Putting a Participating Benefit Projection Together
An integrated dividend-and-bonus problem is a sequence of contractual events, not one calculation. A policy may receive bonuses at different scales, use a dividend to purchase paid-up additions, change to paid-up status and then make a claim. The calculation depends on which events have occurred by the claim date and which policy amount is eligible for each declaration.
The method in this tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The examples use hypothetical contract terms. Each states the relevant eligibility rules explicitly: actual policies can define bonus bases, status-change dates, dividend options and terminal bonuses differently.
For a simple bonus stated as an amount per $1,000, the bonus for one declaration is the rate multiplied by the eligible sum assured measured in thousands. If the eligible amount changes, for example because a policy becomes paid-up, identify the amount in force for that declaration before calculating the bonus. Do not assume that a dividend-funded addition is eligible for a bonus declared before the addition took effect.
A reliable sequence is to record the policy status and the event timing first, then calculate each declared bonus on its eligible amount. Next, calculate any dividend option and its additions. Finally, identify the claim-date benefit and apply any terminal bonus only to its stipulated base.
Mark each bonus declaration, dividend option, status change and claim. Decide whether the claim occurs before or after the relevant declaration.
For every declaration, use its stated rate and the sum assured eligible at that time. Keep bonuses on the basic policy separate from bonuses on additions.
If a dividend buys additions, divide the dividend by the stated purchase price per unit. Include only additions in force by the relevant bonus declaration or claim.
Add the in-force basic or paid-up amount and all declared bonuses. Add a terminal bonus only after confirming its base and rate.
Keep cash dividends already paid separate from the claim benefit. Include them in a total-proceeds comparison only when asked, and use an explicit assumption about interest.
Why the Calculation Order Matters
Suppose the contract declares simple bonuses on eligible amounts \(E_j\), at rates \(r_j\) dollars per $1,000. The declaration in period \(j\) adds \(r_jE_j/1{,}000\) dollars. If the contract says these bonuses are simple, rather than compound, the next declaration is still calculated on its stated eligible amount; previously declared bonuses do not silently become part of that amount.
If the terminal bonus is \(t\) and the contract specifies that its base is the sum of the claim amount before terminal bonus, then the terminal bonus is \(t\) times that pre-terminal amount. Thus, when the pre-terminal amount consists of the basic or paid-up sum assured \(B\), paid-up additions \(A\), and declared bonuses \(R\), the claim amount is:
This identity follows by adding the terminal bonus \(t(B+A+R)\) to the pre-terminal amount \(B+A+R\). It applies only when the contract defines the terminal-bonus base that way. A terminal bonus may instead use a different base, and an undeclared reversionary bonus is not included merely because a future projection illustrates one.
Worked Example: Death Before the Next Bonus Declaration
A hypothetical policy has a basic sum assured of $60,000. It declares simple bonuses at the ends of years 1 and 2 of $7 and $9 per $1,000, respectively. At the end of year 2, a $720 dividend purchases paid-up additions at $240 per $1,000. The additions receive the year 3 bonus scale, but no earlier bonus. The insured dies during year 3, before the year 3 declaration. A 4% terminal bonus applies to the basic sum assured, additions and declared bonuses. Calculate the death benefit.
Assume \(i=0\%\); no cash flow is accumulated or discounted. The death time is stipulated, so no mortality distribution is needed. The dividend is used to buy additions and is not also paid in cash.
Method and timeline. The year 1 and year 2 declarations have occurred, as has the year 2 dividend option. The death occurs before the year 3 declaration, so the year 3 bonus is not declared and is excluded.
The basic sum assured is $60,000, or \(60{,}000/1{,}000=60\) units of $1,000. The year 1 bonus is \(60\times \$7=\$420\); the year 2 bonus is \(60\times \$9=\$540\). Total declared bonuses on the basic policy are \(\$420+\$540=\$960\).
The dividend buys \(\$720/\$240=3\) units of $1,000 in additions, so the additions are \(\$3,000\). Although the additions would be eligible for the year 3 scale if that declaration occurred, it has not occurred by the death. They therefore receive no year 3 bonus in this claim calculation.
The terminal-bonus base is \(\$60{,}000+\$3{,}000+\$960=\$63{,}960\). The terminal bonus is \(0.04\times \$63{,}960=\$2{,}558.40\). The death benefit is \(\$63{,}960+\$2{,}558.40=\$66{,}518.40\).
Check: The total includes the basic sum assured, the additions, the two declared bonuses and the terminal bonus: \(\$60{,}000+\$3{,}000+\$420+\$540+\$2{,}558.40=\$66{,}518.40\). The common mistake avoided is assigning a year 3 bonus when the insured dies before its declaration.
Status Changes and Changing Bonus Scales
A status change can affect the bonus base from a specified effective date. For example, a paid-up conversion may stop future premiums and reduce the sum assured used for later declarations. It does not, by itself, erase bonuses already declared or establish what happens to paid-up additions. Those effects must come from the stated contract terms.
For each declaration, write down the eligible amount before multiplying by the rate. In particular, if a policy converts to paid-up status immediately after a declaration, use the pre-conversion amount for that declaration when the terms say so, and the paid-up amount for later declarations. This prevents applying one amount across all policy years merely because it was the original sum assured.
Worked Example: A Paid-Up Conversion After a Declaration
A hypothetical policy begins with a basic sum assured of $50,000. At the end of year 1 it declares a simple bonus of $4 per $1,000, and a $450 dividend immediately buys paid-up additions at $150 per $1,000. The year 2 bonus is $8 per $1,000 on the basic policy before conversion; the additions also receive that year 2 scale. Immediately after the year 2 declaration, the policy converts to paid-up status at 80% of its original basic sum assured. At the end of year 3, the paid-up basic sum assured and additions each receive a $12 per $1,000 bonus. Maturity is immediately after the year 3 declaration. A 2.5% terminal bonus applies to the paid-up basic amount, additions and all declared bonuses. Find the maturity benefit.
Assume \(i=0\%\); no cash flow accumulation or discounting is needed. The claim event is certain at the stipulated maturity date, so no mortality distribution is needed.
Method and timeline. Calculate year 1 on the original basic amount. The additions start after the year 1 declaration, so they receive no year 1 bonus. Calculate year 2 on the original basic amount and on additions already in force. Convert the basic policy after year 2, then use the paid-up amount for year 3.
The basic policy has \(50{,}000/1{,}000=50\) units. Its year 1 bonus is \(50\times \$4=\$200\), and its year 2 bonus is \(50\times \$8=\$400\).
The dividend buys \(\$450/\$150=3\) units, or $3,000, of paid-up additions. Those additions are in force for the year 2 declaration, so their year 2 bonus is \(3\times \$8=\$24\).
The paid-up basic sum assured is \(0.80\times \$50{,}000=\$40{,}000\). It has 40 units for the year 3 declaration, so its year 3 bonus is \(40\times \$12=\$480\). The $3,000 of additions has 3 units, so its year 3 bonus is \(3\times \$12=\$36\).
All declared bonuses total \(\$200+\$400+\$24+\$480+\$36=\$1{,}140\). The pre-terminal maturity amount is \(\$40{,}000+\$3{,}000+\$1{,}140=\$44{,}140\). The terminal bonus is \(0.025\times \$44{,}140=\$1{,}103.50\), giving a maturity benefit of \(\$44{,}140+\$1{,}103.50=\$45{,}243.50\).
Check: The year 3 bonus uses the $40,000 paid-up amount, not the original $50,000. The total also includes both bonuses earned by the additions. The common mistake avoided is applying the post-conversion base to year 2, even though the conversion takes effect only after that declaration.
Cash Dividends, Additions and Claim-Date Reconciliation
A dividend paid in cash and a dividend used to purchase additions are different benefit choices. A cash dividend paid before the claim is not automatically part of the later claim benefit. If a problem asks for total proceeds over several dates, report the cash payment separately or accumulate it using the specified interest assumption. An addition, by contrast, may increase the claim benefit and may receive later bonuses if the contract says it is eligible.
The following example compares the two options using the same hypothetical policy. It keeps the claim amount distinct from total nominal proceeds, so the dividend is not counted twice.
Worked Example: Comparing Cash and Addition Options at Maturity
A hypothetical policy has a $30,000 basic sum assured and declares simple bonuses of $10, $5 and $9 per $1,000 at the ends of years 1, 2 and 3. At the end of year 2, the policyholder receives a $600 dividend. Under option C it is paid in cash. Under option A it buys paid-up additions at $200 per $1,000; those additions receive only the year 3 bonus. Maturity occurs immediately after the year 3 declaration. A 5% terminal bonus applies to the basic sum assured, additions and all declared bonuses. Compare the maturity benefit under each option and the total nominal proceeds, including the dividend where applicable.
Assume \(i=0\%\), so the earlier cash dividend is not accumulated. No mortality distribution is needed because maturity at the stated time is stipulated.
Method and timeline. The basic policy has \(30{,}000/1{,}000=30\) units. Its three bonuses are \(30\times \$10=\$300\), \(30\times \$5=\$150\) and \(30\times \$9=\$270\), for a total of \(\$720\). Under option C, the dividend remains a separate cash payment. Under option A, additions take effect after year 2 and receive the year 3 bonus only.
Option C: cash dividend. The terminal-bonus base is the basic sum assured plus its declared bonuses: \(\$30{,}000+\$720=\$30{,}720\). The terminal bonus is \(0.05\times \$30{,}720=\$1{,}536\). The maturity benefit is \(\$30{,}720+\$1{,}536=\$32{,}256\). Including the $600 cash dividend paid at the end of year 2, total nominal proceeds are \(\$32{,}256+\$600=\$32{,}856\).
Option A: paid-up additions. The dividend buys \(\$600/\$200=3\) units, or $3,000, of additions. The year 3 bonus on them is \(3\times \$9=\$27\). Thus total declared bonuses are \(\$720+\$27=\$747\), and the terminal-bonus base is \(\$30{,}000+\$3{,}000+\$747=\$33{,}747\). The terminal bonus is \(0.05\times \$33{,}747=\$1{,}687.35\). The maturity benefit and total nominal proceeds are both \(\$33{,}747+\$1{,}687.35=\$35{,}434.35\), because no dividend was paid separately in cash.
Check: Option C total nominal proceeds exceed its maturity benefit by exactly $600, the separately paid dividend. Option A includes the $3,000 of additions and their $27 bonus in the maturity calculation, but does not also add the $600 dividend. The common mistake avoided is adding the dividend to both the additions and the claim benefit.
Common Exam Traps
Integrated problems often reward careful event ordering more than lengthy arithmetic. Before calculating, translate the wording into a timeline and an eligibility rule for each amount. Then check the final benefit by adding its components independently.
- Using a future declaration. A bonus illustrated for a year is not necessarily declared by a death earlier in that year. Include only bonuses declared by the claim date.
- Ignoring the effective date of a status change. If conversion occurs after a declaration, that declaration may still use the former amount. Follow the stated timing rather than applying the new amount retrospectively.
- Giving additions a bonus before they exist. An addition purchased at the end of a year cannot receive an earlier declaration unless the contract explicitly provides otherwise.
- Compounding a simple bonus without instruction. Apply each rate to the eligible amount specified for that declaration. Do not add old bonuses to the next bonus base unless the terms say bonuses compound.
- Misreading the terminal-bonus base. A stated percentage is not enough: identify exactly which amounts the percentage applies to before calculating it.
- Counting a dividend twice. A cash payment and a purchase of additions are alternative treatments of the same dividend, not two benefits from one amount.
Check Your Understanding
For each hypothetical case, calculate the requested amount using the stated terms. Assume \(i=0\%\) where a prior cash payment is included in total nominal proceeds; claims occur at the stated times, so no mortality distribution is needed.
- A $25,000 basic sum assured receives simple bonuses of $6 and $8 per $1,000 at the ends of years 1 and 2. Maturity is immediately after the year 2 declaration. A 3% terminal bonus applies to the basic sum assured and declared bonuses. Find the maturity benefit. (Answer: $26,110.50)
- A $48,000 policy receives a $360 dividend at the end of year 1, used to buy additions at $180 per $1,000. The basic policy receives bonuses of $5 per $1,000 in year 1 and $7 per $1,000 in year 2. The additions receive only the year 2 bonus. Maturity is after year 2, and a 2% terminal bonus applies to the basic sum assured, additions and declared bonuses. Find the maturity benefit. (Answer: $51,601.80)
- A $40,000 policy receives bonuses of $3 and $6 per $1,000 in years 1 and 2, both on the original basic sum assured. It converts to paid-up status at 75% of that amount immediately after year 2. In year 3 the paid-up amount receives a $10 per $1,000 bonus. Maturity is after the year 3 declaration, with a 4% terminal bonus on the paid-up amount and declared bonuses. Find the maturity benefit. (Answer: $31,886.40)
- A $55,000 policy has year 1 and year 2 bonus scales of $4 and $8 per $1,000. The insured dies during year 3 before the year 3 declaration. A 1.5% terminal bonus applies to the basic sum assured and declared bonuses. Find the death benefit. (Answer: $56,494.90)
- A $20,000 policy earns bonuses of $5 and $8 per $1,000 at the ends of years 1 and 2. A $400 dividend at the end of year 1 is either paid in cash or used to buy additions at $200 per $1,000; additions receive the year 2 bonus only. Maturity is after year 2, with a 3% terminal bonus on the basic sum assured, additions and declared bonuses. Compare total nominal proceeds under the cash option with the maturity benefit under the addition option. (Answer: Cash option total nominal proceeds $21,267.80; addition option maturity benefit $22,944.28)