Calculating Simple Reversionary Bonuses

What a Simple Reversionary Bonus Adds

A reversionary bonus is an addition declared on a participating policy. Under the simple method, each declaration is calculated from the policy’s original sum assured. Once declared, that addition is vested according to the assumptions of the calculation. Later simple bonuses do not use earlier vested additions as part of their calculation base.

This distinction makes the calculation straightforward, but it is also a frequent source of error. The amount added in a declaration year depends on that year’s rate and the original sum assured—not on the current total of the sum assured plus previously declared bonuses. This tutorial calculates the vested additions only; it does not calculate a death or maturity claim amount.

Definition: Let $S$ be the original sum assured, and let $r_t$ be the simple reversionary bonus rate declared for year $t$, expressed as a proportion of $S$. The addition declared in year $t$ is $S r_t$. After $n$ declarations, the cumulative vested additions are $S\sum_{t=1}^{n}r_t$. The original sum assured remains the calculation base for every declaration.

A rate might instead be quoted as a dollar amount per $1,000 of original sum assured. In that case, first find the number of $1,000 units in the original sum assured, then multiply by the declared dollar rate. For example, an original sum assured of $240,000 contains 240 units of $1,000. A declaration of $20 per $1,000 therefore adds $240\times \$20=\$4,800$.

The schedule and problem wording determine which declarations apply. In the calculations below, each stated rate applies to one eligible declaration, and the policy is assumed to qualify for every declaration shown. Rates are not prorated for part of a year unless the problem explicitly gives a proration rule. No interest is added to vested bonuses in these calculations.

Calculating the Addition and Cumulative Total

For percentage rates, calculate each year’s amount separately, then add the amounts. Equivalently, add the rates first and multiply their sum by the original sum assured. Keeping the year-by-year amounts visible is useful for checking the declaration schedule and the running cumulative total.

B_t=S r_t,\qquad C_n=\sum_{t=1}^{n}B_t =S\sum_{t=1}^{n}r_t

Here $B_t$ is the addition from declaration $t$, and $C_n$ is the cumulative total after $n$ declarations. If the rate is quoted in dollars per $1,000, the equivalent calculation is

B_t=\left(\frac{S}{1{,}000}\right)q_t

where $q_t$ is the declared number of dollars per $1,000 for that year. The original sum assured must be expressed in dollars in this formula. If it is given in thousands, convert it before using the $1,000-unit count.

Worked Example: Three Percentage Declarations

A hypothetical participating policy has an original sum assured of $240,000. It receives three eligible annual simple reversionary bonus declarations at rates of 1.8%, 1.6%, and 2.1% of the original sum assured, respectively. Calculate each vested addition and the cumulative vested additions after the third declaration.

Method. Apply each year’s rate to the original $240,000, then add the resulting amounts. The declared rates are $0.018$, $0.016$, and $0.021$ as proportions.

First declaration: $ \$240{,}000\times0.018=\$4{,}320$. The cumulative additions after this declaration are $4,320.

Second declaration: $ \$240{,}000\times0.016=\$3{,}840$. The cumulative additions are $ \$4{,}320+\$3{,}840=\$8{,}160$.

Third declaration: $ \$240{,}000\times0.021=\$5{,}040$. The cumulative additions are $ \$8{,}160+\$5{,}040=\$13{,}200$.

Check by summing rates: $1.8\%+1.6\%+2.1\%=5.5\%$, and $ \$240{,}000\times0.055=\$13{,}200$, matching the year-by-year total. No rounding was required; amounts are stated to the nearest cent.

Common mistake avoided: The second year’s rate is not applied to $244,320, which includes the first addition. That would use a growing base and would not be the simple method.

Why the Cumulative Formula Works

The cumulative formula follows directly from the definition of a simple declaration. Start with no declared additions, so $C_0=0$. At declaration $t$, the amount added is $B_t=S r_t$. Therefore the cumulative amount after that declaration is the previous cumulative amount plus the new addition:

C_t=C_{t-1}+S r_t

Repeatedly substituting this recurrence gives $C_1=S r_1$, $C_2=S r_1+S r_2$, and, after $n$ declarations, $C_n=S r_1+S r_2+\cdots+S r_n$. Since $S$ is the same original sum assured in every term, it can be factored out:

C_n=S\sum_{t=1}^{n}r_t

This argument also explains why the total can be calculated either by summing the individual additions or by summing the rates and multiplying once by $S$. The two methods must agree, provided each rate has been interpreted correctly and applies to the same original sum assured.

Worked Example: Rates Quoted per $1,000

A hypothetical policy has an original sum assured of $175,000. Three eligible declarations are quoted as $32, $28, and $35 per $1,000 of original sum assured. Find each addition and the total vested additions after all three declarations.

Method. Convert the original sum assured into $1,000 units. There are $ \$175{,}000/\$1{,}000=175$ units. Multiply each quoted rate by 175.

First declaration: $175\times\$32=\$5{,}600$.

Second declaration: $175\times\$28=\$4{,}900$. The running total is $ \$5{,}600+\$4{,}900=\$10{,}500$.

Third declaration: $175\times\$35=\$6{,}125$. The total is $ \$10{,}500+\$6{,}125=\$16{,}625$.

Check: The quoted rates sum to $\$32+\$28+\$35=\$95$ per $1,000. Thus $175\times\$95=\$16{,}625$, matching the separate calculation. Amounts are exact to the cent.

Common mistake avoided: Multiplying $175,000 by 32 would treat the quoted rate as a percentage or a rate per dollar. The rate is per $1,000, so the correct multiplier is 175 units, not 175,000 dollars.

Using a Known Total to Check a Schedule

Sometimes a problem gives a cumulative vested amount and asks for a missing declaration rate. The same formula can be rearranged. For percentage rates, divide the cumulative amount by the original sum assured to find the sum of the rates. Subtract the known rates to isolate the missing one. This is a reconciliation calculation: the inferred rate should reproduce the given total when substituted back into the schedule.

Worked Example: Finding a Missing Declaration Rate

A hypothetical policy has an original sum assured of $280,000 and four eligible simple reversionary bonus declarations. The first, second, and fourth rates are 0.6%, 1.0%, and 1.4%. The cumulative vested additions after the fourth declaration are $12,600. Find the third declaration rate and its dollar addition.

Method. First convert the cumulative amount to a total rate on the original sum assured. Then subtract the known rates.

The total rate represented by $12,600 is $ \$12{,}600/\$280{,}000=0.045=4.5\%$.

The known rates sum to $0.6\%+1.0\%+1.4\%=3.0\%$. Therefore, the third rate is $4.5\%-3.0\%=1.5\%$.

The third declaration adds $ \$280{,}000\times0.015=\$4{,}200$.

Verification: The four additions are $ \$1{,}680$, $ \$2{,}800$, $ \$4{,}200$, and $ \$3{,}920$. Their sum is $ \$1{,}680+\$2{,}800+\$4{,}200+\$3{,}920=\$12{,}600$, as required. All amounts are exact to the cent.

Common mistake avoided: The known rates must be subtracted from the total rate of 4.5%, not directly from the dollar total. One may instead subtract the known dollar additions from $12,600, but the units must remain consistent throughout.

Checks and Common Exam Traps

A reliable solution makes the base, rate units, and declaration count explicit before calculating. In particular, do not infer a declaration for a year that is not included in the stated schedule. A rate described as “per annum” does not, by itself, establish how many declarations have occurred; use the stated declaration years and eligibility conditions.

The following comparison summarizes the central distinction in the calculation:

Calculation item	Simple reversionary bonus treatment
Base for each declaration	Original sum assured
Amount for a percentage rate	Original sum assured multiplied by that year’s rate
Amount for a rate per $1,000	Number of $1,000 units multiplied by the quoted rate
Cumulative vested additions	Sum of the individual declared additions
Earlier vested additions	Included in the cumulative total, but not in the base for a later simple declaration

A common incorrect calculation is to add the prior vested additions to the sum assured before applying the next rate. For example, after a 2% declaration on $100,000, an incorrect next-year calculation at 2% might be $2\%\times\$102{,}000=\$2{,}040$. Under the simple method, it is $2\%\times\$100{,}000=\$2{,}000$. The prior $2,000 remains part of the cumulative additions, but it does not increase the base for this calculation.

The distinction is not a matter of rounding or presentation: it changes the declared amount. Always label the original sum assured separately from the running vested total, and do not use the running total as a calculation base unless the problem explicitly specifies a different bonus method.

Key takeaway: For simple reversionary bonuses, calculate every declaration on the original sum assured. Add the resulting vested amounts to find the cumulative additions, and check that this total equals the original sum assured multiplied by the sum of the applicable percentage rates.

Check Your Understanding

For each problem, use the stated simple-bonus convention and show the year-by-year amounts before checking the total.

A hypothetical policy has an original sum assured of $190,000 and receives three eligible declarations of 1.2%, 0.8%, and 1.5%. Find the cumulative vested additions. (Answer: $6,650.)
A policy with an original sum assured of $240,000 receives rates of $18, $22, and $25 per $1,000 in three consecutive eligible declaration years. Find the total additions. (Answer: $15,600.)
A $360,000 original sum assured receives simple rates of 0.9%, 1.1%, and 1.4%. Find the addition at the third declaration and the cumulative additions after it. (Answer: third addition $5,040; cumulative additions $12,240.)
A policy with an original sum assured of $150,000 has four eligible declarations. Three rates are 0.8%, 1.2%, and 1.4%; the cumulative additions after all four declarations are $7,800. Find the missing rate and its addition. (Answer: missing rate 1.8%; addition $2,700.)
A $200,000 policy receives three eligible declarations at $15, $24, and $31 per $1,000. A calculation uses the first two additions to form a running total, then applies the third rate to the original sum assured. Find the third addition and final cumulative additions. (Answer: third addition $6,200; cumulative additions $14,000.)

What You'll Learn