Including Vested Bonuses In Death And Maturity Benefits

Introduction

A reversionary bonus can increase the amount payable under a participating policy, but the claim calculation requires more than adding a percentage to the original sum assured. The actuary must identify which bonuses have been declared and vested by the claim date, how those bonuses were calculated, and which basic benefit applies to the claim.

Death and maturity are different insured events. A policy may specify the same basic amount for both, or a different basic death benefit and maturity benefit. The bonus calculation does not, by itself, determine either basic amount or guarantee that every bonus applies to every claim event. Those details come from the policy terms.

Definition: A vested reversionary bonus is a bonus already declared and attached to the policy under its terms. Let $B$ denote the basic amount relevant to a calculation, and let $b_j$ be the amount of the $j$th vested bonus. The total vested bonus is $V_n=\sum_{j=1}^{n}b_j$. When the contract provides that the bonuses are payable on a particular claim, the benefit is the applicable basic benefit plus those eligible vested bonuses.

Set the Claim Amount Before Doing the Arithmetic

At a claim date, begin by identifying the event and the contract’s basic amount for that event. If $D^0_t$ is the basic death benefit at time $t$, and $M^0_T$ is the basic maturity benefit at maturity time $T$, then a contract that pays the vested bonuses on both events has the forms

D_t=D^0_t+\sum_{\text{bonuses vested and payable on death by }t} b_j, \qquad M_T=M^0_T+\sum_{\text{bonuses vested and payable at maturity by }T} b_j.

These expressions are conditional on the contract making the listed bonuses payable for the stated event. If the terms limit bonus eligibility, use only the bonuses that qualify. Likewise, do not substitute the death sum assured for the maturity amount unless the contract says they are the same.

The claim date matters because a bonus is included only after it has been declared and vested under the policy terms. For example, a claim immediately after an anniversary declaration may include that declaration, while a claim immediately before it cannot include a bonus that has not yet vested. A bonus for an incomplete year is not automatically earned; its treatment depends on the contract.

Simple and Compound Bonus Amounts

For a simple reversionary bonus, each declaration is calculated on the fixed base specified by the policy. If the base is the basic sum assured $B$, and the rates for the declarations are $r_1,\ldots,r_n$, then $b_j=Br_j$. The total addition is $B\sum_{j=1}^{n}r_j$.

For a compound reversionary bonus declared on the augmented amount, each new rate applies to the basic amount plus bonuses already declared. Write $C_0=B$ for the amount after no declarations and $C_j$ for the amount after the $j$th declaration. Then $b_j=r_jC_{j-1}$ and $C_j=C_{j-1}+b_j$. The claim benefit, when all these additions are payable, is $C_n$, not $C_n$ plus the same additions again.

The stated bonus base controls the calculation. A rate described as a percentage of the original sum assured is not interchangeable with one applied to the augmented amount. The earlier material on choosing the correct reversionary bonus base is useful for identifying that base; here the key task is to carry the resulting vested amounts into the right claim benefit.

Theorem: If a sequence of compound bonuses is declared at rates $r_1,\ldots,r_n$, each on the amount after previous bonuses, then the augmented amount after the $n$th declaration is $C_n=B\prod_{j=1}^{n}(1+r_j)$. The total vested addition is $C_n-B$. If the bonus rates are instead simple rates on the fixed base $B$, the total addition is $B\sum_{j=1}^{n}r_j$.

Why the Compound Formula Works

The compound formula follows directly from the declaration recurrence. At declaration $j$, the bonus is $r_j$ times the current augmented amount. Adding it to that amount gives

C_j=C_{j-1}+r_jC_{j-1}=C_{j-1}(1+r_j).

Starting with $C_0=B$, the first declaration gives $C_1=B(1+r_1)$. If $C_{j-1}=B\prod_{k=1}^{j-1}(1+r_k)$, then

C_j=C_{j-1}(1+r_j) =B\left(\prod_{k=1}^{j-1}(1+r_k)\right)(1+r_j) =B\prod_{k=1}^{j}(1+r_k).

Thus, by applying the recurrence at each declaration, the product formula holds for every $n$. The total bonus is the difference $C_n-B$, because $C_n$ already contains the basic amount.

Worked Example: Death Just After Two Compound Declarations

A hypothetical policy has a basic death sum assured of $45,000. It declares a compound reversionary bonus of 2% on the augmented amount at the first anniversary and 3% on the augmented amount at the second anniversary. Both declarations are vested and payable on death. The insured dies just after the second declaration. Calculate the death benefit.

Method. Apply each rate to the amount in force immediately before that declaration, then use the resulting augmented amount as the claim benefit.

After the first declaration, the bonus is $0.02(45{,}000)=900$, so the augmented amount is $45{,}000+900=45{,}900$.

The second bonus is $0.03(45{,}900)=1{,}377$. The augmented amount after the second declaration is $45{,}900+1{,}377=47{,}277$.

Therefore, the death benefit is $47,277.00. As a check, $45{,}000(1.02)(1.03)=45{,}900(1.03)=47{,}277$. The result is stated to the nearest cent.

Common mistake avoided. Applying 3% to the original $45,000 would give a second bonus of $1,350 and a total of $47,250. That is wrong here because the second declaration is explicitly based on the augmented amount of $45,900.

Worked Example: A Simple Bonus at Death or Maturity

A hypothetical endowment policy has a basic death sum assured of $80,000 and a basic maturity benefit of $80,000. It declares a simple reversionary bonus of 1.5% of the original sum assured at each policy anniversary. Each declaration is vested and payable on death or maturity. Find (a) the death benefit if death occurs just after the third declaration and (b) the maturity benefit after the fourth declaration.

Method. Since the rate is simple and is based on the original sum assured, each annual bonus has the same amount:

b_j=0.015(80{,}000)=1{,}200.

For death just after the third declaration, three bonuses have vested. Their total is $3(1{,}200)=3{,}600$. The death benefit is $80{,}000+3{,}600=$ $83,600.00.

At maturity after the fourth declaration, four bonuses have vested. Their total is $4(1{,}200)=4{,}800$. The maturity benefit is $80{,}000+4{,}800=$ $84,800.00.

The check is that the maturity claim includes exactly one more declaration than the death claim, so its benefit is higher by $1,200. Both results are stated to the nearest cent.

Common mistake avoided. Using the four-year total for the death claim would include a declaration that had not yet occurred at the stated death date. Count declarations vested by the claim date, not the number of policy years expected to be completed later.

Worked Example: Different Basic Death and Maturity Benefits

A hypothetical policy specifies a basic death benefit of $100,000 and a basic maturity benefit of $105,000. It declares compound bonuses at the end of each of three years, at rates of 2%, 3%, and 1.5%, respectively. Each bonus is calculated on the augmented $100,000 death sum assured, and the resulting vested additions are payable on either death or maturity. Calculate the death benefit for death just after the second declaration and the maturity benefit after the third.

Method. First calculate the augmented amount and vested bonus for each relevant claim date. Then add that vested bonus to the basic amount specified for the claim event.

After declaration one, the amount is $100{,}000(1.02)=102{,}000$, so the vested addition is $2,000. After declaration two, the amount is $102{,}000(1.03)=105{,}060$, so the cumulative vested addition is $105{,}060-100{,}000=5{,}060$. The death occurs after this declaration, so the death benefit is $100{,}000+5{,}060=$ $105,060.00.

For maturity, include the third declaration: its bonus is $0.015(105{,}060)=1{,}575.90$. The augmented amount is $105{,}060+1{,}575.90=106{,}635.90$, and the cumulative vested addition is $6,635.90. The maturity benefit is $105{,}000+6{,}635.90=$ $111,635.90.

The check is that the maturity basic amount exceeds the death basic amount by $5,000, while maturity also includes the third vested bonus of $1,575.90. Its benefit therefore exceeds the death benefit by $5{,}000+1{,}575.90=$$6,575.90, matching $111{,}635.90-105{,}060.00$. Amounts are stated to the nearest cent.

Common mistake avoided. Adding the vested additions to the death sum assured for both claims would produce the wrong maturity benefit. The policy gives a different basic maturity benefit, so that is the amount to which the eligible additions must be added.

Common Claim-Calculation Traps

Using the wrong basic amount. Read the death and maturity provisions separately. They need not specify the same amount.
Including an undeclared bonus. A claim immediately before a declaration does not include that declaration unless the contract provides otherwise.
Using the wrong bonus base. Apply each declared rate to the base specified in the terms, not automatically to the original sum assured or the augmented amount.
Double counting. If you calculate the augmented amount $C_n$, it already includes the basic amount and the bonuses. Do not add $C_n-B$ to $C_n$ again.
Assuming every vested bonus applies to every event. Check the policy’s eligibility provisions; a bonus is included only when payable for the claim being valued.

Key takeaway: For each claim, identify the event, select its contractual basic benefit, count only bonuses vested by the claim date and payable for that event, and apply the declared bonus bases correctly. Add the eligible vested additions once.

Check Your Understanding

For each problem, identify the claim timing and bonus method before calculating.

A policy has a $64,000 death sum assured and declares simple bonuses of 1.25% of that sum assured at each anniversary. Death occurs just after the third declaration, and all declarations are payable on death. Find the death benefit. (Answer: $66,400.00)
A $50,000 policy declares compound bonuses of 2% and then 1%, each on the augmented amount. Death occurs just after the second declaration, and both bonuses are payable. Find the benefit. (Answer: $51,510.00)
An endowment policy has a basic maturity benefit of $94,000. It declares five simple annual bonuses of 0.8% of a $90,000 original sum assured, all payable at maturity. Find the maturity benefit. (Answer: $97,600.00)
A $120,000 death sum assured receives compound bonuses of 1.5%, 2%, and 2.5%, each on the augmented amount. Death occurs just after the third declaration, and all are payable. Find the benefit. (Answer: $127,341.90)
A policy has a $75,000 basic benefit for both death and maturity. It declares compound bonuses of 1%, 1.5%, 2%, and 2.5%, in that order, each payable on either event. Find the death benefit just after the second declaration and the maturity benefit after the fourth. (Answer: Death $76,886.25; maturity $80,384.57)

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