Reading a Cash Dividend Scale
A declared cash dividend scale may state an amount per eligible policy, a rate per unit of sum assured, or a percentage of a specified amount. These are different ways to express a dividend calculation, not interchangeable labels. The calculation depends on identifying both the rate and the base to which it applies.
This tutorial focuses on translating scale wording into a cash amount for an individual policy. Assume throughout that the policy is eligible for the stated declaration and that the sum assured and premium used are the amounts specified for the scale. The calculation does not determine eligibility or declaration timing; those conditions must be established separately.
For a flat per-policy amount, every eligible policy to which that scale entry applies receives the same amount, regardless of its sum assured. For a sum-assured rate, the amount varies with the sum assured. For a percentage rate, the dividend varies with the named percentage base. A percentage without a stated base is incomplete for calculation purposes.
Why Unit Conversion Works
The sum-assured formula follows by counting how many scale units are represented by the policy. If one unit is \(U\) dollars of sum assured, a policy with sum assured \(S\) contains \(S/U\) such units. Multiplying the dividend per unit by the number of units gives the total dividend. Equivalently, the rate can be expressed as a proportion of the sum assured.
Here \(r/U\) is the dividend per dollar of sum assured. Expressed as a percentage of sum assured, the corresponding rate is \(100r/U\%\). For example, a rate of \(r\) dollars per \(U\) dollars of sum assured is not the same as \(r\%\) of sum assured. The denominator \(U\) determines the conversion.
A percentage scale has the same multiplication structure, but its base may be a premium rather than the sum assured. If the declared rate is \(q\%\) of annual premium, then \(D=(q/100)P\). It would be incorrect to substitute \(S\) for \(P\) merely because the policy also has a sum assured.
Record whether each component is per policy, per unit of sum assured, or a percentage. For a percentage, write down its named base.
Express the sum assured in the same units used by a per-unit rate. For example, divide a sum assured in dollars by 1,000 when the scale is dollars per $1,000.
Apply each rate only to its own base. Add the resulting dollar amounts if the scale explicitly includes multiple components.
Confirm that the answer is in dollars, that no base was counted twice, and that the size of the dividend is plausible relative to the inputs.
Worked Calculations
Worked Example: A Rate per $1,000 of Sum Assured
A hypothetical participating policy has a sum assured of $250,000. Its declared cash dividend scale is $6.40 per $1,000 of sum assured. Calculate the dividend, then express the rate as a percentage of sum assured.
Method. Count the number of $1,000 units, multiply by the rate per unit, and then convert that rate to a proportion of the sum assured.
Step 1: Count the units. The number of units is \(250{,}000/1{,}000=250\).
Step 2: Calculate the cash dividend. The dividend is \(250 \times \$6.40=\$1{,}600\).
Step 3: Convert the rate. The rate per dollar of sum assured is \(\$6.40/\$1{,}000=0.0064\). As a percentage, this is \(0.0064 \times 100\%=0.64\%\).
Check. Applying the percentage directly gives \(0.0064 \times \$250{,}000=\$1{,}600\), agreeing with the unit calculation. The dividend is reported to the nearest cent: $1,600.00. The common mistake avoided here is multiplying $250,000 by $6.40 without first converting the sum assured to $1,000 units.
Worked Example: A Percentage of Annual Premium
A hypothetical policy has an annual premium of $2,400 and a sum assured of $120,000. The cash dividend scale is 12.5% of annual premium. Calculate the dividend and express it as an equivalent rate per $1,000 of sum assured for this policy.
Method. First apply the percentage to the stated premium base. Then divide the resulting cash amount by the number of sum-assured units for this particular policy.
Step 1: Calculate the percentage dividend. The percentage as a decimal is \(12.5/100=0.125\). Therefore \(D=0.125 \times \$2{,}400=\$300\).
Step 2: Count the sum-assured units. The number of $1,000 units is \(120{,}000/1{,}000=120\).
Step 3: Convert to a per-unit amount. The equivalent rate is \(\$300/120=\$2.50\) per $1,000 of sum assured.
Check. Applying the converted rate gives \(120 \times \$2.50=\$300\). The dividend is $300.00. This equivalent rate is specific to the policy’s premium-to-sum-assured relationship; it is not necessarily a single rate that can replace the percentage scale for policies with different premiums or sums assured. The common mistake avoided is applying 12.5% to the $120,000 sum assured instead of to the stated annual premium.
Worked Example: Combining Three Scale Components
A hypothetical scale provides $18 per eligible policy, plus $3.75 per $1,000 of sum assured, plus 8% of annual premium. For a policy with sum assured $80,000 and annual premium $1,600, calculate the total cash dividend and express it as an equivalent rate per $1,000 of sum assured.
Method. Calculate the flat amount, sum-assured amount, and premium percentage separately. Add the dollar amounts only after each has been calculated on its stated base.
Step 1: Flat component. The per-policy amount is $18.00.
Step 2: Sum-assured component. The policy has \(80{,}000/1{,}000=80\) units. This component is \(80 \times \$3.75=\$300.00\).
Step 3: Premium component. The decimal form of 8% is 0.08, so this component is \(0.08 \times \$1{,}600=\$128.00\).
Step 4: Add the components. The total is \(\$18.00+\$300.00+\$128.00=\$446.00\).
Step 5: Convert the total to a per-unit equivalent. There are 80 units of $1,000 sum assured, so the equivalent rate is \(\$446.00/80=\$5.575\) per $1,000. This is a conversion for this policy, not a replacement scale for a portfolio with different policy amounts.
Check. Reconstructing the total gives \(80 \times \$5.575=\$446.00\). The cash dividend is $446.00. The common mistake avoided is applying the 8% premium rate to the sum assured, or adding rates before converting them into amounts with consistent units.
Converting Between Scale Bases
A scale expressed on one basis can sometimes be restated on another basis, but the conversion may depend on the policy. A rate per $1,000 of sum assured converts directly to a percentage of sum assured because both use the same base. A per-policy amount can be converted to a per-unit rate by dividing by the number of units, provided the sum assured is positive. A premium-based percentage can be converted to a per-unit sum-assured rate only when both the premium and sum assured are known.
| Original scale wording | Conversion for an individual policy | Important condition |
|---|---|---|
| \(r\) dollars per \(U\) dollars of sum assured | Percentage of sum assured \(=100r/U\%\) | The rate and \(U\) must use the same currency units. |
| \(c\) dollars per policy | Per-\(U\) rate \(=c/(S/U)\) | Requires a positive sum assured; the equivalent rate varies with \(S\). |
| \(q\%\) of annual premium | Per-\(U\) rate \(=(q/100)P/(S/U)\) | Requires the specified annual premium \(P\) and sum assured \(S\). |
These conversions preserve the cash amount for the policy used in the conversion. They do not necessarily produce a scale that preserves amounts for every policy. For example, a flat amount of $100 is $1 per $1,000 for a policy with $100,000 sum assured, but it is $2 per $1,000 for a policy with $50,000 sum assured. A single flat amount therefore cannot generally be represented by one fixed per-unit rate across different sums assured.
Common Errors and Checks
The most frequent errors are unit errors and base errors. A rate stated per $1,000 is multiplied by the number of $1,000 units, not by the dollar sum assured. A percentage is applied to its specified base, not to whichever policy amount is largest. If a scale contains several components, each should be calculated separately before addition.
Also distinguish a premium amount defined by the scale from a payment mode or an instalment. If the scale says a percentage of annual premium, use the annual premium amount specified by the problem. Do not substitute a monthly instalment unless the scale explicitly defines that as its base. Similarly, if the declaration specifies an in-force sum assured, use that amount rather than an original or projected amount.
Check Your Understanding
For each hypothetical scale, calculate the cash dividend or requested equivalent rate. Keep each rate attached to its stated base.
- A policy has sum assured $160,000. Its scale pays $4.25 per $1,000 of sum assured. Calculate the cash dividend. (Answer: $680.00)
- A policy has annual premium $1,850. Its scale pays 9% of annual premium. Calculate the cash dividend. (Answer: $166.50)
- A scale pays $12 per policy plus $3.75 per $1,000 of sum assured. For sum assured $240,000, calculate the total dividend. (Answer: $912.00)
- A scale pays 0.30% of sum assured plus 7.5% of annual premium. For sum assured $90,000 and annual premium $1,350, calculate the total. (Answer: $371.25)
- A policy with sum assured $140,000 receives a dividend at $5.60 per $1,000 of sum assured. Calculate the dividend, the equivalent percentage of sum assured, and the equivalent percentage of annual premium if annual premium is $2,800. (Answer: $784.00; 0.56% of sum assured; 28% of annual premium)