Why Asset-Share Errors Need a Reconciliation
An asset-share projection can look plausible even when its cash-flow timing or policy-count basis is wrong. A premium entered at year-end instead of year-beginning earns too little investment return; dividing an aggregate amount by the opening policy count instead of the surviving count gives the wrong per-policy result. A sign error or a duplicated expense can also shift the answer without making the arithmetic itself look unusual.
A reliable diagnosis begins by writing each cash flow with its amount, sign, timing, and applicable number of policies. The examples below use hypothetical contracts and experience. They focus on checking calculations, not on valuing capital or solvency positions. This material is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
This formula is not a universal cash-flow template: a cash flow at another time must be accumulated only for the period it is actually held, and a benefit paid at a different time needs its own timing adjustment. The policy count must also match the cash flow. For example, a death benefit is multiplied by the number of deaths, whereas a charge applying only to survivors is multiplied by the number of survivors.
A Practical Audit Sequence
Before recalculating the final answer, create a short ledger. Use positive signs for cash entering the fund and negative signs for cash leaving it. Record whether each flow occurs at the beginning or end of the projection year, and identify the policy count to which it applies. Then apply investment return only to amounts held during the investment period.
Decide whether the target is an aggregate asset share or a per-policy asset share, and state the policy count at the opening and closing dates.
Show premiums as inflows and expenses and benefits as outflows. Do not subtract a deduction twice merely because it appears in both a net-premium description and a separate expense line.
For an effective annual return \(i\), a beginning-of-year amount is multiplied by \(1+i\) to reach year-end. A year-end amount is not multiplied by \(1+i\) for that year.
Apply each total cash flow to the policies covered by it. If the requested result is per survivor, divide by the closing survivor count, not the opening count.
Calculate the correct result independently, compare it with the questioned result, and state the numerical effect and cause of each error.
Why the Roll-Forward Formula Holds
The aggregate opening share is \(NA\). At the beginning of the year, each of the \(N\) policies contributes premium \(P\) and incurs expense \(e\), so the net beginning cash flow is \(N(P-e)\). These amounts are present for the whole year. The amount accumulated to year-end before year-end payments is therefore:
At year-end, \(D\) death benefits of \(B\) each are paid, for a total outflow \(DB\). The \(N-D\) survivors incur expenses of \(c\) each, for an outflow \((N-D)c\). Subtracting those outflows gives the aggregate amount remaining for survivors. Dividing by \(N-D\) converts that total to a per-survivor share. This last division is appropriate only if the requested quantity is per survivor; it is not part of the aggregate roll-forward itself.
Worked Examples: Finding the Source of an Error
Worked Example: Beginning Cash Flows Mistaken for Year-End Flows
A hypothetical policy has an opening asset share of $500. A premium of $1,000 and an acquisition expense of $200 are both paid at the beginning of the year. The effective annual investment return is 3%, and a benefit of $100 is paid at year-end. Find the correct year-end asset share, then quantify the error if the net beginning cash flow is mistakenly treated as a year-end amount.
Method. First accumulate the opening share and net beginning cash flow for the full year. The net beginning cash flow is \(1{,}000-200=\$800\). The amount before the year-end benefit is:
After paying the year-end benefit, the correct asset share is \(1{,}339-100=\$1{,}239\).
For the mistaken calculation, the opening share earns the return, but the $800 net beginning flow is incorrectly added at year-end. The result is \(500(1.03)+800-100=515+800-100=\$1{,}215\). The understatement is \(1{,}239-1{,}215=\$24\).
Check. The incorrectly timed $800 misses one year's return of \(800(0.03)=\$24\), exactly matching the understatement. The common mistake is to net beginning cash flows correctly but then omit them from the amount earning investment return.
Worked Example: Aggregate Share and Survivor Count
A hypothetical block has 80 policies, each with an opening asset share of $600. Each policy pays a $900 beginning premium and incurs a $100 beginning expense. The effective annual return is 5%. Two policies die at year-end; each death benefit is $3,000. Each of the survivors incurs a $25 year-end expense. Find the aggregate amount attributable to survivors and the asset share per survivor.
Method. The net beginning cash flow per policy is \(900-100=\$800\). Thus, the aggregate amount invested for the year is \(80(600+800)=80(1{,}400)=\$112{,}000\). After investment return, it is \(112{,}000(1.05)=\$117{,}600\).
The survivor count is \(80-2=78\). The total death benefits are \(2(3{,}000)=\$6{,}000\), and the total survivor expenses are \(78(25)=\$1{,}950\). The aggregate remaining for survivors is \(117{,}600-6{,}000-1{,}950=\$109{,}650\).
The asset share per survivor is \(109{,}650/78=\$1{,}405.7692\) approximately, or $1,405.77 to the nearest cent.
Check. Multiplying the unrounded per-survivor share by 78 returns $109,650. Dividing by the opening count of 80 instead would give $1,370.625, which is not the share per survivor. The common mistake is to confuse a valid aggregate total with the correct closing denominator.
Worked Example: Detecting a Duplicated Expense
A hypothetical policy has an opening asset share of $2,000. At the beginning of the year it receives a gross premium of $600 and incurs an expense of $80. The effective annual return is 4%. A separate maintenance expense of $30 is paid at year-end. Find the correct closing share and quantify the error if the beginning expense is subtracted twice.
Method. The beginning net cash flow is \(600-80=\$520\). The amount invested for the year is \(2{,}000+520=\$2{,}520\), which accumulates to \(2{,}520(1.04)=\$2{,}620.80\). Subtracting the year-end maintenance expense gives \(2{,}620.80-30=\$2{,}590.80\).
If the beginning expense is mistakenly deducted twice, the amount invested becomes \(2{,}000+600-80-80=\$2{,}440\). It accumulates to \(2{,}440(1.04)=\$2{,}537.60\), and after the $30 year-end expense the incorrect result is $2,507.60. The understatement is \(2{,}590.80-2{,}507.60=\$83.20\).
Check. One duplicate $80 deduction at the beginning reduces the closing amount by \(80(1.04)=\$83.20\), because the incorrectly removed amount would have earned 4% over the year. The common mistake is to use a premium already described as net of expenses and then subtract the same expense again; in this example, the stated premium is gross, so the separate $80 expense is deducted once.
Worked Example: Year-End Cash Flows Do Not Earn a Full Year of Return
A hypothetical policy has an opening share of $1,000. The effective annual return is 6%. At year-end, it receives a premium of $400, incurs an expense of $40, and pays a benefit of $100. Find the correct closing share and the overstatement if all three year-end cash flows are incorrectly accumulated for a full year.
Method. The opening share accumulates to \(1{,}000(1.06)=\$1{,}060\). The net year-end cash flow is \(400-40-100=\$260\), which is added without a further year's accumulation. The correct closing share is \(1{,}060+260=\$1{,}320\).
If all flows are incorrectly treated as beginning-of-year amounts, the calculation is \((1{,}000+400-40-100)(1.06)=1{,}260(1.06)=\$1{,}335.60\). The overstatement is \(1{,}335.60-1{,}320=\$15.60\).
Check. The net year-end cash flow is $260, and the erroneous extra return is \(260(0.06)=\$15.60\). The common mistake is to apply the annual accumulation factor to every line simply because all the lines appear in the same annual projection.
Diagnostic Checks and Common Traps
The most useful checks compare the form of the result with the structure of the cash flows. A cash flow moved from beginning to year-end changes the investment accumulation by its amount multiplied by the return it would otherwise have earned. A duplicated beginning expense reduces the closing share by the duplicated expense accumulated to year-end. A wrong denominator changes the per-policy result but does not change the aggregate total already calculated.
| Diagnostic question | What to verify | Typical error signal |
|---|---|---|
| Are the signs consistent? | Premiums enter as inflows; expenses and benefits leave as outflows. | An expense increases the projected share. |
| Is timing explicit? | Beginning amounts earn the year's return; year-end amounts do not. | Every cash flow is multiplied by \(1+i\), regardless of timing. |
| Does each flow use the right count? | Use deaths for death benefits and survivors for survivor-only charges. | A total benefit or expense is multiplied by the opening count without checking who incurs it. |
| Is the requested basis clear? | Keep aggregate totals distinct from per-policy amounts. | The aggregate is divided by the wrong closing or opening count. |
| Is each deduction included once? | Trace every expense from the source data into the ledger. | A stated net amount is reduced again by an expense already removed. |
A useful independent check is to reconstruct the result from an aggregate ledger even when the requested answer is per policy. Compute the invested aggregate, subtract each correctly counted year-end outflow, and only then divide by the relevant policy count. This keeps per-policy arithmetic from hiding errors in totals or counts.
Check Your Understanding
For each hypothetical projection, identify the correct amount and use the stated timing and policy count. Amounts are in dollars unless noted.
- A policy opens with an asset share of $700. A $500 premium and a $50 expense occur at the beginning of the year. The effective annual return is 4%, and a $120 benefit is paid at year-end. Find the closing share. (Answer: $1,076.00)
- A block of 50 policies has an opening share of $400 per policy. Each pays a $500 beginning premium and incurs a $200 beginning expense. The return is 3%. Two policies die at year-end, with a $1,000 benefit per death; each survivor incurs a $10 year-end expense. Find the share per survivor. (Answer: $699.38)
- A policy opens with $1,500. A $250 premium and a $50 expense occur at the beginning of the year. The annual return is 4%, and a $20 expense occurs at year-end. Find the closing share. (Answer: $1,748.00)
- A block of 100 policies has an opening share of $900 each. Each pays a beginning premium of $300 and incurs a beginning expense of $60. The return is 5%. Four policies die at year-end, each with a $5,000 benefit; each survivor incurs a $25 year-end expense. Find the aggregate amount attributable to survivors and the share per survivor. (Answer: $97,300 aggregate; $1,013.54 per survivor)
- A policy opens with $2,500, receives a $700 beginning premium, and incurs a $100 beginning expense. The return is 4%, and a separate $30 expense occurs at year-end. Find the correct closing share and the understatement if the beginning expense is deducted twice. (Answer: $3,194.00; understatement $104.00)