Reserve Basis Consistency Checks

Introduction

A reserve-adjusted profit result is meaningful only when its pieces use a consistent basis. The projected premium and expenses must be placed at the intended times, the benefit probabilities must match the model being tested, and the reserve must be measured at the stated valuation time. A mismatch can create an apparent gain or loss even when the underlying cash flows have not changed.

This tutorial develops a practical consistency check for discrete annual profit projections. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The aim is not to compare alternative reserve methods, but to verify that the assumptions and timing within a given projection fit together.

The Reserve Recurrence and Profit Check

Consider a hypothetical policy at the start of year $t$, before that year's premium and expense are paid. Let $V_t$ be the reserve at that time. Suppose the premium $P_t$ is received and the expense $e_t$ is paid immediately. During the year, the policyholder dies with probability $q_t$ and survives with probability $p_t=1-q_t$. A death benefit $B_{t+1}$ is paid at year-end on death. On survival, the insurer holds reserve $V_{t+1}$, measured at the start of the next year, before the next premium and expense. Let $j$ be the effective annual interest rate used to accumulate funds over the year.

Definition: Under this timing convention, the prospective reserve recurrence is $$ V_t=-P_t+e_t+\frac{q_tB_{t+1}+p_tV_{t+1}}{1+j}. $$ It applies when the premium and expense occur at the start of the year, the benefit and next reserve are measured at year-end, and the valuation basis uses $q_t$, $p_t$, and $j$. Any other cash flows must be added at their actual payment times.

The recurrence says that the reserve, together with the net amount received at the start of the year, must fund the expected year-end claim and the reserve required for survivors. Equivalently, multiplying through by $1+j$ gives the reserve-balance check:

(V_t+P_t-e_t)(1+j)=q_tB_{t+1}+p_tV_{t+1}.

The left side is the year-end value of funds available under the valuation interest assumption. The right side is expected year-end outgo: death claims plus the reserve held for survivors. This check assumes no other cash flows, no initial surplus or deficit beyond $V_t$, and that the same assumptions are used to value the reserve and to calculate the expected profit. If the projection uses a different investment return or different transition probabilities, the expected profit need not be zero.

Why a Matching Basis Produces Zero Expected Profit

Define year-end profit per policy in force at the start of the year as the accumulated funds less the claim paid on death and less the reserve set up for a survivor. On death, the claim is paid; on survival, the next reserve is held. Taking the expectation over the two outcomes gives the following expression under the model assumptions:

\begin{aligned} E[\text{year-end profit}] &=(V_t+P_t-e_t)(1+j)-q_tB_{t+1}-p_tV_{t+1}. \end{aligned}

Substitute the reserve-balance equation into this expression. The accumulated funds equal expected claims plus the expected survivor reserve, so the difference is zero. This is a direct consistency result: a reserve calculated from the same cash flows, probabilities, and interest rate as the profit projection balances that projection in expectation. It does not say that actual profit in either outcome must be zero; it says only that expected profit under the assumed model is zero before any deliberate margins or other items are introduced.

Write down the timing.
State whether each premium, expense, benefit, and reserve is at the start or end of the year.

Separate outcomes.
List the possible transitions and the probability and year-end cash flow for each.

Check the reserve equation.
Accumulate beginning funds at the valuation rate and compare them with expected year-end claims and continuation reserves.

Interpret any residual.
A nonzero expected profit may reflect a genuine basis difference, or an inconsistent assumption, timing convention, or calculation.

Worked Checks

Worked Example: Checking the One-Year Reserve Arithmetic

A hypothetical one-year contract receives a premium of $60 and incurs an expense of $10 at the start of the year. The death probability is 0.20, the year-end death benefit is $400, and the terminal reserve is zero. The valuation interest rate is 25% effective annually. Find the reserve before the premium and verify expected profit on the matching basis.

Method. Apply the reserve recurrence using the stated start-of-year net cash flow and the year-end expected benefit. Here $P_0=60$, $e_0=10$, $q_0=0.20$, $p_0=0.80$, $B_1=400$, $V_1=0$, and $j=0.25$.

\begin{aligned} V_0 &=-60+10+\frac{0.20(400)+0.80(0)}{1.25}\\ &=-50+\frac{80}{1.25}\\ &=-50+64\\ &=14. \end{aligned}

Thus the reserve before the premium is $14.00. For the profit check, funds at the start of the year after the premium and expense are $14+60-10=64$. Accumulated at 25%, they equal $64(1.25)=80$. Expected claims and the terminal reserve equal $0.20(400)+0.80(0)=80$. Therefore expected profit is $80-80=\textbf{$0.00}$.

Common mistake avoided. The quotient $80/1.25$ is $64, not $74. Adding it to $-50$ gives a reserve of $14, not $24. Using $24 would give year-end funds of $(24+60-10)(1.25)=92.50$, and expected profit of $92.50-80=\textbf{$12.50}$, not zero. The wrong reserve fails the balance check.

Worked Example: Separating Valuation and Projection Interest

A hypothetical one-year policy has a start-of-year premium of $60 and expense of $10. The death probability is 0.20, the death benefit is $200 at year-end, and the terminal reserve is zero. The reserve is calculated using a valuation interest rate of 5%, but the profit projection assumes an actual investment return of 3%. Find the reserve and the expected year-end profit, assuming mortality is unchanged.

Method. First calculate the reserve using the valuation rate, not the projection return. Then calculate profit using the return actually assumed in the projection. Here the expected claim is $0.20(200)=40$, while the expected terminal reserve is zero.

\begin{aligned} V_0 &=-60+10+\frac{40}{1.05}\\ &=-50+38.095238\ldots\\ &=-11.904762\ldots \end{aligned}

The reserve is $\textbf{-$11.90}$ to the nearest cent. For the projection profit, funds after the premium and expense are $V_0+60-10=V_0+50=38.095238\ldots$. At the projected 3% return, their year-end value is $38.095238\ldots(1.03)=39.238095\ldots$. Expected year-end outgo is $40, so expected profit is $39.238095\ldots-40=-0.761905\ldots$, or -$0.76 to the nearest cent.

As a check, if the projection instead used the 5% valuation rate, the year-end funds would be $38.095238\ldots(1.05)=40$, exactly matching expected claims. The profit difference arises because the projection return is 3%, while the reserve calculation uses 5%; it is not an arithmetic failure of the reserve recurrence.

Common mistake avoided. Using 3% to calculate the reserve would silently change the valuation basis. The reserve and projected profit answer different questions here: the reserve is defined at 5%, while the profit test measures what happens when funds earn 3%.

Worked Example: Checking Survival Weighting and Year Boundaries

A hypothetical two-year contract charges a premium of $30 and incurs an expense of $5 at the start of each year while the policy is in force. Death benefits of $200 are paid at the end of the year of death. The death probabilities are 0.10 in year 1 and 0.20 in year 2. There is no terminal reserve after year 2, and the effective annual valuation interest rate is 0%. Find the reserves before the start-of-year cash flows in both years and check expected profit in year 1.

Method. Work backward from the terminal reserve. At 0% interest, discounting does not change amounts, but survival weighting still applies: the year 2 reserve is required only if the policy survives year 1.

\begin{aligned} V_2&=0,\\ V_1&=-30+5+0.20(200)+0.80(0)=15,\\ V_0&=-30+5+0.10(200)+0.90(15)\\ &=-25+20+13.5\\ &=8.5. \end{aligned}

The reserves are $V_1=\textbf{$15.00}$ and $V_0=\textbf{$8.50}$. For the year 1 check, the funds after the premium and expense are $8.50+30-5=33.50$. With zero interest, year-end funds remain $33.50. Expected year-end claims are $0.10(200)=20$, and the expected reserve held for survivors is $0.90(15)=13.50$. Their sum is $33.50, so expected year 1 profit is $0.00.

Common mistake avoided. The next reserve is conditional on survival, so its contribution to expected year-end outgo is $0.90(15)$, not $15$. The reserve itself is $V_1=15$; the amount included in the year 1 expected balance is survival-weighted.

Common Consistency Traps

A short audit of the assumptions often identifies a discrepancy before a full projection needs to be rebuilt. Check each item against the reserve definition and the profit calculation rather than relying on labels such as “year 1 reserve” or “annual profit.”

Mixing beginning- and end-of-year reserves. In the recurrence above, $V_t$ is before the current premium and expense, while $V_{t+1}$ is at the next year boundary. Moving either reserve across a cash-flow date changes the equation.
Forgetting survival weighting. A continuation reserve is required only for policies that reach the next year. Its expected contribution is $p_tV_{t+1}$, not $V_{t+1}$.
Using different interest rates without explaining the residual. A valuation rate belongs in the reserve recurrence. The projection return belongs in projected asset accumulation. If they differ, a nonzero expected profit can be appropriate.
Discounting a start-of-year cash flow as though it were paid at year-end. The recurrence assumes $P_t$ and $e_t$ occur at the start of the year. Their net amount is accumulated for the full year.
Comparing totals with different exposure bases. A per-policy expected profit conditional on being in force at the start of the year is not the same quantity as a portfolio total unless the starting number of policies is also applied.

Key takeaway: A reserve-adjusted profit check should reconcile the accumulated beginning funds with expected claims and the reserve required at the next time point. Confirm the timing, transition probabilities, interest assumptions, and exposure basis before interpreting a nonzero residual.

Check Your Understanding

For each problem, use the stated reserve timing: before the start-of-year premium and expense, with benefits and the next reserve at year-end. Round money to the nearest cent.

A one-year contract has premium $50, expense $8, death probability 0.15, year-end death benefit $240, terminal reserve $12, and valuation interest 4%. Find the reserve before the premium. (Answer: $2.42)
A one-year contract has premium $60, expense $10, death probability 0.10, benefit $500, terminal reserve $20, and valuation interest 5%. The projection return is 3%, with the same mortality assumption. Find the valuation reserve and expected projected year-end profit. (Answer: Reserve $14.76; expected profit -$1.30)
A one-year contract has premium $30, expense $4, death probability 0.25, benefit $160, terminal reserve $10, and valuation interest 2%. Find the reserve before the premium and state the expected profit on the matching basis. (Answer: Reserve $20.57; expected profit $0.00)
A one-year contract has premium $45, expense $5, death probability 0.20, benefit $150, terminal reserve $25, and valuation interest 5%. Find the reserve. In the reserve-balance check, what amount represents the expected survivor reserve? (Answer: Reserve $7.62; expected survivor reserve $20.00)
A two-year contract has a premium of $40 and expense of $4 at the start of each year while in force, year-end death benefits of $240, death probabilities 0.10 in year 1 and 0.25 in year 2, terminal reserve zero, and valuation interest 0%. Find $V_1$ and $V_0$. (Answer: $V_1=$$24.00; $V_0=$$9.60)

What You'll Learn