Modified Reserve Calculation Workshop

Why a Calculation Workshop Needs a Common Timeline

Modified reserves can look different from one method to another because the premium schedule or expense treatment changes. The reliable way to compare them is to fix the timing first, calculate the reserve prospectively, and then check the result against the reserve movement over the year. This tutorial uses hypothetical annual contracts and makes those timing conventions explicit.

Assume premiums and any stated expenses are paid at the beginning of a policy year, while death benefits are paid at the end of the year of death. For a policy in force at time $t$, let $V_t$ be the reserve immediately before the premium due at time $t$. Write $p_t$ and $q_t$ for the conditional probabilities of surviving and dying during the following year, so $p_t+q_t=1$. The annual effective interest rate is $i$, and $v=1/(1+i)$. Unless a problem says otherwise, there are no withdrawals, lapses, or benefits other than those specified.

Definition: A modified reserve is a prospective reserve calculated using a specified modified valuation basis or cash-flow schedule rather than the unmodified basis. In the annual model used here, if the valuation premium due at time $t$ is $Q_t$, the death benefit is $B_t$, and there is no valuation expense, then $$ V_t=q_t vB_t+p_t vV_{t+1}-Q_t. $$ This formula is conditional on the policy being in force at time $t$. It applies with the stated beginning-of-year premium and end-of-year benefit timing; a different payment timing requires a different present-value calculation.

A reserve may be negative under a specified modified basis. Unless a minimum reserve or floor is explicitly imposed, do not replace a calculated negative reserve with zero. Also distinguish a valuation premium used to calculate a reserve from the premium actually received by the insurer. They need not be equal.

The Reserve-Movement Check

The prospective definition has a useful equivalent form. Suppose a policy begins the year with reserve $V_t$, receives valuation premium $Q_t$, pays no valuation expense, and is valued using the same mortality and interest assumptions throughout. At the end of the year, a death claim $B_t$ is paid with probability $q_t$; with probability $p_t$, the insurer holds reserve $V_{t+1}$. Therefore the accumulated amount available at year-end must equal the expected claim and surviving-policy reserve.

(V_t+Q_t)(1+i)=q_tB_t+p_tV_{t+1}.

Solving for $V_t$ gives the prospective formula in the definition. This is the core check: if the two sides do not agree, investigate timing, conditional survival, or which premium schedule was used.

For an actual profit test, replace the valuation premium by the actual premium $P_t$, and include an actual beginning-of-year expense $E_t$. The expected profit at the end of the year per policy in force at its start is

\Pi_{t+1}=(V_t+P_t-E_t)(1+i)-q_tB_t-p_tV_{t+1}.

Here the reserves are still the reserves selected for the profit test. A positive result means that, after allowing for the expected claim and the reserve for surviving policies, the accumulated starting funds exceed those amounts. The formula assumes the same annual timing as above and a one-year profit emergence period.

Zillmerization: Amortizing an Initial Allowance

In the version of Zillmerization used in the examples, an initial expense allowance $E_0$ is spread over the expected premium-payment period by adding a level amount $\alpha$ to each valuation premium. If $\ddot{a}_0$ is the expected present value of the valuation premiums of one dollar each, then $\alpha=E_0/\ddot{a}_0$. This definition specifies the method for the calculations below; other reserve systems may define the allowance or apply floors differently.

Q_t^{Z}=Q_t+\alpha,\qquad V_t^{Z}=V_t-\alpha\,\ddot{a}_t, \qquad \alpha=\frac{E_0}{\ddot{a}_0}.

The annuity $\ddot{a}_t$ is the expected present value, at time $t$, of one-dollar premiums due immediately and at later premium dates while the policy remains in force. Since the Zillmer premium is larger, its prospective reserve is lower by the present value of the added future premiums. At issue, that reduction is $E_0$ under this definition. Do not assume the resulting reserve is nonnegative unless the problem imposes a floor.

Worked Example: Zillmer Adjustment on a Two-Year Term Contract

A hypothetical two-year term contract pays $B=\$1{,}000$ at the end of the year of death. The annual mortality probabilities are $q_0=0.10$ and $q_1=0.20$, and interest is $i=10\%$. Premiums are annual and due at the start of each year. Calculate the level net premium, then calculate the Zillmer premium and reserves when the initial expense allowance is $E_0=\$36$. Use no reserve floor.

Method. First calculate expected present values on the stated basis. Here $v=1/1.10=0.9090909$, and survival through the first year is $p_0=0.90$. The expected present value of benefits at issue is

1{,}000(0.10v+0.90(0.20)v^2) =1{,}000(0.0909091+0.1487603) =\$239.67.

The expected present value of a level premium due at issue and, if alive, one year later is

\ddot{a}_0=1+0.90v=1+0.8181818=1.8181818.

Thus the level net premium is $P=\$239.6694/1.8181818=\$131.8182$, or $\$131.82$ to the nearest cent. The added Zillmer amount is $\alpha=\$36/1.8181818=\$19.80$, so $P^Z=\$131.8182+\$19.80=\$151.6182$, or $\$151.62$.

At time 1, immediately before the second premium, the remaining one-year benefit has expected present value $0.20(1{,}000)v=\$181.8182$. The ordinary reserve is therefore $\$181.8182-\$131.8182=\$50.00$. The Zillmer reserve is $\$181.8182-\$151.6182=\$30.20$. At issue, the ordinary reserve is zero because the level net premium was calculated by equivalence. The Zillmer reserve is $\$239.6694-\$151.6182(1.8181818)=-\$36.00$.

Check and trap. The time-1 reserve reduction is exactly the one remaining added premium, $\$19.80$, and the issue reserve reduction is $\$36.00$. A common mistake is to add the Zillmer allowance to the reserve because it is called an expense allowance. Under the specified method, it is added to the valuation premium, so the prospective reserve falls. The negative issue reserve is retained because this example specifies no floor.

Full Preliminary Term: A Special First-Year Premium

Full preliminary term (FPT) separates the first policy year from the remaining contract. The first-year valuation premium is the net single premium for the first-year term benefit. The renewal valuation premium is calculated for the benefits and premium dates remaining after that first year, then used for renewal years as prescribed by the FPT method. Because the first-year premium does not fund a level premium stream from issue, the resulting reserves need not match ordinary net premium reserves.

Worked Example: FPT Reserves With Changing Mortality

A hypothetical three-year term contract pays $\$1{,}000$ at the end of the year of death. The annual death probabilities are $q_0=0.05$, $q_1=0.10$, and $q_2=0.20$. Interest is $0\%$, and premiums are due at times 0, 1, and 2 while the policy is in force. Find the FPT premium schedule and reserves immediately before each premium.

Method. The first-year FPT premium is the one-year net single premium: $Q_0=0.05(\$1{,}000)=\$50.00$. At time 1, conditional on survival through the first year, the expected value of the benefits for the remaining two years is

0.10(\$1{,}000)+0.90(0.20)(\$1{,}000)=\$100+\$180=\$280.

The expected number of renewal premiums, with the first due immediately at time 1, is $1+0.90=1.90$. Therefore the level renewal premium is $R=\$280/1.90=\$147.3684$, or $\$147.37$. The prescribed schedule is $\$50.00$ at time 0 and $\$147.37$ at each renewal date.

At issue, the present value of benefits is $\$50+\$95+\$171=\$316$, since the probabilities of death in years 1, 2, and 3 from issue are $0.05$, $0.95(0.10)$, and $0.95(0.90)(0.20)$. The present value of the FPT premiums is

\$50+0.95(\$147.3684)+0.95(0.90)(\$147.3684) =\$50+\$140+\$126 =\$316.

Thus $V_0=0$. At time 1, the reserve is $\$280-1.90(\$147.3684)=\$0$. At time 2, only the third-year benefit and premium remain, so $V_2=0.20(\$1{,}000)-\$147.3684=\$52.6316$, or $\$52.63$. At expiry $V_3=0$.

Check and trap. In year 2, the reserve identity gives $R=0.10(\$1{,}000)+0.90(\$52.6316)=\$147.3684$. In year 3, $V_2+R=\$52.6316+\$147.3684=\$200$, the expected year-3 claim. A common error is to recalculate a new single-year net premium for year 3 and substitute it for the prescribed FPT renewal premium. That would change the premium schedule and its reserves.

Custom Modifications and Profit-Test Reconciliation

A custom modified basis can prescribe valuation premiums directly, or specify changes to premiums and expenses relative to another basis. The calculation is still prospective: use the stipulated future valuation cash flows and the correct survival probabilities. To test the reserve against an actual profit projection, use actual premiums and expenses in the profit formula, but keep the selected reserves unchanged. The difference between actual net cash flows and valuation cash flows then emerges as profit, with interest and survival applied according to timing.

Fix the timing and basis.
Record when premiums, expenses, and benefits occur, and identify which amounts are valuation assumptions and which are actual profit-test cash flows.

Calculate reserves prospectively.
For each duration, subtract the present value of future valuation premiums from the present value of future benefits, including only cash flows contingent on survival as appropriate.

Check each year’s movement.
Verify that accumulated reserve plus valuation premium, less valuation expense if present, equals expected claims plus the probability-weighted next reserve.

Reconcile actual profit.
Replace valuation premiums and expenses by actual amounts in the profit identity, then compare discounted expected profits with the present value of actual net cash flows in excess of those supporting the modified reserves.

Worked Example: Custom Premium and Expense Basis

A hypothetical two-year term contract pays $\$1{,}000$ at the end of the year of death. Mortality is $q_0=0.10$ and $q_1=0.20$; interest is $10\%$. The modified valuation basis uses premiums $Q_0=\$120$ and $Q_1=\$160$. The actual contract collects $P_0=P_1=\$180$, with actual expenses $E_0=\$40$ and $E_1=\$10$, all paid at the beginning of the applicable year. Calculate the modified reserves and expected actual profits.

Method. With $v=0.9090909$, the time-1 reserve, before the second premium, is

V_1=0.20(\$1{,}000)v-\$160 =\$181.8182-\$160 =\$21.8182.

At issue, the expected present value of benefits is $\$239.6694$, as calculated from $0.10v(\$1{,}000)+0.90(0.20)v^2(\$1{,}000)$. The future valuation premium at time 1 has issue-date expected present value $0.90v(\$160)=\$130.9091$. Therefore

V_0=\$239.6694-\$120-\$130.9091=-\$11.2397.

Check the year-1 reserve movement on the valuation basis. The accumulated amount is $(-\$11.2397+\$120)(1.10)=\$119.6364$. The expected claim plus reserve for survivors is $0.10(\$1{,}000)+0.90(\$21.8182)=\$100+\$19.6364=\$119.6364$. The two sides agree.

For actual profit in year 1, the beginning net cash flow is $P_0-E_0=\$140$. Thus

\Pi_1=(V_0+\$140)(1.10)-\$100-0.90V_1 =\$141.6364-\$100-\$19.6364 =\$22.00.

In year 2, the beginning net cash flow is $P_1-E_1=\$170$, and there is no reserve after expiry. Hence

\Pi_2=(V_1+\$170)(1.10)-0.20(\$1{,}000) =\$211-\$200 =\$11.00.

The expected present value of profit per policy issued is $\$22/1.10+0.90(\$11)/(1.10)^2=\$20+\$8.1818=\$28.18$, rounded to the nearest cent. This also equals the present value of actual net cash flows in excess of the valuation premiums: $(\$140-\$120)+0.90v(\$170-\$160)=\$20+\$8.1818=\$28.18$.

Check and trap. The year-1 profit is the $\$20$ excess in beginning net cash flow accumulated for one year; year-2 profit is the $\$10$ excess accumulated for one year, conditional on surviving year 1. A common mistake is to compare actual premiums with valuation premiums but omit actual expenses, or to discount year-2 profit without multiplying by the probability of surviving to year 2. Both would break the reconciliation.

Checks That Prevent Basis and Timing Errors

The three methods above use different premium patterns, but the same discipline applies to each. Before accepting a modified reserve, confirm that the reserve date is clear, the correct premium is included at that date, and each later premium is weighted by the probability of remaining in force. Then check the one-year reserve movement. For a profit test, additionally confirm that claims and expenses use actual cash flows while reserve amounts use the selected profit-test basis.

The reserve is not itself a profit. A modified basis reallocates expected financial emergence across durations; actual profit depends on actual premiums, expenses, benefits, and the reserves carried at the start and end of the year. The annual identity makes this distinction visible and provides a compact consistency check for linked calculations.

Key takeaway: State the reserve timing and modified cash-flow basis before calculating. Zillmerization changes the valuation premium by an allowance, FPT changes the first-year and renewal premium pattern, and a custom modification follows its prescribed schedule. In every case, verify the prospective reserve using the annual reserve-movement identity before interpreting profit.

Check Your Understanding

Use the same annual timing conventions as in the tutorial. Round money to the nearest cent where needed.

A Zillmer method has an initial allowance of $\$30$, an issue premium annuity value of $2.5$, and a time-2 future premium annuity value of $1.4$. What is the time-2 reserve difference from the unmodified reserve? (Answer: $-\$16.80$)
An FPT renewal premium is being calculated at time 1 for a two-year remaining term. The benefit is $\$2{,}000$, $q_1=0.08$, $q_2=0.12$, and interest is $0\%$. What level renewal premium balances the remaining benefits and premiums? (Answer: $\$198.33$)
Immediately before a premium, a modified reserve covers one remaining year. The death probability is $0.25$, the death benefit is $\$1{,}200$, the interest rate is $5\%$, and the valuation premium due immediately is $\$260$. Calculate the reserve. (Answer: $\$25.71$)
At the start of a year, $V_t=\$50$, $P_t=\$300$, $E_t=\$20$, and $i=4\%$. The death probability is $0.10$, the benefit is $\$2{,}000$, and the reserve for survivors at year-end is $\$100$. Calculate expected profit per policy in force at the start of the year. (Answer: $\$53.20$)
At $0\%$ interest, actual net premiums exceed the modified valuation premiums by $\$15$ in year 1 and by $\$20$ in year 2, conditional on surviving year 1. If the probability of surviving year 1 is $0.90$, what is the issue-date expected present value of the excess cash flows? (Answer: $\$33.00$)

What You'll Learn