Decrement Basis Traps in Guarantee Valuations

Why the Decrement Basis Matters

A guarantee valuation depends not only on the amount payable, but also on which policies remain exposed to it. A death guarantee responds to mortality; a maturity guarantee may be payable only if the policyholder survives and the policy stays in force. Lapse can therefore reduce one expected claim while changing the exposure to another. A calculation can use plausible mortality and lapse rates and still be wrong if those rates come from incompatible decrement bases.

The key distinction is between a single-decrement probability, calculated as though one cause were the only cause of exit, and a multiple-decrement probability, which allows several mutually exclusive causes to remove a policy. Do not add or combine rates from different bases without first specifying how they are related.

Definition: A multiple-decrement model specifies the probabilities of exit from each cause during a period, where only the first exit can occur. For a one-year period, let $q_d$ be the probability of death, $q_l$ the probability of lapse, and $p$ the probability of remaining in force. Then $p+q_d+q_l=1$. A single-decrement probability for cause $j$, denoted here by $q_j^{(1)}$, is the probability of exit from cause $j$ over the period in a hypothetical model where that cause acts alone.

Those definitions describe different quantities. For instance, a net mortality probability from a single-decrement table is not automatically the probability of death in a year when lapse is also operating. A basis must say whether its inputs are cause-specific forces, multiple-decrement probabilities, or net single-decrement probabilities, and any conversion must follow that model.

Competing Risks Under Constant Forces

One useful model assumes constant cause-specific forces over a year. Let $\mu_d$ and $\mu_l$ be the force of death and force of lapse, respectively. The total force is $\mu=\mu_d+\mu_l$. The model assumes that, while the policy is in force, each cause has its stated constant force and the first exit ends exposure to the other cause.

p=e^{-(\mu_d+\mu_l)},\qquad q_d=\frac{\mu_d}{\mu_d+\mu_l}\left(1-e^{-(\mu_d+\mu_l)}\right),\qquad q_l=\frac{\mu_l}{\mu_d+\mu_l}\left(1-e^{-(\mu_d+\mu_l)}\right).

The formulas follow from the cause-specific event density. At time $t$, the probability of still being in force is $e^{-(\mu_d+\mu_l)t}$. The probability of death in the short interval from $t$ to $t+dt$ is approximately this survival probability multiplied by $\mu_d\,dt$. Integrating over the year gives the death probability; integrating the lapse density gives the lapse probability.

q_d=\int_0^1 e^{-(\mu_d+\mu_l)t}\mu_d\,dt =\frac{\mu_d}{\mu_d+\mu_l}\left(1-e^{-(\mu_d+\mu_l)}\right).

The same integration with $\mu_l$ gives $q_l$. Adding the two probabilities gives $1-e^{-(\mu_d+\mu_l)}$, and adding the probability of remaining in force gives one. This formula does not apply unchanged if the forces vary within the year, if the inputs are already multiple-decrement probabilities, or if the causes are not modeled as competing first exits.

Worked Example: One-Year Death Guarantee Under Competing Forces

A hypothetical linked policy has a one-year death top-up of $40,000, payable at the end of the year if death occurs during the year. There is no lapse benefit. Assume constant forces $\mu_d=0.03$ and $\mu_l=0.09$ per year, and an effective annual interest rate $i=4\%$. Find the issue-date expected present value of the top-up per policy.

Method. First calculate the multiple-decrement death probability. Then discount the end-of-year death payment. The payment timing is part of the problem: this is not a continuous payment calculation.

The total force is $0.03+0.09=0.12$. The probability of any exit is $1-e^{-0.12}=0.11307956$, and the probability of remaining in force is $e^{-0.12}=0.88692044$. Allocate the exit probability between causes in proportion to their forces:

q_d=\frac{0.03}{0.12}(0.11307956)=0.02826989,\qquad q_l=\frac{0.09}{0.12}(0.11307956)=0.08480967.

Check: $0.02826989+0.08480967+0.88692044=1.00000000$, subject to displayed rounding. The discount factor is $v=1/1.04=0.96153846$. Thus the expected present value is

40{,}000(0.02826989)(0.96153846)=\$1{,}087.30.

The result is $1,087.30 per policy, rounded to the nearest cent. A common mistake is to use the net single-decrement death probability $1-e^{-0.03}=0.02955447$ as the death probability in this model. That probability assumes lapse is absent, so it overstates the competing-risk death probability. The calculation above uses death and lapse on the same constant-force basis.

Converting Net Probabilities Before Combining Causes

If a basis gives net single-decrement probabilities, the underlying assumptions must be identified before they are combined. Under constant forces, a net probability $q_j^{(1)}$ corresponds to a cause-specific force $\mu_j=-\ln(1-q_j^{(1)})$ for a one-year period. The forces can then be combined using the competing-risk formulas. This is one specific conversion method, not a universal rule for every table or model.

Worked Example: Net Table Inputs and a Death-Only Claim

A hypothetical one-year guarantee pays $30,000 at year-end on death. A valuation file supplies net single-decrement probabilities $q_d^{(1)}=1-e^{-0.02}$ for death and $q_l^{(1)}=1-e^{-0.10}$ for lapse. Assume these probabilities arise from constant forces, both causes operate during the year, and $i=5\%$. Find the expected present value of the death claim per policy.

Method. Recover the cause-specific forces from the net probabilities, combine the forces, and then calculate the multiple-decrement death probability. The lapse itself produces no claim.

The corresponding forces are $\mu_d=-\ln(1-q_d^{(1)})=0.02$ and $\mu_l=-\ln(1-q_l^{(1)})=0.10$. Therefore, the total force is $0.12$, and

q_d=\frac{0.02}{0.12}(1-e^{-0.12}) =\frac{1}{6}(0.11307956) =0.01884659.

The discount factor is $v=1/1.05=0.95238095$. Since the claim is paid at year-end, the expected present value is

30{,}000(0.01884659)(0.95238095)=\$538.47.

The answer is $538.47 per policy, rounded to the nearest cent. Directly using $1-e^{-0.02}=0.01980133$ as the death probability would give about $565.75. That is the net death probability with lapse removed, not death under the stated two-cause model. The common error is not arithmetic; it is using a single-decrement probability as if it were a multiple-decrement probability.

Keep the Basis Consistent in Reserve Projections

A reserve for a guarantee is conditional on the policy being in force at the valuation time. In a projection from issue, however, only the probability of reaching that time in force weights the conditional reserve. This distinction is essential: a lapse probability before the valuation time affects the number of exposed policies, while the future reserve is calculated conditional on survival in force to the valuation time.

Worked Example: Two-Year Death Guarantee and the Year-One Reserve

A hypothetical policy pays a $20,000 top-up at the end of a year in which death occurs. For each of the next two years, assume the same annual competing-decrement probabilities: $q_d=0.04$, $q_l=0.08$, and $p=0.88$. The annual effective interest rate is $i=5\%$. Calculate the issue-date expected present value and the reserve at time 1 for a policy then known to be in force.

Method. At issue, weight the year-two claim by the probability of remaining in force through year one. For the time-1 reserve, condition on being in force at time 1; do not apply the first-year lapse probability again inside that reserve.

For a year-one death, the expected end-of-year payment is $20{,}000(0.04)=\$800$. Its issue-date present value is $800v=800/1.05=\$761.90$. For a year-two death, the probability of reaching year two in force and then dying is $0.88(0.04)=0.0352$. The issue-date present value of that expected payment is

20{,}000(0.88)(0.04)v^2 =704\left(\frac{1}{1.05^2}\right) =\$638.55.

The total issue-date expected present value is $\$761.90+\$638.55=\$1{,}400.45$. At time 1, conditional on being in force, only the second-year claim remains. Its time-1 reserve is

{}_1V=20{,}000(0.04)v=\$761.90.

A consistency check is that the issue-date value of the future reserve, weighted by the probability of reaching time 1 in force, equals the year-two claim contribution: $0.88v(761.90)=\$638.55$, subject to rounding. A common mistake is to omit the year-one in-force probability in the issue-date projection, which incorrectly treats every policy as exposed to the second-year claim. The opposite mistake is to multiply the conditional time-1 reserve by the first-year survival probability and still call the result the reserve for a policy known to be in force at time 1.

Guarantee Type Determines Which Decrements Matter

The same decrement basis can affect different guarantees in different ways. A death top-up is associated with death exits; lapse usually removes future exposure to that payment. A maturity guarantee payable only to a policy that survives to maturity depends on the probability of remaining in force through every preceding period. Higher mortality can reduce maturity exposure, while lower lapse can increase it. Looking at either rate alone can therefore give the wrong direction of change.

The following comparison isolates that interaction. It is an expected present value comparison under two specified deterministic bases, not a complete solvency-capital calculation. Capital requires a defined risk measure, aggregation method, and scenario or distribution framework in addition to the expected cash flows.

Worked Example: Maturity Guarantee Under a Joint Decrement Stress

A hypothetical two-year linked contract pays $15,000 at time 2 only if the policy is still in force at maturity. It pays nothing on death or lapse. The annual effective interest rate is $i=4\%$. Under the base basis, each year has competing probabilities $q_d=0.02$, $q_l=0.10$, and $p=0.88$. Under a stress basis, each year has $q_d=0.03$, $q_l=0.05$, and $p=0.92$. Calculate and compare the expected present values.

Method. The maturity payment occurs only after two successive in-force outcomes. Thus the probability of payment is the product of the annual in-force probabilities, and the payment is discounted for two years.

Under the base basis, the probability of reaching maturity in force is $0.88^2=0.7744$. Since $v^2=1/1.04^2=1/1.0816$, the expected present value is

15{,}000(0.7744)\left(\frac{1}{1.0816}\right) =\$10{,}739.64.

Under stress, the in-force probability to maturity is $0.92^2=0.8464$, so

15{,}000(0.8464)\left(\frac{1}{1.0816}\right) =\$11{,}738.17.

The stress-basis expected present value is higher by $\$11{,}738.17-\$10{,}739.64=\$998.53$, using the displayed rounded amounts. Here mortality increases, but lapse falls by more in the stated probabilities, so the combined probability of remaining in force rises from $0.88$ to $0.92$ each year. A common mistake is to label the scenario a mortality stress and conclude that the maturity guarantee must become less costly. The claim depends on all decrements that determine maturity exposure, not mortality in isolation.

A Practical Basis-Audit Sequence

Before accepting an integrated guarantee valuation, trace the decrement basis from inputs to cash flows. The same assumptions should be used in the exposure projection, the conditional reserve, and the valuation of claims unless a deliberate basis change is being tested. A basis change should be stated explicitly rather than introduced accidentally by mixing table entries.

Identify the input basis.
Record whether each mortality and lapse input is a force, a multiple-decrement probability, or a net single-decrement probability.

State the period and event model.
Specify the time unit, whether exits compete as first events, and any assumption used to convert between bases.

Match decrements to benefits.
Identify which exits trigger a payment and which exits simply remove future exposure.

Separate conditional reserves from projected exposure.
Calculate a reserve conditional on being in force; apply prior survival probabilities only when aggregating or projecting from an earlier time.

Check probabilities and timing.
Verify that probabilities are nonnegative and sum to one, then discount each payment from its stated payment time.

These checks are particularly useful in linked-product guarantee projections, where an account-value shortfall may be payable on death, at maturity, or under both conditions. A clean cash-flow definition prevents the mortality basis, lapse basis, and guarantee trigger from being conflated. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: A net single-decrement probability is not interchangeable with a competing-decrement probability. State the decrement model, convert inputs only under an explicit assumption, and apply survival exposure and discounting at the times the guarantee actually requires.

Check Your Understanding

For each problem, identify the decrement basis before calculating. Unless stated otherwise, payments occur at the end of the indicated year.

Constant forces are $\mu_d=0.01$ and $\mu_l=0.04$ per year. Find the one-year multiple-decrement death probability, lapse probability, and in-force probability. Give probabilities to four decimals. (Answer: $q_d=0.0098,\ q_l=0.0390,\ p=0.9512$.)
A policy has one-year competing probabilities $q_d=0.025$ and $q_l=0.075$. A $10,000 death benefit is paid at year-end, and $i=4\%$. Find the expected present value per policy. (Answer: $240.38.)
Net single-decrement probabilities arise from constant forces $\mu_d=0.02$ and $\mu_l=0.04$. Convert them to one-year multiple-decrement probabilities. Give probabilities to four decimals. (Answer: $q_d=0.0194,\ q_l=0.0388,\ p=0.9418$.)
A two-year maturity guarantee pays $8,000 only if the policy remains in force at time 2. In year 1, $q_d=0.04$ and $q_l=0.06$; in year 2, $q_d=0.03$ and $q_l=0.12$. Assume competing annual probabilities and $i=5\%$. Find the issue-date expected present value. (Answer: $5,551.02.)
A death top-up of $25,000 is paid at the end of year 2 on death in that year. The policy is known to be in force at time 1. In year 2, $q_d=0.04$ and $q_l=0.08$, on a competing-decrement basis, and $i=4\%$. Find the time-1 reserve. If the year-one in-force probability was $0.90$, also find the issue-date present value contribution of the year-two claim. (Answer: time-1 reserve $961.54; issue-date contribution $832.84.)

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