Why Timeline Alignment Matters
An integrated projection can contain several values for the same policy at the same duration: a policy account, units in an investment fund, an actuarial reserve, and an asset share. They are related, but they are not interchangeable. Each has its own definition, assumptions, and timing. A correct calculation can still give a misleading result if, for example, an account deduction is applied after rather than before the fund return, or an end-of-year benefit is treated as if it earned a full year of investment income.
The practical starting point is a dated sequence of events. State what is known at the start of the year, when premiums and deductions occur, when returns are credited, and when benefits and reserves are measured. The conventions below are for a hypothetical annual projection; other product terms may require a different sequence.
Write the Event Order Before the Equations
Under the stated timing, the account immediately before the end-of-year benefit is
The superscript minus indicates the value just before an end-of-year payment. If the benefit is funded from the account up to its available amount, the amount of any additional insurer payment is \(\max(0,B_{t+1}-A_{t+1}^{-})\). That is only the top-up under this particular benefit definition; product terms may define the benefit differently. The account after a claim also depends on the contract terms, so it should not be inferred just from this equation.
An asset share is an accumulation measure, not a policy account or a reserve. To make its timing explicit, suppose premiums, expenses, and charge receipts are all at the start of the year, the benefit is at year-end, and the asset-share accumulation rate is \(j_t\). For the simplified convention used here,
This convention treats \(C_t\) as a receipt in the modeled asset-share cash flows and counts it once. In an actual projection, the asset-share definition and cash-flow perimeter must be stated: an internal transfer or a fund return already represented elsewhere must not be counted again. The rate \(j_t\) is the asset-share accumulation assumption; it need not equal the policy fund return \(r_t\).
A reserve is a liability value on a specified valuation basis. Its change over the year is \(V_{t+1}-V_t\), but that change alone is not a profit measure. For a simplified reconciliation, the movement in modeled surplus \(AS-V\) is
This identity follows by substituting the asset-share recursion and collecting the opening asset-share investment gain, the accumulated start-of-year net cash flows, the end-of-year benefit, and the reserve change. It is a reconciliation on the stated simplified basis, not a universal accounting-profit formula. Different asset-share conventions or cash-flow timing require a corresponding change in the equation.
Record whether each value is measured immediately before or after transactions at the policy anniversary.
Mark premiums, expenses, deductions, returns, benefits, and reserve measurements as start-, during-, or end-of-year events.
Use the account rule for the account, the stated accumulation rule for the asset share, and the valuation basis for reserves.
Compare values at matching dates and check that each cash flow or return has been included once in the relevant measure.
Worked Timeline Calculations
Worked Example: Account Deduction Before the Fund Return
A hypothetical contract has an opening account of $1,000. At the start of the year, a premium of $200 is credited, an expense of $20 is deducted from the account, and a risk charge of $30 is deducted. The account earns a 5% annual fund return. At year-end, the specified benefit is $1,500 and is funded by the account balance plus any insurer top-up. Find the pre-benefit account and the top-up. Use the stated start-of-year deduction and end-of-year benefit timing.
Method. Apply the account transactions in their stated order, then apply the fund return to the balance remaining after the deductions.
After the premium, expense, and charge, the amount invested is
The pre-benefit account is
The insurer top-up is therefore
Check. The account amount plus the top-up equals the stated benefit: \(\$1{,}207.50+\$292.50=\$1{,}500.00\). A common mistake is to calculate \(1{,}000(1.05)+200-20-30\), which gives $1,200. That incorrectly gives the start-of-year premium no investment return. The stated timing requires all four account transactions to be combined before the return.
Worked Example: Two-Year Asset-Share Projection
A hypothetical policy has an initial asset share of $1,000. In year 1, a premium of $120 and expense of $20 occur at the start of the year; the asset-share accumulation rate is 5%; and a benefit of $200 is paid at year-end. In year 2, the start-of-year premium is $80, the expense is $10, the accumulation rate is 4%, and the year-end benefit is $300. Find the asset share at the end of each year under the stated convention.
Method. Accumulate the opening asset share and start-of-year net cash flow for the full year, then subtract the year-end benefit.
For year 1, the starting amount after the premium and expense is
After 5% accumulation, that amount is \(\$1{,}100(1.05)=\$1{,}155\). Subtract the year-end benefit:
For year 2, the starting amount after the premium and expense is
After 4% accumulation, it is \(\$1{,}025(1.04)=\$1{,}066\). Subtract the year-end benefit:
Check. The year-2 opening value is the year-1 closing value, $955, not the original $1,000. The asset share falls from $955 to $766 after the second year’s net cash flows and benefit. The calculation has not applied investment return to either year-end benefit.
Worked Example: Aligning Account, Asset Share, and Reserve
For one hypothetical year, the opening account is $500 and the account receives a $100 premium, has a $10 expense and a $20 account deduction at the start of the year, and earns a 6% fund return. The year-end benefit is $650, funded by the account plus an insurer top-up. Separately, the opening asset share is $800; its start-of-year cash flows are a $100 premium, $10 expense, and $20 charge receipt; its accumulation rate is 4%; and the same $650 benefit is paid at year-end. The reserve is $820 at the start and $840 at the end. Calculate the pre-benefit account, top-up, closing asset share, and change in modeled surplus \(AS-V\).
Method. Project the account and asset share using their own rates and cash-flow rules. Then compare asset share with reserve at matching endpoints. The reserve values are supplied; this example does not derive them.
The account balance after start-of-year transactions is
After the 6% fund return, the pre-benefit account is \(\$570(1.06)=\$604.20\). The insurer top-up is
For the asset share, the start-of-year net amount is \(800+100-10+20=\$910\). Accumulating at 4% gives \(\$910(1.04)=\$946.40\); after the benefit,
The opening modeled surplus is \(AS_0-V_0=800-820=-\$20\). The closing modeled surplus is \(AS_1-V_1=296.40-840=-\$543.60\). Its change is
Check by decomposition. The opening asset-share return is \(800(0.04)=\$32\). The start-of-year net cash flows, after accumulation, contribute \((100-10+20)(1.04)=\$114.40\). Subtract the benefit of $650 and the reserve increase of \(840-820=\$20\):
The two surplus calculations agree. The account’s 6% fund return and the asset share’s 4% accumulation rate have distinct roles; substituting one for the other would change the projection.
Common Timing Traps
The key discipline is to keep each amount attached to both a ledger and a date. The account deduction reduces the account at its specified time; it is not automatically an extra benefit payment. The account’s fund return applies to the balance invested during the period, while the asset-share return applies according to the asset-share convention. A reserve change is a liability movement, not a cash payment. Mixing these items can create a double count or omit an investment period.
| Item | Timeline question | Check |
|---|---|---|
| Premium | Is it received at the start, during, or end of the year? | Apply investment return only for the period it is held. |
| Account deduction | Is it taken before or after the account return? | Use the account balance immediately after the deduction as the invested balance if taken before return. |
| Benefit | Is it paid before or after the period’s return? | Do not give an end-of-year payment a full year’s return. |
| Reserve | At which dates are the opening and closing reserves measured? | Subtract closing minus opening reserve only after matching the dates. |
| Asset share | Which cash flows and accumulation basis define it? | State the convention and avoid counting internal transfers twice. |
These checks are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: integrated problems often test whether the candidate can preserve the timing and meaning of several projected quantities at once. A compact event table is often enough to expose a misplaced return or a mismatched valuation date before the arithmetic begins.
Check Your Understanding
Use the stated timing in each question. All amounts are hypothetical and in dollars.
- An account is $700 before start-of-year transactions. A $100 premium is credited, a $15 expense and a $25 deduction are taken, and the account earns 3%. Find the pre-benefit account. (Answer: $782.80)
- An asset share is $600 at the start of the year. A $80 premium and $10 expense occur at the start, the annual accumulation rate is 2%, and a $150 benefit is paid at year-end. Find the closing asset share. (Answer: $533.40)
- An asset share is $500 and a reserve is $450 at the start of the year. A $100 premium, $10 expense, and $5 charge receipt occur at the start; the asset-share rate is 4%; a $200 benefit is paid at year-end; and the closing reserve is $430. Find the change in \(AS-V\). (Answer: \(-\$61.20\))
- An account of $900 receives a $120 premium and has a $10 expense and $30 deduction at the start of the year. It earns 4%; the year-end benefit is $1,100 and is funded by the account plus a top-up. Find the pre-benefit account and top-up. (Answer: pre-benefit account $1,019.20; top-up $80.80)
- An asset share is $1,000. A $200 premium and $25 expense occur at the start of the year; the annual accumulation rate is 3%; and the year-end benefit is $300. The reserve changes from $950 to $920. Find the closing asset share and the change in \(AS-V\). (Answer: closing asset share $910.25; change in \(AS-V=-\$59.75\))