Modified Reserve Bases in Product Profit Tests

Why Compare Reserve Bases?

A product profit test may show different annual profit patterns depending on the reserve basis used, even when the contractual premiums and benefits are unchanged. A modified basis might, for example, use a different valuation premium or a different expense assumption from the gross basis. The resulting reserve difference affects how profit is allocated between projection years; it does not, by itself, change the policyholder cash flows.

The central discipline is to keep three items separate: contractual cash flows, assumptions used to value the reserve, and the reserve movement used in the profit calculation. Confusing the valuation premium with the premium actually received is a particularly consequential error.

Definition: A gross prospective reserve at time $t$ is the present value, on a specified gross valuation basis, of future benefits and expenses less the present value of future premiums used in that reserve calculation. A modified reserve is calculated in the same prospective way but with one or more valuation inputs changed, such as the valuation premium or expense assumption. Both reserves are conditional on the policy being in the relevant state at time $t$.

For a discrete annual model, let $V_t^b$ denote the reserve just before the premium due at time $t$, where $b$ identifies the basis. Let $\pi_s^b$ be the premium used in the reserve valuation at time $s$, $B_s$ the benefit, and $E_s^b$ the expense valued on basis $b$. The valuation premium $\pi_s^b$ need not equal the contractual premium $P_s$ received in the profit test.

V_t^b = \operatorname{EPV}_b(\text{future benefits and expenses}\mid\text{state at }t) - \operatorname{EPV}_b(\text{future valuation premiums}\mid\text{state at }t).

The expectation and discounting must use the assumptions belonging to basis $b$. For example, changing a valuation expense assumption changes the expense component of the reserve; it does not change the actual expense paid in a profit projection unless the projection itself also changes that cash flow.

Annual Reserve and Profit Relationships

Suppose premiums are payable at the beginning of each year, while benefits and expenses are paid at year-end. Let $C_{t+1}$ be the year-end outgo and let $I_{t+1}$ indicate that the policy remains in a state with a continuation reserve at time $t+1$. With effective annual interest rate $i$, a one-year reserve recursion on basis $b$ is

V_t^b = \frac{\operatorname{E}_b[C_{t+1}+I_{t+1}V_{t+1}^b]}{1+i_b} -\pi_t^b.

This form assumes the valuation premium is paid at time $t$, outgo occurs at time $t+1$, and there is no continuation reserve after termination. More complex state models replace the single continuation term with a sum across possible next states.

For a profit projection, use the actual contractual premium $P_t$ and projected outgo, rather than silently substituting the valuation premium. Define expected end-of-year profit using the reserve held at the start and the reserve held at the end:

\Pi_{t+1}^b = (V_t^b+P_t)(1+i) - \operatorname{E}[C_{t+1}+I_{t+1}V_{t+1}^b].

Here the projection’s interest and cash-flow assumptions are held fixed when comparing bases. In a multi-state model, the expected continuation reserve is the sum of each next-state probability multiplied by that state’s reserve.

Key result: Let $\Delta V_t=V_t^m-V_t^g$ be the modified-minus-gross reserve difference. If the same projected cash flows and interest rate are used for both profit calculations, then $$ \Pi_{t+1}^m-\Pi_{t+1}^g = (1+i)\Delta V_t-\operatorname{E}[I_{t+1}\Delta V_{t+1}]. $$ A reserve difference affects profit through both the accumulated opening reserve and the expected continuation reserve.

Why the Profit Difference Formula Holds

Subtract the gross-basis profit equation from the modified-basis equation. The contractual premium is the same in both calculations, so its contribution cancels. The projected outgo $C_{t+1}$ is also the same and cancels. The remaining terms are the accumulated difference in opening reserves and the expected difference in continuation reserves:

\begin{aligned} \Pi_{t+1}^m-\Pi_{t+1}^g &=[(V_t^m+P_t)-(V_t^g+P_t)](1+i)\\ &\quad-\operatorname{E}[I_{t+1}(V_{t+1}^m-V_{t+1}^g)]\\ &=(1+i)\Delta V_t-\operatorname{E}[I_{t+1}\Delta V_{t+1}]. \end{aligned}

This identity explains why reserve changes alter profit emergence: a reserve basis changes the amount carried forward, so the profit attributed to a year changes even if the projected policy cash flows do not. The identity does not say that total profit must be unchanged under every comparison. That conclusion would require consistent treatment of all periods, including issue profit, terminal reserves, and any differences in valuation assumptions or cash flows.

Worked Example: A Modified Expense Assumption

A hypothetical two-year contract has no mortality or lapse. The effective annual interest rate is 5%. The policyholder pays a contractual premium of $60 at the beginning of each year, and a benefit of $120 is paid at the end of year 2. The actual projected expense is $5 at the end of each year. At time 1, calculate the gross reserve using a $5 annual valuation expense and the modified reserve using a $3 annual valuation expense. Then compare the year-2 profits. There is one premium remaining at time 1.

Method. Calculate each prospective reserve as the discounted value of the remaining benefit and valuation expense, less the valuation premium due immediately. The valuation premium is $60 on both bases in this example.

On the gross basis, the year-2 benefit plus expense is $120+$5=$125. Its time-1 present value is $125/1.05=$119.0476. Subtract the $60 valuation premium:

V_1^g=\frac{120+5}{1.05}-60=59.0476.

On the modified basis, the year-2 benefit plus valuation expense is $120+$3=$123. Its time-1 present value is $123/1.05=$117.1429. Therefore,

V_1^m=\frac{120+3}{1.05}-60=57.1429, \qquad \Delta V_1=57.1429-59.0476=-1.9048.

At time 1 the actual premium of $60 is received. The actual year-end outgo is the $120 benefit plus the $5 actual expense, or $125. There is no reserve after the benefit is paid. Gross-basis profit is $(59.0476+60)(1.05)-125=0.00$. Modified-basis profit is $(57.1429+60)(1.05)-125=-2.00$. The profit difference is $-$2.00$, equal to $(1.05)(-1.9048)$ to the nearest cent.

Check. The modified expense assumption is $2 lower, so its reserve is lower by approximately $2/1.05=$1.90$. With no continuation reserve, that lower opening reserve reduces this year’s profit by approximately $1.05\times$1.90=$2.00$. The common mistake is to use the modified valuation expense of $3 as the actual expense in the profit test; doing so would change the projected cash flow and answer a different question. Rounded to the nearest cent, gross profit is $0.00 and modified-basis profit is a loss of $2.00.

Worked Example: A Modified Valuation Premium

A hypothetical two-year contract has no expenses or decrements. The effective annual interest rate is 10%. A $55 contractual premium is paid at times 0 and 1, and a $110 benefit is paid at the end of each year. Calculate reserves and profit emergence when the gross reserve basis uses a valuation premium of $55 per year, while the modified basis uses $50 per year. Use the actual contractual premium of $55 in both profit calculations.

Method. First find time-1 reserves, then time-0 reserves by discounting future benefits and valuation premiums. Finally apply the annual profit equation. There are two future benefits at time 0, with present values $110/1.10=100$ and $110/1.10^2=90.9091$.

At time 1, the present value of the remaining benefit is $110/1.10=100$. Thus $V_1^g=100-55=45$ and $V_1^m=100-50=50$. At time 0, the present value of benefits is $100+90.9091=190.9091$. The gross valuation premiums have present value $55+55/1.10=105$, while modified valuation premiums have present value $50+50/1.10=95.4545$. Therefore,

V_0^g=190.9091-105=85.9091, \qquad V_0^m=190.9091-95.4545=95.4545.

In year 1, gross-basis profit is $(85.9091+55)(1.10)-110-45=0.00$. Modified-basis profit is $(95.4545+55)(1.10)-110-50=5.50$. In year 2, gross-basis profit is $(45+55)(1.10)-110=0.00$, while modified-basis profit is $(50+55)(1.10)-110=5.50$. The modified basis produces $5.50 more profit in each year.

For the first year, the reserve differences are $\Delta V_0=9.5455$ and $\Delta V_1=5$, so the difference formula gives $1.10(9.5455)-5=$5.50$. For the second year it gives $1.10(5)-0=$5.50$. The calculation checks. The common mistake is to replace the $55 contractual premium with the $50 modified valuation premium in the profit equation. The $50 is only an input to the reserve; the insurer still receives $55.

State-Contingent Continuation Reserves

When a policy can lapse or move between disability, care, or other states, the closing reserve is not necessarily held for every policy in force at the start of the year. A reserve difference in a next state contributes to expected closing reserves in proportion to the probability of reaching that state. Termination benefits, such as a surrender payment, belong in projected outgo; they are not continuation reserves.

Worked Example: Expense Modification with Lapse

A hypothetical two-year policy has annual lapse probability 0.20, with the remaining 0.80 continuing in force. The effective annual interest rate is 5%. A $30 contractual premium is paid at the beginning of each year. At the end of a year, a lapse benefit of $20 is paid if the policy lapses; otherwise the policy remains in force. The maturity benefit is $100 at the end of year 2 if the policy remains in force through that date. An actual expense of $4 is paid at each year-end for every policy in force at the start of that year. Calculate the time-1 reserves on gross and modified expense bases of $4 and $2, respectively, and the corresponding time-0 reserves. Assume valuation lapse probabilities equal the stated probabilities.

Method. At time 1, expected year-end outgo is the lapse benefit with probability 0.20, plus maturity benefit with probability 0.80, plus the expense incurred for the in-force policy. Subtract the time-1 valuation premium of $30 after discounting.

The gross-basis expected outgo is $0.20(20)+0.80(100)+4=88$, giving $V_1^g=88/1.05-30=53.8095$. The modified-basis expected outgo is $0.20(20)+0.80(100)+2=86$, giving $V_1^m=86/1.05-30=51.9048$.

At time 0, the $30 premium is paid immediately. End-of-year outgo consists of the lapse payment, if any, and the year-1 expense. Only the 0.80 probability of continuing policies carries a time-1 reserve. Hence

\begin{aligned} V_0^g &=\frac{0.20(20)+4+0.80(53.8095)}{1.05}-30 =18.6168,\\ V_0^m &=\frac{0.20(20)+2+0.80(51.9048)}{1.05}-30 =15.2608. \end{aligned}

The reserve differences are $\Delta V_1=-1.9048$ and $\Delta V_0=-3.3560$. Comparing year-1 profits, the difference formula gives $1.05(-3.3560)-0.80(-1.9048)=-2.00$. The 0.80 factor matters because only continuing policies have a time-1 reserve; lapsed policies receive the same surrender benefit on both bases. This verifies the calculation. A common error is to subtract the full time-1 reserve difference rather than its probability-weighted amount.

Interpreting Modified-Basis Results

A modified basis can change the level and timing of reported profits without changing contractual benefits or premiums. A lower valuation expense generally reduces the reserve, while a lower valuation premium generally increases it, all else equal. However, the annual profit effect depends on both opening and continuation reserve differences, as the profit identity shows.

Before interpreting a comparison, check that the calculation has kept these items aligned:

Valuation versus contractual premium: reserve premiums determine the prospective liability; contractual premiums are the cash receipts in the profit test.
Expense timing: an expense paid at year-end is discounted as a year-end amount, not as an immediate payment.
State probabilities: continuation reserves are weighted by the probabilities of reaching their corresponding states.
Comparison basis: the profit-difference identity assumes the same projected cash flows and interest accumulation in both profit calculations. If those also change, their effects must be separated rather than attributed solely to the reserve basis.
Sign convention: $\Delta V_t=V_t^m-V_t^g$. A negative difference means the modified reserve is lower; it does not automatically mean every year’s profit is lower.

These comparisons are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a product profit test, the useful conclusion is not merely that one reserve is larger. The important question is how the reserve difference propagates through the projection and changes profit by year and state.

Check Your Understanding

For each problem, distinguish the valuation basis from the contractual cash flows and use the stated timing conventions.

A certain one-year contract has an effective annual interest rate of 4%, a $70 contractual and valuation premium at time 1, a $90 benefit at year-end, and no decrement. The gross valuation expense is $2 and the modified valuation expense is $5, both paid at year-end. Find the modified-minus-gross time-1 reserve difference. (Answer: the modified reserve is $2.88 higher.)
A certain two-year contract has an effective annual interest rate of 5%, identical benefit assumptions on both bases, and two remaining valuation premiums payable at times 0 and 1. The gross valuation premium is $60 each year; the modified valuation premium is $56 each year. Find $V_0^m-V_0^g$. (Answer: $7.81.)
At time 1, an in-force policy has a 0.10 lapse probability and 0.90 continuation probability. At year-end it pays $30 on lapse or $80 on continuation, and incurs a $3 expense regardless of the transition. There are no future premiums or continuation reserves after year-end. With effective annual interest 4%, calculate the modified-minus-gross reserve difference if the modified expense assumption is $1 instead of $3. (Answer: the modified reserve is $1.92 lower.)
In a one-year profit comparison, the effective annual interest rate is 5%, $\Delta V_t=$6$, and there are two next states: continuation with probability 0.80 and $\Delta V_{t+1}=$4$, or termination with probability 0.20 and no continuation reserve. Find modified-minus-gross expected profit for the year. (Answer: $3.10.)

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