Introduction
A participating policy may include more than one kind of bonus in its claim payment. Reversionary bonuses declared during the policy term can add to the benefit, while a terminal bonus may provide an additional amount when an eligible claim occurs. The calculation is manageable if each component is kept separate until its contractual basis is clear.
The main point is that a terminal-bonus rate does not have one universal base. It may apply to the basic sum assured, to the sum assured plus vested additions, or to another expressly specified amount. The benefit wording determines the calculation. Applying the right rate to the wrong base can produce a materially incorrect claim.
The formula separates three amounts that can be easy to conflate. The basic sum assured is the original contractual amount. Vested reversionary additions are bonuses already added to the benefit under the applicable bonus scale. The terminal bonus is a further amount calculated at the claim date if the claim event qualifies. This tutorial treats eligibility as given in each problem; a terminal bonus is not included merely because reversionary additions exist.
Set the Bases Before Calculating
First calculate the vested additions using the declared reversionary-bonus method and the stated bonus scale. Then write down the terminal-bonus base in words or as an equation. For example, a clause might define the terminal bonus as 8% of the basic sum assured, or as 8% of the basic sum assured plus vested additions. Those clauses have different results even though both quote an 8% rate.
| Terminal-bonus wording | Terminal-bonus calculation | Claim amount |
|---|---|---|
| Rate \(r\) of basic sum assured | \(T=rS\) | \(S+R+rS\) |
| Rate \(r\) of basic sum assured plus vested additions | \(T=r(S+R)\) | \(S+R+r(S+R)\) |
| Specified amount \(K\) per unit of basic sum assured | Convert the rate to a monetary amount using its stated unit | \(S+R+T\) |
A terminal-bonus rate is not automatically a reversionary-bonus rate. The first is applied at the claim date to its specified base; the second creates additions according to its own declaration and method. Do not add a terminal bonus to the vested additions and then apply the terminal rate again unless the contract explicitly defines such a calculation.
Apply the stated simple or compound reversionary-bonus scale through the relevant policy years. Use the basic sum assured or other specified base for that scale.
For example, record \(B_T=S\) if the rate applies to the basic sum assured, or \(B_T=S+R\) if it applies to the augmented benefit.
Multiply the stated rate by its base, or convert a per-unit rate using the unit in the terms.
Calculate \(C=S+R+T\), then check that each contractual component appears exactly once.
Why the Components Are Added
The claim formula follows from the way the amounts are defined. The basic sum assured is one part of the benefit. Vested additions increase that benefit by \(R\), so the benefit before any terminal bonus is \(S+R\). If the contract also provides a terminal amount \(T\), that amount is additional. Therefore:
The derivation does not prescribe the base for \(T\). That comes from the terminal-bonus terms. If the terms specify a percentage of the benefit before terminal bonus, substitute \(T=r(S+R)\). If they specify a percentage of the basic sum assured, substitute \(T=rS\). In either case, the other components remain in the final sum.
Worked Calculations
Worked Example: Simple Additions and a Basic-Sum-Assured Terminal Base
A hypothetical policy has a basic sum assured of $80,000. It receives a simple reversionary bonus of $35 per $1,000 of basic sum assured at the end of each of eight policy years. All eight additions are vested. An eligible claim occurs after the eighth declaration. The terminal bonus is 12% of the basic sum assured only. Calculate the claim amount.
Method. Calculate the simple additions on the basic sum assured, identify the terminal-bonus base from the wording, and then add the three components. This avoids treating the terminal rate as if it applied to the augmented benefit.
The number of $1,000 units in the basic sum assured is:
The annual simple addition is \(80\times \$35=\$2{,}800\). Over eight declarations, the vested additions are:
The terminal-bonus wording specifies the basic sum assured as its base, so:
The final claim amount is:
Check. The amount before terminal bonus is $102,400. Adding the $9,600 terminal bonus gives $112,000. The terminal bonus is not 12% of $102,400 because the stated base is the basic sum assured.
Worked Example: Compound Additions and an Augmented-Benefit Base
A hypothetical policy has a basic sum assured of $50,000. A 2% compound reversionary bonus is declared at each of five annual anniversaries, and every declaration vests. At an eligible claim after the fifth declaration, a terminal bonus of 10% applies to the basic sum assured plus all vested additions. Calculate the claim amount to the nearest cent.
Method. Accumulate the compound additions by applying each annual factor to the benefit including earlier additions. Then apply the terminal rate to \(S+R\), as expressly stated. This avoids applying 10% only to the basic sum assured.
After five declarations, the benefit including reversionary additions is:
Therefore, the vested additions are:
The terminal-bonus base includes those additions, so:
Thus:
Rounded to the nearest cent, the claim is $60,724.44. As a check, the same total is \(1.10\times \$55{,}204.04016\), because the terminal bonus is 10% of the benefit including additions.
Worked Example: Per-Unit Terminal Bonus on Vested Additions Only
A hypothetical participating policy has a basic sum assured of $120,000. It receives a simple reversionary bonus of $25 per $1,000 of basic sum assured at each of six anniversaries; all additions are vested. The policy then has an eligible claim. The terminal bonus is $60 per $1,000 of vested reversionary additions, not per $1,000 of total benefit. Find the claim amount.
Method. Convert both per-unit scales using their stated bases. In particular, do not use the basic sum assured as the terminal-bonus base merely because it was the base for the reversionary additions.
The basic sum assured contains \( \$120{,}000/\$1{,}000=120 \) units. Each annual reversionary addition is:
After six declarations:
The terminal-bonus base is the vested additions, which contain \( \$18{,}000/\$1{,}000=18 \) units. Thus:
The claim is:
Check. The terminal bonus is small relative to the basic sum assured because the contract applies its per-unit amount only to the $18,000 of vested additions. Applying $60 to 120 units would use the wrong base.
Common Mistakes and Exam Checks
A reliable solution makes the base visible before any percentage is multiplied. A common wrong calculation is to calculate the augmented benefit and then apply a terminal rate to it automatically, even where the terms specify the basic sum assured. For the first worked example, that incorrect calculation would be:
The correction is to use the stated base \(S=\$80{,}000\), giving:
The incorrect amount overstates the claim by $2,688, exactly 12% of the $22,400 vested additions that were not part of the specified terminal-bonus base. Conversely, when the terms do include vested additions in the terminal base, applying the rate to \(S\) alone understates the terminal bonus.
- Do not omit vested additions. If they are vested and payable under the stated claim conditions, include \(R\) in the benefit even when the terminal-bonus base excludes them.
- Do not count additions twice. If \(R\) has already been calculated as a dollar amount, add it once to \(S\). A terminal bonus on \(S+R\) is additional to that amount; it does not replace it.
- Respect the unit. A rate “per $1,000” must be multiplied by the number of $1,000 units in its stated base, not by the base amount as though it were a count of units.
- Keep the claim conditions separate. Apply a terminal bonus only if the problem states that the claim event is eligible. The formula for combining amounts does not establish eligibility.
These checks matter because the same numerical rate can generate different claim amounts under different contract wording. A concise solution should name the reversionary-bonus base, state the terminal-bonus base, calculate each amount, and show the final addition. This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the terminal-bonus base before calculating the claim amount.
- A hypothetical policy has a basic sum assured of $40,000 and receives a simple reversionary bonus of $30 per $1,000 of basic sum assured at each of four anniversaries. All additions vest. An eligible claim receives a terminal bonus of 8% of the basic sum assured. Find the claim amount. (Answer: $48,000)
- A hypothetical policy has a basic sum assured of $25,000, with a 3% compound reversionary bonus declared and vested at each of three annual anniversaries. The eligible-claim terminal bonus is 6% of the basic sum assured only. Calculate the claim to the nearest cent. (Answer: $28,818.18)
- A hypothetical policy has a basic sum assured of $90,000 and a simple reversionary bonus of $20 per $1,000 of basic sum assured for five declarations. The terminal bonus is 4% of the basic sum assured plus vested additions. Calculate the final claim. (Answer: $102,960)
- A hypothetical policy has a basic sum assured of $70,000 and four annual compound reversionary-bonus declarations of 1.5%, all vested. The terminal bonus is 7% of the basic sum assured plus vested additions. Find the claim to the nearest cent. (Answer: $79,496.13)
- A hypothetical policy has a basic sum assured of $100,000 and receives a simple bonus of $15 per $1,000 of basic sum assured at each of eight anniversaries. All additions are vested. The terminal bonus is $40 per $1,000 of vested additions only. Find the claim amount. (Answer: $112,480)