Why Timing Matters
A policy-year surplus calculation depends on more than the amounts received and paid. A premium received at the start of a year can earn investment income throughout the year; a claim paid halfway through the year cannot. Similarly, an expense paid at the start of the year reduces the funds available to invest, while an expense paid at year-end does not reduce investment earnings earned earlier in that year. The closing reserve also matters: assets remaining after claims and expenses support both that reserve and any surplus above it.
The objective here is to trace the cash flows in chronological order and determine how much surplus emerges by the end of a policy year. The examples use hypothetical contracts and explicitly state their timing assumptions. Unless a problem says otherwise, assume a deterministic annual effective investment rate, no taxes or reinsurance, and no initial free assets beyond those needed to equal the opening reserve.
A Policy-Year Timeline and the Basic Equation
First consider a simple annual convention: opening assets are available at time 0; the premium \(P\) is received and expense \(E\) is paid at time 0; a claim \(B\) is paid at the end of the year; and the closing reserve \(V_1\) is measured immediately after that claim. The annual effective investment rate is \(i\). With no other cash flows, the amount invested during the year is \(A_0+P-E\), so the closing assets are
Substituting this expression into the definition of \(S_1\) gives the period surplus equation. The equation makes the timing explicit: the start-of-year cash balance earns interest for the full year, while the year-end claim does not.
When \(A_0=V_0\), the same formula applies. The term \(P-E\) is the net amount added at the start of the year; \(i(A_0+P-E)\) is the investment income on the funds invested; \(B\) is the claim paid; and \(V_1-V_0\) is the increase in the required reserve. A reserve decrease makes \(-(V_1-V_0)\) positive, all else equal.
Record opening assets and the opening reserve. If they are equal, there are no initial free assets to carry into the period.
Record when premiums arrive and when expenses and claims are paid. Do not assign a full year of interest to cash that arrives or leaves partway through the year.
Apply the investment return only to the balance held during each part of the year.
Subtract the closing reserve from closing assets, then compare with opening free assets to obtain the surplus that emerged during the year.
Worked Policy-Year Cash-Flow Examples
The annual equation is convenient when all premiums and expenses occur at the start of the year and claims occur at the end. If a cash flow happens during the year, divide the year into intervals and accumulate each interval's balance separately. Use the annual effective rate to determine the rate for each interval when needed.
Worked Example: Start-of-Year Premium and Expense, Year-End Claim
A hypothetical participating policy has opening assets of $1,000 and an opening reserve of $1,000. The insurer receives a $120 premium and pays $20 of expenses at the start of the year. It pays a $30 claim at year-end and holds a $1,120 closing reserve. The annual effective investment rate is 5%. Calculate the surplus emerging during the year.
Method. Apply the one-year accumulation to the opening assets plus the start-of-year premium less the start-of-year expense. The opening free assets are zero because \(A_0=V_0\).
The amount invested is \(1{,}000+120-20=\$1{,}100\). After one year at 5%, that amount is \(1{,}100(1.05)=\$1{,}155\). After paying the year-end claim, closing assets are \(1{,}155-30=\$1{,}125\).
Surplus is closing assets less the closing reserve: \(1{,}125-1{,}120=\$5\). As a check, the expanded period equation gives net start-of-year cash of \(120-20=\$100\), investment income of \(0.05(1{,}100)=\$55\), a claim of \(\$30\), and reserve growth of \(1{,}120-1{,}000=\$120\). Thus \(100+55-30-120=\$5\), matching the asset calculation.
Common mistake avoided: Do not count the $30 claim as if it were paid at the start of the year. Under the stated year-end timing, the full $1,100 earns interest before the claim is paid.
Worked Example: A Claim Paid Halfway Through the Year
A hypothetical policy has opening assets and reserve of $500. At the start of the year, the insurer receives a $100 premium and pays $20 of expenses. It pays a $50 claim exactly halfway through the year and holds a $620 closing reserve. The annual effective investment rate is 21%. Calculate the emerging surplus, using the equivalent half-year rate.
Method. Divide the year into two equal intervals. Since \((1.10)^2=1.21\), an annual effective rate of 21% corresponds to an effective half-year rate of 10%.
The amount invested at the start is \(500+100-20=\$580\). At midyear, it has accumulated to \(580(1.10)=\$638\). After the $50 claim is paid, the balance is \(638-50=\$588\). That balance earns 10% in the second half of the year, giving closing assets of \(588(1.10)=\$646.80\).
The surplus is \(646.80-620=\$26.80\). For a check that identifies the investment income, the first half earns \(638-580=\$58\), and the second half earns \(646.80-588=\$58.80\). Total investment income is \(\$116.80\). The net start-of-year cash is \(\$80\), the claim is \(\$50\), and the reserve increases by \(620-500=\$120\). Therefore \(80+116.80-50-120=\$26.80\), as calculated from closing assets.
If the claim were incorrectly treated as a year-end payment, closing assets would be \(580(1.21)-50=\$651.80\), and the calculated surplus would be \(\$31.80\). This overstates surplus by \(\$5\), which is the additional second-half investment income of \(50(0.10)=\$5\) that the wrongly timed claim appears to earn.
Worked Example: The Effect of Expense Timing
A hypothetical policy has opening assets and reserve of $2,000. The insurer receives a $300 premium at the start of the year, pays a $100 claim at year-end, and holds a $2,250 closing reserve. The annual effective investment rate is 5%. Compare the surplus if a $40 expense is paid at the start of the year with the surplus if that expense is paid at year-end.
Method. Calculate closing assets separately for each timing. The start-of-year expense reduces the balance invested for the full year. The year-end expense is paid after that balance earns investment income.
Expense paid at the start. The amount invested is \(2{,}000+300-40=\$2{,}260\). It accumulates to \(2{,}260(1.05)=\$2{,}373\). After the $100 claim, closing assets are \(2{,}373-100=\$2{,}273\). Surplus is \(2{,}273-2{,}250=\$23\).
Expense paid at year-end. The amount invested is \(2{,}000+300=\$2{,}300\), which accumulates to \(2{,}300(1.05)=\$2{,}415\). After both the $100 claim and the $40 expense, closing assets are \(2{,}415-100-40=\$2{,}275\). Surplus is \(2{,}275-2{,}250=\$25\).
The year-end expense produces $2 more surplus in this example. Both cases have the same total premium, expense, claim, and closing reserve, but the year-end expense leaves an additional $40 invested during the year. That generates \(40(0.05)=\$2\) more investment income. The asset calculations confirm the difference: \(\$25-\$23=\$2\).
Common mistake avoided: Do not use the same investment base for both expense timings. Applying 5% to \(\$2{,}300\) when the expense is actually paid at the start would incorrectly give that $40 a year of investment income.
Justifying the Cash-Flow Accumulation
The calculation method follows from accumulating each balance between the times when cash flows occur. With the simple annual timing convention, the time-0 balance after the premium and expense is \(A_0+P-E\). At the end of the year, that balance has grown by the factor \(1+i\), so it is \((A_0+P-E)(1+i)\). Paying the year-end claim leaves \((A_0+P-E)(1+i)-B\) in assets. Subtracting the closing reserve gives closing free assets.
To calculate the change in free assets rather than just closing free assets, subtract opening free assets, \(A_0-V_0\). Expanding the result separates the initial cash flows, investment income, claim, and reserve change:
This derivation depends on the stated timing. If a claim or expense is paid midyear, the one-step accumulation no longer describes the account correctly. Accumulate the balance to the payment date, make the payment, and then accumulate the remaining balance to year-end. The same principle applies to any additional cash flow: its investment effect depends on how long the funds are held.
Interpreting Surplus Emergence
Surplus for a policy year is the result of the entire financial position at the measurement date, not a measure of cash income alone. A positive premium margin can be offset by claims, expenses, investment losses, or an increase in reserves. Conversely, a reserve decrease can contribute positively to the period's measured surplus. These effects must be evaluated together, using consistent opening and closing measurement dates.
When applying the calculation to participating business, first determine the surplus that emerges under the stated assumptions and timing. Do not treat that amount automatically as a dividend or other distribution. The amount available for allocation and decisions about participation are separate questions. For this tutorial's purpose, surplus means the change in assets above the required reserve; no distribution is assumed.
The main exam trap is to calculate from annual totals while ignoring when those totals occur. Equal annual totals do not guarantee equal surplus: a start-of-year expense, a midyear claim, and a year-end expense produce different investment earnings. Write the timeline before using a formula, and check the final result by comparing closing assets with the closing reserve. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, track the timing of the cash flows before calculating surplus.
- A hypothetical policy begins with assets and reserve of $800. A premium of $90 is received and an expense of $10 is paid at the start of the year. The annual effective investment rate is 4%, a $20 claim is paid at year-end, and the closing reserve is $900. What surplus emerges? (Answer: \(-\$4.80\))
- A policy begins with assets and reserve of $1,200. At the start of the year it receives a $150 premium and pays $30 of expenses. The annual effective investment rate is 5%; a $40 claim is paid at year-end, and the closing reserve is $1,350. Calculate the surplus. (Answer: \(-\$4.00\))
- A policy begins with assets and reserve of $400. It receives an $80 premium at the start of the year, pays a $20 claim halfway through the year, and has a closing reserve of $500. The annual effective investment rate is 21%, with an equivalent half-year rate of 10%. Calculate the surplus. (Answer: \(\$58.80\))
- A policy begins with assets and reserve of $1,000, receives a $200 premium at the start of the year, and pays a $50 claim at year-end. The annual effective rate is 10%, and the closing reserve is $1,150. Compare surplus when a $20 expense is paid at the start of the year with surplus when it is paid at year-end. (Answer: Start-of-year expense: \(\$98\); year-end expense: \(\$100\); the year-end timing produces \(\$2\) more surplus.)
- A policy begins with assets and reserve of $700. At the start of the year it receives a $100 premium and pays a $20 expense. A $30 claim is paid halfway through the year, the annual effective investment rate is 21% (10% per half-year), and the closing reserve is $800. Calculate the surplus using the account balance at each half-year boundary. (Answer: \(\$110.80\))