Using Dividends To Buy Paid-Up Additions

Introduction

A participating policyholder may elect, or be required by the contract, to use a declared cash dividend to buy paid-up additional insurance. Instead of receiving the allocated amount as cash, the policyholder exchanges it for additional coverage that requires no further premium. To calculate the resulting benefit, identify the amount used for the purchase, the applicable purchase rate, and the benefits promised by each addition.

The purchase rate is supplied by the hypothetical contract or problem. It is not necessarily the dividend scale rate, an annual premium rate, or an interest rate. It converts a stated amount of money into a stated amount of additional insurance. Once an addition is purchased, its benefit is included in a claim only if the contract’s benefit terms say it is payable for that claim event.

Definition: A paid-up addition is additional insurance purchased by a single payment, such as an allocated dividend, with no further premiums due for that additional coverage. A purchase rate states the single-payment cost of a specified unit of additional insurance. For example, if the purchase rate is $r$ dollars per $K$ dollars of additional benefit, an allocated dividend $D$ buys $D/r$ units, or $K(D/r)$ dollars of benefit.

Converting a Dividend into Additional Benefit

Suppose the purchase rate is $r$ dollars for each $K$ dollars of additional benefit. If the whole dividend $D$ is used to buy additions, then the number of benefit units purchased is $D/r$, and the added benefit amount is

B_{\text{add}}=K\frac{D}{r}.

For example, a rate of $90 per $1,000 of additional benefit means that each $90 buys $1,000 of insurance. A $360 dividend therefore buys $360/90=4$ units, or $4,000 of additional benefit. The rate’s unit matters: dividing by a rate quoted per $1,000 gives a number of $1,000 units, not a dollar amount.

If only a fraction $w$ of the declared dividend is allocated to additions, use $D_{\text{add}}=wD$ in the conversion. Any remaining amount is treated according to the stated dividend option, such as being paid in cash. The portion paid in cash is not part of the additional insurance benefit.

Model assumptions: The purchase rate is treated as applicable to the policy’s current duration or attained age as specified. The calculation assumes that the allocated amount is available at the stated purchase time and that the contract permits the resulting amount of insurance. Do not infer a purchase rate from an annual dividend scale. Unless a question gives a mortality or interest model for deriving the rate, use the stated rate directly; it already summarizes the pricing basis for this conversion.

Worked Example: One Dividend Buys Additional Endowment Insurance

A hypothetical participating endowment policy has a $40,000 basic benefit, payable on death before maturity or on survival to maturity. At the fifth anniversary, the policyholder uses the entire declared $525 dividend to buy paid-up additional endowment insurance. The stated purchase rate is $105 per $1,000 of additional benefit. Calculate the total benefit if death occurs after the purchase but before maturity, and if the policy reaches maturity.

Assumptions and method. The dividend and claim timing are fixed as stated; no mortality distribution is imposed. No interest accumulation is applied after the dividend is used. The given rate is the single-payment cost per $1,000 of additional benefit.

First calculate the number of $1,000 units purchased: $525/105=5$ units. The additional benefit is therefore $5\times \$1,000=\$5,000$.

For death after the purchase, the basic death benefit is $40,000 and the paid-up addition contributes $5,000. The total death benefit is $\$40,000+\$5,000=\$45,000$.

For maturity, the basic maturity benefit is $40,000 and the additional endowment benefit is $5,000. The total maturity benefit is also $\$40,000+\$5,000=\$45,000$.

Check. Five units at $105 each cost $5\times \$105=\$525$, exactly the dividend allocated. The calculation avoids treating the purchase rate as a benefit rate or adding the $525 purchase payment itself to the claim benefit.

Why the Conversion Formula Holds

The formula follows directly from the meaning of a purchase rate. If $n$ units are bought and each unit costs $r$, the total cost is $nr$. This cost must equal the allocated dividend. Solving for the number of units gives $n=D/r$; each unit provides $K$ dollars of benefit, so the total additional benefit is $Kn=K(D/r)$.

D=nr \quad\Longrightarrow\quad n=\frac{D}{r} \quad\Longrightarrow\quad B_{\text{add}}=Kn=K\frac{D}{r}.

This is a price-to-quantity conversion, not an accumulation calculation. If the dividend is used immediately to purchase additions, do not first accumulate it at interest. Conversely, if the contract specifies that an amount is retained as cash and accumulates with interest, that is a different dividend option and requires the stated accumulation terms.

Several Purchases at Different Durations

A purchase rate may vary by policy duration or attained age. In that case, calculate each addition using the rate that applies when that particular dividend is used. Add the resulting benefit amounts; do not divide the total dividends by one rate unless the same rate applies to every purchase.

B_{\text{add,total}} = \sum_{j=1}^{m} K\frac{D_{\text{add},j}}{r_j},

where $D_{\text{add},j}$ is the amount allocated to additions at purchase $j$, and $r_j$ is the purchase rate then applicable, quoted per $K$ dollars of benefit. This sum assumes that the additions remain in force and that their benefit amounts are measured on the same basis.

Worked Example: Three Purchases at Different Rates

A hypothetical participating policy has a $60,000 death benefit. At three successive anniversaries, the policyholder uses each entire dividend to purchase paid-up additions. The dividends are $360, $390, and $440. The respective rates are $90, $97.50, and $110 per $1,000 of additional benefit. Calculate the death benefit immediately after the third purchase.

Assumptions and method. The claim is assumed to occur after the third purchase, and all three additions remain in force. The scenario is deterministic; no mortality distribution or interest accumulation is used. Apply each rate only to the dividend for its own anniversary.

At the first anniversary, $360/90=4$ units are purchased, giving $4\times \$1,000=\$4,000$.

At the second anniversary, $390/97.50=4$ units are purchased, giving another $\$4,000$.

At the third anniversary, $440/110=4$ units are purchased, giving another $\$4,000$. Thus the cumulative additional benefit is $\$4,000+\$4,000+\$4,000=\$12,000$.

The death benefit after the third purchase is $\$60,000+\$12,000=\$72,000$.

Check. The cost of each four-unit addition is respectively $4\times \$90=\$360$, $4\times \$97.50=\$390$, and $4\times \$110=\$440$. Each matches its dividend. The method avoids applying one anniversary’s rate to purchases made at other durations.

Timing, Allocation, and Claim Benefits

A purchased addition affects benefits only from the time it is in force, according to the policy terms. If a claim occurs before a dividend is declared or before the purchase is completed, that not-yet-purchased addition is not automatically included. For each claim calculation, establish the sequence of events: declaration, allocation and purchase, then claim.

Also distinguish the amount used to purchase additions from the amount of insurance purchased. A $400 dividend allocated at a rate of $80 per $1,000 does not create $400 of additional insurance; it creates five $1,000 units, or $5,000 of benefit.

Worked Example: Partial Allocation and a Claim After the Purchase

A hypothetical endowment policy has a $75,000 basic death benefit and a $75,000 basic maturity benefit. At an anniversary, a $640 dividend is declared. Under the policyholder’s selected option, 75% buys paid-up additions and the rest is paid in cash. The purchase rate is $80 per $1,000 of additional benefit. Calculate the additional benefit, the death benefit after the purchase, and the cash paid.

Assumptions and method. A claim is assumed to occur after the anniversary purchase. The contract’s additions pay the same face amount on death before maturity and on maturity. No mortality distribution or interest accumulation is assumed; the stated allocation and purchase rate are applied directly.

The amount allocated to additions is $0.75\times \$640=\$480$. The remaining cash amount is $\$640-\$480=\$160$.

The number of $1,000 units purchased is $480/80=6$. Therefore, the additional benefit is $6\times \$1,000=\$6,000$.

The death benefit after the purchase is $\$75,000+\$6,000=\$81,000$. Under the stated endowment terms, the maturity benefit after the purchase is also $\$75,000+\$6,000=\$81,000$. The $160 was paid as cash at the anniversary and is not an additional claim benefit.

Check. Six units cost $6\times \$80=\$480$; adding the $160 cash portion gives the original $640 dividend. This checks both the allocation and the purchase conversion.

Common Mistakes and Exam Traps

A frequent error is to treat “$80 per $1,000” as if the addition were $80 for each $1,000 of dividend. The rate instead says that $80 buys $1,000 of insurance. For a $480 allocation, the correct conversion is $480/80=6$ units, or $6,000 of benefit. A mistaken calculation such as $480\times 80/1,000=\$38.40$ reverses the rate and gives the wrong quantity.

Other errors arise from timing and benefit design. A dividend declared after the claim cannot ordinarily purchase an addition for that claim. An addition that is payable at death does not necessarily have a maturity benefit unless the contract provides one. Always use the death and maturity terms stated in the problem rather than assuming that all forms of additional insurance pay identically.

Key takeaway: For every dividend purchase, identify the amount actually allocated, use the purchase rate applicable at that purchase time, convert to additional benefit in the rate’s stated units, and then include only in-force additions payable under the relevant claim benefit.

Check Your Understanding

For each problem, use the stated purchase rate and benefit terms. Assume any claim occurs after the specified purchases unless the question says otherwise.

A $28,000 endowment benefit is supplemented by a paid-up addition bought with a $315 dividend at $105 per $1,000. The addition pays its face amount on death or maturity. What are the resulting death and maturity benefits? (Answer: $31,000 each.)
A policyholder buys additions using $270 at $90 per $1,000 in year one and $360 at $120 per $1,000 in year two. The basic death benefit is $45,000. What is the death benefit after the second purchase? (Answer: $51,000.)
A $500 dividend is split so that 60% buys additions at $75 per $1,000 and the balance is paid in cash. How much additional benefit is purchased, and how much cash is paid? (Answer: $4,000 additional benefit and $200 cash.)
A $50,000 basic maturity benefit has two in-force additions: $3,000 bought at the first anniversary and $2,500 bought at the second. The additions each pay their face amount at maturity. What is the maturity benefit after the second purchase? (Answer: $55,500.)
A $420 dividend is used to buy additions at $105 per $1,000. The policy has a $32,000 death benefit, and the claim occurs before the next anniversary’s dividend declaration. The purchased addition pays its face amount on death. What death benefit is payable, excluding any future dividend? (Answer: $36,000.)

What You'll Learn