Allowing for Declared Bonuses in Asset-Share Cash Flows

Why Bonus Timing Matters

A declared bonus can affect an asset-share projection even when no money leaves the fund at the moment of declaration. For example, a declared reversionary bonus may increase the amount payable when a policyholder later dies or reaches maturity. The additional amount is then part of the benefit cash flow. By contrast, a cash bonus paid now is immediate outgo. An undeclared, contingent bonus is not yet a guaranteed benefit and should not be treated as a contractual claim payment merely because it might be awarded later.

The essential task is to match each bonus with its form, declaration status, payment date, and payment condition. This is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A declared reversionary bonus is an addition to policy benefits that, once declared, is generally guaranteed and payable on a specified event such as death or maturity. A cash bonus is a bonus paid directly in cash at its stated payment date. A contingent or undeclared bonus has not yet become a guaranteed benefit; its payment depends on a future declaration or other specified condition.

Declaration and payment are different events. Declaring a reversionary bonus increases the benefit amount that may be paid later. Unless the projection explicitly models a transfer out of the asset-share fund at declaration, declaration itself is not a cash outflow. If the policy later makes a claim, the declared addition belongs in the claim amount. A bonus paid immediately, on the other hand, is an outflow when paid.

A Cash-Flow Framework for Bonuses

Consider a cohort of $N_t$ policies at the start of year $t$, with aggregate assets $A_t$. Suppose premiums and any immediate bonus payments occur at the start of the year, investment return is earned at effective annual rate $i$, and expenses and claims are paid at year-end. If $P_t$ is the premium per policy, $C_t$ is the total cash bonus paid at the start, $E_t$ is year-end expense outgo, $q_t$ is the probability of death during the year, and $B_t$ is the death benefit including declared additions, the expected-value projection is

A_{t+1} = \bigl(A_t+N_tP_t-C_t\bigr)(1+i)-E_t-N_tq_tB_t.

This recurrence assumes the cohort remains in force at the start of the year and uses expected death counts. If claims or expenses occur at a different time, their investment accumulation must be adjusted to match that timing. If cash bonuses are payable only to policies that survive to a particular date, the number eligible for payment must also reflect that condition.

For a simple reversionary bonus of $b$ dollars declared each year on an original benefit $S$, the accumulated declared addition after $k$ declarations is $kb$, if each declaration is a fixed dollar amount. The benefit payable on death is then $S+kb$, assuming all $k$ declarations apply to that policy and no other benefit adjustments are specified. Other bonus scales may use a percentage of the original benefit or a percentage of the benefit already increased; the projection must use the actual stated basis.

Key distinction: Include a declared addition in a claim amount when the policy event makes that addition payable. Include an immediate cash bonus as outgo when it is paid. Do not deduct an attached declared bonus from assets at declaration and then deduct it again when the claim is paid.

Worked Projections

Worked Example: Declared Addition in Expected Death Claims

A hypothetical cohort has 100 policies and aggregate assets of $1,000,000 at the start of a year. Each policy pays a $500 premium immediately. The fund earns 4% for the year. Year-end expenses are $5,000. The assumed annual death probability is 0.02, and the projection uses expected deaths. The death benefit is $12,000 plus a $600 reversionary bonus already declared before the year begins. Find the expected assets allocated per surviving policy at year-end.

Method. Project aggregate assets first, allowing for expected claims that include the declared bonus, then divide by the expected number of survivors. The bonus is attached to the death benefit; it is not an immediate payment at the start of the year.

There are $100(0.02)=2$ expected deaths and $100-2=98$ expected survivors. The claim amount per death is

$$ 12{,}000+600=12{,}600. $$

Premium income at the start of the year is $100(500)=\$50{,}000$. Assets before investment are therefore

1{,}000{,}000+50{,}000=\$1{,}050{,}000.

After investment return, assets are $1{,}050{,}000(1.04)=\$1{,}092{,}000$. Expected death claims are $2(12{,}600)=\$25{,}200$. After claims and expenses, the assets remaining for survivors are

1{,}092{,}000-25{,}200-5{,}000=\$1{,}061{,}800.

Thus the expected asset share per survivor is

\frac{1{,}061{,}800}{98} = \$10{,}834.6939\ldots \approx \boxed{\$10{,}834.69}.

Check. Without the declared addition, expected claims would be $2(12{,}000)=\$24{,}000$, leaving $\$1{,}063{,}000$. The declared bonus therefore reduces aggregate survivor assets by $2(600)=\$1{,}200$, as expected. The common mistake is to deduct the $600 at declaration and also include it in the death claim; here it was already declared, so only the claim payment is an outflow in this year's projection.

Worked Example: Which Bonuses Belong in This Year's Claims?

A hypothetical group has 100 policies and $1,000,000 of aggregate assets at the start of year 3. It earns 5% during the year, has $10,000 of year-end expenses, and pays no premiums. The base death benefit is $20,000. Two fixed reversionary bonuses of $600 each were declared in years 1 and 2. A third $600 bonus will be considered for declaration at the end of year 3. Deaths are assumed to occur before that year-end declaration, and the annual death probability is 0.01. Find the expected year-end assets per survivor, excluding any bonus that is not yet declared.

Method. Apply only bonuses declared before the claim event. The two existing declarations add $2(600)=\$1{,}200$ to each death benefit. The possible year-3 declaration is not included in claims that occur before it is declared.

The claim amount for a year-3 death is

20{,}000+1{,}200=\$21{,}200.

Expected deaths are $100(0.01)=1$, so the expected number of survivors is 99 and expected claims are $1(21{,}200)=\$21{,}200$. Investment increases the opening assets to $1{,}000{,}000(1.05)=\$1{,}050{,}000$. After claims and expenses, the assets remaining are

1{,}050{,}000-21{,}200-10{,}000=\$1{,}018{,}800.

The asset share per expected survivor is

\frac{1{,}018{,}800}{99} = \$10{,}290.9090\ldots \approx \boxed{\$10{,}290.91}.

Check. The declared additions account for $1(1{,}200)=\$1{,}200$ of the expected claim outgo. A year-3 bonus that is only under consideration is not a declared benefit for these pre-declaration claims. If it is later declared, it may increase benefits payable on subsequent events, but that does not make it a cash claim payment in this projection period. The common mistake is to include a possible future declaration in the current claim amount without respecting the stated event timing.

Worked Example: An Immediate Cash Bonus and an Attached Bonus

A hypothetical policy cohort begins the year with $10,000 of assets per policy. Each of 100 in-force policies pays a $500 premium at the start of the year. A declared cash bonus of $100 per policy is also paid immediately. The fund earns 3%, and year-end expenses are $5,000 for the cohort. The annual death probability is 0.02. A declared reversionary addition of $400 is payable on death, in addition to a $12,000 base death benefit. Find the expected assets per survivor at year-end.

Method. Deduct the cash bonus when it is paid, before investment. Include the attached $400 addition in expected death claims, not as a second immediate outflow at declaration.

Opening aggregate assets are $100(10{,}000)=\$1{,}000{,}000$. Premium income is $100(500)=\$50{,}000$, while the immediate cash bonus costs $100(100)=\$10{,}000$. Assets invested are therefore

1{,}000{,}000+50{,}000-10{,}000=\$1{,}040{,}000.

After the 3% investment return, assets are $1{,}040{,}000(1.03)=\$1{,}071{,}200$. Expected deaths are $100(0.02)=2$, leaving 98 expected survivors. The death benefit per claim is $12{,}000+400=\$12{,}400$, so expected claims are $2(12{,}400)=\$24{,}800$. Subtracting the year-end expenses and claims gives

1{,}071{,}200-5{,}000-24{,}800=\$1{,}041{,}400.

The expected asset share per survivor is

\frac{1{,}041{,}400}{98} = \$10{,}626.5306\ldots \approx \boxed{\$10{,}626.53}.

Check and trap. The total bonus-related outgo in this projection is the $10,000 cash payment plus $2(400)=\$800$ in expected claim additions. Deducting another $400 per policy at the start because the reversionary bonus was declared would subtract $40,000 even though no such cash payment occurs then. That incorrect treatment would leave $(1{,}041{,}400-40{,}000)/98=\$10{,}218.37$ per survivor. It double counts the attached bonus by treating declaration as payment and later including it in claims.

Derivation and Practical Checks

The cohort recurrence follows by tracking cash through the year in its actual order. At the start, the fund has $A_t$, receives $N_tP_t$, and pays immediate cash bonuses $C_t$. The balance exposed to investment is therefore $A_t+N_tP_t-C_t$. Earning return $i$ gives $(A_t+N_tP_t-C_t)(1+i)$. Year-end expenses reduce this by $E_t$, and expected death claims reduce it by $N_tq_tB_t$. Combining these terms gives

A_{t+1} = \bigl(A_t+N_tP_t-C_t\bigr)(1+i)-E_t-N_tq_tB_t.

The term $N_tq_tB_t$ is expected death-claim outgo because each of the $N_t$ policies has probability $q_t$ of a claim, and each claim pays $B_t$. If the benefit consists of a base amount $S_t$ and declared additions $R_t$, then $B_t=S_t+R_t$. This is the point at which an attached reversionary bonus affects the cash flow. An amount that remains undeclared is not part of $R_t$, unless the projection deliberately includes an assumed future payment as a separate projected outgo.

This formula is not a universal timing rule. If premiums arrive at year-end, they do not earn a full year's investment return. If a cash bonus is paid at year-end, it is deducted after investment rather than before it. If benefits are paid at the start of the year or at other times, the projection must apply the appropriate accumulation to each cash flow. A reliable calculation states the timing assumption before applying the recurrence.

Identify the bonus form and status.
Separate cash paid now from additions attached to future benefits and from amounts still contingent or undeclared.

Set the event and timing.
Determine which policies are eligible and whether the declaration occurs before or after the projected claim event.

Put each amount in the right cash-flow term.
Deduct an immediate cash payment when paid; include a declared addition in benefits when payable.

Reconcile the cohort.
Check that ending assets equal opening assets plus income and investment return, less expenses, claims, and other outgo; then divide by the appropriate survivor count if a per-survivor amount is required.

A useful final check is to isolate the incremental cost of a declared addition. If $D$ expected claims each include an additional declared amount $R$, the incremental claim outgo is $DR$. This amount should agree with the difference between claim outgo with and without that addition. It should not also appear as an immediate deduction unless the scenario explicitly specifies an actual transfer or payment at declaration.

Common Traps

Equating declaration with payment. A declaration can increase a future benefit without causing an immediate cash outflow.
Omitting declared additions from claims. If an attached bonus is payable on death or maturity, omitting it understates claim outgo.
Double counting. Do not deduct the same attached addition at declaration and again as part of the claim unless the projection describes two distinct cash flows.
Paying an undeclared bonus as if guaranteed. A contingent amount is not automatically a contractual claim payment. Include it only when the stated assumptions make it payable or explicitly project it.
Ignoring eligibility and timing. A bonus declared after a claim event cannot be included in that claim unless the contract or scenario expressly gives it that effect.
Mixing payment dates. A bonus paid at the start of the year changes the amount invested; a payment at year-end does not reduce the year's investment base.

Check Your Understanding

For each problem, use the stated timing and distinguish immediate payments from additions included in claims.

A hypothetical cohort has $400,000 of assets at the start of the year, earns 4%, and has year-end expenses of $2,000. There are 40 policies, with death probability 0.05. Each death benefit is $9,000 plus a $500 declared addition. Use expected deaths and find the year-end asset share per survivor. (Answer: $10,394.74 per survivor)
A cohort of 200 policies has a $15,000 base death benefit and three previously declared fixed additions of $250 each. The annual death probability is 0.02. Find the expected death-claim outgo attributable to the declared additions alone. (Answer: $3,000)
A fund has $720,000 at the start of the year for 60 policies. It earns 5%, has $6,000 of year-end expenses, and has annual death probability 0.05. The death benefit is $10,000 plus two declared additions of $300 each. Find the expected asset share per survivor. (Answer: $12,600 per survivor)
For 80 in-force policies, a declared cash bonus of $75 is paid immediately to every policy. The expected number of deaths during the year is 2, and each death claim includes a declared attached addition of $250. Find the total bonus-related outgo in the year, counting both types of payment. (Answer: $6,500)
A hypothetical cohort has 50 policies and expected deaths of 2. The base benefit is $8,000, with a declared attached addition of $200. A further $400 terminal bonus is only contingent and has not been declared; no assumption makes it payable in the projection year. Find expected claim outgo and state whether the contingent amount is included. (Answer: $16,400; exclude the undeclared $400 per policy)

What You'll Learn