Why Project a Bonus Schedule?
A reversionary bonus schedule turns a stated annual bonus scale into a sequence of benefit amounts. For each policy year, the schedule shows the basic sum assured, the bonus declared for that year, and the guaranteed benefit after that declaration. This makes it easier to check both the arithmetic and the timing: a bonus declared at an anniversary is not part of the benefit immediately before that declaration.
The calculations in this tutorial concern a hypothetical participating policy. Assume that the policy remains in force through the anniversaries being projected, that each stated bonus is declared at the end of its policy year, and that a declared reversionary addition becomes guaranteed under the policy terms. No terminal bonus, cash dividend, interest adjustment, or probability of survival is included. A schedule alone does not establish which claims receive a benefit; the contract’s claim provisions determine that.
Set Up the Schedule Before Calculating
Begin by identifying the bonus scale’s unit and its base. A scale may state a percentage of the current benefit including earlier additions, which produces a compound bonus. Alternatively, it may state a fixed dollar amount per $1,000 of basic sum assured, which produces a simple bonus when the scale applies only to that original sum assured. These are different instructions, not interchangeable calculation choices.
Use one row per policy year. A useful layout is: policy year, opening guaranteed benefit, rate for that year, declared addition, and closing guaranteed benefit. The closing amount for one year becomes the opening amount for the next. If the scale changes by duration, use the rate assigned to the year in which the declaration is made.
| Schedule item | What it records |
|---|---|
| Basic sum assured | The original sum assured, shown separately so it is not confused with accumulated additions. |
| Opening guaranteed benefit | The benefit amount after the previous year’s declaration; at time zero this is usually the basic sum assured. |
| Bonus rate | The rate or dollar scale applicable to the current policy year, including its stated base. |
| Declared addition | The amount added to the guaranteed benefit at the year-end declaration. |
| Closing guaranteed benefit | The opening amount plus the current year’s declared addition. |
For a compound percentage scale, let \(B_0\) be the basic sum assured, \(r_t\) the rate declared at the end of year \(t\), \(\Delta_t\) the addition declared in that year, and \(B_t\) the closing guaranteed benefit. If the scale explicitly applies to the opening benefit including previously declared additions, then
For a simple scale stated as \(c_t\) dollars per $1,000 of basic sum assured, the addition is instead
The second formula uses \(B_0\), not the opening benefit \(B_{t-1}\). If a policy’s scale states a different base, follow that wording rather than assuming either formula. The scale’s unit, base, duration, and rounding rule must all be read before any arithmetic begins.
Why the Compound Recurrence Works
The compound recurrence follows directly from the definition of the benefit base. Immediately before the year-\(t\) declaration, the opening guaranteed amount is \(B_{t-1}\). If the declared rate is \(r_t\) of that opening amount, the new addition is \(r_tB_{t-1}\). Adding it to the opening amount gives \(B_t=B_{t-1}+r_tB_{t-1}=B_{t-1}(1+r_t)\). The next year uses this new closing amount as its opening base, so earlier declared additions participate in later compound bonuses.
This reasoning depends on the stated base being the opening benefit including vested additions. It does not apply to a scale that says the rate is applied only to the original sum assured, or to a scale that specifies a fixed dollar addition. In those cases the annual addition must be calculated using the stated alternative base.
Record the rate for each policy year, its unit, the benefit base, and any rounding instruction.
Apply that year’s rate to the specified base. Do not carry forward a rate or base unless the scale says to do so.
Add the declared amount to the opening benefit to obtain the closing benefit for that year.
Use the closing benefit as the next opening amount, then reconcile the final benefit to the basic sum assured plus all additions.
Worked Schedules
Worked Example: Compound Scale With Different Annual Rates
A hypothetical policy has a basic sum assured of $50,000. Its declared compound reversionary bonus rates are 2% at the end of year 1, 3% at the end of year 2, and 2% at the end of year 3. Each rate applies to the opening guaranteed benefit, including additions already declared. No intermediate rounding is required. Project the schedule.
Method. Use the compound recurrence year by year. The opening amount in year 1 is the basic sum assured, $50,000.
Year 1. The addition is \(0.02(50{,}000)=\$1{,}000\). The closing benefit is \(50{,}000+1{,}000=\$51{,}000\).
Year 2. The opening benefit is $51,000, not $50,000. The addition is \(0.03(51{,}000)=\$1{,}530\). The closing benefit is \(51{,}000+1{,}530=\$52{,}530\).
Year 3. The opening benefit is $52,530. The addition is \(0.02(52{,}530)=\$1{,}050.60\). The closing benefit is \(52{,}530+1{,}050.60=\$53{,}580.60\).
The projected schedule is therefore: year 1, addition $1,000 and closing benefit $51,000; year 2, addition $1,530 and closing benefit $52,530; year 3, addition $1,050.60 and closing benefit $53,580.60.
Check. Total additions are \(1{,}000+1{,}530+1{,}050.60=\$3{,}580.60\). Adding this to the original $50,000 gives $53,580.60, matching the final closing benefit.
Common mistake avoided. Applying the year-2 rate to only the original sum assured would give \(0.03(50{,}000)=\$1{,}500\), understating the addition by $30. The scale specifies a compound base, so the correct year-2 base is $51,000.
Worked Example: Simple Bonuses Stated Per $1,000
A hypothetical endowment policy has a basic sum assured of $80,000. Its scale declares simple reversionary additions of $18, $20, and $22 per $1,000 of basic sum assured in years 1, 2, and 3, respectively. Each addition is declared at the end of its year. Calculate each annual addition and the benefit after year 3.
Method. Convert the original basic sum assured into units of $1,000. There are \(80{,}000/1{,}000=80\) units. Since the scale is simple and explicitly uses basic sum assured, the number of units remains 80 in every year.
Year 1. The addition is \(18(80)=\$1{,}440\). The closing benefit is \(80{,}000+1{,}440=\$81{,}440\).
Year 2. The addition is \(20(80)=\$1{,}600\). The closing benefit is \(81{,}440+1{,}600=\$83{,}040\).
Year 3. The addition is \(22(80)=\$1{,}760\). The closing benefit is \(83{,}040+1{,}760=\$84{,}800\).
Check. The cumulative additions are \(1{,}440+1{,}600+1{,}760=\$4{,}800\). The final amount is \(80{,}000+4{,}800=\$84{,}800\), as in the schedule.
Common mistake avoided. A simple scale does not apply each later rate to the previous closing benefit. For example, calculating year 2 as \(20(81.44)\) would incorrectly use 81.44 units. The stated base is the original $80,000, so year 2 has 80 units.
Worked Example: Compound Scale With Annual Rounding
A hypothetical policy has a basic sum assured of $62,500. Its compound reversionary bonus rates are 1.8%, 2.4%, 2.4%, and 1.2% for years 1 through 4. Each rate applies to the opening guaranteed benefit, including prior additions. Each annual addition is rounded to the nearest cent before it is added to the benefit. Project the four-year schedule.
Method. Apply each rate to the previous closing amount, round the declared addition to cents, and use the resulting closing amount as the next opening amount.
Year 1. The opening benefit is $62,500. The addition is \(0.018(62{,}500)=\$1{,}125.00\). The closing benefit is $63,625.00.
Year 2. The addition is \(0.024(63{,}625)=\$1{,}527.00\). The closing benefit is \(63{,}625+1{,}527=\$65{,}152.00\).
Year 3. The unrounded addition is \(0.024(65{,}152)=\$1{,}563.648\), which rounds to $1,563.65. The closing benefit is \(65{,}152+1{,}563.65=\$66{,}715.65\).
Year 4. The unrounded addition is \(0.012(66{,}715.65)=\$800.5878\), which rounds to $800.59. The closing benefit is \(66{,}715.65+800.59=\$67{,}516.24\).
Check. The rounded additions total \(1{,}125.00+1{,}527.00+1{,}563.65+800.59=\$5{,}016.24\). Adding them to $62,500 gives $67,516.24, equal to the year-4 closing benefit.
Common mistake avoided. Rounding only the final benefit, rather than each declared addition as instructed, can produce a different later-year base. Here the year-4 rate is applied to the specified year-3 closing benefit of $66,715.65.
Timing, Guarantees, and Schedule Checks
A schedule must identify whether each amount is before or after the declaration. Under the assumptions used here, the year-\(t\) addition is declared at the end of year \(t\). Immediately before that declaration, the amount built up from prior declarations is \(B_{t-1}\); immediately after it, the amount is \(B_t\). If a claim occurs between declarations, the schedule alone does not establish whether an additional interim amount is payable. That depends on the policy provisions for the claim date.
A declared addition is described as guaranteed only to the extent the contract makes it part of the relevant benefit. For a death or maturity amount, confirm that the contract includes the vested additions for that event. Do not add an undeclared future bonus to a guaranteed schedule, and do not treat a projected future scale as a declaration already made. This schedule also does not include terminal bonuses or other benefit components.
A reliable projection has several independent checks:
- For each year, the closing benefit equals the opening benefit plus the current addition.
- The next year’s opening benefit equals the prior year’s closing benefit.
- For a simple scale, each addition uses the prescribed fixed base; for a compound scale, the opening benefit is used only when the scale specifies that base.
- The final benefit equals the basic sum assured plus the sum of all declared additions, subject to the stated rounding convention.
- The rate used belongs to the policy year of declaration, not automatically to the year in which a later claim occurs.
The most consequential exam trap is to calculate an addition correctly but attach it to the wrong base or year. A neat table does not correct a misread scale. Before calculating, annotate the scale—for example, “2.4% of opening benefit” or “$20 per $1,000 original sum assured”—and then carry that exact instruction through the rows. These disciplined checks are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each hypothetical policy, use the stated declaration timing and base. Assume no rounding unless a question specifies it.
- A $40,000 basic sum assured earns compound bonuses of 2% in year 1 and 3% in year 2, each applied to the opening guaranteed benefit. Find both additions and the benefit after year 2. (Answer: additions $800 and $1,224; year-2 benefit $42,024.)
- A policy has a $75,000 basic sum assured and a simple scale of $12 per $1,000 of basic sum assured for each of four years. Find the annual addition and the benefit after year 4. (Answer: $900 each year; benefit $78,600.)
- A $100,000 policy has compound rates of 1.5%, 1.5%, and 2% in years 1–3, applied to the opening benefit. Find the year-3 addition and closing benefit. (Answer: year-3 addition $2,060.45; closing benefit $105,082.95.)
- A $120,000 policy has simple additions of $8, $10, and $12 per $1,000 of basic sum assured in years 1–3. Find the total declared additions and the benefit after year 3. (Answer: total additions $3,600; benefit $123,600.)
- A $40,000 policy earns compound rates of 1% and 1.25% in years 1 and 2. Each addition is rounded to the nearest cent before being added. Find the year-2 addition and closing benefit. (Answer: year-2 addition $505.00; closing benefit $40,905.00.)