Why Units and Rounding Matter
A declared bonus rate is not yet a policy-level bonus amount. It must be applied to the correct base, expressed in the correct units, and rounded according to the contract or declaration. A rate of $18.75 per $1,000 of sum assured, for example, does not mean 18.75% of the sum assured. Treating it that way makes the calculated bonus ten times too large.
The arithmetic can also depend on whether fractional units are permitted and whether rounding occurs after each declaration or only after amounts are combined. Those details matter when a bonus becomes part of a policy benefit: rounding too early can change the amount ultimately recorded. The calculations below concern declared rates and stated contractual rules; they do not assume that a future bonus rate has been declared.
Read the Scale Before Calculating
Before multiplying, identify four items: the declared rate, its base, the unit denominator, and the rounding rule. A scale might state a dollar amount per $1,000 of sum assured, a percentage of the original sum assured, or a percentage of a current benefit base. These are different instructions, even if their numerical rates look similar.
For a rate of \(r\) dollars per \(K\) dollars of sum assured, and a policy sum assured \(S\), the number of rate units is \(S/K\). If fractional units are allowed, the policy-level amount for one declaration is:
If a scale gives a percentage \(p\) of a stated base \(B\), express \(p\) as a decimal before applying it. For example, 1.6% is 0.016, so the amount is \(0.016B\). The base must still be read carefully: a percentage of original sum assured is not the same as a percentage of the benefit including previously vested additions.
| Scale wording | Policy-level calculation | Check before calculating |
|---|---|---|
| $r per $K of sum assured | \(rS/K\) | Whether fractional units are allowed |
| \(p\)% of original sum assured | \((p/100)S\) | That the original sum assured is the stated base |
| \(p\)% of current bonus-bearing benefit | \((p/100)B\) | Which additions are included in \(B\) |
| $r per policy unit | \(r\) times the permitted unit count | How the contract defines and counts a unit |
Simple and Compound Amounts Use Different Bases
For a simple reversionary bonus, each declared amount is calculated on the specified fixed base, commonly the original sum assured. If the same annual rate applies for \(n\) declarations and the base remains \(S\), the unrounded total additions are \(n\) times the amount for one declaration. This does not mean that the rate is applied to the benefit after earlier additions.
For a compound reversionary bonus, the contract specifies a base that includes vested additions. In the common case where each declaration applies the same rate \(r\) to the current bonus-bearing benefit, the recurrence is \(B_j=B_{j-1}(1+r)\), with \(B_0=S\). The resulting benefit after \(n\) declarations is \(S(1+r)^n\). The formula is appropriate only when the rate and base rule remain as stated for all \(n\) declarations; a different base or rate requires a different calculation.
Rounding Is Part of the Calculation
Do not assume that a single rounding convention applies to every contract. A stated rule may require rounding each annual policy-level addition to the nearest cent before it vests. Another may require carrying full precision through the calculation and rounding only the combined amount. If the governing rule is not given, state the convention used rather than silently choosing one.
In the examples, “nearest cent, half up” means that a third decimal digit of 5 or more increases the second decimal digit by one. Unless an example says otherwise, fractional units are permitted and intermediate amounts are not rounded. A rounded currency amount is reported to two decimal places.
Record the rate and the exact base to which it applies.
Divide by the stated unit size, or use the percentage as a decimal.
Use a fixed base for the stated simple bonus, or update the base when the contract specifies compounding.
Round at the required stage and check the final policy amount against the scale.
Worked Example: A Simple Rate Per $1,000
A hypothetical participating policy has a sum assured of $87,500. It receives four annual simple reversionary bonus declarations, each at $18.75 per $1,000 of original sum assured. Fractional units are permitted. The contract says to carry full precision through the calculation and round the total addition to the nearest cent, half up. Find the total additions and the benefit including those additions. Method: convert the sum assured to scale units, calculate one declaration, then multiply by the number of declarations. This avoids treating a dollar-per-unit rate as a percentage.
Step 1: Find the number of units. The unit size is $1,000, so:
Step 2: Calculate one annual addition.
Step 3: Add four simple declarations. The base stays at the original sum assured, so:
The benefit including these additions is \(\$87{,}500+\$6{,}562.50=\$94{,}062.50\). Check: dividing the total addition by four gives $1,640.625 per declaration, the amount obtained from the stated rate. The result is reported to the nearest cent. A common mistake would be to apply 18.75% to $87,500; that would give $16,406.25 for one year, ten times the correct amount.
Worked Example: A Compound Percentage
A hypothetical policy has an initial sum assured of $80,000. It receives three annual compound bonuses, each at 2.5% of the current bonus-bearing benefit. Fractional cents are retained during calculation, and the final benefit is rounded to the nearest cent, half up. Find the total bonus additions and the final benefit. Method: apply the recurrence using the benefit after the preceding declaration. This avoids repeatedly calculating the rate on the original sum assured.
Initial benefit: \(B_0=\$80{,}000\).
First declaration: the bonus is \(0.025\times \$80{,}000=\$2{,}000\), so \(B_1=\$82{,}000\).
Second declaration: the bonus is \(0.025\times \$82{,}000=\$2{,}050\), so \(B_2=\$84{,}050\).
Third declaration: the bonus is \(0.025\times \$84{,}050=\$2{,}101.25\), so \(B_3=\$86{,}151.25\).
The additions total \(\$2{,}000+\$2{,}050+\$2{,}101.25=\$6{,}151.25\). Check: \( \$80{,}000(1.025)^3=\$80{,}000(1.076890625)=\$86{,}151.25\), matching the recurrence. Applying 2.5% to the original sum assured three times would instead produce simple additions of $6,000 and would not follow the stated compound base.
Worked Example: Rounding Each Annual Addition
A hypothetical policy has a sum assured of $64,375 and receives three simple annual bonuses at $24.60 per $1,000 of sum assured. Fractional units are permitted. The contract requires each annual policy-level addition to be rounded to the nearest cent, half up, before it is added to the policy. Find the recorded total additions. Method: calculate and round each declaration separately, then sum the recorded amounts. This follows the stated timing of rounding.
The policy has \(\$64{,}375/\$1{,}000=64.375\) units. The unrounded amount for one declaration is:
Rounding each annual amount to the nearest cent, half up, gives $1,583.63. Therefore the recorded total for three declarations is:
Check: the total is three times the amount actually recorded for one declaration, as the contract requires. If full precision were carried until after summing instead, the calculation would be \(3\times \$1{,}583.625=\$4{,}750.875\), which rounds to $4,750.88. That one-cent difference shows why the rounding stage must be followed rather than presumed.
Common Mistakes and an Exam Check
A reliable calculation begins with the wording of the scale, not with the calculator. For a policy-level amount, keep the scale denominator, the benefit base, the number of declarations, and the rounding rule visible in the working. This makes the calculation auditable and helps reveal whether a result is the right order of magnitude.
- Confusing a rate per unit with a percentage. A rate of $18.75 per $1,000 is equivalent to 1.875% of that base, not 18.75%.
- Using the wrong base. A simple rate on original sum assured is not recalculated on vested additions. A compound rate uses the updated base only when the contract says so.
- Ignoring fractional units. Do not round the number of units to a whole number unless the contract specifies how units are counted.
- Rounding at the wrong stage. “Round each declaration” and “round the total” can produce different recorded amounts.
- Adding a bonus twice. If a stated current benefit already includes vested additions, do not add those additions again when calculating the benefit after the next declaration.
For work aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus, a concise check is to restate the scale in words: “this rate gives this many dollars for each stated unit of this stated base.” Then verify that the resulting policy amount is consistent with those words and that rounding occurred at the specified stage.
Check Your Understanding
For each problem, follow the stated base, unit convention, and rounding instruction.
- A hypothetical policy with sum assured $52,000 receives two simple annual bonuses of $13.40 per $1,000 of original sum assured. Fractional units are permitted, and only the total is rounded to cents. Find the total additions. (Answer: $1,393.60)
- A hypothetical policy begins with a benefit of $90,000 and receives two compound bonuses of 1.2% of the current benefit. No intermediate rounding is required. Find the final benefit. (Answer: $92,172.96)
- A hypothetical policy with sum assured $48,750 receives two simple annual bonuses of $31.25 per $1,000. Each annual amount is rounded to the nearest cent, half up, before recording. Find the recorded total additions. (Answer: $3,046.88)
- A hypothetical policy with sum assured $136,000 receives four simple annual bonuses of 0.85% of original sum assured. Find the total additions. (Answer: $4,624.00)
- A hypothetical policy has sum assured $62,250. Its scale gives $7.80 per $500 unit per year, and the contract permits only complete units, ignoring any remainder. It receives three simple declarations, with no intermediate rounding needed. Find the total additions. (Answer: $2,901.60)