Choosing The Correct Reversionary Bonus Base

Why the Bonus Base Matters

A reversionary bonus adds to a policy’s guaranteed benefit when declared. The quoted percentage or dollar amount does not, by itself, tell you how much is added: you must also identify the base to which the scale applies. A percentage of the original sum assured produces a different addition from the same percentage of the benefit after earlier additions. A fixed dollar amount per policy is different from both.

The calculation therefore begins with the scale’s wording, not with a calculation. In particular, do not assume that every percentage bonus is calculated on the current guaranteed benefit. This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $B$ denote the original sum assured. Let $G_{k-1}$ denote the guaranteed benefit immediately before the $k$th bonus declaration, and let $b_k$ be the bonus addition at that declaration. A scale’s bonus base is the amount or unit to which its quoted rate applies. For the calculations here, each rate applies at the stated declaration, and an addition becomes part of the guaranteed benefit for later declarations only if the scale specifies the current guaranteed benefit as the later base.

Three common descriptions indicate three distinct calculation bases:

Scale wording	Calculation of the addition at declaration $k$	What changes over time?
$r_k$ of original sum assured	$b_k=r_kB$	The rate may change; the base $B$ does not.
$r_k$ of current guaranteed benefit	$b_k=r_kG_{k-1}$	Both the rate and the base may change.
A fixed amount $c_k$ per policy	$b_k=c_k$	The amount may change; it is not a percentage of the sum assured.

For a percentage scale, the rate must be converted to a decimal before multiplication: for example, $2\%=0.02$. “Per policy” means the specified monetary amount is added once for each policy covered by the scale. It does not mean “per dollar of sum assured.” If a scale instead specifies an amount per $1{,}000$ of sum assured, that is a different basis and the number of $1{,}000$-unit blocks must be counted.

How the Bases Affect Accumulation

The guaranteed benefit after a declaration is the amount before the declaration plus the addition just declared. Thus, for a percentage of original sum assured, the update is $G_k=G_{k-1}+r_kB$. For a percentage of current guaranteed benefit, the update is $G_k=G_{k-1}+r_kG_{k-1}$. For a fixed amount per policy, it is $G_k=G_{k-1}+c_k$.

These updates follow directly from what each phrase says the scale multiplies. The “original” base is fixed at issue, while “current” means the guaranteed benefit just before the declaration. Using the benefit after the declaration as its own base would be circular and would overstate the addition.

Derivation: If the $k$th rate $r_k$ is applied to the current guaranteed benefit before declaration, then the addition is $b_k=r_kG_{k-1}$. Adding that amount to the pre-declaration benefit gives $$ G_k=G_{k-1}+b_k=G_{k-1}+r_kG_{k-1}=G_{k-1}(1+r_k). $$ This multiplication occurs because the addition itself is a percentage of the amount already guaranteed. By contrast, if the base is the original sum assured $B$, then $b_k=r_kB$, so $$ G_k=G_{k-1}+r_kB. $$ There is no multiplication of the previous guaranteed benefit in that update: the original sum assured, rather than the accumulated benefit, is the stated base.

A useful check is to ask what the rate would be multiplied by at each declaration. If the answer is the same original amount every year, it is an original-sum-assured scale. If the answer includes earlier additions, it is a current-guaranteed-benefit scale. If the addition is already stated in dollars per policy, no benefit amount is multiplied by a percentage.

Worked Example: A Percentage of Original Sum Assured

A hypothetical participating policy has an original sum assured of $B=\$100{,}000$. It receives a reversionary bonus of $2\%$ of original sum assured at the end of each of three policy years. Assume the stated rate applies at each declaration and no other additions or changes occur. Find the guaranteed benefit after the third declaration.

Method. The phrase “of original sum assured” fixes the base at $\$100{,}000$ for all three declarations. Each annual addition is $$ b_k=0.02(\$100{,}000)=\$2{,}000. $$

Start with $G_0=\$100{,}000$. After year 1, $$ G_1=\$100{,}000+\$2{,}000=\$102{,}000. $$ After year 2, the base remains $\$100{,}000$, so $$ G_2=\$102{,}000+\$2{,}000=\$104{,}000. $$ After year 3, $$ G_3=\$104{,}000+\$2{,}000=\$106{,}000. $$

Check. Three additions of $\$2{,}000$ total $\$6{,}000$, and $\$100{,}000+\$6{,}000=\$106{,}000$. The calculation avoids the common mistake of applying the $2\%$ rate to the increasing guaranteed benefit when the scale explicitly says original sum assured.

Worked Example: Rates on the Current Guaranteed Benefit

A hypothetical policy has an initial guaranteed benefit of $\$100{,}000$. Its scale declares $2\%$ of the current guaranteed benefit after year 1, $3\%$ after year 2, and $4\%$ after year 3. Each rate applies to the benefit immediately before that declaration. Calculate the additions and the benefit after year 3.

Method. Use $G_k=G_{k-1}(1+r_k)$, with each rate expressed as a decimal. Initially $G_0=\$100{,}000$.

At the first declaration, $r_1=0.02$, so $$ b_1=0.02(\$100{,}000)=\$2{,}000,\qquad G_1=\$100{,}000+\$2{,}000=\$102{,}000. $$ At the second declaration, the pre-declaration benefit is now $\$102{,}000$, so $$ b_2=0.03(\$102{,}000)=\$3{,}060,\qquad G_2=\$102{,}000+\$3{,}060=\$105{,}060. $$ At the third declaration, use $\$105{,}060$ as the base: $$ b_3=0.04(\$105{,}060)=\$4{,}202.40,\qquad G_3=\$105{,}060+\$4{,}202.40=\$109{,}262.40. $$

Check. The additions total $\$2{,}000+\$3{,}060+\$4{,}202.40=\$9{,}262.40$. Adding that total to $\$100{,}000$ gives $\$109{,}262.40$. The common mistake is to calculate every addition as a percentage of the initial $\$100{,}000$, which would ignore the scale’s stated current-benefit base.

Worked Example: A Fixed Amount Per Policy

A hypothetical policy has a guaranteed benefit of $\$80{,}000$ before the first declaration. The scale provides a fixed reversionary addition of $\$1{,}250$ per policy at each of four annual declarations. Determine the total additions and the benefit after the fourth declaration.

Method. Because the scale specifies a dollar amount per policy, add $\$1{,}250$ once at each declaration. The percentage of the original sum assured is not needed.

After year 1, $G_1=\$80{,}000+\$1{,}250=\$81{,}250$. After year 2, $G_2=\$81{,}250+\$1{,}250=\$82{,}500$. After year 3, $G_3=\$82{,}500+\$1{,}250=\$83{,}750$. After year 4, $$ G_4=\$83{,}750+\$1{,}250=\$85{,}000. $$ The total additions are $4(\$1{,}250)=\$5{,}000$, and $\$80{,}000+\$5{,}000=\$85{,}000$.

Check. The final benefit agrees with the starting benefit plus four fixed additions. Multiplying $\$1{,}250$ by a percentage of the sum assured would introduce a percentage that the scale does not contain.

Choosing the Base When Descriptions Are Similar

A scale should be read for both its rate and its base. “Two per cent of the original sum assured” and “two per cent of the current guaranteed benefit” use the same rate but are not interchangeable. Likewise, “$\$1{,}250$ per policy” is a monetary addition, not a percentage. If wording does not identify the base clearly, the calculation is not determined by the rate alone; the terms of the scale must be clarified rather than silently assuming a base.

Locate the base phrase.
Identify whether the scale says original sum assured, current guaranteed benefit, or a fixed amount per policy.

Write the addition before calculating.
Use $b_k=r_kB$, $b_k=r_kG_{k-1}$, or $b_k=c_k$, as indicated by the wording.

Update only after finding the addition.
Set $G_k=G_{k-1}+b_k$. Use the updated amount as a later base only when the scale specifies the current guaranteed benefit.

The base matters whenever additions are projected over multiple declarations: a small difference in the amount added at one declaration can affect later additions under a current-benefit scale. Keeping the base explicit at each step makes the calculation auditable and helps expose mismatches between the scale wording and the arithmetic.

Common Mistakes

Using the current benefit for every percentage scale. This is wrong when the scale says “of original sum assured.” Apply the percentage to the original amount on each occasion.
Reusing the original amount for every percentage scale. This is wrong when the scale says “of current guaranteed benefit.” Use the amount immediately before the declaration, including prior additions.
Treating a fixed amount per policy as a rate. Add the stated dollars once per policy per declaration; do not multiply them by the sum assured.
Using the post-declaration benefit as the base for that same declaration. The current benefit means the pre-declaration amount here. Calculate the addition first, then update the benefit.

For example, in the current-benefit example, wrongly applying the second-year $3\%$ to the original $\$100{,}000$ gives an addition of $\$3{,}000$, rather than the correct $\$3{,}060$. The resulting second-year benefit would be $\$105{,}000$, not $\$105{,}060$. The correction is to take $3\%$ of the pre-declaration current benefit, $\$102{,}000$.

Key takeaway: The quoted rate does not determine the addition until its base is identified. Write the base in the addition formula first, calculate the addition, and only then update the guaranteed benefit.

Check Your Understanding

For each problem, identify the scale’s base before calculating. Answers are rounded to the cent where needed.

A hypothetical policy has an original sum assured of $\$50{,}000$. A scale declares $1.5\%$ of original sum assured at each of four declarations. Find the benefit after the fourth declaration. (Answer: $\$53{,}000.00$)
A hypothetical policy starts with a guaranteed benefit of $\$60{,}000$. It receives $1\%$, $2\%$, and $3\%$ of current guaranteed benefit at successive declarations. Find the final benefit. (Answer: $\$63{,}666.36$)
A scale adds $\$900$ per policy at each of five declarations to an initial guaranteed benefit of $\$72{,}000$. Find the total additions and final benefit. (Answer: total additions $\$4{,}500.00$; final benefit $\$76{,}500.00$)
A hypothetical policy has original sum assured $\$100{,}000$ and current guaranteed benefit $\$125{,}000$. Compare the addition under a $2.4\%$ scale based on original sum assured with the addition under a $2.4\%$ scale based on current guaranteed benefit. (Answer: $\$2{,}400.00$ on original sum assured; $\$3{,}000.00$ on current guaranteed benefit)
A hypothetical policy starts at $\$90{,}000$. The first declaration is $2\%$ of original sum assured; the second is $2\%$ of current guaranteed benefit; the third is a fixed $\$2{,}000$ per policy. Find the benefit after all three declarations. (Answer: $\$95{,}636.00$)

Scale wording	Calculation of the addition at declaration \(k\)	What changes over time?
\(r_k\) of original sum assured	\(b_k=r_kB\)	The rate may change; the base \(B\) does not.
\(r_k\) of current guaranteed benefit	\(b_k=r_kG_{k-1}\)	Both the rate and the base may change.
A fixed amount \(c_k\) per policy	\(b_k=c_k\)	The amount may change; it is not a percentage of the sum assured.

What You'll Learn