Introduction
An annual profit calculation depends not only on which premiums, expenses, and benefits occur, but also on when each amount is received or paid. A premium received at the start of a policy year can earn investment income for nearly the whole year. A claim paid at the end of the year has little or no time to accumulate before the closing valuation. Treating both amounts as though they occurred at the same time can distort the year-end position.
A policy-year timeline makes these dates explicit. It places the opening reserve at the start of the year, records each cash flow at its contractual or assumed payment time, and places the closing reserve at the end. This tutorial focuses on that timing map. It does not derive the complete annual profit identity or prescribe a full sign convention for profit vectors.
Reading the Policy-Year Timeline
It is convenient to measure time within one policy year, using \(t=0\) for its beginning and \(t=1\) for its end. The opening reserve is therefore at \(t=0\), and the closing reserve is at \(t=1\). A cash flow made six months after the start is at \(t=0.5\), if the model treats six months as half a year.
The timeline is determined by the contract and the valuation assumptions. A premium might be payable in advance at \(t=0\), in arrears at \(t=1\), or in installments during the year. An expense may be incurred when a policy is issued, spread through the year, or paid at renewal. A benefit may be paid when a covered event occurs or after a specified claim-settlement delay. If the question states a payment date, use that date; do not substitute the event date.
| Item | Typical timeline position | Timing question to check |
|---|---|---|
| Opening reserve | \(t=0\) | At what instant is the opening liability valued? |
| Premium | As stated by the premium schedule | Is it paid in advance, in arrears, or by installments? |
| Expense | When incurred or paid, as specified | Is it an issue expense, renewal expense, or recurring expense? |
| Claim benefit | At the stated payment time | Is payment immediate, delayed, or at a fixed date? |
| Closing reserve | \(t=1\) | Does the policy remain in force at the valuation instant? |
An event and its cash flow need not share a date. For example, a death may occur at \(t=0.7\), with the benefit paid at \(t=0.8\). The death occurs during the policy year, but the insurer's payment is placed at \(t=0.8\) for a cash-flow accumulation calculation. Whether the policy has a closing reserve depends on its status and the valuation definition at \(t=1\); a reserve is not itself a claim payment.
Worked Example: Building a Timeline for a Policy That Terminates
A hypothetical one-year policy begins at time \(t=0\). Its opening reserve is $500. A premium of $1,200 is paid at issue, and an acquisition expense of $300 is paid at the same time. A covered death occurs at \(t=0.7\); the $10,000 death benefit is paid immediately. Assume an effective annual interest rate of 4%, with fractional-year accumulation at the equivalent force \(\delta=\ln(1.04)\). The stated death path is deterministic: no mortality distribution or probability calculation is needed. Place the items on the timeline and identify the closing-reserve status.
Step 1: Locate the opening items. The opening reserve of $500 is at \(t=0\). The $1,200 premium receipt and $300 acquisition-expense payment are also at \(t=0\), because both are specified as occurring at issue.
Step 2: Locate the death benefit. Death occurs at \(t=0.7\), and payment is immediate, so the $10,000 benefit is placed at \(t=0.7\). If payment instead were delayed by one month, the payment time—not the death time—would be used to accumulate the benefit.
Step 3: Determine the closing-reserve status. The policy terminates on death at \(t=0.7\). It is therefore not an in-force policy at \(t=1\), and there is no closing reserve for this policy as an in-force contract at that time. This does not mean the death benefit is omitted: it is a cash flow at \(t=0.7\).
Check. The timeline contains the opening reserve and issue-date transactions at \(t=0\), the death payment at \(t=0.7\), and no in-force closing reserve at \(t=1\). The interest assumption is stated, but no accumulation is needed to answer the timing question.
Method and trap avoided. Assign each item its own date before deciding what remains at year end. A common error is to place the benefit at \(t=1\) simply because the calculation covers a full policy year, or to assign a closing reserve to a policy that terminated during the year.
Accumulating Cash Flows to the Closing Date
When amounts occur at different times, a useful comparison is to express them at one common date. Under a constant effective annual interest rate \(i\), assume fractional-year accumulation follows the factor \((1+i)^t\). Thus a cash flow at time \(t\) accumulates from \(t\) to the year end by the factor \((1+i)^{1-t}\). This is a timing calculation; it is not, by itself, the full annual profit formula.
The reason for the factor is the compound-interest accumulation law. A unit invested for one year accumulates to \(1+i\). Under the stated fractional-year assumption, a unit invested for \(1-t\) years accumulates to \((1+i)^{1-t}\). Multiplying by the amount \(C\) gives its year-end equivalent. The factor is 1 for a cash flow at \(t=1\), and \(1+i\) for a cash flow at \(t=0\), as expected.
For a reserve roll-forward, the opening reserve also accumulates over the full year. If the only items are an opening reserve \(V_0\) and cash flows \(C_j\) at times \(t_j\), the amount represented at year end before any separate closing-reserve interpretation is
This expression is an accumulation of the listed items, not a substitute for specifying the policy's benefit, expense, and reserve definitions. In particular, a cash-flow list must be complete before it is accumulated.
Worked Example: A Midyear Receipt and a Year-End Benefit
For a hypothetical policy, the opening reserve is $1,000. The effective annual interest rate is 4%, with fractional-year accumulation using \((1.04)^t\). At \(t=0.5\), a premium of $220 is received and an expense of $60 is paid, giving a net midyear receipt of $160. A benefit of $300 is paid at \(t=1\), after which the policy remains in force. Treat this stated path as deterministic; no mortality distribution is needed. Calculate the closing reserve by accumulating each item to year end.
Step 1: Accumulate the opening reserve. It is invested for the full year:
The opening reserve therefore contributes $1,000 of principal and $40 of interest at the year-end basis.
Step 2: Combine the midyear receipt and expense. Both occur at \(t=0.5\), so their net amount is
This net receipt accumulates for half a year. The half-year accumulation factor is
Thus its year-end value is
The midyear receipt contributes $160 of principal and approximately $3.17 of investment gain.
Step 3: Include the year-end benefit. The $300 benefit is paid at \(t=1\), so it has no time to accumulate before the closing date. Subtract it from the accumulated amounts:
Rounded to the nearest cent, the closing reserve is $903.17.
Check and trap avoided. The same result can be displayed by separating principal, interest, receipt, and benefit:
A tempting but incorrect calculation is \(1{,}000+40+3.1686-300=743.17\). It includes the opening reserve and the interest earned, but omits the $160 midyear receipt itself. The corrected decomposition includes both the receipt and its investment gain. This check helps distinguish a missing principal cash flow from a missing interest amount.
Method. Accumulate each dated item to the same closing date, then combine the year-end equivalents. Do not apply a half-year factor to a year-end benefit, and do not forget the principal amount when calculating the investment gain on a receipt.
Payment Timing Can Change the Year-End Amount
Two scenarios can have the same benefit amount and the same event but different year-end cash-flow values if the payment dates differ. Likewise, premium and expense amounts at the start of the year must be accumulated for longer than amounts at the end. The timeline makes these distinctions visible before any annual profit vector is assembled.
Worked Example: Claim Payment During the Year Versus at Year End
A hypothetical in-force policy has an opening reserve of $800. At \(t=0\), a premium of $240 is received and an expense of $40 is paid. A $500 claim payment is made at \(t=0.4\). The policy remains in force through \(t=1\), and the closing reserve is valued then. Assume an effective annual interest rate of 5%, with fractional-year accumulation \((1.05)^t\). The claim payment time is specified and deterministic; no mortality distribution is required. Find the accumulated year-end amount, and compare it with the amount if the same claim were paid at \(t=1\).
Step 1: Accumulate the opening reserve.
Step 2: Accumulate the start-of-year net receipt. The premium less the expense is \(240-40=200\), received net at \(t=0\). It accumulates for one full year:
Step 3: Accumulate the claim payment from \(t=0.4\) to \(t=1\). It accumulates for \(1-0.4=0.6\) years:
Step 4: Combine the year-end equivalents. The opening reserve and start-of-year net receipt total \(840+210=1{,}050\) at year end before the claim payment. After allowing for the claim payment's accumulated amount:
Rounded to the nearest cent, the resulting year-end amount is $535.15.
If the $500 claim were instead paid at \(t=1\), its year-end value would be $500, and the corresponding amount would be
The difference is approximately \(550-535.15=\)$14.85, reflecting the accumulation on the claim payment made at \(t=0.4\). The claim amount is unchanged; only its payment time differs.
Check and trap avoided. The claim paid at \(t=0.4\) is multiplied by \((1.05)^{0.6}\), not by \((1.05)^{0.4}\). The exponent is the time remaining until year end. The comparison also confirms that an earlier outgo has a larger year-end accumulated value than the same outgo paid at the closing date.
A Reliable Timeline Procedure
Before calculating an annual result, make a timeline and assign each item its date. The following sequence separates timing decisions from later profit-vector sign conventions.
Mark \(t=0\) and \(t=1\), and identify the valuation instants for the opening and closing reserves.
Include every premium, expense, claim payment, other benefit, and reserve specified for the year.
Use the stated receipt or payment date. Keep the date of an insured event separate from a delayed benefit-payment date.
Determine whether the policy is still in force at \(t=1\), and whether a closing reserve is applicable under the stated valuation basis.
Under the stated interest model, apply \((1+i)^{1-t}\) to an amount at time \(t\) to express it at year end.
A timeline also clarifies what a calculation has not yet done. It does not determine the probability of a claim, select the exposure basis, or establish the full signs and components of annual profit. Those are separate modeling choices. For a stochastic model, the timeline is still built first; probabilities or expected values are then attached to the dated events as required.
Check Your Understanding
For each problem, build the timeline before checking the final answer. Unless otherwise stated, use fractional-year accumulation under a constant effective annual interest rate.
- A hypothetical policy has an opening reserve of $700. At \(t=0\), it receives a $150 premium and pays a $30 expense. A $100 benefit is paid at \(t=1\), and the policy remains in force. Identify the opening reserve, the dated cash flows, and the closing-reserve time. (Answer: Opening reserve at \(t=0\); net start-of-year receipt $120 at \(t=0\); benefit $100 at \(t=1\); closing reserve valued at \(t=1\).)
- An opening reserve of $600 earns interest at an effective annual rate of 3%. A net receipt of $80 occurs at \(t=0.5\), and a $150 benefit is paid at \(t=1\). What is the resulting year-end amount after accumulating the reserve and receipt? (Answer: \(600(1.03)+80\sqrt{1.03}-150=\)$549.19, rounded to cents.)
- An opening reserve is $400. At \(t=0\), the insurer receives $90. A $120 claim payment is made at \(t=0.25\), and a $15 expense is paid at \(t=1\). At an effective annual interest rate of 4%, calculate the year-end amount. (Answer: \(400(1.04)+90(1.04)-120(1.04)^{0.75}-15=\)$371.02, rounded to cents.)
- A $200 claim payment is made at \(t=0.5\) rather than at \(t=1\). If the effective annual interest rate is 5%, what is the difference between the two payments' accumulated year-end amounts? (Answer: \(200\sqrt{1.05}-200=\)$4.94, rounded to cents; the midyear payment has the larger year-end accumulated amount.)
- A hypothetical policy has a $900 opening reserve and a $2,000 benefit payable immediately upon a covered event at \(t=0.6\). The event occurs, and the policy terminates. What is the benefit-payment time, and is there an in-force closing reserve at \(t=1\)? (Answer: The benefit is paid at \(t=0.6\); there is no in-force closing reserve for that terminated policy at \(t=1\).)