Worked Example Of Simple And Compound Bonuses

Introduction

A reversionary bonus can increase a policy’s guaranteed claim benefit each time an addition is declared. The key calculation is not only the percentage rate: it is also the benefit base to which that rate applies. Under a simple scale the base remains the original sum assured, while under a compound scale it includes earlier additions. As a result, the same annual rate can produce different guaranteed benefits over the same duration.

This tutorial compares simple and compound bonus scales for hypothetical policies. Each example tracks the addition in every policy year and then calculates the guaranteed amount payable on a stated claim date. The calculations are conditional benefit projections: they do not discount future payments or calculate the probability of a claim.

Definition: Let $S$ be the original basic sum assured and let $r_k$ be the reversionary bonus rate declared for policy year $k$. Under a simple bonus scale, the year-$k$ addition is $r_kS$. Under a compound bonus scale, the year-$k$ addition is $r_kG_{k-1}$, where $G_{k-1}$ is the guaranteed benefit immediately before that year’s addition. In either case, the claim benefit after $n$ declarations is the original sum assured plus all additions credited by then.

Set the Benefit Base Before Calculating

For a simple scale, the original sum assured is the base in every year. If the rates are constant at $r$, the guaranteed amount after $n$ years is

G_n^{\text{simple}}=S+n(rS)=S(1+nr).

For a compound scale, the guaranteed amount grows recursively. The addition for the next year is based on the amount already guaranteed at the start of that year:

G_0^{\text{compound}}=S,\qquad G_k^{\text{compound}}=G_{k-1}^{\text{compound}}(1+r_k).

The recurrence makes the timing explicit: first identify the guaranteed amount before the year’s declaration, calculate the addition using the applicable rate and base, and then add it to obtain the amount after declaration. A prior bonus is included once in the current guaranteed amount; it is not separately added again.

Assumptions for these projections: Each quoted bonus rate is applied once for its stated policy year and becomes guaranteed when declared. No terminal bonus, cash dividend, interest accumulation, tax, or other benefit adjustment applies. The claim is assumed to occur after the stated year’s declaration. Because the examples calculate a conditional guaranteed amount rather than an expected present value, no mortality distribution or claim probability is needed; no discounting is performed.

Worked Calculations

For a fair comparison, use the same original sum assured, annual rates, and claim timing under both scales. Only the rule for selecting the bonus base changes.

Worked Example: Constant Rate Over Four Years

A hypothetical participating policy has an original sum assured of $80,000. It declares a 2.5% reversionary bonus at the end of each of four policy years. Calculate the addition in every year and the guaranteed claim benefit immediately after the fourth declaration under (a) a simple scale and (b) a compound scale.

Assumptions and method. The claim occurs at the end of year 4, after that year’s declaration. The annual bonus rate is 2.5% in each year. The policy’s claim timing is fixed for this calculation; no mortality probability, mortality distribution, or interest rate is used because this is not a present-value calculation. Under the simple scale, multiply the original $80,000 by 0.025 each year. Under the compound scale, multiply the guaranteed amount at the start of each year by 0.025.

(a) Simple scale. In every year, the addition is $0.025\times \$80{,}000=\$2{,}000$. The guaranteed amounts after each declaration are:

\begin{aligned} G_0&=\$80{,}000,\\ G_1&=\$80{,}000+\$2{,}000=\$82{,}000,\\ G_2&=\$82{,}000+\$2{,}000=\$84{,}000,\\ G_3&=\$84{,}000+\$2{,}000=\$86{,}000,\\ G_4&=\$86{,}000+\$2{,}000=\$88{,}000. \end{aligned}

Thus the four additions are $2,000, $2,000, $2,000, and $2,000, for total additions of $8,000. The guaranteed claim benefit is $88,000.00.

(b) Compound scale. The first addition is $0.025\times \$80{,}000=\$2{,}000$, giving $82,000 after year 1. Each subsequent addition uses the amount at the start of that year:

\begin{aligned} \text{Year 1 addition}&=0.025(\$80{,}000)=\$2{,}000, & G_1&=\$82{,}000,\\ \text{Year 2 addition}&=0.025(\$82{,}000)=\$2{,}050, & G_2&=\$84{,}050,\\ \text{Year 3 addition}&=0.025(\$84{,}050)=\$2{,}101.25, & G_3&=\$86{,}151.25,\\ \text{Year 4 addition}&=0.025(\$86{,}151.25)=\$2{,}153.78125, & G_4&=\$88{,}305.03125. \end{aligned}

The additions total $2,000 + $2,050 + $2,101.25 + $2,153.78125 = $8,305.03125. Rounding only the final claim benefit to cents gives $88,305.03. The compound-scale benefit exceeds the simple-scale benefit by $88,305.03125 − $88,000 = $305.03125, or $305.03 to the nearest cent.

Check. The compound additions increase because each year’s base includes prior additions. The final compound amount also equals $\$80{,}000(1.025)^4=\$88{,}305.03125$, which verifies the recurrence. A common mistake is to use $2,000 as the compound addition in every year; that incorrectly keeps the original sum assured as the base.

Worked Example: Different Rates in Successive Years

A hypothetical policy has an original sum assured of $50,000. Its declared bonus rates are 3% in year 1, 2% in year 2, and 4% in year 3. Find each annual addition and the guaranteed amount after year 3 for both bonus scales.

Assumptions and method. The claim is at the end of year 3, after the third declaration. Use the stated rate for each year. As in the first example, the claim time is fixed, no interest or mortality model is required, and the result is an undiscounted guaranteed benefit. Apply each rate to $50,000 for the simple scale; apply it to the current guaranteed amount for the compound scale.

(a) Simple scale. The base remains $50,000:

\begin{aligned} \text{Year 1 addition}&=0.03(\$50{,}000)=\$1{,}500,\\ \text{Year 2 addition}&=0.02(\$50{,}000)=\$1{,}000,\\ \text{Year 3 addition}&=0.04(\$50{,}000)=\$2{,}000. \end{aligned}

The total additions are $1,500 + $1,000 + $2,000 = $4,500. Therefore the guaranteed amount is $50,000 + $4,500 = $54,500.00.

(b) Compound scale. Calculate the guaranteed amount after each declaration:

\begin{aligned} G_0&=\$50{,}000,\\ \text{Year 1 addition}&=0.03(\$50{,}000)=\$1{,}500, & G_1&=\$51{,}500,\\ \text{Year 2 addition}&=0.02(\$51{,}500)=\$1{,}030, & G_2&=\$52{,}530,\\ \text{Year 3 addition}&=0.04(\$52{,}530)=\$2{,}101.20, & G_3&=\$54{,}631.20. \end{aligned}

The compound additions total $1,500 + $1,030 + $2,101.20 = $4,631.20, so the final guaranteed claim benefit is $54,631.20. It is $131.20 greater than the simple-scale amount.

Check. The final amount can also be verified by multiplying the successive growth factors: $ \$50{,}000(1.03)(1.02)(1.04)=\$54{,}631.20 $. A common mistake is to apply the 4% year-3 rate to the original $50,000 under the compound scale; the correct year-3 base is $52,530.

Worked Example: Tracking a Mixed Rate Schedule

A hypothetical policy has an original sum assured of $120,000 and declares bonuses of 2%, 3%, 2%, and 4% in policy years 1 through 4, respectively. A claim occurs immediately after the fourth declaration. Calculate the annual additions, the final guaranteed amount under each scale, and the difference between the two amounts.

Assumptions and method. All four declarations are guaranteed, the claim is after year 4, and no other benefit is payable. No interest, mortality assumption, or probability distribution is used: the requested quantities are conditional benefit amounts, not expected values. First calculate each simple addition from $120,000; then recursively update the compound base after each declaration.

(a) Simple scale. The annual additions are $0.02(\$120{,}000)=\$2{,}400$, $0.03(\$120{,}000)=\$3{,}600$, $0.02(\$120{,}000)=\$2{,}400$, and $0.04(\$120{,}000)=\$4{,}800$. Their total is $2,400 + $3,600 + $2,400 + $4,800 = $13,200. The guaranteed amount after each year is $122,400, $126,000, $128,400, and $133,200, respectively. Thus the final simple-scale benefit is $133,200.00.

(b) Compound scale. Apply each rate to the amount at the start of the corresponding year:

\begin{aligned} G_0&=\$120{,}000,\\ \text{Year 1 addition}&=0.02(\$120{,}000)=\$2{,}400, & G_1&=\$122{,}400,\\ \text{Year 2 addition}&=0.03(\$122{,}400)=\$3{,}672, & G_2&=\$126{,}072,\\ \text{Year 3 addition}&=0.02(\$126{,}072)=\$2{,}521.44, & G_3&=\$128{,}593.44,\\ \text{Year 4 addition}&=0.04(\$128{,}593.44)=\$5{,}143.7376, & G_4&=\$133{,}737.1776. \end{aligned}

The compound additions total $2,400 + $3,672 + $2,521.44 + $5,143.7376 = $13,737.1776, which added to $120,000 gives $133,737.1776. Rounded only at the final step, the guaranteed claim benefit is $133,737.18. The difference from the simple-scale benefit is $133,737.1776 − $133,200 = $537.1776, or $537.18 to the nearest cent.

Check. The year-4 addition is 4% of $128,593.44, not 4% of $120,000 and not 4% of the amount after the year-4 addition. The former would incorrectly treat the scale as simple; the latter would apply the bonus to itself as well.

Why Compound Additions Use the Updated Amount

The compound-scale formula follows directly from the definition of the bonus base. Immediately before declaration $k$, the guaranteed amount is $G_{k-1}^{\text{compound}}$. The declaration adds the rate $r_k$ times that amount. Therefore,

\begin{aligned} G_k^{\text{compound}} &=G_{k-1}^{\text{compound}}+ r_kG_{k-1}^{\text{compound}}\\ &=G_{k-1}^{\text{compound}}(1+r_k). \end{aligned}

Applying this relation in sequence gives $G_n^{\text{compound}}=S\prod_{k=1}^{n}(1+r_k)$. For a constant rate $r$, this becomes $S(1+r)^n$. By contrast, the simple additions sum to $S\sum_{k=1}^{n}r_k$, so the final simple amount is $S(1+\sum_{k=1}^{n}r_k)$. These formulas describe benefit accumulation under the stated scales; they are not investment-return formulas.

If all rates are nonnegative, the compound amount is at least the simple amount when both use the same original sum assured, rates, and duration. To see why, the compound base before each declaration is at least $S$; hence its addition $r_kG_{k-1}^{\text{compound}}$ is at least $r_kS$. Summing the annual additions preserves that inequality. If a rate is negative or the scale has special rules, this conclusion should not be assumed without checking the actual terms.

Common Exam Traps and a Reliable Check

The most frequent error is choosing the wrong base. A simple scale uses the original sum assured every year, even though previous bonuses have increased the guaranteed claim amount. A compound scale uses the guaranteed amount immediately before the current declaration. The labels describe the rule for the base, not whether a bonus is guaranteed or whether the policy earns investment interest.

Using the compound base for a simple scale: this incorrectly increases later simple additions.
Keeping the original base for a compound scale: this omits the effect of prior guaranteed additions.
Applying a declaration twice: after calculating an addition, update the guaranteed amount once; do not add the same bonus again at claim.
Ignoring the claim date: include only declarations made by the specified claim date. If the claim precedes a declaration, that declaration is not included.
Rounding at each step: retain exact intermediate amounts when possible and round the final money amount to cents, unless the policy terms prescribe a different rounding rule.

A practical check is to list, for every year, the rate, the base, the resulting addition, and the amount after declaration. For a simple scale the base column should always equal the original sum assured. For a compound scale the base in each row should equal the previous row’s amount after declaration. The sum of all additions plus the original sum assured must equal the final guaranteed benefit.

Key takeaway: Under a simple reversionary bonus scale, calculate each addition from the original sum assured. Under a compound scale, calculate each addition from the guaranteed amount at the start of that year. State the claim timing, record every annual addition, and reconcile the additions to the final guaranteed benefit.

Check Your Understanding

For each hypothetical policy, identify the bonus base in each year and calculate the requested guaranteed amount. Assume each claim occurs immediately after the last stated declaration; no interest, mortality probabilities, or other benefit adjustments are involved.

A $60,000 policy receives a simple bonus of 1.5% at the end of each of three years. Find each annual addition and the final guaranteed amount. (Answer: additions $900, $900, and $900; final amount $62,700.00)
A $40,000 policy receives a compound bonus of 2% in year 1 and 3% in year 2. Find both additions and the amount after year 2. (Answer: additions $800 and $1,224; final amount $42,024.00)
A $75,000 policy has simple bonus rates of 2%, 1%, and 3% in years 1–3. Find the final guaranteed amount. (Answer: additions $1,500, $750, and $2,250; final amount $79,500.00)
A $90,000 policy has compound bonus rates of 1%, 2%, and 2% in years 1–3. Find each addition and the final amount. (Answer: additions $900, $1,818, and $1,854.36; final amount $94,572.36)
A $100,000 policy has rates of 2%, 3%, and 1% in years 1–3. Compare the final amounts under simple and compound scales using these same rates. (Answer: simple additions $2,000, $3,000, and $1,000, final $106,000.00; compound additions $2,000, $3,060, and $1,050.60, final $106,110.60; compound exceeds simple by $110.60)

What You'll Learn