Why a Reserve Movement Appears in Profit
A reserve is a liability value, not a cash payment made when it changes. Yet a change in the amount held for future obligations affects the profit left over at the end of a policy year. The sign can be understood directly from the insurer’s balance reconciliation: after the year’s cash flows, the insurer must still hold enough assets to support the closing reserve.
This tutorial derives the reserve movement contribution under a clearly specified annual timing convention. The main result is simple: opening reserve minus closing reserve. A closing reserve larger than the opening reserve uses up part of the year’s resources; a smaller closing reserve releases resources. This term is one component of profit, not a substitute for premiums, claims, expenses, or investment income.
Deriving the Term from the Balance Reconciliation
Begin with one policy or a block of policies. Assume the assets at the start of the year equal the opening reserve, \(A_0=V_0\). For this derivation, premiums and expenses are paid at the beginning of the year, earn the stated annual interest rate along with the opening assets, and claims are paid at year-end. Any expenses paid at year-end can instead be included with the year-end outgo. Let \(P\) be premiums, \(E_0\) beginning-of-year expenses, \(B\) year-end benefits, and \(i\) the effective annual interest rate.
The assets available immediately after the beginning-of-year cash flows are \(V_0+P-E_0\). These assets earn interest \(i(V_0+P-E_0)\). After the year-end claim is paid, the assets are therefore:
At year-end, the insurer must retain \(V_1\) to support future obligations. The amount remaining after this liability is provided for is the profit in this simplified reconciliation:
The opening reserve appears because it was part of the assets available at the start. The closing reserve is subtracted because those assets are still required for future benefits. Grouping these two reserve amounts produces the movement term \(V_0-V_1\). The derivation does not say that the reserve change itself is a cash flow; it says that the change in the liability affects the assets left after the closing liability is recognized.
Under the stated assumption, opening assets equal the opening reserve, \(A_0=V_0\).
Add beginning premiums, subtract beginning expenses, include investment income using the specified timing, and subtract benefits and other outgo.
Subtract \(V_1\) from the assets remaining at year-end to find the surplus after reserving.
The reconciliation contains \(+V_0-V_1\), so the reserve movement contribution is opening reserve minus closing reserve.
If the closing policy status is uncertain, the reconciliation can be applied separately to each outcome and then probability-weighted. For example, a death benefit may be paid and no future reserve held on death, while a survivor has a positive closing reserve. For expected profit per policy issued, the relevant reserve amount is the expected closing reserve across those outcomes. If profit is instead being calculated conditional on a policy being in force at year-end, use the reserve conditional on that status.
The algebraic sign is important. If \(V_1>V_0\), then \(V_0-V_1<0\): the reserve movement reduces profit. If \(V_1<V_0\), then \(V_0-V_1>0\): the reserve movement adds to profit. These are contributions to the profit reconciliation, not claims about whether the entire year’s profit must be negative or positive.
Worked Applications
Worked Example: Expected Profit with an Increase in the Closing Reserve
A hypothetical one-year policy has an opening reserve of $1,000. The policyholder pays a $300 premium and the insurer pays $20 of expenses at the start of the year. Both amounts are invested with the opening assets at an effective annual interest rate of 4%. A death benefit of $2,500 is paid at year-end if death occurs during the year. Assume annual mortality is \(q=0.02\), with \(D\sim\operatorname{Bernoulli}(0.02)\), where \(D=1\) denotes death. The death claim is paid at year-end; the closing reserve is $0 on death and $1,050 on survival. Find expected profit per policy issued.
Method. Reconcile expected year-end assets against the probability-weighted closing reserve. This method avoids treating the survivor’s reserve as if it applied to every policy issued.
1. Find the beginning-of-year investment amount. The opening assets plus premium less expense are \(1{,}000+300-20=1{,}280\).
2. Calculate investment income. At 4%, income is \(0.04(1{,}280)=\$51.20\). Assets before the year-end claim are \(1{,}280+51.20=\$1{,}331.20\).
3. Calculate expected claims. The death probability is 0.02, so expected claim outgo is \(0.02(2{,}500)=\$50.00\). Expected assets after claims are \(1{,}331.20-50.00=\$1{,}281.20\).
4. Calculate the expected closing reserve. Survival probability is \(1-0.02=0.98\). Thus the expected closing reserve is \(0.98(1{,}050)+0.02(0)=\$1{,}029.00\).
5. Find profit. Expected profit is \(1{,}281.20-1{,}029.00=\$252.20\). As a check using the components, the reserve movement is \(1{,}000-1{,}029=-\$29.00\), and \(300-20-50+51.20-29.00=\$252.20\).
The reserve increases in expectation by $29, so its contribution is negative $29. Reversing the subtraction would incorrectly add $29 and produce profit of $310.20 rather than $252.20.
Worked Example: A Release Despite a Death Claim
A hypothetical policy has an opening reserve of $1,000. A $300 premium and $20 expense are paid at the beginning of the year. Assets earn an effective annual interest rate of 4%. Annual mortality is \(q=0.05\), with \(D\sim\operatorname{Bernoulli}(0.05)\). A death benefit of $1,000 is paid at year-end if death occurs. The closing reserve is $0 on death and $800 on survival. Calculate expected profit per policy issued.
Method. First reconcile the assets after expected claims, then subtract the expected state-contingent closing reserve. The common error avoided here is assuming that a year with claim outgo cannot also have a reserve release.
1. Accumulate beginning assets and cash flows. The amount invested is \(1{,}000+300-20=\$1{,}280\). Investment income is \(0.04(1{,}280)=\$51.20\), giving \(1{,}280+51.20=\$1{,}331.20\) before claims.
2. Allow for expected claims. Expected claim outgo is \(0.05(1{,}000)=\$50.00\). Expected assets after claims are \(1{,}331.20-50.00=\$1{,}281.20\).
3. Find the expected closing reserve. The probability of survival is \(0.95\), so the expected closing reserve is \(0.95(800)+0.05(0)=\$760.00\).
4. Reconcile profit. Expected profit is \(1{,}281.20-760.00=\$521.20\). The reserve movement is \(1{,}000-760=\$240.00\), and the component calculation checks: \(300-20-50+51.20+240=\$521.20\).
Although the insurer pays expected claims of $50, the expected closing reserve is $240 below the opening reserve. That release contributes positively to the reconciliation. It does not cancel or erase the claim outgo; both amounts appear as separate components.
Worked Example: An Increase in the Reserve Can Reduce Profit
A hypothetical policy has an opening reserve of $1,000. At the beginning of the year, the insurer receives a $500 premium and pays $50 in expenses. These amounts, together with the opening assets, earn an effective annual interest rate of 4%. Annual mortality is \(q=0.20\), with \(D\sim\operatorname{Bernoulli}(0.20)\). The insurer pays a $2,000 death benefit at year-end if death occurs. The closing reserve is $0 on death and $1,500 on survival. Find expected profit per policy issued.
Method. Calculate expected year-end assets and expected closing reserves separately, then reconcile them. This prevents confusing a high reserve for surviving policies with the expected reserve for the original cohort.
1. Calculate the invested amount and income. Beginning assets after premium and expense are \(1{,}000+500-50=\$1{,}450\). Investment income is \(0.04(1{,}450)=\$58.00\), so assets before claims are \(1{,}450+58=\$1{,}508.00\).
2. Calculate expected claims and assets after claims. Expected claims are \(0.20(2{,}000)=\$400.00\). Expected assets after claims are \(1{,}508-400=\$1{,}108.00\).
3. Calculate the expected closing reserve. Survival probability is \(0.80\), giving expected closing reserve \(0.80(1{,}500)+0.20(0)=\$1{,}200.00\).
4. Reconcile profit and check the sign. Expected profit is \(1{,}108-1{,}200=-\$92.00\). The reserve movement is \(1{,}000-1{,}200=-\$200.00\), and the component calculation gives \(500-50-400+58-200=-\$92.00\). The expected reserve increase reduces profit by $200.
Conditions and Common Exam Traps
The formula \(V_0-V_1\) follows from the asset reconciliation, but its use depends on matching the reserve basis and cash-flow timing. In the examples, opening assets equal the opening reserve, and the investment income calculation explicitly reflects the assumed beginning-of-year premiums and expenses. If cash flows occur at other times, their investment income must be adjusted to those times. The reserve movement term itself remains the opening liability less the closing liability; the other cash-flow and investment-income terms must reflect the actual timing convention.
For a policy with multiple possible year-end states, use the closing reserve appropriate to each state before calculating an expectation. If death leaves no future liability, the death-state reserve is zero in these examples. For a portfolio, the same principle applies across policies and states: reconcile the relevant opening assets and liabilities with year-end assets and closing liabilities on a consistent basis.
| Reserve pattern | Movement term | Effect on profit |
|---|---|---|
| Closing reserve exceeds opening reserve | \(V_0-V_1<0\) | Reduces profit |
| Closing reserve is below opening reserve | \(V_0-V_1>0\) | Increases profit |
| Opening and closing reserves are equal | \(V_0-V_1=0\) | No reserve movement contribution |
Common mistake: reverse the sign. Suppose the opening reserve is $1,200 and the closing reserve is $1,350. The wrong calculation \(1{,}350-1{,}200=+\$150\) treats the increase as a profit addition. The correct term is \(1{,}200-1{,}350=-\$150\), so the reserve increase reduces profit by $150. A useful check is to ask what the closing liability does: if the insurer must retain more at year-end, less is left over as profit.
Another trap is to apply a survivor’s closing reserve to every policy in an expected-profit calculation. If the probability of survival is \(p\), the survivor reserve is \(R\), and the reserve on death is zero, then the expected closing reserve is \(pR\), not \(R\). A further timing trap is to mix an investment-income amount calculated on opening reserves with a reserve movement calculation that has silently included interest. Under the convention used here, investment income is calculated separately and the movement term is simply \(V_0-V_1\).
Check Your Understanding
For each problem, use the stated timing convention and report the requested amount. Money is reported to the nearest cent.
- A hypothetical policy has opening reserve $2,400, beginning-of-year premium $600, and beginning expense $100. There are no claims or decrements during the year. Assets earn 3% effective annual interest, and the closing reserve is $2,600. Find profit using the balance reconciliation. (Answer: $387.00)
- A hypothetical policy has opening reserve $1,500, beginning premium $200, and beginning expense $20. At 4% effective annual interest, these amounts are invested for the full year. Mortality is \(q=0.10\), with a $1,000 year-end death benefit; the closing reserve is $1,200 on survival and $0 on death. Find expected profit per policy issued. (Answer: $567.20)
- A hypothetical policy has opening reserve $800, beginning premium $300, and beginning expense $50. The invested amount earns 5% effective annual interest. Mortality is \(q=0.25\), with a $1,200 year-end death benefit; the closing reserve is $1,400 on survival and $0 on death. Find expected profit per policy issued. (Answer: -$247.50)
- A hypothetical block has opening reserve $900. Annual mortality is \(q=0.04\), with a Bernoulli death indicator; the closing reserve is $950 on survival and $0 on death. Find the expected closing reserve and the reserve movement contribution. (Answer: Expected closing reserve $912.00; movement contribution -$12.00)