Applying a Declared Dividend to a Premium
A declared cash dividend may be used to reduce a premium due. The calculation is simple only after the contract’s terms are clear: the declared dividend is an amount available for allocation, but the premium credit is limited by the premium eligible for the offset. If the dividend exceeds that eligible amount, the excess must be handled as the contract specifies; it is not automatically a cash payment.
This tutorial treats the declared dividend as given. It focuses on applying it to a premium at the same due date, rather than calculating the dividend scale or accumulating a dividend over time. All amounts in the examples are hypothetical and deterministic.
If the whole premium is eligible, then \(E=G\). If the dividend is smaller than the premium, it reduces the amount payable but does not clear the full premium. If it equals the premium, the premium is fully credited. If it is larger, the credit still cannot exceed the premium; the excess is separate from the premium calculation.
For all worked examples below, assume a hypothetical contract with a declared dividend and premium due on the same date. The effective annual interest rate is \(i=4.0\%\), but no interest adjustment applies because there is no time between the dividend’s availability and its application. No mortality-contingent payment is being valued, so mortality assumptions and a probability distribution do not enter the arithmetic.
Why the Credit Is Limited
A premium credit cannot be larger than either the dividend available or the eligible premium. It is therefore the smaller of those two amounts. The remaining total premium is the original total premium less the credit. The following steps also show why the amount due cannot become negative just because the dividend exceeds the eligible premium.
Separate the premium eligible for the offset, \(E\), from any ineligible part, \(G-E\). Do not apply the dividend to an ineligible component unless the contract permits it.
The credit cannot exceed either amount available, so \(A=\min(D,E)\).
Subtract the credit from the total premium: \(N=G-A\). Since \(A\leq E\leq G\), this amount is nonnegative.
If \(D>E\), the excess is \(R=D-E\). Its disposition—such as cash payment or another contractually specified treatment—cannot be inferred from the premium-offset calculation alone.
To justify the expression for \(N\), write the total premium as the ineligible part \(G-E\) plus the eligible part \(E\). After applying the dividend, the unpaid eligible part is \(E-A\). If \(D\leq E\), then \(A=D\), so \(E-A=E-D\). If \(D>E\), then \(A=E\), so \(E-A=0\). These two cases give \(E-A=\max(E-D,0)\). Thus \(N=(G-E)+\max(E-D,0)\), which is always nonnegative.
The excess amount \(R\) is not a premium credit. When the contract requires the entire excess to be paid immediately in cash, the cash excess \(C\) equals \(R\), and \(D=A+C\). Without that condition, keep \(R\) separate and describe its disposition under the contract. In particular, do not label an excess as cash merely because it is larger than the premium.
Worked Calculations
Worked Example: Dividend Smaller Than the Premium
A hypothetical policy has a total premium of $1,840, all eligible for the declared cash dividend. The dividend is $1,265. Find the premium credit and the remaining amount due. Assume the same-date 4.0% effective annual rate and deterministic amounts described above; no mortality assumption or probability distribution is needed.
Method. Since the full premium is eligible, \(E=G=\$1{,}840\). The credit is the smaller of the dividend and eligible premium:
The remaining premium is:
The dividend does not exceed the eligible premium, so \(R=\max(1{,}265-1{,}840,0)=\$0\). The policyholder owes $575 after the credit, and there is no excess to allocate. Check: the premium is accounted for as $1,265 credited plus $575 still due, totaling $1,840. The common mistake avoided is treating the dividend as if it automatically paid the entire premium.
Worked Example: Dividend Exactly Equals the Premium
A hypothetical policy has a $1,260 premium, all of which is eligible for a declared dividend of $1,260. Determine the credit, remaining premium, and excess. As above, \(i=4.0\%\) is not used for a same-date offset; the amounts are fixed, with no mortality contingency or probability distribution.
Method. Here \(E=G=\$1{,}260\), so:
The excess is \(R=\max(D-E,0)=\max(1{,}260-1{,}260,0)=\$0\). The premium is fully satisfied by the credit; no further premium payment is due and no excess remains. Check: the credit equals the premium exactly. The equality case does not create either an unpaid premium or an excess.
Worked Example: Dividend Larger Than the Premium
A hypothetical contract has a $1,475 premium, all eligible for a declared dividend of $1,620. Its terms require any amount of the dividend beyond the premium credit to be paid immediately in cash. Find the credit, premium due, and cash excess. The amounts are same-date and deterministic; \(i=4.0\%\) creates no adjustment, and no mortality assumption or probability distribution is involved.
Method. With \(E=G=\$1{,}475\), the premium credit is capped at the eligible premium:
The premium remaining due is:
The excess is calculated in the dividend-minus-eligible-premium order:
Because the contract explicitly requires the excess to be paid in cash, \(C=R=\$145\). Check: the full dividend is accounted for as a $1,475 premium credit plus a $145 cash payment, totaling $1,620. The policyholder owes no premium. The common mistake avoided is using the credit cap as a reason to discard the excess, or reversing the subtraction when finding it.
Worked Example: Only Part of the Premium Is Eligible
A hypothetical policy has a total premium of $2,200, comprising $1,800 eligible for the dividend offset and $400 ineligible for it. The declared dividend is $2,050. The contract says any dividend amount beyond the eligible premium is carried forward under its terms, not paid immediately in cash. Find the premium credit, amount due, and excess. Assume same-date amounts, \(i=4.0\%\) with no interest adjustment, and no mortality contingency or probability distribution.
Method. Here \(G=\$2{,}200\), \(E=\$1{,}800\), and \(D=\$2{,}050\). The credit cannot exceed the eligible premium:
The total amount still due is:
The excess is:
The $400 due is precisely the ineligible part of the premium; the eligible portion has been fully credited. The $250 excess is carried forward as specified, and is not described as a cash payment. Check: $1,800 credited plus $400 due accounts for the $2,200 premium; separately, the $2,050 dividend consists of $1,800 applied and $250 carried forward. The mistake avoided is assuming that a dividend larger than the eligible amount necessarily clears the entire premium or produces cash.
Common Exam Traps
The comparison must be made against the eligible premium, not automatically against the total premium. When the whole premium is eligible, the comparison is between \(D\) and \(G\). When only part is eligible, it is between \(D\) and \(E\), while the amount due is still calculated from the total premium \(G\).
- Crediting more than the eligible amount. If \(D>E\), the credit remains \(A=E\); it does not become \(D\).
- Calling every excess a cash payment. \(R\) measures the amount beyond the eligible premium. It is cash only if the contract says it is paid in cash.
- Subtracting in the wrong order. The excess is \(D-E\) when \(D>E\). The remaining premium is \(G-A\). These are different calculations with different meanings.
- Ignoring ineligible premium components. A dividend can fully cover the eligible part while an ineligible premium amount remains payable.
- Confusing a premium credit with the declared dividend. A credit is limited by eligibility; a declared amount in excess of that credit must be accounted for separately.
A reliable exam calculation states the total premium, the eligible amount, the dividend, the credit, and the remaining premium in that order. It then identifies any excess and applies only the disposition explicitly given by the hypothetical contract. This keeps the premium accounting distinct from any separate treatment of the excess dividend.
Check Your Understanding
For each hypothetical case, calculate the premium credit and amount still due. State any excess separately and follow the stated contract treatment. All amounts are same-date deterministic amounts; no interest adjustment, mortality contingency, or probability distribution applies.
- A $1,835 premium is fully eligible for a declared dividend of $725. Find the premium credit and amount due. (Answer: $725 credit; $1,110 due; no excess.)
- A $1,460 premium is fully eligible for a dividend of $1,460. Find the premium credit and amount due. (Answer: $1,460 credit; $0 due; no excess.)
- A $1,725 premium is fully eligible for a dividend of $1,910, and the contract requires the excess to be paid immediately in cash. Find the credit, premium due, and cash excess. (Answer: $1,725 credit; $0 due; $185 cash excess.)
- A $2,400 total premium includes $1,900 eligible for the dividend offset. A declared dividend of $1,650 is applied to the eligible amount. Find the credit and total amount due. (Answer: $1,650 credit; $750 due.)
- A $2,150 premium includes $1,750 eligible for a dividend credit. The dividend is $1,980, and the contract carries any excess forward rather than paying it in cash. Find the premium credit, amount due, and excess disposition. (Answer: $1,750 credit; $400 due; $230 excess carried forward.)