Comparing Cash Dividends With Reversionary Bonuses

Why the Form of a Dividend Matters

A declared amount of $300 can be delivered in different ways. It might be paid in cash when declared, left in a cash account under the policy’s terms, or used as a reversionary addition that increases a future claim benefit. The stated amount alone does not tell us when the policyholder receives value, whether payment depends on a future claim, or how much is payable on death or maturity.

The comparison in this tutorial is about equivalent stated amounts: the same dollar amount is declared at the same time, but the contract delivers it differently. That does not make the options equivalent in value or in claim effect. The contract terms govern, including whether a declared amount is vested, whether a cash account is payable on surrender or lapse, and which claims qualify for an addition. The examples below specify those terms explicitly.

Definition: A cash dividend is a declared amount delivered in cash, either immediately or through a specified cash-account option. A reversionary addition is a declared amount added to the policy benefit and paid with an eligible claim, as specified by the contract. For the examples here, an addition is vested once declared and its stated amount is guaranteed for the eligible claim events described. This does not imply that every dividend or bonus arrangement has the same terms.

What to Compare

A sound comparison separates three questions. First, when is the amount paid? An immediate cash dividend is paid at declaration; a reversionary addition is paid at a later claim. A retained cash dividend may accumulate interest before payment. Second, what is guaranteed? Under a contract where a declared addition is vested, its specified amount is guaranteed for eligible claims. A cash dividend already paid is no longer contingent on a later claim, while a balance left in an account is subject to that account’s stated terms. Third, what is included in the claim benefit? A reversionary addition increases that benefit; a dividend paid earlier does not automatically do so.

Present value also depends on payment timing and event probabilities. Let $C$ be the amount declared at time $s$, let $i$ be the effective annual discount rate, and let $v=1/(1+i)$. If the amount is paid at time $t$, its value at time 0 is $Cv^t$. If payment occurs only on certain claim events, the expected present value weights each discounted payment by the probability of its event. A cash account may have a different amount at payment because of credited interest.

Key comparison: Equal stated amounts do not imply equal claim benefits or equal expected present values. Compare the payment date, the events that trigger payment, any accumulation credited to cash, and the contractual benefit base.

Worked Comparisons

Worked Example: Immediate Cash, a Retained Account, and a Vested Addition

A hypothetical policy declares $300 at time 1. The policyholder can take $300 in cash immediately, leave it in an account earning an effective annual rate of 4%, or receive a vested $300 reversionary addition on an eligible claim. An eligible claim occurs at time 2 with probability 0.25 or at time 3 with probability 0.65; the policy lapses at time 2.5 with probability 0.10. The cash account is forfeited on lapse, and the addition is not payable on lapse. Use an effective annual discount rate of 4% from time 0. Compare expected present values and the claim-benefit increments.

Method. Discount each payment to time 0 and weight it by the probability of the event that produces it. The immediate cash option pays regardless of a later claim, so its present value is not weighted by claim probabilities:

\operatorname{EPV}_{\text{cash}} =300v =\frac{300}{1.04} =288.461538\ldots

Thus the expected present value of immediate cash is $288.46, rounded to the nearest cent. It does not increase the claim benefit under the terms in this example.

For the reversionary addition, the payment is $300 only on an eligible claim. Its expected present value is

\begin{aligned} \operatorname{EPV}_{\text{addition}} &=0.25\left(\frac{300}{1.04^2}\right) +0.65\left(\frac{300}{1.04^3}\right)\\ &=0.25(277.3668639\ldots) +0.65(266.6989076\ldots)\\ &=69.3417160\ldots+173.3542900\ldots\\ &=242.6960060\ldots \end{aligned}

The expected present value of the addition is $242.70. On either eligible claim, it increases the claim benefit by $300; on lapse, it pays nothing.

For the retained account, the balance is $300(1.04)=312$ at time 2 and $300(1.04)^2=324.48$ at time 3. Its expected present value is

\begin{aligned} \operatorname{EPV}_{\text{account}} &=0.25\left(\frac{312}{1.04^2}\right) +0.65\left(\frac{324.48}{1.04^3}\right)\\ &=0.25(288.461538\ldots) +0.65(288.461538\ldots)\\ &=259.615384\ldots \end{aligned}

The account’s expected present value is $259.62. The account pays $312 or $324.48 on the respective claims, whereas the reversionary addition pays $300. The account and the addition both pay nothing on lapse, but their claim amounts differ because the account earns interest. The immediate cash option has the highest expected present value here because it is paid even if the policy later lapses.

Check. The probabilities sum to $0.25+0.65+0.10=1$. The account balance is greater at time 3 than at time 2, as expected from positive credited interest. The three options therefore differ in both timing and contingencies, despite the same $300 amount at declaration. The common mistake avoided is treating the stated amount as if it were necessarily paid at the same time, or on the same events, under every option.

Worked Example: Claim Amount Versus Earlier Cash

A hypothetical endowment policy has a base benefit of $20,000 and declares $500 at time 1. Under option C, the $500 is paid to the policyholder immediately. Under option R, the $500 becomes a vested reversionary addition payable on either death at time 2 or maturity at time 5. Assume death at time 2 has probability 0.60 and maturity at time 5 has probability 0.40. Use an effective annual discount rate of 5%. Compare the claim benefits and the expected present value of the $500 component under each option.

Method. First separate what is paid at declaration from what is included in the claim. Under option C, the claim benefit remains $20,000, and the policyholder has already received $500. Under option R, the claim benefit is $20{,}000+500=\$20{,}500$ at either eligible event.

The cash dividend is paid at time 1 regardless of which later claim event occurs. Its present value is

\operatorname{EPV}_{C} =\frac{500}{1.05} =476.190476\ldots

For option R, discount the addition to each possible claim time and weight by the claim probability:

\begin{aligned} \operatorname{EPV}_{R} &=0.60\left(\frac{500}{1.05^2}\right) +0.40\left(\frac{500}{1.05^5}\right)\\ &=0.60(453.514739\ldots) +0.40(391.763083\ldots)\\ &=272.108843\ldots+156.705233\ldots\\ &=428.814076\ldots \end{aligned}

The expected present values of the $500 components are $476.19 for option C and $428.81 for option R. The difference is caused by timing and contingencies: option C pays at time 1, while option R pays later and only at an eligible claim. In claim-benefit terms, option R increases either claim by $500; option C does not. The common mistake avoided is adding the $500 cash dividend to the later claim benefit as well as counting it as already paid. That would count the same dividend twice.

Check. The claim probabilities sum to 1. The reversionary addition is $500 in both claim outcomes, but the present value differs because time 2 and time 5 are different payment dates. The conclusion is specific to the stated contract terms; a cash option payable only on an eligible claim would require different event weights.

Worked Example: Interest on a Cash Account Does Not Equal a Fixed Addition

A hypothetical contract declares $200 at time 1. Under option A, a cash account earns an effective annual rate of 2% and is paid only on an eligible claim. A claim occurs at time 2 with probability 0.20 or time 3 with probability 0.70; lapse has probability 0.10, and the account is forfeited on lapse. Under option B, a vested $200 reversionary addition is paid on either eligible claim and not on lapse. Use an effective annual discount rate of 5% from time 0. Calculate and compare the expected present values.

Method. Accumulate the account from its declaration at time 1 to the payment date, then discount the payment to time 0. The account balances are

200(1.02)=204 \quad\text{at time 2},\qquad 200(1.02)^2=208.08 \quad\text{at time 3}.

Therefore,

\begin{aligned} \operatorname{EPV}_{A} &=0.20\left(\frac{204}{1.05^2}\right) +0.70\left(\frac{208.08}{1.05^3}\right)\\ &=0.20(185.0340136\ldots) +0.70(179.747328\ldots)\\ &=37.0068027\ldots+125.8231296\ldots\\ &=162.8299323\ldots \end{aligned}

Thus option A has expected present value $162.83. For option B, the amount is $200 at either eligible claim, so

\begin{aligned} \operatorname{EPV}_{B} &=0.20\left(\frac{200}{1.05^2}\right) +0.70\left(\frac{200}{1.05^3}\right)\\ &=0.20(181.4058957\ldots) +0.70(172.7675197\ldots)\\ &=36.2811791\ldots+120.9372638\ldots\\ &=157.2184429\ldots \end{aligned}

Thus option B has expected present value $157.22. The cash account’s value is greater because it earns 2% interest before payment; the discount rate is 5%. It is not correct to treat the account as a flat $200 at every claim date, nor to add account interest to the reversionary addition. Under option B, the claim-benefit increment remains $200 as specified.

Check. The eligible-claim probabilities sum to $0.20+0.70=0.90$, leaving the stated 0.10 lapse probability. Both options pay nothing on lapse. The account’s claim payments, $204 and $208.08, exceed $200, consistent with its positive credited rate. The calculation avoids the common mistake of discounting a cash-account payment without first accumulating it to the payment date.

Why the Expected-Present-Value Formula Works

The calculation method follows directly from partitioning possible outcomes. Suppose a benefit pays $C_j$ at time $t_j$ in outcome $j$, and the mutually exclusive outcomes have probabilities $p_j$ that sum to 1. The present value in outcome $j$ is $C_jv^{t_j}$. Taking the expectation over the outcomes gives the sum of each present value multiplied by its probability:

\operatorname{EPV} =E[\text{present value}] =\sum_j p_j C_jv^{t_j}.

If some outcomes produce no payment, set $C_j=0$ for those outcomes. For a cash account declared at time $s$ that earns an effective annual rate $j$ until a claim at time $t$, its balance is $C(1+j)^{t-s}$, assuming annual effective accumulation at that rate and no withdrawals, fees, or other adjustments. Its contribution to expected present value is therefore

p\,C(1+j)^{t-s}v^t,

where $p$ is the probability of that claim outcome. This expression does not apply unchanged if the account has different crediting terms, is accessible on surrender, or is paid on events other than those specified. Those features must be represented in the outcome payments and probabilities.

A useful special case explains when accumulation can offset later payment. If an amount $C$ is declared at time $s$, accumulates at the same effective annual rate $i$ used for discounting, and is paid with certainty at time $t$, then

C(1+i)^{t-s}v^t =C(1+i)^{t-s}(1+i)^{-t} =Cv^s.

Its present value equals the value of receiving $C$ at time $s$. This equality requires certainty of payment and matching accumulation and discount rates. A reversionary addition contingent on an eligible claim, or a cash account forfeited on lapse, does not meet that certainty condition.

Common Exam Traps

Putting an earlier cash dividend into the claim benefit. If cash has already been paid and the contract does not add it to the insured benefit, the later claim amount is unchanged. Keep prior cash receipts separate from claim benefits.
Ignoring event contingencies. A cash amount paid unconditionally at declaration should not be multiplied by the probability of a later claim. Conversely, an addition paid only on an eligible claim must be weighted by the probability of that claim.
Using the wrong payment date. Discount a reversionary addition from the claim date, not from its declaration date. For an interest-bearing cash account, first calculate the balance at the payment date.
Assuming every declared amount has identical guarantees. State which amount is vested and which events trigger payment. Contract terms, not the label “dividend,” determine the guarantee and payment conditions in a particular hypothetical scenario.

Key takeaway: Compare cash dividends and reversionary additions by tracing each payment from declaration to receipt: identify its amount, date, triggering event, guarantee, and effect on the claim benefit. Equal stated amounts can produce different cash flows and different expected present values.

Check Your Understanding

For each problem, distinguish the claim increment from any cash paid earlier, and use the stated payment conditions.

A hypothetical policy declares $100 at time 1. A cash dividend is paid immediately; alternatively, a vested addition pays $100 only on a certain claim at time 2. The effective annual discount rate is 10%. Find the present value of each $100 component. (Answer: Cash: $100/1.10 = $90.91; addition: $100/1.10² = $82.64.)
A $300 cash account is declared at time 1, earns 5% effective annually, and is forfeited on lapse. An eligible claim at time 3 has probability 0.80; otherwise the policy lapses. The discount rate is 5%. Find the account’s expected present value. (Answer: $0.80[300(1.05)^2/1.05^3]=\$228.57$.)
A $400 dividend is paid immediately regardless of later events. Alternatively, a vested $400 addition is payable only on an eligible claim, whose probability is 0.75; assume a zero discount rate. Compare expected values and state the addition to the claim benefit under each option. (Answer: Cash expected value $400 and no claim increment; addition expected value $300 and a $400 claim increment if the claim is eligible.)
A policy has a $15,000 base claim benefit. At time 1, the policyholder receives a $250 cash dividend. At time 3, an eligible claim occurs, and the contract does not add prior cash dividends to the insured benefit. What is the claim benefit, and what are the total undiscounted receipts from the cash dividend and claim? (Answer: Claim benefit $15,000; total receipts $15,250.)

What You'll Learn