Introduction
A participating policy may have one declared reversionary bonus scale for earlier policy years and a different scale for future years. A lower replacement rate does not ordinarily remove additions that have already vested. The projection must therefore keep two things separate: what has already been declared and what might be added under the new scale.
This tutorial considers hypothetical policies with reversionary bonuses. The calculations assume that each declaration vests at the stated policy anniversary, that the policy remains in force, and that no claim occurs before the projection date. A projection of future bonuses is conditional on the assumed future scale; it is not itself a declaration or a guarantee. The calculation excludes terminal bonuses and other benefits.
Separate Vested Additions From Future Declarations
Before calculating, establish the policy-year timing and the basis of each rate. A rate may be stated as a fixed addition for each $1,000 of original sum assured, or as a percentage applied to the current vested benefit. These bases are not interchangeable. A projection should state which one applies rather than infer it from the word “bonus.”
For a simple bonus stated as \(r\) dollars per $1,000 of original sum assured, a policy with original sum assured \(S\) receives an addition of \(rS/1000\) at each declaration. For a compound bonus stated as a percentage \(c\) of the current vested benefit \(B\), the next vested benefit is \(B(1+c)\). A change of scale changes the future rate, not the historical declarations.
Worked Example: A Compound Scale Falls After Two Declarations
A hypothetical participating policy has an original sum assured of $100,000. It receives a compound reversionary bonus of 4% at each of the first two anniversaries. From the third anniversary onward, the assumed compound rate is 2% per year, applied to the current vested benefit. Find the vested benefit immediately after the fifth anniversary. Assume no claim before then.
Method. Apply the old scale to the first two declarations, then apply the replacement scale to the benefit already vested at the end of year 2. The rates are annual bonus rates, not interest rates.
At issue, the benefit is $100,000. After the first declaration:
After the second declaration, still using the original 4% scale:
The replacement rate applies at anniversaries 3, 4, and 5. Thus:
Rounded to the nearest cent at the final step, the projected vested benefit is $114,780.26. The total additions are $114,780.26 − $100,000 = $14,780.26.
Check. The first two declarations increase the benefit by a factor of \(1.04^2=1.0816\), and the next three by \(1.02^3=1.061208\). Their product is \(1.0816(1.061208)=1.1478025728\), giving $100,000 × 1.1478025728 = $114,780.25728 before rounding. This agrees with the year-by-year calculation.
Common mistake avoided. Applying 2% to all five years would discard the old declarations and give \(100{,}000(1.02)^5=\$110{,}408.08\). The replacement rate applies only from year 3, so that calculation is incorrect.
Why the Compound Projection Works
A compound declaration increases the benefit that is already vested. If the benefit immediately before a declaration is \(B\), and the declared rate is \(c\), then the bonus addition is \(cB\). The benefit immediately after declaration is the old benefit plus that addition:
Applying this recurrence repeatedly over a period with a constant rate gives \(B_n=B_0(1+c)^n\). When the rate changes, split the recurrence at the change date. If the first \(m\) declarations use rate \(c_1\) and the next \(n\) declarations use \(c_2\), the result is \(B_0(1+c_1)^m(1+c_2)^n\). This follows because the second sequence starts from the benefit reached after the first sequence; it does not replace that benefit.
The recurrence depends on the stated compound basis. If a scale instead states fixed additions per $1,000 of original sum assured, each addition is calculated from that original amount. Those additions are accumulated, not multiplied by the current vested benefit.
Worked Example: A Simple Scale Replaces an Earlier Simple Scale
A hypothetical policy has an original sum assured of $240,000. For each of the first three anniversaries, the declared simple reversionary bonus is $45 per $1,000 of original sum assured. At the fourth anniversary, a new simple scale of $30 per $1,000 applies, and it is assumed to continue for two declarations. Find the benefit immediately after the fifth anniversary.
Method. Calculate each annual addition from the original sum assured, using the rate for that declaration. Because both scales are simple and based on the original sum assured, the additions do not themselves earn further bonus.
There are \(240{,}000/1{,}000=240\) units of $1,000. Under the old scale, each of the first three declarations adds:
The total vested additions from those declarations are \(3(10{,}800)=\$32{,}400\). Under the replacement scale, each of the fourth and fifth declarations adds:
The two new declarations add \(2(7{,}200)=\$14{,}400\). The total additions are $32,400 + $14,400 = $46,800, so the benefit is:
Check. The five additions are $10,800, $10,800, $10,800, $7,200, and $7,200. Their sum is $46,800, consistent with the total benefit.
Common mistake avoided. Multiplying the $240,000 sum assured by 4% or 3% would treat the per-$1,000 rates as percentages. The rate is a dollar amount per $1,000, so the correct annual additions are $10,800 and $7,200.
The New Scale’s Benefit Base Matters
At a scale change, the policy’s vested benefit may already exceed its original sum assured. The new rate might apply to that current benefit, or it might continue to apply to the original sum assured. The resulting projections differ. The wording of the scale determines the calculation; a rate alone is not enough to identify its base.
| Replacement scale wording | Calculation of each future addition | Effect of earlier vested additions |
|---|---|---|
| Simple rate per $1,000 of original sum assured | Rate × original sum assured ÷ 1,000 | Preserved, but not included in the new scale’s base |
| Compound percentage of current vested benefit | Current vested benefit × percentage | Preserved and included in the base for future declarations |
| Other specified benefit base | Use the base stated in the scale or contract | Follow the stated definition; do not assume either basis above |
Worked Example: New Compound Rate on Existing Vested Additions
At a change date, a hypothetical policy has an original sum assured of $150,000 and $12,000 of already vested additions, giving a current vested benefit of $162,000. The replacement scale declares a compound bonus of 3% at each of the next two anniversaries, calculated on the current vested benefit. Find the projected benefit after the second new declaration.
Method. Preserve the $12,000 already vested, then use the stated current-benefit base for each new declaration.
After the first new declaration:
The second declaration is calculated on $166,860, not on the original sum assured:
Thus the projected benefit after two declarations is $171,865.80. The two new additions total $171,865.80 − $162,000 = $9,865.80.
Check. The same calculation in one step is \(162{,}000(1.03)^2=162{,}000(1.0609)=\$171{,}865.80\). If the scale had instead stated a simple rate of 3% of original sum assured, each addition would have been $4,500 and the two-year benefit would have been $171,000. That is a different basis and not the one specified here.
Common mistake avoided. Applying 3% only to the $150,000 original sum assured would ignore the explicit compound base. Conversely, applying the current-benefit calculation when the scale specifies a simple original-sum-assured base would overstate future additions.
A Reliable Projection Process
A disciplined calculation makes the effective date and the base visible. This is particularly useful when a projection contains several changes in rate or a mixture of simple and compound scales.
Count the declarations that have already occurred and identify the first anniversary to which the replacement scale applies.
Use the original sum assured plus all additions already declared. Do not remove earlier additions when the scale changes.
Determine whether the new rate is a fixed addition per unit of original sum assured, a percentage of the current vested benefit, or another expressly stated base.
Apply the rate in force for that declaration only. For a compound percentage, update the benefit after each declaration; for a simple addition, use the specified fixed base.
Reconcile the final benefit to the original benefit plus vested and projected additions. Confirm that no declaration was omitted, duplicated, or recalculated under the wrong scale.
Common Traps and Interpretation
The most important distinction is between a change in future scale and a revision of past declarations. A future rate change does not, by itself, imply that earlier vested additions are removed or recalculated. In a compound projection, preserving those additions also means they form part of the base for future increases when the new scale specifies the current vested benefit as its base.
- Do not apply the replacement rate retroactively. Match each rate to its effective policy years.
- Do not confuse a percentage with a per-$1,000 amount. Convert a per-unit rate using the number of $1,000 units in the stated base.
- Do not assume that every new scale is compound. Use the exact base and method given.
- Do not count old additions twice. If the current vested benefit already includes them, do not add them again after projecting future compound declarations.
- Do not treat an assumed future scale as already declared. The projection depends on its stated assumption; actual future declarations may differ.
A benefit projection is useful for comparing assumptions and tracing the guaranteed amount already built up under a hypothetical contract. The arithmetic is only reliable when its dates, vesting assumptions, and benefit bases are explicit. These methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each hypothetical policy, identify the effective scale and stated benefit base before calculating. Round monetary answers to the nearest cent where needed.
- A $80,000 policy receives simple bonuses of $25 per $1,000 of original sum assured for each of the first two anniversaries. A replacement simple rate of $18 per $1,000 applies at each of the next three anniversaries. What is the benefit after the fifth declaration? (Answer: $88,320.00)
- A policy has an original sum assured of $50,000. It receives compound bonuses of 5% for two declarations, then 3% for two declarations, with each rate applied to the current vested benefit. What is the benefit after four declarations? (Answer: $58,482.11)
- At a scale-change date, a policy’s current vested benefit is $124,000. The new scale is a compound 2.5% per year on the current vested benefit for three declarations. Find the projected benefit after the third declaration. (Answer: $133,534.44)
- A $90,000 policy has received three simple additions at $40 per $1,000 of original sum assured. A new simple rate of $22 per $1,000 applies for each of the next four declarations. Find the benefit after those four new declarations. (Answer: $108,720.00)
- A $120,000 policy receives one compound declaration of 4%. The rate then changes to 2% for each of the next two declarations, each applied to the current vested benefit. Find the benefit after the third declaration. (Answer: $129,841.92)