Reserve Recalculation After an Assumption Change

Why Reserves Must Be Recalculated Consistently

A valuation assumption change can alter a reserve even when no premium has been received and no claim has occurred. That immediate change is a remeasurement of the liability, not profit earned from the period’s experience. To assess experience profit fairly, use reserves calculated on one consistent basis at both the opening and closing of the period. If a basis change takes effect at either boundary, show its impact separately.

The distinction matters because a reserve increase reduces surplus, while a reserve decrease releases surplus. If an opening reserve is calculated on one basis and a closing reserve on another, the profit result mixes experience with a change in measurement. The resulting number may be arithmetically correct but difficult to interpret.

Definition: A reserve-basis change is a change in the assumptions used to value the liability, such as the valuation mortality or valuation interest rate. At a valuation date, the reserve impact per policy is the revised reserve minus the previous-basis reserve. A positive reserve impact increases the liability and reduces surplus by the same amount; a negative reserve impact decreases the liability and increases surplus.

A Simple Reserve Model and Profit Convention

To make the recalculation explicit, consider a hypothetical $n$-year term insurance with benefit $B$, payable at the end of the year of death. Assume annual periods, level conditional death probability $q_v$ on the valuation basis, $p_v=1-q_v$, and effective annual valuation interest rate $i_v$, with $v_v=1/(1+i_v)$. There are no premiums or expenses in the reserve formula below. For a policy still in force at time $k$, the reserve is the expected present value of covered future claims under that valuation basis.

V_k = B\sum_{r=1}^{n-k} p_v^{\,r-1}q_vv_v^r, \qquad k=0,1,\ldots,n.

This formula uses the probability, on the valuation basis, of surviving to each future year and then dying in that year. It is conditional on the policy being in force at time $k$; it is not multiplied by the probability that the policy was still in force at time $k$. The formula is for this specific term-insurance model. Other benefit patterns, premiums, expenses, or contract states require the corresponding cash flows and probabilities to be valued.

For a one-year profit-test period, suppose a premium $P$ is received at the start of the year, the opening reserve earns the actual investment return $j$, claims are paid at year-end, and the closing reserve is held only for policies surviving to year-end. Let $q_a$ and $p_a=1-q_a$ be actual annual death and survival probabilities. With no expenses, expected profit per policy in force at the opening is:

\text{Expected period profit} =(V_0+P)(1+j)-q_aB-p_aV_1.

Here $V_0$ and $V_1$ must be calculated on the same valuation basis if the purpose is to measure experience profit on that basis. The actual probabilities determine expected claims and the number of policies requiring a closing reserve; the valuation probabilities and interest rate determine the reserve amounts.

Why the Profit Formula Works

At the start of the year, the opening reserve and premium provide funds of $V_0+P$. Both earn the actual return $j$, so the year-end fund before outgo is $(V_0+P)(1+j)$. Expected claim outgo is $q_aB$. A policy survives with probability $p_a$, and its closing liability is then $V_1$, giving expected closing reserve $p_aV_1$. Subtracting both claim outgo and the liability that must remain gives the formula.

\underbrace{(V_0+P)(1+j)}_{\text{year-end funds}} - \underbrace{q_aB}_{\text{expected claims}} - \underbrace{p_aV_1}_{\text{expected closing liability}} = \text{expected period profit}.

This is an expected per-opening-policy calculation. For an actual portfolio, use actual opening and surviving policy counts, actual claims, and the actual cash-flow amounts. The timing assumptions also matter: if premiums arrive later in the year or claims are paid at a different time, the investment accumulation in the formula must be adjusted.

Worked Example: Recalculating Both Reserves Under a New Mortality Basis

A hypothetical two-year term policy pays $B=\$1{,}000$ at the end of the year of death. On the old valuation basis, $q_v=0.10$; on the revised basis, $q_v=0.20$. The valuation effective annual interest rate is $i_v=5\%$ on both bases. Calculate the old and revised opening reserves, the old and revised reserve at time 1 for a surviving policy, and the immediate opening reserve impact.

Method. Use the two-year version of the term-insurance reserve formula at time 0 and the one-year remaining liability at time 1. The calculation assumes annual death probabilities, year-end claim payment, and no premiums or expenses. The common mistake avoided is changing the opening reserve but retaining the old-basis closing reserve in a profit test.

First, $v_v=1/1.05=0.95238095$. For the old basis, the opening reserve is:

V_0^{\text{old}} =1{,}000\left[0.10(0.95238095) +0.90(0.10)(0.95238095)^2\right] =\$176.87.

The two terms are the discounted claim in year 1 and the discounted claim in year 2, respectively. For the revised basis, the opening reserve is:

V_0^{\text{new}} =1{,}000\left[0.20(0.95238095) +0.80(0.20)(0.95238095)^2\right] =\$335.60.

At time 1, a surviving policy has one year of cover remaining. Therefore:

V_1^{\text{old}}=1{,}000(0.10)(0.95238095)=\$95.24, \qquad V_1^{\text{new}}=1{,}000(0.20)(0.95238095)=\$190.48.

The opening liability increases by $\$335.60-\$176.87=\$158.73$ per policy. Thus, the immediate impact on surplus is a decrease of $\$158.73$ per policy. The closing reserve must also be recalculated on the revised basis if the period’s experience profit is to be measured using that basis. Values are rounded to the nearest cent.

Worked Example: Separating the Opening Basis Impact from Expected Profit

Use the same hypothetical two-year policy and assumptions as in the preceding example. At time 0, the reserve basis changes from $q_v=0.10$ to $q_v=0.20$, with $i_v=5\%$. During the following year, the actual death probability is $q_a=0.15$, the actual investment return is $j=8\%$, and a premium of $\$100$ is received at the start of the year. Calculate (1) the immediate surplus impact and (2) expected profit for the year on the revised basis.

Method. First measure the basis change using the difference between the old and new opening liabilities. Then calculate period profit with the revised opening and closing reserves. Assume claims are paid at year-end and no expenses apply. The common mistake avoided is including the opening remeasurement inside the experience-profit calculation without identifying it.

From the preceding reserve calculations, the old-basis opening reserve is $\$176.87$, the revised opening reserve is $\$335.60$, and the revised closing reserve per survivor is $\$190.48$. The immediate liability increase is $\$158.73$, so the surplus impact is $-\$158.73$.

For the year’s expected profit, $p_a=1-0.15=0.85$. The opening reserve and start-of-year premium accumulate to:

$$ (335.6009+100)(1.08)=470.4490. $$

Expected claim outgo is $0.15(1{,}000)=\$150$. The expected closing reserve is $0.85(190.4762)=\$161.9048$. Therefore:

470.4490-150-161.9048=\$158.54.

Expected experience profit on the revised basis is $\$158.54$ per opening policy, rounded to the nearest cent. The opening basis change separately reduces surplus by $\$158.73$. Keeping those amounts distinct makes clear that one is an immediate liability remeasurement and the other is the expected result of the following year’s cash flows and experience.

A Closing-Date Basis Change

The same separation is needed if assumptions change at the end of the period. First calculate the period’s profit using the closing reserve on the basis used for that period. Then calculate the additional surplus impact of remeasuring that closing liability on the new basis. If the new basis is intended to apply throughout the profit-test period instead, recalculate both its opening and closing reserves on that basis and identify the opening remeasurement separately.

Worked Example: Experience Profit Followed by Closing Reserve Remeasurement

A hypothetical portfolio has 100 two-year term policies, each with a $\$1{,}000$ death benefit. At the start of the year the valuation basis has already been changed to $q_v=0.20$, with $i_v=5\%$. The opening reserve is $\$335.6009$ per policy. A premium of $\$100$ per policy is received at the start of the year, and the actual investment return is $8\%$. During the year, 12 claims occur and 88 policies survive. At year-end, the closing valuation mortality assumption changes from $q_v=0.20$ to $q_v=0.25$, while the valuation interest rate remains $5\%$. Calculate the period profit before the closing remeasurement and the closing remeasurement impact.

Method. Calculate the period result using the closing reserve under the $q_v=0.20$ basis, then compare that liability with the new-basis closing liability. The common mistake avoided is treating the increase in the closing reserve as claim experience.

Before remeasurement, the closing reserve per surviving policy is $1{,}000(0.20)/1.05=\$190.4762$. The total closing reserve is $88(190.4762)=\$16{,}761.90$. The opening reserve and premiums, accumulated at the actual return, are:

100(335.6009+100)(1.08)=\$47{,}044.90.

The 12 claims cost $12(1{,}000)=\$12{,}000$. Thus, period profit before the closing remeasurement is:

47{,}044.90-12{,}000-16{,}761.90=\$18{,}283.00.

On the new closing basis, the reserve per survivor is $1{,}000(0.25)/1.05=\$238.0952$, giving a total closing liability of $88(238.0952)=\$20{,}952.38$. The liability increases by $\$20{,}952.38-\$16{,}761.90=\$4{,}190.48$, so the closing remeasurement reduces surplus by $\$4{,}190.48$. It is reported separately from the $\$18{,}283.00$ period profit. All dollar amounts are rounded to the nearest cent.

Practical Checks and Common Traps

A reserve-basis change is not itself an experience deviation. Experience profit compares actual or expected cash flows with the reserve and pricing framework used for the test. A basis change changes the measurement framework. Keeping a short reserve bridge makes the two effects visible:

Record the old-basis reserve.
Identify the valuation assumptions and the date to which the existing reserve applies.

Recalculate on the new basis.
Use the revised assumptions for the reserve at the relevant valuation date. The difference is the liability remeasurement.

Calculate period profit consistently.
Use opening and closing reserves on the same basis, with actual assumptions for actual cash flows and experience.

Show any boundary remeasurement separately.
If the basis changes at the closing date, quantify its surplus impact after calculating period profit on the pre-change basis.

A frequent exam error is to combine a revised opening reserve with an old-basis closing reserve, or vice versa, without explaining that a basis change occurred between those dates. Another is to give the liability increase as a positive profit impact: the liability change is positive, but its direct surplus effect is negative. Always state whether the figure is a change in reserve or a change in surplus.

Key takeaway: Recalculate opening and closing reserves on a consistent valuation basis for the experience-profit calculation. Then show the immediate surplus effect of each basis change as a separate item. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, keep the reserve remeasurement separate from the experience calculation where the question asks for both.

A one-year hypothetical term policy pays $\$2{,}000$ at year-end on death. The valuation interest rate is $5\%$; valuation mortality changes from $q_v=0.04$ to $q_v=0.06$. Find the old and revised opening reserves and the immediate surplus impact. (Answer: old reserve $\$76.19$, revised reserve $\$114.29$, surplus impact $-\$38.10$ per policy.)
A two-year term policy has $B=\$1{,}000$, valuation $q_v=0.10$, and valuation interest $i_v=5\%$. Find its opening reserve and its reserve at time 1 conditional on survival. (Answer: opening reserve $\$176.87$; time-1 reserve $\$95.24$.)
For a one-year test, $V_0=\$200$, $P=\$50$, $j=4\%$, actual $q_a=0.10$, $B=\$1{,}000$, and $V_1=0$. Find expected period profit per opening policy. (Answer: $\$160.00$.)
At a closing valuation date, 80 policies survive. Each has a $\$5{,}000$ one-year death benefit. Under the new basis, $q_v=0.08$ and $i_v=4\%$. Find the aggregate closing reserve. (Answer: $\$30{,}769.23$.)
A basis change increases the opening reserve from $\$80$ to $\$100$ per policy. For the following period, $P=\$20$, $j=5\%$, expected claim outgo is $\$30$, and the expected closing reserve is $\$70$, all per opening policy. Find the immediate surplus impact and the period profit using the revised opening reserve. (Answer: immediate surplus impact $-\$20.00$; period profit $\$26.00$.)

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