Premium Income in an Annual Profit Vector
A premium-income entry in an annual profit vector represents premium amounts attributable to the policy-year under the stated profit-test convention. The calculation starts with the contractual premium, not an assumed premium rate: identify the policies eligible to pay, the amount due per payment, and how many payments are scheduled in the year. These details matter when a block contains different durations, premium modes, or premium-paying statuses.
This tutorial calculates amounts only. It does not determine the exact point within the year when each receipt occurs, or the investment income on those receipts. Those questions require a separate cash-flow timing convention. For the calculations here, use the annual amount specified by the problem and record premium income as a positive insurer cash flow.
First Identify the Premium-Paying Policies
The number of policies in force is not automatically the number that generates premium income. A policy may be in force but paid-up, within a premium holiday, beyond its premium-paying term, or otherwise not required to pay a premium under its contract. Conversely, a policy may be premium-paying even though its premium amount differs from that of other policies in the block.
Use the status and exposure basis supplied in the problem. If a problem states that \(N\) policies are eligible to pay the premium for the policy-year, use that count. If it gives a total in-force count and identifies a subset that is paid-up or not premium-paying, exclude that subset. Do not apply a further survival or lapse adjustment to an already stated eligible count.
For a group with a common premium and frequency, the annual amount is found by multiplying the eligible policy count by the amount payable per policy during the year. For several groups, calculate each group separately and add the results. A group is defined by the characteristics that affect the calculation, particularly premium amount, payment frequency, duration, and eligibility.
If group \(j\) pays \(m_j\) level installments of \(p_j\) during a complete policy-year, its annual premium amount per policy is \(m_jp_j\). If the contract instead specifies an annual premium \(P_j\), use \(P_j\) directly. The distinction between these two descriptions is essential.
Worked Example: Annual Premiums and Premium-Paying Status
A hypothetical insurer has 240 policies in force during a policy-year. Of these, 18 are paid-up and do not owe a premium for the year, while the remaining policies are eligible to pay one annual contractual premium of $840 each. Calculate the premium income attributable to the year.
Method. First determine the eligible count, then multiply by the annual amount per eligible policy. The problem gives an annual premium, not a modal installment.
The number eligible to pay is \(240-18=222\) policies. Each contributes one annual premium of $840, so the total is
The premium-income amount is $186,480. As a check, the 18 paid-up policies contribute zero premium, and the 222 eligible policies each contribute exactly one annual premium. A common mistake would be to multiply the $840 annual premium by a payment frequency; no such additional multiplier applies here.
Annualizing Modal Installments
A modal premium is the amount of one scheduled installment. Common modes include annual, semiannual, quarterly, and monthly payment. For a complete policy-year with level installments, the number of installments is respectively 1, 2, 4, or 12. The annual amount is the installment amount times that number. This calculation uses the contractual installment amount provided; it does not assume that a modal premium is exactly an annual premium divided by the number of installments.
| Payment mode | Installments in a complete policy-year | Annual amount per eligible policy if installments are level |
|---|---|---|
| Annual | 1 | 1 times the annual installment |
| Semiannual | 2 | 2 times the semiannual installment |
| Quarterly | 4 | 4 times the quarterly installment |
| Monthly | 12 | 12 times the monthly installment |
The installment count applies only when the problem establishes that the policies are eligible for the full set of scheduled payments in the year. If the number of payable installments is stated directly, use that number. If a premium ceases partway through a year or a payment is missed, the amount must follow the problem’s stated contractual and profit-test rules rather than an automatic full-year multiplier. The exact timing of receipts within the year is a separate issue.
Worked Example: Combining Quarterly Premium Groups
A hypothetical block has 120 eligible policies at duration 2, each paying quarterly installments of $180, and 40 eligible policies at duration 5, each paying quarterly installments of $225. Both groups pay all four scheduled installments during the policy-year. Calculate their combined premium income.
Method. Annualize each group’s installment separately, multiply by that group’s eligible count, and then add. Both groups are quarterly, but their installment amounts differ.
For the duration 2 group, the annual amount per policy is \(4\times \$180=\$720\). Its total contribution is
For the duration 5 group, the annual amount per policy is \(4\times \$225=\$900\). Its contribution is
Therefore, combined premium income is
Check. The group contributions are $86,400 and $36,000, so their sum is $122,400. The common error of multiplying the first group correctly but omitting the quarterly multiplier for the second would understate the total; apply the frequency to each group before adding.
Premiums That Change by Duration
A contract may specify different premiums at different policy durations. In that case, the premium applicable to a policy-year is determined by its duration in that year. Do not apply the issue-duration premium to every year unless the contract says it remains level. Nor should a duration change be applied to the entire block when only some policies have reached that duration.
A practical method is to partition the block by duration and premium mode, identify the contractual installment for each group, and calculate each group’s annual amount. A duration-specific schedule might specify a quarterly installment of $210 in duration 1, $230 in duration 2, and $250 in duration 3. For each duration, use the installment shown for that duration and the number of eligible policies at that duration.
Make a distinct group for each eligible policy count, duration-specific premium, and payment frequency that differs.
Decide whether the stated amount is an annual premium or one installment. Do not apply an installment multiplier to an annual amount.
For a modal installment, multiply the installment by the number scheduled in the policy-year; for an annual premium, use the annual amount directly.
Multiply each per-policy annual amount by its eligible count, add the group totals, and verify that each duration and frequency was used once and only once.
Worked Example: A Quarterly Premium Schedule by Duration
A hypothetical policy has quarterly premiums that change by duration: $210 per installment in duration 1, $230 in duration 2, and $250 in duration 3. In the corresponding policy-years, the insurer has 75, 64, and 50 eligible policies at durations 1, 2, and 3, respectively. Each eligible policy pays all four quarterly installments in its policy-year. Find total premium income across these three duration groups.
Method. Use the duration-specific installment for each group, annualize it using four quarterly payments, multiply by the group’s eligible count, and then sum the contributions.
At duration 1, the annual amount per policy is \(4\times \$210=\$840\). The group contributes \(75\times \$840=\$63{,}000\).
At duration 2, the annual amount per policy is \(4\times \$230=\$920\). The group contributes \(64\times \$920=\$58{,}880\).
At duration 3, the annual amount per policy is \(4\times \$250=\$1{,}000\). The group contributes \(50\times \$1{,}000=\$50{,}000\).
Adding the three amounts gives
Check. The duration 3 group contributes exactly $50,000 because each of 50 policies pays $1,000 for the year. The other two contributions total $121,880, and \( \$121{,}880+\$50{,}000=\$171{,}880\). The mistake to avoid is applying one duration’s installment to all 189 policies.
Why the Group Formula Works
The calculation follows from adding the contractual payments. If each of \(N\) eligible policies pays \(m\) installments of \(p\) during the policy-year, one policy contributes \(p+p+\cdots+p\), with \(m\) terms. That sum is \(mp\). Since all \(N\) policies in the group have the same stated premium and schedule, summing their contributions gives \(N(mp)\).
For multiple groups, the total is the sum of each group’s total because the policies belong to separate groups and each contractual premium is counted in its own group. This is an amount calculation, not a probability calculation: if the eligible counts already reflect the required exposure basis, no additional probability factor belongs in the multiplication.
In an annual profit vector, the resulting premium amount is an inflow and therefore has a positive sign under the usual insurer cash-flow convention. The amount must be consistent with the policy-year being tested and the premium status specified for that year. Whether a particular receipt is assigned to the beginning, middle, or another point in the year is not determined by annualizing the premium; it depends on a separate timing assumption.
Common Mistakes and a Sign Check
The most frequent errors arise from confusing three separate quantities: the number of eligible policies, the premium per installment, and the annual premium amount. A reliable calculation labels all three before multiplying.
- Multiplying an annual premium by the mode. If $900 is the annual premium, the per-policy annual amount is $900, not four times $900 just because payments are quarterly.
- Forgetting to annualize an installment. If $900 is a quarterly installment and four installments are scheduled, the per-policy annual amount is \(4\times \$900\).
- Using all in-force policies as premium-paying. Exclude policies explicitly identified as paid-up or otherwise not eligible to pay.
- Using one duration’s premium for every policy. Apply the amount for each group’s own duration.
- Applying a frequency multiplier twice. If a problem already gives total premium per policy for the year, do not multiply that total by the installment count again.
- Adding groups before annualizing them. If groups have different installments or payment modes, calculate each annual contribution separately before summing.
As a final check, compare the total with the group-level amounts. Each group’s total must equal its eligible count times its annual amount per policy. Also check the units: policies multiplied by dollars per policy gives dollars, while installments per year must be included only when starting from a per-installment amount.
Check Your Understanding
For each problem, use only the stated eligible policies and contractual premium information. Treat the amounts as premium income for a complete policy-year.
- A block has 140 eligible policies, each paying one annual premium of $720. What is the annual premium income? (Answer: $100,800)
- A hypothetical block has 90 eligible policies at duration 1 with a monthly installment of $48, and 60 eligible policies at duration 3 with a quarterly installment of $275. Calculate total premium income for the year, applying the correct frequency to each duration group. (Answer: $117,840)
- At duration 4, 32 eligible policies pay an annual premium of $1,125. At duration 5, 28 eligible policies pay an annual premium of $1,200. Find their combined premium income. (Answer: $69,600)
- A block has 45 eligible policies paying monthly installments of $62 and 80 eligible policies paying semiannual installments of $410. Each group makes all scheduled payments in the year. Calculate the combined amount. (Answer: $99,080)
- In a hypothetical product, 50 eligible policies at duration 2 pay quarterly installments of $165, and 30 eligible policies at duration 6 pay monthly installments of $95. Another 12 policies are in force but paid-up. Find the premium income for the year. (Answer: $67,200)
Accurate premium amounts are a necessary component of a profit vector because every later profit calculation depends on correctly identifying the cash inflows assigned to the policy-year. This treatment of premium amounts is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.