being active at time 1, are:

$$ q_d(500)+q_l(60)+p(250) =0.0190325(500)+0.0761301(60)+0.9048374(250) =9.5163+4.5678+226.2094 =240.2934. $$

The net premium after expense at time 1 is $100-10=\$90$. Discounting the expected payment and subtracting that net premium gives:

V_1=\frac{240.2934}{1.05}-90 =228.8509-90 =\$138.85.

For the first-year profit, the net premium at issue accumulates to $90(1.05)=\$94.50$. The insurer pays the expected death and surrender benefits, and sets up $V_1$ only on policies still active. Thus:

\Pi_1=94.50-q_d(500)-q_l(60)-pV_1 =94.50-9.5163-4.5678-0.9048374(138.8509) =-\$45.22.

The calculation rounds only at the final step to cents. The negative profit is not a calculation error: under these hypothetical terms, the first-year accumulated premium less expense is insufficient to cover the expected transition payments and the reserve held for continuing policies. Common mistake avoided: subtracting $V_1$ for every policy active at issue would ignore that only the fraction $p$ remains active at year-end.

Annual Profit and the Effect of a Changed Lapse Assumption

For a discrete annual profit test, let $V_0$ be the reserve per active policy at the start of the year and $V_1$ the reserve per policy active at the end. With a start-of-year premium and expense, and year-end transition payments, expected profit per policy active at the start is:

\Pi_1=(V_0+P-E)(1+i)-q_dB_d-q_lB_l-pV_1.

The factor $p$ multiplies the ending reserve because that reserve is established only for policies continuing in force. If lapse or death has already triggered a payment and terminated coverage, the corresponding terminating policy does not also receive an active-state reserve.

A useful check is consistency between the reserve and the profit projection. If the prospective reserve uses the same transition assumptions, benefit amounts, interest rate, and timing as the profit test, a future year's expected profit can be zero even when that year's cash flows are substantial. The opening reserve is part of the funds available to meet those cash flows. When actual lapse differs from the lapse assumption used to set reserves, this consistency no longer holds automatically.

Key result: In a profit projection, lapse affects results through both the probability of reaching the active state and the surrender payment when lapse occurs. A changed lapse assumption can therefore change transition benefits, the number of reserves held, and future profit emergence. State clearly whether the projected transition probabilities also determine the reserve basis.

Comparing Lapse Intensities Through NPV

To compare assumptions, keep the product cash flows and interest rate fixed, change the lapse force, and recompute the transition probabilities. If reserves are also changed to match each assumption, the reserve itself changes. Alternatively, a profit test can use a fixed reserve basis while projecting different actual experience. These are different comparisons and should not be mixed.

For a two-year projection, let $\Pi_1$ denote expected profit at time 1 per policy active at issue, and $\Pi_2$ conditional expected profit at time 2 per policy active at time 1. Under a constant one-year continuation probability $p$, the NPV of expected profits at issue is:

\operatorname{NPV}=v\Pi_1+v^2p\Pi_2,\qquad v=\frac{1}{1+i}.

The factor $p$ converts conditional second-year profit to an expectation per policy active at issue. The factor $v^2$ discounts that expected profit from time 2 to issue. Do not multiply first-year profit by $p$: it is already calculated per policy active at the start of year 1 and includes the probabilities of that year's transitions.

Worked Example: Higher Lapse Experience with a Fixed Reserve Basis

Use the hypothetical two-year policy from the preceding worked example. The valuation reserve at time 1 remains $V_1=\$138.8509$, based on $\lambda=0.08$. Now project actual lapse at $\lambda=0.18$, with mortality force $\mu=0.02$, interest rate $i=0.05$, and the same premiums, expenses, and benefits. Calculate first-year expected profit, conditional second-year profit, and NPV.

Method. Recalculate the actual transition probabilities but keep the valuation reserve fixed. In year 2, include that reserve as opening funds; the reserve is not an extra benefit payment.

The total force is $0.20$, so:

p=e^{-0.20}=0.8187308,\qquad q_d=\frac{0.02}{0.20}(1-p)=0.0181269,\qquad q_l=\frac{0.18}{0.20}(1-p)=0.1631423.

The first-year expected death and surrender payments are $0.0181269(500)=\$9.0635$ and $0.1631423(60)=\$9.7885$. The ending reserve is set up only for the $p$ fraction still active. Therefore:

\Pi_1=94.50-9.0635-9.7885-0.8187308(138.8509) =94.50-9.0635-9.7885-113.6815 =-\$38.03.

For a policy active at time 1, the expected year-2 payments under the actual lapse force are:

0.0181269(500)+0.1631423(60)+0.8187308(250) =9.0635+9.7885+204.6827 =\$223.5347.

The year-2 funds before those payments are the accumulated opening reserve and the accumulated net premium: $(138.8509+90)(1.05)=\$240.2934$. The conditional year-2 profit is $240.2934-223.5347=\$16.7587$. Expected year-2 profit per policy active at issue is $0.8187308(16.7587)=\$13.7209$. Consequently:

\operatorname{NPV} =\frac{-38.0335}{1.05}+\frac{13.7209}{1.05^2} =-36.2224+12.4452 =-\$23.78.

All money amounts are rounded to cents only in the reported final results. The second-year profit is positive because the fixed reserve basis anticipated lower lapse than the actual projection; the reserve and accumulated net premium exceed expected year-2 payments. Common mistake avoided: using the fixed reserve but also recomputing year-2 profit as though it were a matched prospective reserve would erase the very basis difference being measured.

Common Lapse-Model Traps

Ignoring competing exits. With both mortality and lapse active, calculate each transition probability using its share of the total force, not by treating each cause as if it operated alone.
Omitting the surrender payment. A lapse removes future coverage but can create an immediate or year-end cash outflow. Include it once, at the assumed payment time.
Reserving for terminated policies. The active-state reserve is multiplied by the probability of remaining active. A terminating state has no continuing active reserve after its settlement.
Mixing actual and valuation assumptions. A fixed valuation reserve combined with changed actual lapse produces a different profit pattern from a reserve recalculated on the new lapse basis.
Discounting from the wrong date. A time-2 profit is discounted by $v^2$ to issue; its probability of reaching year 2 must also be included when profit is conditional on surviving year 1.

Careful treatment of lapse matters because surrender benefits and lost future premiums can make the direction of a profit change non-obvious. A higher lapse rate may reduce future active claims and reserves, but it can also increase surrender payments and reduce future premium income. The transition probabilities, payment timing, reserve basis, and profit definition must all be aligned before interpreting the result. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the stated timing assumptions and round probabilities to four decimals and money to cents.

A hypothetical contract has lapse force $\lambda=0.10$ and no mortality during a year. Find the one-year active probability and lapse probability. (Answer: $p=0.9048,\ q_l=0.0952$.)
With $\mu=0.05$ and $\lambda=0.10$, calculate the one-year death and lapse probabilities under competing forces. (Answer: $q_d=0.0464,\ q_l=0.0929$.)
A one-year policy has $\mu=0.02,\lambda=0.10,i=0.04$, start-of-year premium $80, expense $8, year-end death benefit $400, lapse payment $50, and maturity benefit $200 if active. Find the prospective reserve at the start of the year. (Answer: $110.34.)
A policy has lapse force $0.05$, no mortality, and no changes in force over two years. What is the probability of remaining active for both years and the probability of lapsing by year 2? (Answer: $p_{\text{2-year}}=0.9048$, probability of lapse by year 2 $=0.0952$.)
At $i=0.05$, a profit of $120 occurs at time 1, and conditional profit at time 2 is $40 if the one-year continuation probability is 0.90. Calculate NPV at issue. (Answer: $\frac{120}{1.05}+\frac{0.90(40)}{1.05^2}=\$146.94$.)

What You'll Learn