Reserves for Reduced Paid-Up Participating Policies

What Changes When a Policy Becomes Reduced Paid-Up?

A policyholder who stops paying premiums may, under the policy terms, retain insurance with a smaller guaranteed benefit instead of continuing the original contract or ending the cover. This is called reduced paid-up insurance. The reduced benefit is fixed by the contract or by an agreed conversion basis; it is not automatically equal to the original sum assured.

For a participating policy, conversion does not by itself remove benefits already declared or make future discretionary bonuses certain. Valuation must reflect the reduced guaranteed benefit, any vested bonuses included in the claim amount, and any future bonuses or terminal bonus allowed for on the valuation basis. Because the policy is paid-up, no future premiums are included. The examples below use hypothetical policy terms and assumptions.

Definition: A reduced paid-up participating policy is a participating policy that continues after future premiums cease, with its guaranteed benefits reduced according to the contract or conversion method. Its prospective reserve is the expected present value of covered future benefits and any included future expenses, less the expected present value of future premiums. For a fully paid-up policy, the future-premium term is zero.

The reduced amount may be determined by a contractual scale, by a formula based on premiums paid, or, in a specified conversion arrangement, by equating the value of the revised benefits to a value available at conversion. Those methods need not produce the same amount. The valuation method should therefore begin with the actual reduced benefits, rather than assume a universal conversion formula.

Projecting the Benefits and Claim Times

For a simple annual model, suppose the policy is valued at time 0 and claims are paid at the end of the year of death. Let $q_k$ be the conditional probability of death during year $k$, given survival to its start, and let ${}_{k}p$ be the probability of surviving $k$ years from the valuation date. Then the probability of a death claim in year $k$ is ${}_{k-1}p q_k$. If a maturity benefit is payable at the end of year $n$, its probability is ${}_{n}p$.

Let $D_k$ denote the claim amount for a death in year $k$, and $M_n$ the maturity amount. With an effective annual interest rate $i$, $v=1/(1+i)$. The prospective value of these benefits is:

\operatorname{EPV}(\text{benefits}) = \sum_{k=1}^{n} v^k\,{}_{k-1}p q_k D_k + v^n\,{}_{n}p M_n.

If death and maturity benefits have different amounts, use the applicable amount for each event. If the contract has no maturity payment, omit the maturity term; survival to the end of the term then produces no claim payment. Any expenses included in the valuation basis are added to expected present value of benefits, while future premiums, if any, are subtracted.

Previously declared reversionary bonuses are generally vested under the stated policy terms and should be included in the applicable guaranteed claim benefit. A future reversionary bonus is not yet vested: include it only as a projection consistent with the valuation basis and the policy’s bonus rules. A terminal bonus may be payable only at claim and should be included only where the assumed scale provides for it. These distinctions matter because the timing and probability of payment differ.

Why the Prospective Formula Works

The formula follows by partitioning possible outcomes according to when a claim occurs. The events “death during year 1,” “death during year 2,” and so on are mutually exclusive. The event “survival to maturity” is also separate from deaths during the term. Each event’s claim amount is multiplied by its probability and discounted from its payment time; summing these expected present values gives the total.

Find each claim probability.
For death in year $k$, use survival to the start of that year multiplied by the conditional probability of death: ${}_{k-1}p q_k$.

Determine the claim amount at that time.
Include the reduced guarantee, bonuses vested by then, and any projected claim bonus applicable under the stated basis.

Discount and add the event values.
Multiply each event probability and amount by the discount factor for its payment time. Add any maturity value separately.

This reasoning also explains why a reduced paid-up reserve is not obtained simply by multiplying the original reserve by the ratio of the reduced sum assured to the original sum assured. The payment timing, vested bonuses, future bonus projection, and any expenses may not scale in the same proportion as the guaranteed sum.

Worked Example: Two-Year Endowment With a Projected Bonus

A hypothetical two-year reduced paid-up endowment has a guaranteed amount of $10,000. Death or survival to maturity produces a payment. The effective annual interest rate is 5%. The probability of death in year 1 is 0.10; conditional on surviving year 1, the probability of death in year 2 is 0.20. If the policy survives to the end of year 1, a reversionary bonus of 2% of the original reduced amount, or $200, is added. A terminal bonus of 5% of the amount including vested reversionary bonuses is paid on death or maturity. Find the reserve at time 0.

Method. Project the claim amount and probability at each payment time, then discount the expected payments. There are no future premiums.

A year-1 death occurs with probability 0.10, before the year-1 bonus is added. The claim basis is $10,000, so the terminal bonus is $0.05(10{,}000)=\$500$. The year-1 claim amount is $10,500.

The probability of surviving year 1 is $1-0.10=0.90$. If alive, the $200 reversionary bonus is added, making the amount for a year-2 death or maturity $10,200. The terminal bonus is $0.05(10{,}200)=\$510$, so the year-2 claim amount is $10,710.

The year-2 death probability from the original valuation date is $0.90(0.20)=0.18$. The maturity probability is $0.90(0.80)=0.72$. Both events pay $10,710, so the total probability of a year-2 payment is $0.18+0.72=0.90$.

V_0 = \frac{0.10(10{,}500)}{1.05} + \frac{(0.18+0.72)(10{,}710)}{1.05^2} = 1{,}000+\frac{9{,}639}{1.1025} = 1{,}000+8{,}742.8571 = \$9{,}742.86.

The reserve is $9,742.86, rounded to the nearest cent. As a check, the undiscounted expected year-1 payment is $1,050 and the undiscounted expected year-2 payment is $9,639; discounting each at its actual payment time gives the stated reserve. The common mistake avoided here is applying the year-1 bonus to a year-1 death claim, even though the policy must survive to the bonus date.

Conversion Value and Later Reserves

A conversion basis may use the value of the benefits available at the conversion date to determine the reduced amount. If the conversion rule is an equivalence rule, it sets the value of the revised benefits equal to the specified amount available for conversion. This is a rule for determining the revised benefit, not a universal property of every reduced paid-up policy. Contractual conversion terms may use a different method.

After conversion, a reserve at a later duration is prospective from that duration. It reflects only benefits and expenses still possible from that point, using the conditional mortality assumptions and current bonus status. A bonus declared since conversion may increase vested benefits, while projected future bonuses remain assumptions rather than certain payments.

Worked Example: Reduced Amount From an Equivalence Rule

At conversion, a hypothetical two-year endowment has an amount of $$B$ to be determined. Its valuation basis has an effective annual interest rate of 4% and a 0.20 conditional death probability in each year. Death or survival to maturity pays the amount then applicable. If the policy survives year 1, a projected reversionary bonus equal to 1% of $B$ is added for a year-2 claim. The conversion rule equates the benefit value to the available reserve, set at exactly $10{,}000(635/676)$ dollars. Find $B$.

Method. Express the expected present value of claims as a multiple of $B$, then apply the conversion equivalence. No future premiums are payable.

A year-1 claim has probability 0.20 and amount $B$. Its present value is $0.20B/1.04$. If the policy survives year 1, with probability 0.80, the year-2 amount is $1.01B$. A year-2 claim is certain conditional on surviving year 1 because the contract pays either on death or at maturity. Its present value from time 0 is $0.80(1.01B)/1.04^2$.

Since $0.20/1.04=5/26=130/676$, and $0.80(1.01)/1.04^2=505/676$, the total value is $635B/676$. Equating this to the specified available reserve gives:

\frac{635B}{676} = 10{,}000\frac{635}{676} \quad\Longrightarrow\quad B=\$10{,}000.

The reduced guaranteed amount is $10,000. The reserve amount is approximately $9,393.49, but the exact fraction in the question makes the equivalence exact. The common mistake avoided is valuing the year-2 benefit as $B$ rather than $1.01B$, or applying the bonus even when the policy has not survived to its declaration date.

Worked Example: A Death-Only Reserve After a Bonus Is Vested

At a valuation date, a hypothetical reduced paid-up term policy has one year of cover remaining. Its guaranteed amount is $15,000, and a previously declared $1,000 reversionary bonus is vested. A terminal bonus of 4% of the combined guaranteed amount and vested bonus is payable on death. There is no maturity benefit. The effective annual interest rate is 5%, and the conditional probability of death during the remaining year is 0.25. Find the reserve.

Method. Value only the claim that can occur during the remaining term. The amount on death is $15{,}000+1{,}000=\$16{,}000$ before terminal bonus. The terminal bonus is $0.04(16{,}000)=\$640$, giving a total claim of $16,640.

The probability of that claim is 0.25, and it is paid at the end of the year. Survival produces no payment because this is death-only cover with no maturity benefit. Therefore:

V = \frac{0.25(16{,}640)}{1.05} = \frac{4{,}160}{1.05} = \$3{,}961.90.

The reserve is $3,961.90, rounded to the nearest cent. The common mistake avoided is treating survival as if it produced a claim payment: valuing $16,640 as certain would give $16,640/1.05, which is not the value of this death-only benefit.

Checks That Keep the Valuation Consistent

A reliable projection keeps the conversion terms, bonus timing, and valuation basis aligned. In particular, distinguish the amount that is guaranteed now from amounts projected to be declared later. If a projected bonus scale is used, it should be applied consistently to the benefit base specified by the policy; do not silently switch between the original sum assured and the reduced sum assured.

Use the actual reduced benefit. Do not assume that the original benefit remains in force or that a simple proportion of an earlier reserve determines the new reserve.
Include vested bonuses at the appropriate claim dates. Confirm whether a declared bonus is already part of the guaranteed claim amount.
Apply future bonuses only when the policy is eligible for them. A survival condition or declaration date can determine whether a bonus is included in a claim.
Separate death and maturity payments. Survival produces a payment only if the contract provides a maturity benefit.
Set future premiums to zero. A paid-up contract has no future premium income to subtract from its prospective benefit value.
Use a consistent basis. Mortality, interest, bonus projections, expenses, and payment timing must all refer to the same valuation basis.

The central valuation task is therefore to project the reduced paid-up contract as it now stands, event by event. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each question, identify the applicable claim amounts and timings before calculating.

A one-year reduced paid-up death-only policy has a $20,000 claim amount, including all vested benefits. A 3% terminal bonus is applied to that amount at death. The effective annual interest rate is 4%, and the probability of death is 0.08. What is the reserve? (Answer: $1,584.62)
A two-year reduced paid-up endowment has a $5,000 guaranteed amount. The effective annual interest rate is 5%; $q_1=0.10$ and $q_2=0.20$. A 2% bonus on $5,000 is added if the policy survives year 1. Death or maturity pays the applicable amount. Find the time-0 reserve. (Answer: $4,639.46)
A three-year reduced paid-up death-only policy has a $12,000 guarantee and a vested bonus of $600. The conditional death probability is 0.10 in each year and the effective annual interest rate is 4%. A $120 reversionary bonus is added at each anniversary survived, and no terminal bonus applies. Find the time-0 reserve. (Answer: $3,194.56)
A one-year reduced paid-up endowment pays $8,000 on death during the year or on survival to maturity. The effective annual interest rate is 5%, and the death probability is 0.20. There are no additional bonuses or expenses. Find the reserve and state why it does not depend on the mortality probability. (Answer: $7,619.05; a claim of $8,000 is paid at year-end whether death occurs or the policy survives.)

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