Introduction
A premium stated as an annual amount does not by itself specify when the insurer receives cash. A policyholder might pay once at the start of the policy year, or make several installments during the year. Those schedules can have the same scheduled total but different cash flows, especially if the policy terminates before the year ends.
For annual profit calculations, the essential steps are to identify the amount due at each payment date, place each receipt in the correct policy year, and distinguish the scheduled premium total from the amount received if the policy does not remain in force. The examples below treat premiums as positive insurer cash inflows.
From an Annual Amount to Dated Cash Flows
Let \(G\) be the premium for a full policy year on an annual-payment basis. Under a simple equal-installment convention, with no modal loading, each of \(m\) installments is \(G/m\). If installments are due in advance, the payment times, measured in years from the start of the policy year, are
Thus an annual payment is due at time 0; half-yearly payments are due at times 0 and \(1/2\); quarterly payments at 0, \(1/4\), \(1/2\), and \(3/4\); and monthly payments at \(j/12\), for \(j=0,1,\ldots,11\). The installment amounts sum to \(G\), but the receipts are spread across the year. This description assumes payments are due in advance. A contract with payments due in arrears has different payment dates, which must be used instead.
An installment need not equal \(G/m\). A contract may apply a modal loading, or state a specific installment amount. In that case, use the amount actually specified by the contract to construct the schedule. Do not assume that dividing an annual premium by the frequency recovers a loaded modal installment.
| Frequency | Number of installments | Advance payment times | Installment if there is no loading |
|---|---|---|---|
| Annual | 1 | 0 | \(G\) |
| Half-yearly | 2 | 0, \(1/2\) | \(G/2\) |
| Quarterly | 4 | 0, \(1/4\), \(1/2\), \(3/4\) | \(G/4\) |
| Monthly | 12 | \(j/12,\ j=0,\ldots,11\) | \(G/12\) |
Worked Example: Building Four Schedules for One Policy-Year Premium
A hypothetical policy has an annual-payment premium of $1,800. Assume the policy remains in force throughout the year, installments are level and due in advance, and there is no modal loading. Find the installments, their payment times, and the scheduled annual total for annual, half-yearly, quarterly, and monthly payment.
Method. For each frequency \(m\), divide $1,800 by \(m\), then list the advance payment times. This separates the total amount from its timing.
For annual payment, \(m=1\). The installment is \(1{,}800/1=\)$1,800, due at time 0. The scheduled total is $1,800.
For half-yearly payment, \(m=2\). Each installment is \(1{,}800/2=\)$900, due at times 0 and \(1/2\). The total is \(2(900)=\)$1,800.
For quarterly payment, \(m=4\). Each installment is \(1{,}800/4=\)$450, due at times 0, \(1/4\), \(1/2\), and \(3/4\). The total is \(4(450)=\)$1,800.
For monthly payment, \(m=12\). Each installment is \(1{,}800/12=\)$150, due at times \(j/12\), for \(j=0,\ldots,11\). The total is \(12(150)=\)$1,800.
The scheduled total is $1,800 in all four cases, but the receipt dates differ. For example, the quarterly schedule has $450 due at each of four dates, whereas the annual schedule has the full $1,800 due at time 0. Check: each installment amount multiplied by its number of installments gives $1,800. The common mistake avoided here is to treat equal annual totals as identical dated cash flows.
Scheduled Premiums Versus Expected Receipts
A scheduled annual total assumes every installment falls due while the policy is in force. If the contract requires premiums only while the policy is in force, a termination before a later installment date can reduce the amount received. The appropriate expected amount depends on the probability the policy is still in force at each due date.
Suppose a policy begins in force at age \(x\), premiums are due in advance at times \(\tau_1,\ldots,\tau_m\), and installment \(j\) is \(c_j\). Assume each due premium is collected if the policy is in force immediately before that due date, there is no refund or pro-rata adjustment on termination, and there is no point probability of termination exactly at a due date. If termination is by death only, the probability of being in force at time \(\tau_j\) is \({}_{\tau_j}p_x\). The expected premium receipts during the policy year, per policy in force at its start, are
If other forms of exit, such as lapse, also stop premiums, replace \({}_{\tau_j}p_x\) with the probability the policy is in force at \(\tau_j\). Mortality and lapse probabilities should not be combined by simply multiplying separate probabilities unless their relationship justifies that calculation. The formula also relies on the stated premium rule; a contract that refunds or prorates premium at termination requires a different cash-flow model.
Why the Expected-Receipt Formula Holds
For each due date \(\tau_j\), define an indicator \(I_j\) that equals 1 if the premium is collected and 0 otherwise. The receipt at that date is \(c_j I_j\), so the total receipts are the sum of the dated receipts. By linearity of expectation, the expected total is the sum of their expectations:
Under the assumptions above, \(P(I_j=1)={}_{\tau_j}p_x\) for death-only termination. Therefore the expected-receipt expression follows. It does not require the indicators at different dates to be independent: the sum-of-expectations identity holds whether or not they are dependent.
Worked Example: Applying a Quarterly Modal Loading
A hypothetical contract has an annual-mode premium of $2,400. The insurer charges a 5% modal loading on the total annual amount for quarterly payment. Installments are level and due in advance, and the policy remains in force for the full year. Find the quarterly installment, scheduled total, and dated cash flows.
Method. Apply the stated loading to the annual-mode amount to find the full-year modal total, then divide that total by four. This avoids applying the loading separately to an already divided installment.
The modal full-year total is
There are four quarterly installments, so each installment is
The insurer receives $630 at each of times 0, \(1/4\), \(1/2\), and \(3/4\). The scheduled annual total is \(4(630)=\)$2,520. The 5% increase over the annual-mode premium is \(2{,}520-2{,}400=\)$120, which is also \(0.05(2{,}400)=\)$120.
A common error is to calculate four installments of \(2{,}400(1.05)=\)$2,520 each, giving an incorrect total of $10,080. The 5% loading applies to the annual amount once, not to each installment as though each were a full annual premium. Check: four installments of $630 give the loaded annual total of $2,520.
Worked Example: Expected Monthly Premiums Under Uniform Termination
A hypothetical policy is in force at the start of a policy year and has an annualized premium of $1,200, payable as twelve monthly installments of $100 in advance. Assume there is no modal loading. Termination is the only reason premiums stop; the probability of termination during the year is 12%, and conditional on termination occurring in the year, its time is uniform over that year. There is no premium refund. Find the expected premium receipts during the year. Compare this with quarterly payments of the same annualized amount.
Method. Calculate the probability the policy is still in force at each due date, multiply each installment by that probability, and add. Under the stated uniform timing assumption, the probability of termination by time \(t\) is \(0.12t\) for \(0\leq t\leq1\). Thus the probability of being in force at time \(t\) is \(1-0.12t\).
Monthly installments are due at \(t=j/12\), for \(j=0,\ldots,11\). At time \(j/12\), the in-force probability is
The twelve probabilities are \(1.00, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.90,\) and \(0.89\). Their sum is an arithmetic series:
Therefore expected monthly receipts are \(100(11.34)=\)$1,134. The scheduled total, if the policy remains in force at all twelve due dates, is still \(12(100)=\)$1,200.
For quarterly payment, the installment is \(1{,}200/4=\)$300, due at times 0, \(1/4\), \(1/2\), and \(3/4\). The corresponding in-force probabilities are \(1.00, 0.97, 0.94,\) and \(0.91\), with sum 3.82. Expected quarterly receipts are \(300(3.82)=\)$1,146.
The quarterly expected amount is $12 greater than the monthly expected amount, even though both schedules have a scheduled total of $1,200. Under these assumptions, more of the quarterly premium is due earlier, before some terminations occur. The annual premium, due at time 0, would produce $1,200 if the policy is in force at inception. Check: each expected receipt is no greater than its $1,200 scheduled total, and the quarterly amount exceeds the monthly amount because its later installments are less frequent. The common mistake avoided is multiplying every installment by 1, ignoring that later due dates may not be reached while the policy is in force.
Common Traps in Policy-Year Premium Cash Flows
Premium entries in an annual profit vector should reflect the cash-flow convention being used, not just the annualized premium stated in the contract. A useful process is to write down the due dates first, then enter the installment amounts on those dates, and only then apply any in-force probabilities required by the model.
- Confusing annualized amount and modal installment. If $1,200 is payable monthly with no loading, the installment is $100, not $1,200. If a loading is stated, apply it according to its stated basis.
- Assuming the scheduled total is always received. A full-year scheduled total is not the same as expected receipts when later payments depend on the policy remaining in force.
- Using the wrong payment dates. Advance and arrears schedules differ. Do not place an arrears installment at the start of the year just because its premium amount is known at that time.
- Counting a premium after termination. Under a pay-while-in-force assumption, only include an installment when the policy is in force at its due date.
- Mixing payment timing with discounting. The calculations here identify nominal receipts and their dates. Any investment accumulation or discounting requires an explicitly specified interest convention and valuation date.
These distinctions matter because profit calculations use cash flows, not merely annual premium labels. Two policies with the same annualized premium may have different premium inflows by date, and policies with the same modal schedule may have different expected receipts if their probabilities of remaining in force differ. State the schedule, the in-force condition, and whether the result is scheduled or expected.
Check Your Understanding
For each problem, use the stated advance-payment and termination assumptions; report expected receipts to the nearest cent where needed.
- A policy has a $1,800 annualized premium, payable half-yearly in equal installments without loading. State the installment amount, payment times, and scheduled annual total. (Answer: $900 at times 0 and \(1/2\); scheduled total $1,800.)
- A hypothetical policy has an annualized premium of $960 payable monthly in equal installments, with a 12% loading applied once to the full annualized amount. Find the installment and loaded scheduled annual total. (Answer: $89.60 per month; scheduled total $1,075.20.)
- A quarterly schedule has a $1,600 annual-mode premium and a 4% loading applied to the full annual amount. Find the installment and list its four due dates. (Answer: $416 at times 0, \(1/4\), \(1/2\), and \(3/4\); scheduled total $1,664.)
- A $2,000 annualized premium is paid in two equal installments in advance. Termination probability is 10% over the year, with termination time uniform conditional on termination in the year; there is no refund. Find expected receipts. (Answer: $1,950.)
- A policy pays a $1,200 annualized premium in four equal quarterly installments in advance. Termination probability is 8% over the year, with termination time uniform conditional on termination in the year; there is no refund. Find expected receipts. (Answer: $1,164.)