Why Lapse and Interest Must Be Considered Together
An interest-rate change can affect a long-term policy through more than one channel. It can change investment income and the value assigned to future payments. It can also change policyholders’ incentives to surrender: for example, a hypothetical policyholder may be more likely to lapse when newly available market rates make an existing contract less attractive. When a policy has a valuable guarantee, however, surrendering also gives up that guarantee. The remaining policyholders may therefore be a less numerous but more guarantee-exposed group.
The direction and size of the result depend on the contract. A higher lapse rate can reduce expected payments when the surrender value is below the benefit payable to a persisting policyholder. It can instead increase payments if surrender values are high. Likewise, a higher investment return can reduce a guarantee shortfall, but a higher rate used to discount future payments also reduces their present value. Those effects should be identified separately before they are combined.
The lapse probability is an assumption about behavior under a scenario, not a mechanical consequence of the interest rate. A model might specify a higher lapse probability when market rates rise, but that relationship should be supported by the product design and the scenario assumptions. In particular, lapse behavior can differ between policies with valuable guarantees and policies without them.
Expected Payments When Lapse Rates Change
For a cohort of \(N\) identical in-force policies, the expected year-end payments are the expected number of lapses times the surrender payment plus the expected number of persisting policies times the benefit. With a common lapse probability \(q\), the expected payment per initial policy is \(qS+(1-q)B\). This is a cash-flow calculation before discounting.
If \(S<B\), increasing \(q\) reduces expected payments, since more policies receive the smaller surrender payment instead of the larger persistence benefit. If \(S>B\), increasing \(q\) raises expected payments. The lapse rate alone does not determine the direction; the relative amounts payable on lapse and persistence matter.
Worked Example: Higher Lapses and a Fixed Maturity Guarantee
A hypothetical insurer has 1,000 identical one-year policies. Each policy has a maturity guarantee of $112 and a surrender payment of $104 if it lapses during the year. Premium assets of $100 per policy earn 6% in the baseline scenario and 9% in a higher-rate scenario. The assumed lapse probabilities are 10% and 35%, respectively. Ignore expenses, taxes, and all other cash flows. Calculate expected payments and profit per initial policy under each scenario.
Method. Calculate asset accumulation per policy, then calculate expected payments using the mutually exclusive lapse and persistence outcomes. Define one-year profit per initial policy as accumulated assets less expected payments.
Baseline scenario. Assets accumulate to \(100(1.06)=\$106.00\). Expected payments are \(0.10(104)+0.90(112)=10.40+100.80=\$111.20\). Thus profit is \(106.00-111.20=-\$5.20\) per initial policy.
Higher-rate scenario. Assets accumulate to \(100(1.09)=\$109.00\). Expected payments are \(0.35(104)+0.65(112)=36.40+72.80=\$109.20\). Thus profit is \(109.00-109.20=-\$0.20\) per initial policy.
The profit improves by \(-0.20-(-5.20)=\$5.00\) per policy. Of this improvement, $3.00 comes from the additional asset accumulation. The increased lapse probability replaces persistence benefits of $112 with surrender payments of $104 for an additional 25% of policies, reducing expected payments by \(0.25(112-104)=\$2.00\). These effects sum to the $5.00 improvement. For 1,000 policies, the corresponding profits are -$5,200 and -$200.
Check. In each scenario, expected payments lie between the two possible payments, $104 and $112. The higher-rate scenario has greater asset accumulation and more lapses, and both changes improve profit here because the surrender payment is below the guarantee. The calculation does not imply that every higher-rate scenario improves profit; different surrender terms or lapse assumptions can reverse the result.
Present Values and Reserve Effects
A reserve calculation also reflects the timing of payments. In a one-year model with a deterministic effective annual valuation rate \(i\), a payment at year-end has present value equal to its amount multiplied by \(v=1/(1+i)\). If the lapse payment \(S\) and persistence benefit \(B\) are paid at the same year-end, and \(q\) is the lapse probability during that year, the expected present value per in-force policy is:
This simplified reserve is the expected present value of the specified future benefit payments. It excludes future premiums, expenses, and other cash flows; a product reserve must include the terms required by its valuation basis. The discount rate \(i\) need not be the same as the asset yield that influences lapse behavior. They are set equal in the next example only as a stated simplifying assumption.
Worked Example: Separating the Lapse and Discount Effects on a Reserve
A hypothetical one-year policy pays $102 on lapse and $110 on persistence at the year-end. Under the baseline assumptions, \(q=0.10\) and the valuation rate is 4%. Under a higher-rate scenario, \(q=0.40\) and the valuation rate is 8%. Assume both payments occur at year-end and that the rates and lapse probabilities are scenario inputs. Calculate the reserve in each scenario and show one sequential decomposition of the change.
Baseline. Expected payment is \(0.10(102)+0.90(110)=10.20+99.00=\$109.20\). Therefore, \(V_0=109.20/1.04=\$105.00\).
Higher-rate scenario. Expected payment is \(0.40(102)+0.60(110)=40.80+66.00=\$106.80\). Therefore, \(V_0=106.80/1.08=\$98.8889\), or $98.89 to the nearest cent.
To separate the effects in sequence, first change the lapse probability while keeping the valuation rate at 4%. The intermediate reserve is \(106.80/1.04=\$102.6923\), or $102.69. The lapse change reduces the reserve by \(105.00-102.6923=\$2.3077\). Next change the valuation rate from 4% to 8%, holding the higher lapse probability fixed. The reserve falls by \(102.6923-98.8889=\$3.8034\). The total decrease is \(105.00-98.8889=\$6.1111\), which agrees with the sum of the two sequential changes before rounding.
Check. The lapse effect is downward because the surrender payment of $102 is less than the persistence benefit of $110. The higher discount rate also lowers the present value of a positive future payment. The numerical decomposition depends on the order in which the two assumptions are changed; it is a transparent sequential attribution, not a claim that the two effects are independent.
Guarantee Exposure Among Persisting Policies
For a maturity guarantee linked to an account, the guarantee cost is not necessarily the whole maturity benefit. It is the amount by which the guaranteed benefit exceeds the account value available to fund it. In a simple model with account value \(A\) and guaranteed maturity amount \(G\), the top-up per persisting policy is \(\max(G-A,0)\). A lapse may avoid that maturity top-up, but the policy may still require a surrender payment.
The expected guarantee top-up per initial policy is \((1-q)\max(G-A,0)\) when all persisting policies have the same account value and all lapsing policies leave the guarantee before maturity. This expression shows two distinct channels: the rate scenario can change the top-up per persisting policy through \(A\), and it can change the number of persisting policies through \(q\). The formula does not cover contracts where a guarantee is payable on surrender or where account values and lapse behavior vary across policies.
Worked Example: Market Returns, Lapse, and a Maturity Floor
A hypothetical one-year contract receives a single $100 premium at issue. At the year-end, a policy that lapses receives a $101 surrender payment. A policy that persists receives the greater of its account value and a guaranteed $108. In a low-rate scenario, the account earns 2% and the lapse probability is 5%. In a high-rate scenario, the account earns 8% and the lapse probability is 40%. Assume no other payments, expenses, taxes, or capital costs. For each scenario, calculate the account value, guarantee top-up per persisting policy, expected total payment per initial policy, and one-year profit per initial policy.
Low-rate scenario. The account value is \(100(1.02)=\$102\). A persisting policy receives \(\max(102,108)=\$108\), so the guarantee top-up is \(108-102=\$6\). Expected top-up per initial policy is \(0.95(6)=\$5.70\). Expected total payment is \(0.05(101)+0.95(108)=5.05+102.60=\$107.65\). Assets accumulated per initial policy are $102, so profit is \(102-107.65=-\$5.65\).
High-rate scenario. The account value is \(100(1.08)=\$108\). A persisting policy receives \(\max(108,108)=\$108\), so there is no guarantee top-up. Expected total payment is \(0.40(101)+0.60(108)=40.40+64.80=\$105.20\). Profit is \(108-105.20=\$2.80\).
The low-rate scenario has a $6 shortfall for each persisting policy, whereas the high-rate scenario has none. The low-rate lapse probability also leaves 95% of the initial policies exposed to that shortfall. The higher-rate scenario has both a higher account value and more lapses, but the effects should not be confused: the account return eliminates the guarantee shortfall, while lapses change the mix of surrender and maturity payments. In this example, the calculation treats all accumulated assets as available against the stated payments and includes no separate cost of capital.
Check. In the low-rate scenario, the guarantee adds $6 only for the 95% who persist, giving expected top-up of $5.70. In the high-rate scenario, the account already equals the guarantee, so the top-up is zero. The expected total payment includes surrender payments in both scenarios; it is not equal to the expected guarantee top-up.
Common Mistake: Applying Lapse to the Wrong Cash Flow
A frequent error is to multiply the guaranteed maturity payment by the lapse probability, or to assume that a policy which lapses generates no payment. In the first worked example, the correct higher-rate expected payment is \(0.35(104)+0.65(112)=\$109.20\). A mistaken calculation of \(0.65(112)=\$72.80\) omits the surrender payments on 35% of policies. It understates expected payments by \(0.35(104)=\$36.40\) per initial policy.
The correction is to write down all mutually exclusive outcomes before applying probabilities: lapse means a surrender payment; persistence means the applicable benefit, including any guarantee. Then apply discounting only after the expected cash flows and their payment dates are clear. For a multi-year contract, lapse probabilities must also be applied at the appropriate durations; a single one-year lapse probability cannot be reused as if it were an annual probability over the whole term.
Check Your Understanding
For each problem, treat lapse and persistence as mutually exclusive outcomes and use the stated assumptions only.
- A one-year hypothetical policy pays $110 on lapse and $120 on persistence, with lapse probability 25%. What is the expected year-end payment per initial policy? (Answer: $117.50)
- A policy pays $105 on lapse and $125 on persistence at year-end. The lapse probability is 20%, and the effective annual valuation rate is 5%. Calculate the one-year expected present value of the payments. (Answer: $115.24)
- A cohort has $200 of assets per initial policy, earning 3% for one year. A lapse pays $205, persistence pays a guaranteed $220, and the lapse probability is 10%. Ignoring all other cash flows, calculate profit per initial policy. (Answer: -$12.50)
- A one-year contract pays $90 on lapse and $130 on persistence. In one scenario, \(q=0.15\) and \(i=3\%\); in another, \(q=0.45\) and \(i=7\%\). Calculate the expected present value in each scenario. (Answer: $120.39 in the first scenario and $104.67 in the second)
- A hypothetical one-year policy has $105 of accumulated assets per initial policy, a $98 lapse payment, and a $110 persistence benefit. What lapse probability makes expected payments equal accumulated assets, so that profit is zero before expenses? (Answer: \(5/12\), or 41.67%)