Diagnosing Reserve Timing Errors

Why Timing Must Be Explicit

A profit test can use reasonable assumptions and still produce an incorrect result if its reserve and cash-flow dates are inconsistent. An opening reserve supports the policy at the start of a projection year; a closing reserve is the liability held at the end of that year. They generally differ. Similarly, a benefit payable at year-end does not earn investment return during that year, while a beginning-of-year premium does. Treating these amounts as though they occurred on the same date changes the reported profit.

This tutorial uses an annual projection with a beginning reserve, stated cash-flow dates, and a required end-of-year reserve for continuing policies. All examples are hypothetical. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: The opening reserve $R_t$ is the reserve per policy in force at the beginning of projection year $t$. The closing reserve $R_{t+1}$ is the reserve per continuing policy required at the end of that year. If a proportion $p_t$ of policies continue and the remaining proportion exits with no additional reserve liability, the expected closing reserve for the original cohort is $p_tR_{t+1}$. A benefit payable at the end of the year is deducted as an end-of-year cash flow, not accumulated or discounted across the year.

For the annual examples below, premiums and expenses explicitly described as beginning-of-year flows are available at the start of the year. They, together with $R_t$, earn the full annual effective interest rate $i$. End-of-year benefits are paid after that accumulation. The profit measure is the accumulated fund after cash flows, less the required closing reserve for continuing policies. Other timing conventions are possible, but the dates and accumulation factors must be stated and used consistently.

Profit formula for the stated convention: Suppose a proportion $q_t$ exits during the year, with $p_t=1-q_t$, and each exiting policy receives benefit $B_t$ at year-end. If there is no reserve liability for an exited policy after that payment, and $P_t$ and $E_t$ are beginning-of-year premium and expense per opening policy, then expected year-end profit per opening policy is $$ \Pi_t=(R_t+P_t-E_t)(1+i)-q_tB_t-p_tR_{t+1}. $$

Why the Reserve in Profit Is the Closing Reserve

The reserve deduction in the formula represents the liability remaining at the end of the projection year. It is therefore based on the reserve appropriate at that date, applied only to policies that continue. The opening reserve has already entered the calculation as part of the assets carried into the year. Subtracting it again in place of the closing liability does not measure the end-of-year position.

The formula follows by tracing the expected fund forward from the beginning of the year. The opening reserve, premium, and expense are beginning-of-year amounts, so their net amount is $R_t+P_t-E_t$. Investment at annual effective rate $i$ makes this $(R_t+P_t-E_t)(1+i)$ at year-end. At that date, expected claim payments are $q_tB_t$, and the required reserve for continuing policies is $p_tR_{t+1}$. Subtracting both obligations gives the formula. This derivation depends on the stated dates: if a cash flow occurs at midyear, it cannot be accumulated for a full year under this convention.

Worked Example: Replacing the Closing Reserve with the Opening Reserve

A hypothetical block begins the year with a reserve of $50 per policy. Each policy pays a beginning-of-year premium of $12 and incurs a beginning-of-year expense of $2. The annual effective interest rate is 8%. During the year, 10% of policies make a year-end claim of $30 and exit; the other 90% continue and require a closing reserve of $45 each. Find the correct expected profit per opening policy, then calculate the result if the opening reserve is incorrectly used for continuing policies.

Method. First accumulate beginning-of-year funds for the full year. Then deduct the expected year-end claim and the expected closing reserve for continuing policies.

The beginning-of-year net amount is $$ 50+12-2=60. $$

At year-end, after interest, the fund is $$ 60(1.08)=64.80. $$

The expected claim payment is $$ 0.10(30)=3.00, $$ and the expected closing reserve is $$ 0.90(45)=40.50. $$

Therefore, correct expected profit is $$ 64.80-3.00-40.50=\$21.30. $$

If the opening reserve is incorrectly used as the reserve for continuing policies, the expected reserve deduction becomes $$ 0.90(50)=45.00. $$ The resulting incorrect profit is $$ 64.80-3.00-45.00=\$16.80. $$

The error understates profit by $21.30-16.80=\$4.50$, exactly $0.90(50-45)$. Common mistake avoided: using the opening reserve as though it were the closing liability. The correct reserve is the reserve required at the valuation date at year-end.

Year-End Benefits Must Stay at Year-End

When a claim is payable at the end of the year, it is deducted after investment accumulation on beginning-of-year funds. Deducting the claim before accumulation implicitly treats the payment as if it left the fund at the beginning of the year. At a positive interest rate, that wrong ordering also removes the investment return that would have been earned on the claim amount until year-end.

For a year-end benefit with expected amount $q_tB_t$, deducting it before accumulation rather than after accumulation changes the result by $i q_tB_t$, provided the other cash flows and reserve deductions are unchanged. This difference is a useful diagnostic: it is the interest on the expected benefit over the period by which it was moved.

Worked Example: Detecting a Benefit Deducted Too Early

A hypothetical block has an opening reserve of $80 per policy. A premium of $15 and an expense of $5 are both paid at the beginning of the year. The annual effective interest rate is 5%. Ten-year? No: during this one-year projection, 20% of policies claim a year-end benefit of $40 and exit; the remaining policies require a closing reserve of $50 each. Find the correct expected profit and the profit obtained by deducting the expected benefit before accumulating the fund.

Method. Use the stated one-year projection and preserve the year-end date of the benefit. The opening funds after the beginning-of-year premium and expense are $$ 80+15-5=90. $$

After one year of investment, these funds are $$ 90(1.05)=94.50. $$ The expected year-end claim payment is $$ 0.20(40)=8.00, $$ and the expected closing reserve is $$ 0.80(50)=40.00. $$ Thus the correct profit is $$ 94.50-8.00-40.00=\$46.50. $$

If the expected benefit is incorrectly deducted before investment accumulation, the fund at year-end is treated as $$ (90-8)(1.05)=82(1.05)=86.10. $$ After the same closing reserve deduction, the incorrect profit is $$ 86.10-40.00=\$46.10. $$

The understatement is $46.50-46.10=\$0.40$. The diagnostic calculation gives the same result: $$ i q_tB_t=0.05(8)=\$0.40. $$ Common mistake avoided: deducting a year-end claim before interest accumulation. The claim is paid at year-end, so it does not reduce the fund earning interest earlier in the year.

The phrase “deduct before accumulation” has a precise consequence: it assigns the benefit an earlier payment date. It is not merely an algebraic rearrangement. In a timing audit, write the date beside each flow before simplifying the calculation.

Midyear Cash Flows Need a Midyear Accumulation Factor

A midyear expense is not normally accumulated for a full year or treated as though it were paid at year-end. Under an annual effective rate $i$ with compound interest over fractional years, a payment at the midpoint accumulates for one-half year by the factor $(1+i)^{1/2}$. A beginning-of-year amount accumulates for a full year by $1+i$, and a year-end amount accumulates for zero years by 1.

Timing rule: Under annual effective interest $i$ and fractional-year compound accumulation, an amount $C$ paid at time $s$ years after the beginning of a one-year period has year-end value $C(1+i)^{1-s}$. In particular, the factors are $1+i$ at the beginning, $(1+i)^{1/2}$ at midyear, and 1 at year-end. This rule assumes that the effective rate applies consistently over fractional years; a different contractual crediting convention requires its own accumulation factors.

For example, if $i=21\%$, the half-year factor is $(1.21)^{1/2}=1.10$. A midyear expense of $4 therefore has year-end value $4.40. Treating it as a beginning-of-year expense gives it the full-year factor 1.21 and a year-end value of $4.84. The difference is $0.44, not $0.40: $$ 4(1.21-1.10)=\$0.44. $$

Worked Example: Correctly Accumulating a Midyear Expense

A hypothetical policy block has an opening reserve of $100 per policy and receives a beginning-of-year premium of $10. The annual effective interest rate is 21%. An expense of $4 is paid at midyear. At year-end, 10% of policies receive a benefit of $30 and exit; the other 90% continue and require a closing reserve of $80 each. Find the correct expected profit and the profit if the expense is mistakenly accumulated for a full year.

Method. Accumulate the opening reserve and premium for a full year, the midyear expense for one-half year, and the benefit for no time because it is paid at year-end.

The opening fund plus premium accumulates to $$ (100+10)(1.21)=110(1.21)=133.10. $$

Since $(1.21)^{1/2}=1.10$, the midyear expense has year-end value $$ 4(1.10)=4.40. $$ The expected year-end benefit is $$ 0.10(30)=3.00, $$ and the expected closing reserve is $$ 0.90(80)=72.00. $$

The correct expected profit is therefore $$ 133.10-4.40-3.00-72.00=\$53.70. $$

If the expense is incorrectly treated as a beginning-of-year payment, its accumulated value is $$ 4(1.21)=4.84. $$ The incorrect profit is then $$ 133.10-4.84-3.00-72.00=\$53.26. $$

The profit understatement is $53.70-53.26=\$0.44$, which equals $4(1.21-1.10)$. Common mistake avoided: assigning a midyear expense a full-year accumulation factor. The correct factor follows from the expense date and the specified annual effective rate.

A Practical Timing Audit

When a projected profit appears inconsistent with the cash flows or reserve basis, do not begin by changing assumptions. First reconcile dates. The following sequence separates reserve errors from cash-flow timing errors.

Label the valuation dates.
Write down the start and end of the projection period and identify whether each reserve is measured at the beginning or end.

Attach a date to every flow.
Mark premiums, expenses, benefits, and other payments as beginning, midyear, or year-end flows, using a more precise date if the model requires it.

Apply the matching accumulation factor.
For the stated fractional-year convention, use $1+i$, $(1+i)^{1/2}$, or 1 for beginning, midyear, or year-end amounts, respectively.

Apply the closing liability to the correct group.
Multiply the reserve per continuing policy by the proportion continuing. Add any separate obligations for exiting policies if the model specifies them.

Reconcile the error.
Compare the corrected calculation with the reported one. Attribute the difference to the specific reserve substitution or date shift, rather than treating it as an unexplained residual.

A useful check is to compare the year-end fund before obligations with the end-of-year liabilities being deducted. Every item in that comparison must be valued at the same date. A reserve error can be isolated as the difference between the correct and substituted closing-reserve deductions. A cash-flow timing error can be isolated by comparing the correct accumulation factor with the factor actually used.

Common Timing Traps

Incorrect treatment	Why it is wrong	Correction
Using $R_t$ for continuing policies at year-end	$R_t$ is the opening liability, not necessarily the liability required at the closing date.	Use $p_tR_{t+1}$ when only continuing policies hold the stated closing reserve.
Accumulating a year-end benefit with beginning-of-year funds	This moves the payment earlier and removes investment return that should have been earned.	Deduct the benefit at year-end, without a further accumulation or discount factor.
Accumulating a midyear expense for a full year	The cash leaves the fund halfway through the year, not at the beginning.	Use the half-year factor under the stated fractional-year interest convention.
Applying a reserve per continuing policy to the whole opening cohort	Policies that exit do not remain in the continuing group under the assumptions used here.	Multiply by the continuing proportion, and separately allow for any exit liability that applies.

These corrections depend on the model’s stated assumptions. If a claim is paid at a different time, a reserve remains payable after termination, or the fund follows a different crediting convention, the corresponding cash flows and liabilities must be valued on that basis instead. The general discipline is unchanged: state the dates, identify which policies remain liable, and value each item at the projection endpoint before combining them.

Key takeaway: A timing check is a date check. The opening reserve belongs to the beginning of the year; the closing reserve belongs at the end. A benefit or expense must be accumulated only from its actual payment date to the valuation date. When dates are aligned, the source and direction of most reserve-timing errors become visible.

Check Your Understanding

For each problem, use the stated payment dates and calculate profit per opening policy. Unless stated otherwise, exiting policies have no reserve liability after their year-end benefit.

A block has opening reserve $40, beginning premium $6, beginning expense $1, annual effective interest 5%, exit proportion 25%, year-end benefit $20 per exit, and closing reserve $30 per continuing policy. Find expected profit. (Answer: $19.75)
A block has opening reserve $120, beginning premium $10, beginning expense $2, annual effective interest 4%, exit proportion 5%, year-end benefit $60 per exit, and closing reserve $100 per continuing policy. Find expected profit. (Answer: $35.12)
A block has opening reserve $50 and beginning premium $10. The annual effective interest rate is 44%, so the half-year accumulation factor is 1.20. A $10 expense is paid at midyear. At year-end, 20% exit with a benefit of $20 and the continuing 80% require a closing reserve of $30. Find correct profit and the profit if the expense is incorrectly accumulated for a full year. (Answer: Correct profit $46.40; incorrect profit $44.00)
A block has opening reserve $70, beginning premium $8, beginning expense $3, annual effective interest 10%, exit proportion 10%, year-end benefit $50 per exit, and closing reserve $60 per continuing policy. Find correct profit and profit if the opening reserve is substituted for the closing reserve. (Answer: Correct profit $23.50; incorrect profit $14.50)
A block has opening reserve $200 and beginning premium $20. The annual effective interest rate is 21%. A $6 expense is paid at midyear. At year-end, 10% exit with a benefit of $100 and the continuing 90% require a closing reserve of $150. Find correct profit, then find the profit if the expense is accumulated for a full year. (Answer: Correct profit $114.60; incorrect profit $113.94)

Incorrect treatment	Why it is wrong	Correction
Using \(R_t\) for continuing policies at year-end	\(R_t\) is the opening liability, not necessarily the liability required at the closing date.	Use \(p_tR_{t+1}\) when only continuing policies hold the stated closing reserve.
Accumulating a year-end benefit with beginning-of-year funds	This moves the payment earlier and removes investment return that should have been earned.	Deduct the benefit at year-end, without a further accumulation or discount factor.
Accumulating a midyear expense for a full year	The cash leaves the fund halfway through the year, not at the beginning.	Use the half-year factor under the stated fractional-year interest convention.
Applying a reserve per continuing policy to the whole opening cohort	Policies that exit do not remain in the continuing group under the assumptions used here.	Multiply by the continuing proportion, and separately allow for any exit liability that applies.

What You'll Learn