What Makes a Reversionary Bonus Compound?
A reversionary bonus is an addition to a policy benefit that is declared at a specified date and, under the hypothetical contract terms used here, becomes part of the benefit base for later declarations. With a compound reversionary bonus, each new bonus is calculated on the sum assured plus previously declared additions. The base therefore grows from one declaration to the next.
The focus is the arithmetic of applying each annual rate to that changing base. Unless a problem states otherwise, assume the annual bonus is declared at the end of each policy year, is expressed as a percentage of the sum assured plus previously declared bonuses, and has no additional interest credited to it between declarations. A declaration is not a cash payment in these calculations: it increases the stated benefit amount.
This calculation depends on the contract basis given in the question. If the stated bonus basis differs, follow that basis rather than assuming it is compound. In the examples below, the basis is specified directly so the calculation can concentrate on the year-by-year effect.
The Calculation at Each Declaration
Maintain two quantities: the bonus declared so far and the current benefit base. At a declaration, multiply the stated rate by the current base to find the new addition. Add that amount to the accumulated bonuses, then use the increased base for the next declaration. Keeping these quantities visible makes it easier to check each step.
Before the first declaration, the base is the original sum assured \(S\).
Multiply the rate for that declaration by the current base.
Add the new addition to the base. This updated amount is used for the next declaration.
After the final declaration, the benefit amount is the original sum assured plus all declared additions.
The reason the base grows in this way is straightforward. If the current base is \(G\) and the next declared rate is \(r\), the addition is \(rG\). Including that addition makes the next base \(G+rG=G(1+r)\). Thus, each declaration increases the current base by the stated percentage; it does not merely add the same percentage of the original sum assured every year.
For a sequence of different annual rates, apply this step using the rate for each declaration in turn. Do not apply a later rate to an earlier base or treat a rate as an amount in dollars.
Worked Example: Three Declarations at a Fixed Rate
A hypothetical participating policy has an original sum assured of $100,000. A compound reversionary bonus of 3% is declared at the end of each of the next three policy years. Calculate each addition and the benefit amount immediately after the third declaration. Use the stated 3% rate at each declaration; no other interest applies.
Method. Update the base after every declaration, then add the three additions to the original sum assured. The main error to avoid is calculating 3% of $100,000 each year.
Before year 1: the base is $100,000. The first addition is \(0.03\times \$100{,}000=\$3{,}000\). The updated base is \(\$100{,}000+\$3{,}000=\$103{,}000\).
Before year 2: the base is $103,000. The second addition is \(0.03\times \$103{,}000=\$3{,}090\). The updated base is \(\$103{,}000+\$3{,}090=\$106{,}090\).
Before year 3: the base is $106,090. The third addition is \(0.03\times \$106{,}090=\$3{,}182.70\). The updated base is \(\$106{,}090+\$3{,}182.70=\$109{,}272.70\).
The total additions are \(\$3{,}000+\$3{,}090+\$3{,}182.70=\$9{,}272.70\). The benefit after the third declaration is \(\$100{,}000+\$9{,}272.70=\$109{,}272.70\). All amounts are exact to the cent, so no intermediate rounding was needed.
Check: each addition is 3% of the base immediately before its declaration, and each updated base equals the previous base plus that addition.
When the Declared Rates Change
A different rate can apply at each declaration. The process is unchanged: use the current year’s rate on the current base, then update the base. The rate sequence and the base sequence are separate inputs; recording both prevents an easy but consequential mix-up.
Worked Example: A Different Rate in Each Year
A hypothetical policy has a sum assured of $80,000. Its compound reversionary bonus rates are 2% at the end of year 1, 3% at the end of year 2, and 4% at the end of year 3. Find the additions and the benefit after the third declaration. Assume the rates apply to the sum assured plus bonuses already declared.
Method. Proceed chronologically, pairing each rate with the base at that declaration. The common mistake is to use the latest rate on the original sum assured, or to apply it before updating the previous year’s base.
Year 1: the starting base is $80,000. The addition is \(0.02\times \$80{,}000=\$1{,}600\), giving an updated base of \(\$80{,}000+\$1{,}600=\$81{,}600\).
Year 2: use the year 2 rate of 3% on $81,600. The addition is \(0.03\times \$81{,}600=\$2{,}448\), giving an updated base of \(\$81{,}600+\$2{,}448=\$84{,}048\).
Year 3: use the year 3 rate of 4% on $84,048. The addition is \(0.04\times \$84{,}048=\$3{,}361.92\). The final base is \(\$84{,}048+\$3{,}361.92=\$87{,}409.92\).
The total additions are \(\$1{,}600+\$2{,}448+\$3{,}361.92=\$7{,}409.92\). Adding them to the original sum assured gives \(\$80{,}000+\$7{,}409.92=\$87{,}409.92\). The two ways of checking the final amount agree.
Check: the year 3 addition is 4% of $84,048, not 4% of $80,000 or $81,600.
Compound Versus Simple Additions
The word “compound” matters because previous additions affect later bonus amounts. Under a simple bonus basis, each year’s addition would instead be calculated on the original sum assured. Comparing the two can be a useful arithmetic check, but the calculation must follow the basis named in the hypothetical contract.
Worked Example: Comparing Compound and Simple Results
A hypothetical policy has a $60,000 sum assured and declares a 2.5% annual reversionary bonus at each of four year-end declarations. Under the compound basis, each rate applies to the sum assured plus previous additions. Calculate the compound benefit after year 4, then compare it with the benefit if the same four declarations were simple bonuses on the original sum assured. Assume the rate and timing stated here apply throughout.
Method. First calculate each compound addition using the updated base. Then calculate the simple additions separately on $60,000 each year. The trap is to use the simple basis for one or more years in the compound calculation.
Year 1: \(0.025\times \$60{,}000=\$1{,}500\). The updated base is $61,500.
Year 2: \(0.025\times \$61{,}500=\$1{,}537.50\). The updated base is \(\$61{,}500+\$1{,}537.50=\$63{,}037.50\).
Year 3: \(0.025\times \$63{,}037.50=\$1{,}575.9375\). The updated base is \(\$63{,}037.50+\$1{,}575.9375=\$64{,}613.4375\).
Year 4: \(0.025\times \$64{,}613.4375=\$1{,}615.3359375\). The final base is \(\$64{,}613.4375+\$1{,}615.3359375=\$66{,}228.7734375\).
The compound benefit rounds to $66,228.77. For the simple comparison, each addition is \(0.025\times \$60{,}000=\$1{,}500\), so four additions total \(\$4\times \$1{,}500=\$6{,}000\), and the simple benefit is $66,000. The compound result is \(\$66{,}228.7734375-\$66{,}000=\$228.7734375\), or $228.77 more when rounded to cents.
Check: every compound addition after year 1 exceeds $1,500 because it is calculated on a base larger than $60,000. The compound total must therefore exceed the simple total.
Common Errors and Checks
A reliable calculation should make the declaration date, rate, and base explicit at every step. The following incorrect calculation illustrates the central exam trap.
For the $100,000 policy with three annual declarations at 3%, an incorrect calculation would use \(0.03\times \$100{,}000=\$3{,}000\) in every year, leading to a claimed total of \(\$100{,}000+3(\$3{,}000)=\$109{,}000\). That is a simple-bonus calculation, not a compound-bonus calculation. The correction is to apply year 2’s rate to $103,000 and year 3’s rate to $106,090, producing additions of $3,090 and $3,182.70, respectively, and a final benefit of $109,272.70.
- Do not confuse a rate with a dollar amount. A 3% rate must be converted to 0.03 before multiplication.
- Do not reuse the original sum assured as the base. Under the compound basis specified here, previously declared additions are included.
- Do not swap the rate sequence. When rates vary, apply each rate at its stated declaration date.
- Do not add interest unless the terms say to do so. The increases in these examples result from the declared bonus basis, not a separate investment return.
- Check the total two ways. The final base should equal the original sum assured plus the sum of all additions.
For exam calculations, keep full precision in intermediate amounts and round the final money result to cents unless the question specifies another convention. If the question supplies rates to a particular precision, use those rates as given; do not round an updated base prematurely.
Check Your Understanding
For each hypothetical policy, apply the stated compound basis and use the rate at each declaration. Answers are rounded to cents where needed.
- A $50,000 sum assured receives a 2% bonus at the end of each of two years. What are the two additions and the benefit after the second declaration? (Answer: additions $1,000 and $1,020; benefit $52,020.)
- A $120,000 sum assured receives rates of 1.5%, 2%, and 3% at the ends of years 1, 2, and 3. What is the benefit after year 3? (Answer: additions $1,800, $2,436, and $3,727.08; benefit $127,963.08.)
- A $75,000 sum assured receives a 4% compound bonus at the end of each of three years. Calculate the total additions and final benefit. (Answer: additions $3,000, $3,120, and $3,244.80; total additions $9,364.80; benefit $84,364.80.)
- A $40,000 sum assured receives rates of 2%, 2.5%, and 3% at the ends of years 1, 2, and 3. Calculate the final benefit. (Answer: additions $800, $1,020, and $1,254.60; benefit $43,074.60.)
- A $10,000 sum assured has two equal annual compound declarations at an unknown rate. The benefit immediately after the second declaration is $10,609. Find the annual rate. (Answer: 3% per year.)