Why Acquisition Costs Matter at Issue
Commissions and other acquisition expenses are often concentrated at the start of a long-term policy. Premium income may be received at issue as well, but the net amount available to earn interest can still be small or negative. A profit test must therefore keep the timing of each cash flow clear: an issue expense paid at the beginning of the year is not accumulated in the same way as a claim paid at year-end.
An opening reserve can provide funds to support these initial outgoes. Its effect is not simply to “cancel” the acquisition expense: under an annual model, the reserve and issue cash flows accumulate with interest, while claims and the closing reserve are valued at the end of the year. The calculations below use a one-year discrete-time model and state the timing assumptions explicitly. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
This form assumes that the premium and the issue expenses are paid at the beginning of the year, that interest is earned on the resulting net balance, and that death benefits are paid at year-end. The survivor expense and closing reserve apply only to policies surviving to year-end. If a cash flow is paid at a different time, or if a benefit is paid at the moment of death, its accumulation or discounting must reflect that timing instead.
Cash Profit and Reserve-Adjusted Profit
Cash profit records the expected accumulated cash available after the year’s benefits, before setting aside a closing reserve. With the timing above, cash profit is \((V_0+P-C-M_0)(1+i)-qB-pE_1\). Profit after providing the closing reserve is lower by \(pV_1\), because that reserve is required only for survivors. This distinction makes it possible to see whether a first-year loss is caused by immediate cash outgoes, by the reserve requirement, or by both.
The general formula also shows how an acquisition cost affects first-year profit. Holding all other terms fixed, an extra issue cost \(\Delta C\) reduces profit by \((1+i)\Delta C\), because the cost is paid at the beginning and would otherwise have earned interest for the year. The same reasoning applies to an extra issue expense \(\Delta M_0\).
Worked Example: Measuring First-Year Acquisition Strain
A hypothetical one-year profit test assumes a 4% effective annual interest rate. Each policy receives a $500 premium at issue, pays a commission of $125 and an additional acquisition expense of $80 at issue, and pays a $15 maintenance expense at issue. The death probability is 0.0100, the year-end death benefit is $20,000, and the closing reserve is $200 for each survivor. The opening reserve is zero. Calculate cash profit and profit after the closing reserve.
Method. Apply the annual profit formula, putting the premium and all issue-time expenses into the opening balance before accumulating interest. The survivor reserve is then weighted by the probability of survival.
The total issue-time outgo is \(125+80+15=220\). Thus the amount accumulated is
The probability of survival is \(p=1-0.0100=0.9900\). The expected death benefit is \(0.0100(20{,}000)=200.00\). Therefore cash profit, before the closing reserve, is
The expected closing reserve is \(0.9900(200)=198.00\). Profit after providing that reserve is
The check is \(291.20-200.00-198.00=-106.80\), agreeing with the direct calculation. The first-year cash result is positive, but the reserve requirement makes reserve-adjusted profit negative. Common mistake avoided: subtracting the full $200 closing reserve rather than the expected reserve of $198, which applies only to survivors.
The Opening Reserve Needed for Zero First-Year Profit
In some calculations, the opening reserve is treated as an unknown and is chosen to make first-year profit equal to a target, such as zero. This is a one-year break-even calculation under the stated profit-test assumptions. It does not imply that a prescribed valuation reserve must equal that amount: an actual reserve may be set on a different basis or for a different purpose.
To find the break-even opening reserve, set \(\Pi_1=0\) and solve the profit equation for \(V_0\). Keeping the timing in the definition gives
The expected year-end outgoes are divided by \(1+i\) because the accumulated opening balance must equal those outgoes. The issue-time premium is subtracted, while issue-time expenses are added: a larger premium reduces the reserve needed, and larger acquisition costs increase it. The result can be negative. A negative mathematical break-even amount indicates that the other first-year cash flows already exceed expected year-end outgoes at \(V_0=0\); it is not automatically an allowable reserve.
Classify each item as paid at issue or at year-end, and identify which year-end amounts apply only to survivors.
Accumulate the opening reserve and issue-time net cash flow, then subtract expected year-end benefits, expenses, and closing reserves.
For zero first-year profit, set \(\Pi_1=0\) and isolate \(V_0\); then substitute the answer back into the original equation.
Worked Example: Solving for a Break-Even Opening Reserve
A hypothetical policy receives a $200 premium at issue and pays $50 of commission and acquisition expense at issue. The effective annual interest rate is 5%. The probability of death during the year is 0.0200, and the year-end death benefit is $4,000. Each survivor incurs a $15 year-end expense and requires a $100 closing reserve. Find the opening reserve that makes expected first-year profit zero.
Method. Use survival weighting for both year-end survivor items, then solve the reserve-adjusted profit equation. Here \(p=1-0.0200=0.9800\), so
The expected death benefit is \(0.0200(4{,}000)=80.00\). The expected survivor expense is \(0.9800(15)=14.70\), and the expected closing reserve is \(0.9800(100)=98.00\). Total expected year-end outgoes are therefore
Set profit equal to zero:
so
To the nearest cent, the break-even opening reserve is $33.52. Check using the unrounded reserve: \(1.05(33.5238095\ldots+150)=192.70\), so calculated profit is zero, apart from displayed rounding. Common mistake avoided: solving after omitting the issue-time $50 cost or failing to multiply the survivor expense and closing reserve by 0.9800.
How the Strain Can Change in Later Durations
Acquisition costs are often concentrated at issue, so they do not recur in the same form in later policy years. A later duration can therefore show positive profit even when the first year showed a loss. That change does not mean the acquisition cost has been reversed: it means later-year premiums, claims, expenses, and reserve movements produce a different result.
For each duration, the profit formula must use the opening reserve for that duration and the closing reserve for survivors at the end of that duration. A duration’s profit is usually stated per policy in force at its start. If profits are combined across a cohort measured from issue, later-duration profit must also be weighted by the probability that a policy remains in force to that duration. The following example separates those two statements.
Worked Example: First-Year Strain and Later-Year Profit
A hypothetical two-year contract has a 5% effective annual interest rate and a $300 premium at the beginning of each year. At issue, it incurs a $250 acquisition cost and a $20 expense. The opening reserve at issue is zero. In year 1, the death probability is 0.0100, the year-end death benefit is $10,000, and the closing reserve for each survivor is $100. In year 2, the death probability is 0.0200, the year-end death benefit remains $10,000, and the closing reserve is zero. A $20 expense is paid at the beginning of year 2. Calculate profit per policy in force at the start of each year, then weight year 2 profit back to the original cohort.
Method. Apply the annual reserve-adjusted profit formula separately by duration. The issue cost appears only in year 1; the year 2 result is conditional on being in force at the start of year 2.
For year 1, the amount accumulated from issue is
The expected year 1 death benefit is \(0.0100(10{,}000)=100.00\), and the expected closing reserve is \(0.9900(100)=99.00\). Thus year 1 profit per policy in force at issue is
For year 2, the opening reserve is $100. The net issue-time cash flow for that year is \(100+300-20=380\), which accumulates to \(380(1.05)=399.00\). Expected year 2 death benefits per policy in force at the start of year 2 are \(0.0200(10{,}000)=200.00\). There is no year-end reserve. Therefore year 2 profit per policy in force at the start of that year is
The probability of surviving year 1 is 0.9900. Hence year 2 profit weighted per original policy issued is \(0.9900(199.00)=197.01\). The two duration profits, when expressed per original policy, sum to \(-167.50+197.01=29.51\). The check distinguishes the year 2 conditional profit of $199.00 from its cohort-weighted amount of $197.01. Common mistake avoided: adding duration profits directly without survival-weighting the later-year amount.
Interpretation and Common Exam Traps
An opening reserve increases profit by \((1+i)\) for each additional dollar of reserve under this timing model. An extra dollar of acquisition cost decreases profit by the same amount. Thus the reserve can offset acquisition strain in the profit test, but only after respecting the timing and the other expected outgoes. A zero-profit opening reserve is a model result, not a universal measure of the economic cost of acquiring a policy.
| Item | How it enters this model | Frequent error |
|---|---|---|
| Commission or acquisition expense at issue | Subtract from the opening balance before interest | Subtract it at year-end without accumulation |
| Death benefit at year-end | Subtract \(qB\) | Multiply by survival probability |
| Survivor expense or closing reserve | Subtract \(p\) times the amount | Apply it to every policy in force at the start |
| Later-duration profit | Calculate per policy in force at that duration | Combine across durations without cohort survival weighting |
A reliable check is to reconstruct the profit from its cash-flow timing: accumulate the opening reserve and beginning-of-year net cash flow, subtract expected death outgoes, then subtract survivor expenses and reserves. If solving for \(V_0\), substitute the result into that original equation. If comparing acquisition-cost scenarios, keep all other assumptions fixed; the difference should equal the extra issue cost accumulated at interest.
Check Your Understanding
For each problem, use the stated timing assumptions and distinguish a per-in-force result from a cohort-weighted result where required.
- A policy has \(V_0=0\), a $180 premium, $40 of issue expense, and a $10 expense also paid at issue. At 4% interest, \(q=0.0300\), the year-end death benefit is $3,000, and the closing reserve is $50 per survivor. Find first-year profit. (Answer: -$3.30)
- A policy receives a $250 premium and pays $60 of acquisition expense at issue. At 5% interest, \(q=0.0100\), the year-end death benefit is $5,000, and each survivor incurs a $12 year-end expense and has a closing reserve of $80. Find the opening reserve for zero first-year profit. (Answer: -$55.64)
- A hypothetical policy receives a $400 premium at issue. Commission is 15% of premium, and a separate $30 acquisition expense and $10 expense are paid at issue. At 3% interest, \(q=0.0200\), the year-end benefit is $6,000, and the closing reserve is $120 per survivor. With zero opening reserve, find first-year profit. (Answer: $71.40)
- All assumptions in a profit test are fixed except that the acquisition expense paid at issue increases by $75. The effective annual interest rate is 4%. By how much does first-year profit change? (Answer: It decreases by $78.00)
- A policy receives a $300 premium and pays $120 of acquisition cost plus $15 of other expense at issue. At 5% interest, \(q=0.0200\), the year-end death benefit is $8,000, and each survivor incurs a $20 year-end expense and requires a $150 closing reserve. Find the opening reserve for zero first-year profit. (Answer: $146.05)