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Exam ALTAM · Dividend and Bonus Systems · Tutorial 739 of 999

Calculating Terminal Bonuses

Calculate a terminal bonus from its stated rate and eligible base, and determine the total claim payable at the specified claim date.

Advanced 10 min read

What You'll Learn

  • Define a terminal bonus and distinguish its rate from its eligible benefit base.
  • Calculate a bonus expressed as a percentage or as an amount per $1,000.
  • Separate the base used to calculate the bonus from the benefit payable before the bonus.
  • Add the terminal bonus to the claim amount at the specified maturity or death date.
  • Identify common errors involving excluded amounts, rate conversions, and double counting.

What a Terminal Bonus Calculation Requires

A terminal bonus is an additional amount payable at a specified claim event under the terms of a participating policy. In a calculation, the key inputs are the stated terminal bonus rate and the eligible benefit base to which that rate applies. The base is not automatically the same as the total claim payable before the bonus: the policy terms or problem statement determine which amounts are included.

The calculation is usually direct once those inputs are identified. First calculate the terminal bonus from the eligible base. Then add it to the benefit payable before the terminal bonus, provided the statement says the bonus is additional to that benefit. All amounts must be measured at the stated claim date.

Definition: Let \(B\) be the eligible benefit base at the claim date and let \(r\) be a terminal bonus rate stated as a proportion of that base. The terminal bonus is \(T=rB\). If \(C\) is the amount payable at the claim date before the terminal bonus, the total claim is \(C+T\), when the terminal bonus is payable in addition to \(C\).

A rate may instead be quoted as a number of dollars per $1,000 of eligible base. If the quoted rate is \(q\) dollars per $1,000, then the bonus is \(q(B/1{,}000)\). For example, a rate of $25 per $1,000 is equivalent to a rate of 2.5% because \(25/1{,}000=0.025\).

The rate and base must refer to the same claim and valuation date. In the examples below, all benefit amounts and rates are stipulated at the claim date, and the claim is assumed to occur as described. There is no need to calculate a probability of death or survival to find the conditional amount payable. Any interest rate is stated for context; no accumulation or discounting is required when the values are already at the claim date and the bonus is paid immediately.

Calculating and Adding the Bonus

The most reliable method is to keep the bonus calculation separate from the claim total. This makes it easier to see whether the right base and rate have been used and prevents adding the bonus twice.

1
Identify the claim-date base.
Use only the amounts specified as eligible for the terminal bonus. Do not infer the base from the total benefit if the statement gives a separate eligible base.
2
Express the rate consistently.
Convert a percentage to a decimal, or calculate the number of $1,000 units when the rate is quoted per $1,000.
3
Calculate the terminal bonus.
Multiply the eligible base by the rate, using the specified units.
4
Form the claim total.
Add the bonus to the pre-bonus benefit payable at the claim date, without adding any component twice.

Worked Example: Terminal Bonus at Maturity

A hypothetical participating endowment matures at the end of policy year 20. Its basic sum assured is $80,000, and $12,000 of reversionary bonuses have vested. The maturity benefit before any terminal bonus is $92,000. The terminal bonus is stated as 18% of the basic sum assured plus vested reversionary bonuses. Calculate the terminal bonus and total maturity claim.

Assumptions. The stipulated amounts and rate apply at the maturity date, and the terminal bonus is payable immediately in addition to the maturity benefit. For context, assume an effective annual interest rate of 4%; it is not used because all amounts are already valued at the claim date. Maturity at the end of year 20 is given, so no mortality assumption or probability distribution is required for this conditional claim calculation; the data are deterministic.

Step 1: Find the eligible base. The rate applies to basic sum assured plus vested bonuses:

$$ B=80{,}000+12{,}000=92{,}000. $$

Step 2: Calculate the terminal bonus. Convert 18% to 0.18 and multiply by the eligible base:

$$ T=0.18(92{,}000)=16{,}560. $$

Step 3: Add it to the maturity benefit. The pre-bonus maturity benefit is $92,000, so:

$$ \text{Total maturity claim}=92{,}000+16{,}560=108{,}560. $$

The answer is a terminal bonus of $16,560 and a total maturity claim of $108,560. As a check, the bonus is 18% of the stated $92,000 base, and the total is exactly the pre-bonus benefit plus that bonus. The method avoids treating the 18% as a rate on some other amount, such as the total after the bonus.

Why the Calculation Has This Form

The percentage rate describes the terminal bonus per dollar of eligible base. If the rate is \(r\), each dollar of base contributes \(r\) dollars of bonus. With \(B\) eligible dollars, there are \(B\) such units, so the total bonus is \(rB\). If the bonus is paid in addition to a pre-bonus claim \(C\) at the same date, the total payment is the sum of those two amounts.

$$ T=\underbrace{r+r+\cdots+r}_{B\text{ dollars of eligible base}}=rB, \qquad \text{claim total}=C+T=C+rB. $$

This addition is appropriate only when the terminal bonus is specified as an additional claim payment and \(C\) excludes it. If a problem instead defines a benefit amount that already includes the terminal bonus, adding the bonus again would double count it. The wording and the definitions of \(B\) and \(C\) therefore matter as much as the arithmetic.

Worked Example: Rate Quoted per $1,000 at Death

A hypothetical participating policy pays a death benefit of $75,500 before a terminal bonus. The eligible base for the terminal bonus is separately stated as $72,000. The terminal bonus rate at the claim date is $24 per $1,000 of eligible base. Find the terminal bonus and total death claim.

Assumptions. The death claim is assumed to occur on the stated date and the bonus is paid immediately in addition to the $75,500. Assume an effective annual interest rate of 3%; there is no discounting because both amounts are claim-date values. The claim is treated conditionally as having occurred, so no mortality distribution is needed. The stated data are deterministic.

Step 1: Count the $1,000 units in the eligible base.

$$ \frac{72{,}000}{1{,}000}=72\text{ units}. $$

Step 2: Multiply by the rate per unit.

$$ T=72(24)=1{,}728. $$

Step 3: Add the bonus to the pre-bonus death benefit.

$$ \text{Total death claim}=75{,}500+1{,}728=77{,}228. $$

The terminal bonus is $1,728, and the total death claim is $77,228. As a rate check, $24 per $1,000 is \(24/1{,}000=0.024\), or 2.4%; applying 2.4% to $72,000 also gives $1,728. Using $75,500 as the base would be incorrect because the problem explicitly identifies a different eligible base.

Base and Payable Benefit Can Differ

An amount can be payable at the claim date without being eligible for the terminal bonus. Conversely, the stated base may include a component that is not presented as a separate extra payment in the claim total. Follow the definition of the eligible base, rather than assuming that every payable amount earns the bonus.

Worked Example: Excluding a Separately Payable Dividend

At the maturity date of a hypothetical policy, the basic sum assured is $110,000, vested reversionary additions are $15,000, and a cash dividend of $2,000 is payable. The stated terminal bonus is 14.5% of the basic sum assured plus vested additions; the cash dividend is explicitly excluded from the eligible base. Calculate the bonus and the full maturity payment.

Assumptions. All listed amounts apply at maturity, the cash dividend is payable in addition to the basic sum assured and vested additions, and the terminal bonus is immediately payable in addition to all three. Assume an effective annual interest rate of 4%; it is not used because there is no time between the claim-date amounts. Maturity is given, so no mortality model is needed for this conditional calculation. The inputs are deterministic.

Step 1: Calculate only the eligible base.

$$ B=110{,}000+15{,}000=125{,}000. $$

The $2,000 cash dividend is excluded by the stated terms and is not part of \(B\).

Step 2: Calculate the terminal bonus.

$$ T=0.145(125{,}000)=18{,}125. $$

Step 3: Calculate the pre-bonus amount payable.

$$ C=110{,}000+15{,}000+2{,}000=127{,}000. $$

Step 4: Add the terminal bonus.

$$ \text{Total maturity payment}=127{,}000+18{,}125=145{,}125. $$

The terminal bonus is $18,125 and the total maturity payment is $145,125. The amount payable before the bonus includes the dividend, but the bonus base does not. This distinction is essential: the same amount need not belong to both calculations.

Common Exam Trap: Applying the Rate to the Wrong Amount

In the preceding example, a tempting but incorrect calculation is \(0.145(127{,}000)=18{,}415\). That applies the rate to the entire pre-bonus payment, including the $2,000 cash dividend that the problem excludes. It overstates the bonus by:

$$ 18{,}415-18{,}125=290=0.145(2{,}000). $$

The correction is to use the eligible base of $125,000, giving a bonus of $18,125. Then add that bonus once to the pre-bonus payment of $127,000. A separate, equally important check is that the stated pre-bonus payment must not already include the terminal bonus; if it does, do not add the same bonus again.

Key takeaway: Treat the eligible base, the terminal bonus, and the pre-bonus claim amount as separate quantities. Calculate the bonus from the stated base and rate; add it to the claim amount only when the terms make it an additional payment.

Check Your Understanding

For each problem, use the stated claim-date amounts and include the terminal bonus only once.

  1. A hypothetical endowment has a $54,000 eligible base, including all vested additions. Its pre-bonus maturity payment is $54,000, and the terminal bonus rate is 12% of the base. Find the bonus and maturity payment. (Answer: terminal bonus $6,480; maturity payment $60,480)
  2. A death claim has a pre-bonus amount of $88,500 and an explicitly stated eligible base of $85,000. The terminal bonus rate is $19 per $1,000. Find the bonus and total claim. (Answer: terminal bonus $1,615; total claim $90,115)
  3. A maturity payment before the terminal bonus consists of a $100,000 basic sum assured, $9,000 vested additions, and a $1,200 cash dividend. The dividend is excluded from the $100,000-plus-vested-additions eligible base. The bonus rate is 16%. Find the bonus and total payment. (Answer: terminal bonus $17,440; total payment $127,640)
  4. A policy has a $48,000 eligible base, a pre-bonus claim of $50,000, and a terminal bonus rate of 2.5% of the eligible base. Calculate the claim total. (Answer: terminal bonus $1,200; claim total $51,200)
  5. A hypothetical claim has a pre-bonus benefit of $73,600 and an eligible base of $70,000. The terminal bonus is quoted as $32 per $1,000 of base. Calculate the bonus and total amount payable. (Answer: terminal bonus $2,240; total amount payable $75,840)

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