Worked Example With Multiple Bonus Mechanisms

Why Multiple Bonus Mechanisms Need Separate Treatment

A participating policy may provide more than one kind of bonus. Reversionary bonuses, once declared and vested, increase the guaranteed benefit. A terminal bonus may be added at a qualifying claim, with its amount determined by the contract’s stated rate and benefit base. Cash dividends may have been paid separately during the policy term, or may remain unpaid and be payable as a distinct accumulated balance.

The arithmetic is usually manageable. The difficulty is interpreting the contract correctly: which additions are guaranteed, whether the claim event qualifies for a terminal bonus, what amount that bonus is calculated on, and whether cash dividends are still payable. The method below treats each component as a separate item before assembling the claim. All products and values in the examples are hypothetical. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For a claim under a participating policy, the basic benefit is the contract’s stated benefit before bonus additions. A vested reversionary bonus is a declared addition that remains payable under the contract’s terms. A terminal bonus is an additional amount payable only when the claim satisfies the contract’s eligibility conditions. An accumulated cash dividend is a cash dividend that was declared but not paid out, together with any accumulation credited under the contract.

A Component-by-Component Claim Calculation

Suppose the contract specifies a basic benefit $B$, vested reversionary additions $R$, a terminal-bonus amount $T$ if the claim qualifies, and an unpaid cash-dividend balance $C$. If the contract makes all four items payable at the claim, and there are no deductions such as a policy loan, the claim amount $Q$ is

$$ Q = B + R + T + C. $$

This is a bookkeeping identity, not a rule that determines the contract terms. In particular, it does not say how $T$ is calculated. The terminal-bonus rate, its benefit base, and the claim events on which it is payable must come from the policy rules stated in the problem. Likewise, $C$ is included only if the dividends remain unpaid and the contract makes their accumulated value payable at the claim.

A useful sequence is to establish eligibility first, calculate each amount on its specified base, then add the separate payable components. That order helps keep the terminal bonus and cash dividends out of one another’s bases unless the contract explicitly includes them.

Identify the claim event and date.
For example, distinguish a maturity claim from a death claim and determine which declared bonuses exist by that date.

Establish eligibility.
Apply any minimum-duration, claim-event, or other terminal-bonus conditions before calculating a terminal-bonus amount.

Calculate each component once.
Use the stated base for the terminal bonus and accumulate any unpaid dividends only as the contract specifies.

Assemble and check the claim.
Add the payable components and verify that no item has been included in more than one component.

Why the Addition Works

The claim formula follows from the meaning of the contract components. If, at the claim date, the insurer owes a basic benefit $B$, vested additions $R$, an eligible terminal bonus $T$, and an unpaid dividend balance $C$, then each amount is a separate payable cash amount. The total owed is their sum. For example, when $T=rH$, where $r$ is the specified terminal-bonus rate and $H$ is the contract-defined base, substitution gives

$$ Q = B + R + rH + C, $$

provided the claim is eligible and $C$ is payable. If eligibility fails, set $T=0$. If there is no unpaid dividend balance, set $C=0$. These substitutions do not change the other components unless the contract says they are linked.

For an unpaid dividend $D_j$ declared at the end of year $j$, accumulated to a claim at the end of year $n$ at an effective annual rate $i$, the accumulated value is $D_j(1+i)^{n-j}$, provided the contract credits that rate and uses annual accumulation over those periods. This follows by applying the one-year accumulation factor $1+i$ once for each of the $n-j$ complete years. Thus, for several unpaid dividends,

C = \sum_j D_j(1+i)^{n-j}.

This expression does not apply if the contract uses a different crediting rule, different timing, or no interest. Use the terms given rather than imposing an assumed accumulation convention.

Worked Example: Maturity Claim With All Three Bonus Mechanisms

A hypothetical endowment policy matures at the end of year 12. Its basic maturity benefit is $60,000. Twelve annual reversionary bonuses of $1,200 each have been declared and vested. The policy qualifies for a terminal bonus of 8% of the sum of the basic benefit and vested reversionary bonuses. Three cash dividends of $300 each, declared at the ends of years 4, 8, and 12, remain unpaid; the contract credits no interest on them. Calculate the maturity claim.

Assumptions. The claim is known to be a maturity claim at the end of year 12; no mortality probability is needed to calculate this conditional claim. The contract’s dividend accumulation rate is 0% per year. No policy loan or other deduction applies. No probability distribution is used.

Method: calculate the benefit base, then each payable item separately. The total vested reversionary additions are

R = 12 \times \$1{,}200 = \$14{,}400.

The terminal-bonus base specified by the contract is

H = \$60{,}000 + \$14{,}400 = \$74{,}400.

Therefore the terminal bonus is

T = 0.08 \times \$74{,}400 = \$5{,}952.

The three unpaid dividends total

C = 3 \times \$300 = \$900.

Adding the four amounts once gives

Q = \$60{,}000 + \$14{,}400 + \$5{,}952 + \$900 = \$81{,}252.

Check. The basic benefit plus reversionary additions is $74,400; adding the terminal bonus gives $80,352; adding the separate dividend balance gives $81,252. The common mistake avoided here is to include the dividends in the terminal-bonus base even though the contract defines that base as the basic benefit plus reversionary additions only.

Eligibility and Benefit Bases Are Contract-Specific

A terminal-bonus rate by itself is not enough information to calculate a terminal bonus. A complete calculation needs both the applicable rate and the benefit base to which it applies. The base might be the basic benefit alone, or it might include some specified bonus additions. Cash dividends are not automatically part of that base.

The claim event also matters. A contract may provide a terminal bonus at maturity but not on a death claim, or it may make different provisions by claim event or duration. A rate should not be applied before checking the eligibility condition. In particular, a terminal bonus that is not payable is not a zero-rate bonus that can be calculated on an assumed base; its amount is simply zero for that claim.

Component	What to establish	Common calculation error
Basic benefit	The amount payable before bonus additions	Adding an amount already included in the quoted basic benefit
Reversionary additions	Which declared additions are vested by the claim date	Including future or undeclared bonuses
Terminal bonus	Eligibility, rate, and contract-defined base	Applying a rate to the wrong base or claim event
Cash dividends	Whether they remain unpaid and how they accumulate	Counting paid dividends again, or adding the balance to the terminal-bonus base without authority

Worked Example: Death Claim With a Restricted Terminal-Bonus Base

A hypothetical participating policy has a basic death benefit of $40,000. By the end of policy year 9, nine reversionary additions of $1,000 each have vested. The contract pays a 6% terminal bonus on death claims after policy year 5, calculated on the basic death benefit alone. Two cash dividends of $250 each, declared at the ends of years 3 and 6, remain unpaid and earn no interest. The insured dies at the end of year 9. Find the death claim.

Assumptions. The death in year 9 is given; no mortality model or probability is required for this conditional benefit calculation. The interest credited to unpaid dividends is 0% per year. There are no deductions, and both dividends remain payable.

Method: check eligibility before applying the rate. The claim occurs after year 5, so the death claim qualifies. The vested reversionary additions are

R = 9 \times \$1{,}000 = \$9{,}000.

The contract specifies the basic death benefit alone as the terminal-bonus base, so

T = 0.06 \times \$40{,}000 = \$2{,}400.

The unpaid dividend balance is

C = \$250 + \$250 = \$500.

Therefore,

Q = \$40{,}000 + \$9{,}000 + \$2{,}400 + \$500 = \$51{,}900.

Check. The terminal bonus is 6% of $40,000, or $2,400, exactly as required. A common incorrect calculation uses $49,000 (the basic benefit plus reversionary additions) as the base, giving $2,940. That calculation is wrong because this contract explicitly limits the base to the basic benefit.

Accumulating Unpaid Dividends to the Claim Date

An unpaid dividend may be recorded as a balance that accumulates under a stated interest convention. Its claim-date value is separate from the reversionary additions and terminal bonus unless the policy terms explicitly say otherwise. Pay attention to the dates: a dividend declared at the end of year 1 has two full years to accumulate to the end of year 3, while one declared at the end of year 2 has one.

Worked Example: Maturity Claim With Interest on Cash Dividends

A hypothetical policy pays a $25,000 basic maturity benefit at the end of year 3. Three annual reversionary bonuses of $500 each have vested. The terminal bonus is 20% of the basic benefit plus vested reversionary additions. Cash dividends of $200, $300, and $400 are declared at the ends of years 1, 2, and 3, respectively, and remain unpaid. The contract accumulates each unpaid dividend at an effective annual rate of 4%. Calculate the maturity claim.

Assumptions. The claim occurs at the end of year 3 and is eligible for the terminal bonus. Dividend accumulation is annual at $i=0.04$, with each dividend accumulated from its declaration date to the claim date. No mortality distribution is used because the claim event is specified. No deductions apply.

Method: accumulate each dividend for its own number of complete years. The vested additions are

R = 3 \times \$500 = \$1{,}500.

The terminal-bonus base and amount are

H = \$25{,}000 + \$1{,}500 = \$26{,}500, \qquad T = 0.20 \times \$26{,}500 = \$5{,}300.

The year-1 dividend accumulates for two years:

\$200(1.04)^2 = \$200(1.0816) = \$216.32.

The year-2 dividend accumulates for one year:

\$300(1.04) = \$312.00.

The year-3 dividend is declared at the claim date, so it earns no accumulation interest:

\$400(1.04)^0 = \$400.00.

Thus the accumulated cash-dividend balance is

C = \$216.32 + \$312.00 + \$400.00 = \$928.32.

The final claim is

Q = \$25{,}000 + \$1{,}500 + \$5{,}300 + \$928.32 = \$32{,}728.32.

Check. The claim-date dividend balance exceeds the original $900 total by $28.32, which is the credited accumulation. The common timing error is to accumulate all three dividends for two years; only the first dividend has two full years to accumulate.

Common Traps and a Final Reconciliation

When a problem contains several bonus mechanisms, write down the contract rule beside each component before doing arithmetic. This prevents a correct calculation from being applied to the wrong amount or at the wrong time.

Applying the terminal-bonus rate before checking eligibility. Verify the claim event and duration condition first. If the condition fails, the terminal bonus is zero.
Assuming a standard terminal-bonus base. Use the base stated in the contract; do not automatically include reversionary additions or cash dividends.
Counting a dividend twice. A dividend already paid is not also an unpaid balance due at claim. An unpaid dividend is included as a separate amount only once.
Accumulating every dividend for the same period. Determine each dividend’s declaration date and the number of accumulation periods before the claim.
Adding an undeclared bonus. Include only vested additions that exist by the claim date, not projected future bonuses.

A final reconciliation is to list the four potential components and mark each as included, excluded, or zero under the contract rules. Then add only the payable amounts. For example, a nonqualifying terminal bonus is recorded as zero, while a dividend already paid is excluded rather than added to the unpaid balance. This simple check is especially useful when a policy has several declaration dates or distinct rules for maturity and death claims.

Key takeaway: A multiple-bonus claim is assembled from distinct payable components. Determine eligibility, use the specified terminal-bonus base, accumulate only unpaid dividends under the stated convention, and add each amount once.

Check Your Understanding

For each hypothetical policy, calculate the claim amount from the stated contract rules. Amounts are in dollars, and there are no deductions unless stated.

A maturity benefit is $30,000. Five vested reversionary additions of $600 each are payable. The terminal bonus is 5% of the basic benefit plus reversionary additions. Two unpaid cash dividends of $250 and $350 earn no interest. (Answer: $35,250.00)
A death claim has a basic benefit of $50,000 and vested additions of $7,500. The contract allows a 10% terminal bonus on the basic benefit alone, and this claim qualifies. Unpaid dividends total $800 with no interest. (Answer: $63,300.00)
A policy matures at the end of year 3 with a basic benefit of $20,000 and vested additions of $2,000. Its terminal bonus is 8% of the basic benefit plus additions. Unpaid dividends of $100, $200, and $300 were declared at the ends of years 1, 2, and 3 and accumulate at 5% effective annually. (Answer: $24,380.25)
A policy has a basic death benefit of $45,000 and vested additions of $6,000. The contract pays a terminal bonus only on maturity, not on death. The insured dies in year 7. Unpaid cash dividends total $600 and earn no interest. (Answer: $51,600.00)
A qualifying maturity claim has a basic benefit of $30,000 and vested additions of $5,000. Its terminal bonus is 12% of the basic benefit plus additions; cash dividends are excluded from this base. The accumulated unpaid cash-dividend balance is $400. (Answer: $39,600.00)

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