Reserves Conditional on Policy Persistence

Why the Conditioning Matters

A reserve is attached to a policy’s state at a valuation date. For a universal life policy, that state can include its account value, benefit design, attained duration, and whether it is still in force. These details matter because a reserve calculated for a policy that is active is not the same quantity as an average across everyone who originally purchased the policy, including those who have since lapsed.

This distinction is especially important when interpreting portfolio results. A reserve conditional on being in force answers, “How much is held for an active policy with this state?” An average per original policy also reflects the chance that the policy is still active. Those quantities have different denominators and should not be interchanged.

Definition: Let $I_t=1$ indicate that a policy is in force at valuation time $t$, and let $Z_t$ denote its observed policy state at that time. The reserve conditional on persistence to time $t$ is the prospective value of future net outgo, given $I_t=1$ and the policy’s state $Z_t$. Under a positive-liability sign convention, $$ V_t^{\mathrm{inforce}}(Z_t) = E\!\left[ \operatorname{PV}_t(\text{future benefits and expenses} -\text{future premiums and charges}) \mid I_t=1,\ Z_t \right]. $$

The conditioning concerns the policy’s status at the valuation date. It does not, by itself, mean that the policy is assumed to remain in force forever after that date. A valuation may assume no future lapses, or may explicitly model future lapse-contingent cash flows. The examples below assume no future lapses, so the only persistence condition is that the policy is active at time $t$. A reserve based on a specified future-lapse assumption would use that assumption in its future cash flows.

Account Value and the Guarantee Component

A universal life valuation may separate the account-value obligation from an additional insurer-funded guarantee obligation. The account value is not automatically identical to the total reserve: the total depends on the valuation basis and on the future benefits, charges, expenses, and premiums included. To make the calculations transparent, the worked examples use a stated two-part convention: the account-value component is held at its current value, and an additional reserve is calculated for a specified death-benefit amount above the account value. The supplemental amount is paid by the insurer when death occurs.

Under that convention, the total reserve is the account-value component plus the prospective reserve for the supplemental benefit. The supplemental reserve is conditional on the policy being in force at the valuation date. In these examples, no future premiums, charges, expenses, or lapses affect that component.

Worked Example: One-Year Supplemental Death Benefit

Method. Calculate the expected supplemental benefit at the end of the year, discount it to the valuation date, then add the account-value component. A hypothetical policy is in force at time $t$, with account value \$80,000 and a supplemental death benefit of \$20,000 payable at the end of the year if death occurs during that year. The annual mortality probability is $q=0.010$, and the annual effective valuation interest rate is $i=0.04$. Assume no future lapse, premiums, charges, or expenses. The account-value component of the reserve is \$80,000.

The probability of death during the year is $0.010$, so the expected supplemental payment at year-end is $0.010(20{,}000)=\$200$. The discount factor is $v=1/(1.04)=0.96153846\ldots$. Therefore, the supplemental reserve is

V_t^{\mathrm{supp}} = \frac{0.010(20{,}000)}{1.04} = \frac{200}{1.04} = 192.30769\ldots

The total reserve under the stated two-part convention is $80{,}000+192.30769\ldots=\$80{,}192.31$, rounded to the nearest cent.

Check. The supplemental reserve is less than the \$200 expected year-end payment because the payment is discounted. The calculation is conditional on the policy being in force at time $t$; it does not multiply by the probability that the policy survived or avoided lapse before $t$. The common mistake avoided here is applying a past persistence probability to a reserve for a policy already known to be active.

Deriving a Multi-Year Conditional Reserve

For a policy already in force at time $t$, future mortality probabilities are conditional on being alive and in force at the start of each future period. Suppose supplemental benefits $G_k$ are payable at the end of year $k$ if death occurs in that year. Let $q_k$ be the probability of death during year $k$, conditional on being alive and in force at its start. If there are no future lapses, the probability of surviving the first $k-1$ years and then dying in year $k$ is the product of the relevant conditional survival probabilities and the year-$k$ death probability.

With a constant annual effective valuation rate $i$, the reserve for these supplemental benefits is the sum of each expected payment discounted from its payment date. In the two-year case, the probability of a year-two payment is $(1-q_1)q_2$: the policyholder must survive year one and then die in year two.

V_t^{\mathrm{supp}} = \frac{q_1G_1}{1+i} + \frac{(1-q_1)q_2G_2}{(1+i)^2}

This expression follows by partitioning possible death times. A year-one death produces $G_1$ with probability $q_1$; a year-two death produces $G_2$ with probability $(1-q_1)q_2$. Multiplying each payment by its probability gives its expected value at the payment date, and discounting by the corresponding power of $1+i$ brings it back to time $t$. The terms are then added because only one of these mutually exclusive death outcomes occurs.

Worked Example: Two-Year Benefit with Changing Amounts

Method. Use conditional death probabilities for each policy year, discount each year’s expected payment separately, and add the account-value component. At time $t$, a hypothetical in-force policy has an account value of \$50,000. Its supplemental death benefit is \$10,000 if death occurs in year one and \$8,000 if death occurs in year two. Assume $q_1=q_2=0.020$, an annual effective valuation rate $i=0.04$, no future lapse, and no premiums, charges, or expenses. The account-value reserve component is \$50,000.

The probability of a year-one death payment is $0.020$. Its expected payment at the end of year one is $0.020(10{,}000)=\$200$, with present value $200/1.04=\$192.30769\ldots$.

The probability of surviving year one and dying in year two is $(1-0.020)(0.020)=0.98(0.020)=0.0196$. The expected year-two payment is therefore $0.0196(8{,}000)=\$156.80$. Its present value is $156.80/(1.04)^2=156.80/1.0816=\$144.9704\ldots$.

Thus the supplemental reserve is $192.30769\ldots+144.9704\ldots=\$337.2781\ldots$. Adding the \$50,000 account-value component gives a total reserve of \$50,337.28, rounded to the nearest cent.

Check. The year-two probability includes survival through year one, and the year-two payment is discounted for two years. The answer is slightly above \$50,000 because the supplemental benefits have positive expected present value. A common error is to use $q_2G_2$ alone for year two, omitting the probability of reaching that year.

Conditional Reserves and Portfolio Averages

A portfolio average depends on which policies are included in the denominator. If terminated policies are assigned zero reserve and the in-force reserve is $V_t$, then the average reserve per original policy is not generally $V_t$. For a homogeneous group with persistence probability $p_t=P(I_t=1)$, the relationship is

E[I_tV_t] = P(I_t=1)E[V_t\mid I_t=1] = p_t V_t^{\mathrm{inforce}}.

The first equality follows because $I_tV_t=0$ when $I_t=0$, while $I_tV_t=V_t$ when $I_t=1$. Taking expectations therefore retains only the in-force cases. The conditional expectation among those cases is then multiplied by the probability of being in force. This calculation is an average at the valuation date, not a replacement for valuing a particular active policy.

For a heterogeneous portfolio, one conditional reserve cannot usually be multiplied by a single persistence probability unless it represents the appropriate average among current in-force policies. If the reserve levels differ by policy type, first calculate or identify the in-force average using the actual composition of that group. The following example shows both denominators.

Worked Example: In-Force Average Versus Original-Cohort Average

Method. Total the reserves for active policies, then divide by the relevant population count. A hypothetical cohort began with 1,000 policies. At a valuation date, 600 policies of type A remain in force, each with a reserve of \$500; 200 policies of type B remain in force, each with a reserve of \$900; and the other 200 policies have lapsed and are assigned zero reserve.

The total reserve for the in-force policies is $600(500)+200(900)=300{,}000+180{,}000=\$480{,}000$. There are $600+200=800$ in-force policies, so the average reserve per in-force policy is $480{,}000/800=\$600$.

There are 1,000 policies in the original cohort. The average reserve per original policy, including the zero reserves for lapsed policies, is $480{,}000/1{,}000=\$480$. The cohort persistence proportion is $800/1{,}000=0.80$, and $0.80(\$600)=\$480$, which checks the result.

Check. The \$600 amount describes the average conditional reserve among policies known to be active. The \$480 amount describes the average per original policy. The common mistake avoided is dividing the total reserve by 1,000 and then calling the result the reserve for an in-force policy.

What the Conditioning Does—and Does Not—Mean

A conditional reserve is based on the policy state actually being valued. If an individual policy is known to be active, its reserve is not reduced because some other policies in the original cohort have lapsed. Likewise, a lapsed contract is not assigned the reserve of an active contract merely because it belonged to the same cohort.

The conditioning also does not erase the need for a valuation basis. The reserve still depends on assumptions about future mortality, interest, benefits, expenses, and any future premiums or charges included in the model. If future lapses are included, the model must specify their effect on future payments and termination cash flows. If no future lapses are included, as in the examples here, the reserve is conditional on being active now but assumes continuation for the projected periods unless another termination event occurs.

Use the following sequence to keep the quantities distinct:

Identify the valuation population.
Decide whether the calculation is for a specified active policy, the group of in-force policies, or the original cohort.

State the policy state and valuation assumptions.
Record the account value, benefit design, mortality and interest assumptions, and any future lapse, expense, premium, or charge assumptions.

Calculate the conditional reserve.
Value future net outgo given the policy is in force at the valuation date and has the specified state.

Aggregate only after defining the denominator.
For an original-cohort average, include terminated policies consistently, often with zero reserve after termination; for an in-force average, divide only by the active count.

Common Exam Traps

Applying past persistence twice. If the policy is observed to be in force at the valuation date, do not multiply its conditional reserve by the probability that it reached that date.
Confusing current conditioning with future continuation. “Conditional on being in force at time $t$” says nothing by itself about future lapses. State the future lapse assumption separately.
Using the wrong denominator. Total reserve divided by in-force count gives an in-force average; divided by original count it gives an original-cohort average when terminated contracts have zero reserve.
Ignoring policy mix. If active policy types have different reserve amounts, the in-force average must reflect their actual counts and reserve levels.
Calling account value the total reserve without checking the model. Account value may form one component, but additional guarantee, expense, or other obligations can make the total different.

Key takeaway: A reserve conditional on persistence to the valuation date is a value for a policy known to be active, given its current state. An average that includes policies that have lapsed is a different quantity, with a different population and denominator.

Check Your Understanding

For each problem, identify whether the requested amount is conditional on being in force or averaged across an original cohort, and show the key calculation.

A policy is in force with an account-value component of \$40,000 and a supplemental death benefit of \$6,000 payable at the end of one year. The annual death probability is 0.015 and the effective annual valuation rate is 0.05. There are no other cash flows or future lapses. Find the total reserve under the two-part convention used above. (Answer: \$40,085.71)
A policy is in force at time $t$. It pays a supplemental \$4,000 at the end of year one on death, or \$3,000 at the end of year two on death. The conditional death probabilities are $q_1=0.010$ and $q_2=0.020$, and the effective annual valuation rate is 0.03. Assume no future lapses. Find the supplemental reserve. (Answer: \$94.83)
A homogeneous cohort has a 0.75 probability of being in force at the valuation date. The reserve conditional on being in force is \$720. Terminated policies have zero reserve. Find the average reserve per original policy. (Answer: \$540)
A cohort began with 500 policies. At valuation, 300 active policies have reserves of \$250 each and 100 active policies have reserves of \$550 each; the remaining 100 have lapsed and have zero reserve. Find the average per in-force policy and the average per original policy. (Answer: \$325 per in-force policy; \$260 per original policy)
A cohort began with 500 policies. At valuation, 200 active policies each have a reserve of \$400, 100 active policies each have a reserve of \$900, and 200 policies have lapsed with zero reserve. Find the average conditional reserve among in-force policies and the average per original policy, then check using the persistence proportion. (Answer: \$566.67 per in-force policy; \$340 per original policy)

What You'll Learn