Reconciling Cash-Flow Profit and Reserve Profit

Why Two Profit Calculations Should Agree

An annual profit projection can be described in two ways. A cash-flow calculation accumulates the opening reserve and the year's cash flows to the end of the year, then allows for claims and the reserve required for surviving policies. A reserve-profit calculation instead separates the investment gain on the opening reserve from the change in expected reserve. When both calculations use the same cash flows, timing, and reserve basis, they must give the same expected profit.

This reconciliation is useful because the two forms expose different errors. The cash-flow form makes it easier to check what is paid and when. The reserve-profit form makes it easier to see how a change in reserve affects profit. The calculations below use a simple annual model for a hypothetical death-benefit contract.

Definition: Consider a policy in force at the start of year $t$. Let $V_t$ be its opening reserve, $P$ its premium paid at the start of the year, and $e_0$ its expense paid at the start of the year. Let $i$ be the effective annual interest rate, $q$ the probability of death during the year, and $p=1-q$ the probability of survival. A death benefit $B$ and any end-of-year expense $e_1$ are paid at year-end. A policy surviving to year-end has reserve $V_{t+1}$; a policy that dies has zero terminal reserve after its death benefit is paid. The expected end-of-year profit, conditional on being in force at the start of the year, is $G_t$.

The model assumes that the start-of-year premium and expense are paid regardless of whether death occurs later in the year. Interest is earned at rate $i$ on the opening reserve and the net start-of-year cash flow. The death benefit and end-of-year expense do not earn interest in this model. The reserve $V_{t+1}$ applies to a survivor, not to a policy that has terminated on death. These timing conventions are part of the model, not details that can be changed without changing the formula.

The Cash-Flow Identity

At the start of the year, the insurer has the opening reserve $V_t$, receives $P$, and pays $e_0$. The amount accumulated to year-end is therefore $(V_t+P-e_0)(1+i)$. At year-end, the insurer pays the death benefit with probability $q$, pays the end-year expense, and sets aside the reserve for survivors with probability $p$.

G_t=(V_t+P-e_0)(1+i)-qB-e_1-pV_{t+1}.

This is an expected profit per policy in force at the start of the year. It is not the profit conditional on survival or conditional on death. For example, the expected death claim is $qB$, not $B$, because the benefit is paid only in the death outcome.

Define the expected net reserve change as expected terminal reserve less opening reserve. Under the model above, a death terminates the policy with zero terminal reserve, while a survivor has reserve $V_{t+1}$. Thus:

\Delta V_t=pV_{t+1}-V_t.

Theorem: Under the stated annual timing and reserve assumptions, cash-flow profit can be written as reserve profit: $$ G_t=(P-e_0)(1+i)-qB-e_1+iV_t-\Delta V_t. $$

Why the Identity Holds

The result follows by expanding the accumulated opening resources and then grouping the reserve terms. This is an algebraic reconciliation: it does not require a separate assumption about how profit is allocated between investment earnings and reserve movement.

Expand accumulated resources.
$(V_t+P-e_0)(1+i)=V_t(1+i)+(P-e_0)(1+i)$.

Separate the interest on opening reserve.
Since $V_t(1+i)=V_t+iV_t$, the opening reserve returns its original amount $V_t$ plus interest $iV_t$.

Replace expected terminal reserve.
From $\Delta V_t=pV_{t+1}-V_t$, rearrange to obtain $pV_{t+1}=V_t+\Delta V_t$.

Cancel the opening reserve.
Substituting into the cash-flow equation gives $V_t+iV_t+(P-e_0)(1+i)-qB-e_1-V_t-\Delta V_t$, which simplifies to the reserve-profit identity.

The derivation also clarifies the sign of the reserve change. If expected terminal reserves exceed the opening reserve, then $\Delta V_t$ is positive and reduces current profit. If expected terminal reserves are lower, then $\Delta V_t$ is negative, and subtracting it increases current profit. The amount $iV_t$ is the interest earned on the opening reserve; it is not the interest earned on the premium, which is already included in $(P-e_0)(1+i)$.

Worked Example: Start-of-Year Premium and Expense

A hypothetical contract has an opening reserve of $250. A premium of $1,000 and an expense of $40 are both paid at the start of the year. The annual effective interest rate is $i=4\%$. The probability of death is $q=0.10$, so the survival probability is $p=0.90$. The death benefit of $5,000 and the reserve for a survivor of $700 are both measured at year-end. Calculate expected annual profit by both methods. Method: first project accumulated resources and expected outgo; then reconcile using the expected reserve change.

Cash-flow method. The opening reserve plus start-of-year net cash flow is $250+1{,}000-40=1{,}210$. Accumulated to year-end, this is $1{,}210(1.04)=1{,}258.40$. The expected death claim is $0.10(5{,}000)=500$, and the expected survivor reserve is $0.90(700)=630$. There is no end-year expense. Therefore:

G_t=1{,}258.40-500-630=\$128.40.

Reserve-profit method. The expected net reserve change is $0.90(700)-250=630-250=380$. The accumulated start-of-year net premium is $(1{,}000-40)(1.04)=998.40$, and interest on the opening reserve is $250(0.04)=10$. Therefore:

G_t=998.40-500+10-380=\$128.40.

The two calculations agree. As an additional check, the death-outcome profit is $1{,}258.40-5{,}000=-\$3{,}741.60$, and the survivor-outcome profit is $1{,}258.40-700=\$558.40$. Weighting them gives $0.10(-3{,}741.60)+0.90(558.40)=-374.16+502.56=\$128.40$. The common mistake this example avoids is subtracting the full death benefit instead of its probability-weighted expected amount in the expected-profit calculation.

Expense Timing Must Match the Formula

The identity does not require every expense to occur at the start of the year. It requires each expense to be placed at its actual payment time. A start-of-year expense $e_0$ is deducted before accumulation, so its effect at year-end is $e_0(1+i)$. An end-of-year expense $e_1$ is deducted after accumulation, so it is not multiplied by $1+i$. Treating an end-year expense as a start-year expense understates its year-end cost by $e_1i$.

Worked Example: An Expense Paid at Year-End

A hypothetical policy has an opening reserve of $100 and a premium of $600 paid at the start of the year. An expense of $30 is paid at year-end. The effective annual interest rate is $i=5\%$, the death probability is $q=0.05$, and the survival probability is $p=0.95$. The year-end death benefit is $2,000, and the survivor reserve is $250. Calculate the expected reserve change and annual profit. Method: keep the expense outside the accumulation because it is paid at year-end.

The expected terminal reserve is $0.95(250)=237.50$, so the expected net reserve change is $237.50-100=\$137.50$. Accumulated opening reserve and premium are $(100+600)(1.05)=735$. The expected death claim is $0.05(2{,}000)=100$. Subtracting the end-year expense and expected terminal reserve gives:

G_t=735-30-100-237.50=\$367.50.

The reserve-profit form checks the result: $600(1.05)-30-100+0.05(100)-137.50=630-30-100+5-137.50=\$367.50$. The common timing error is to subtract the $30 before accumulation. That gives $(100+600-30)(1.05)-100-237.50=\$366.00$, which is $1.50 too low. The difference is $30(0.05)=\$1.50$, the interest incorrectly charged on an expense that is not paid until year-end.

Using the Identity to Check Profit or a Reserve Change

Once the identity is established, it can be rearranged to answer questions about profit thresholds or implied reserve movement. In particular, for fixed premiums, expenses, interest, and benefit assumptions, profit falls dollar for dollar as $\Delta V_t$ rises. That is a statement about the expected reserve change defined here; it does not mean that every reserve movement in a more complex product has this simple form.

Worked Example: Maximum Reserve Increase for Nonnegative Profit

A hypothetical contract has opening reserve $V_t=\$300$, start-of-year premium $P=\$900$, start-of-year expense $e_0=\$50$, and interest rate $i=4\%$. The death probability is $q=0.10$, the survival probability is $p=0.90$, and the year-end death benefit is $B=\$4{,}000$. Find the largest expected net reserve change consistent with nonnegative annual profit. Then calculate profit if the survivor reserve is $800. Method: isolate $\Delta V_t$ in the reserve-profit identity, then compare with the change implied by the stated survivor reserve.

First calculate the part of profit not including reserve change. Accumulated start-of-year net premium is $(900-50)(1.04)=850(1.04)=884$. Expected claims are $0.10(4{,}000)=400$, and interest on the opening reserve is $300(0.04)=12$. Thus:

G_t=884-400+12-\Delta V_t=496-\Delta V_t.

For nonnegative profit, $496-\Delta V_t\geq 0$, so $\Delta V_t\leq \$496$. The largest allowable expected net reserve change is therefore $496.00. With survivor reserve $800, the expected terminal reserve is $0.90(800)=720$, giving $\Delta V_t=720-300=\$420$. Profit is $496-420=\$76.00$.

A direct cash-flow check gives $(300+900-50)(1.04)=1{,}196$ accumulated resources, less expected claims of $400 and expected survivor reserves of $720: $1{,}196-400-720=\$76.00$. The common mistake is to use $800 as the expected terminal reserve instead of weighting it by survival probability. That would give the wrong reserve change and a profit that does not reconcile to the cash flows.

Common Traps and Limits of the Reconciliation

The identity is a reliable check only after the model has been stated precisely. For the examples here, death is the only decrement, death benefits are paid at year-end, and the reserve on death is zero after the claim is paid. If there are other exits, payments at different times, or a terminal amount on a terminating policy, those cash flows and terminal values must be represented explicitly. The simple expression $\Delta V_t=pV_{t+1}-V_t$ is not automatically valid when the terminal reserve differs by outcome.

When reconciling a calculation, check each of the following before investigating arithmetic:

Probability weighting: expected claims use $qB$, and expected survivor reserves use $pV_{t+1}$.
Interest timing: start-of-year flows accumulate for the full year; end-of-year flows do not.
Reserve-change sign: define $\Delta V_t$ as expected terminal reserve minus opening reserve, so it enters profit with a minus sign.
Consistent basis: both forms must use the same probabilities, benefit amounts, expenses, interest rate, and reserve definition.

A fast reconciliation is to calculate $G_t$ once from accumulated resources less expected year-end outgo, and once from accumulated net premium, opening-reserve interest, expected claims, expenses, and reserve change. A mismatch usually points to an omitted probability, a sign error, or a cash flow placed at the wrong time.

Key takeaway: With start-of-year premium and expense, end-of-year death benefits and expenses, and zero terminal reserve on death, expected annual profit is $G_t=(P-e_0)(1+i)-qB-e_1+iV_t-\Delta V_t$, where $\Delta V_t=pV_{t+1}-V_t$. The cash-flow and reserve-profit forms agree when their timing and assumptions match.

Check Your Understanding

For each hypothetical contract, use the stated annual timing and compare the cash-flow and reserve-profit forms where requested.

A contract has opening reserve $150, start-of-year premium $500, start-of-year expense $20, $i=4\%$, $q=0.08$, $p=0.92$, year-end death benefit $3,000, and survivor reserve $400. Find expected annual profit. (Answer: $47.20)
A contract has opening reserve $200, start-of-year premium $750, end-of-year expense $40, $i=3\%$, $q=0.04$, $p=0.96$, year-end death benefit $2,500, and survivor reserve $500. Find the expected reserve change and annual profit. (Answer: reserve change $280.00; profit $358.50)
A contract has opening reserve $400, start-of-year premium $1,000, start-of-year expense $60, $i=5\%$, $q=0.02$, $p=0.98$, year-end death benefit $6,000, and survivor reserve $900. Find expected annual profit. (Answer: $405.00)
A contract has opening reserve $250, start-of-year premium $600, start-of-year expense $20, $i=4\%$, $q=0.10$, and year-end death benefit $3,000. What is the largest expected net reserve change consistent with nonnegative annual profit? (Answer: $313.20)
A contract has opening reserve $100, start-of-year premium $800, start-of-year expense $30, $i=5\%$, $q=0.05$, and year-end death benefit $4,000. If expected annual profit is $113.50, find the survivor reserve, given that death leaves zero terminal reserve. (Answer: $631.58)

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