Explaining Differences Between Reserve Methods

Why Two Reserve Calculations Can Disagree

A prospective reserve values the future net obligations on policies in force at a valuation time. A retrospective reserve starts with cash flows since issue, accumulates them to that time, and expresses the resulting amount per policy still in force. Under consistent assumptions and a matching initial funding basis, the two methods agree. A disagreement is therefore a useful diagnostic signal: the calculations may not describe the same contracts, assumptions, timing, or group of policies.

The most important practical distinction is between cash flows per policy issued and a reserve per policy surviving to the valuation time. Past premiums and benefits are first accounted for across the original cohort. The resulting net amount is then divided by the probability of surviving to the valuation time. Removing past claims on policies that died, or dividing by the wrong survival probability, changes the quantity being calculated.

Definition: A prospective reserve at duration $t$ is the expected present value, at $t$, of future benefits and expenses less future premiums, conditional on the policy being in force at $t$. A retrospective reserve at duration $t$ is the accumulated expected net past inflows per policy issued, divided by the probability of being in force at $t$. The valuation instant—such as immediately before or after a premium—is part of the reserve definition.

For an annual model, suppose premiums are received while a policy is in force, death benefits are paid at the end of the year of death, and $i$ is the effective annual interest rate. At duration $t$, the prospective calculation includes only future cash flows and conditions on survival to $t$. The retrospective calculation includes all past premiums, benefits, and expenses, including payments associated with policies that did not survive to $t$. Both methods must use the same benefit and premium schedule, mortality assumptions, interest rate, expense basis, and timing convention.

A Cohort Balance Explains the Agreement

The retrospective calculation is not obtained by looking only at the policies that survive. Instead, it begins with expected accumulated cash flows for the original cohort. Past premiums increase that amount; past benefits and expenses reduce it. Dividing by the probability of remaining in force converts the cohort amount into an amount per surviving policy.

{}_tV_x^{\text{ret}} = \frac{\text{expected accumulated past premiums} -\text{expected accumulated past benefits and expenses}} {{}_tp_x}

This expression assumes the numerator is calculated per policy issued, with every past cash flow accumulated to the same valuation instant and using the same interest basis. A future premium due at the valuation instant belongs in the prospective or retrospective calculation only if the reserve is defined just after that premium. If the reserve is just before payment, it is still a future cash flow prospectively and has not yet been received retrospectively.

The reason the two methods can agree is the equivalence balance at issue. Under an equivalence premium, expected present value of premiums equals expected present value of benefits and expenses over the contract. Split that balance at duration $t$: accumulate the past part to $t$, and value the future part at $t$. The whole balance remains zero. The past net amount per issued policy therefore matches the present value of future net outgo per issued policy. Since future obligations occur only for policies still in force, that latter amount is the survival probability to $t$ multiplied by the prospective reserve per survivor. Dividing the past net amount by that same survival probability gives the prospective reserve.

\text{accumulated past net inflows per policy issued} = {}_tp_x \times \text{prospective reserve per policy in force at }t

This argument requires a consistent valuation basis and a matching initial balance. If the premium is not an equivalence premium under the stated assumptions, or if an initial reserve or surplus is present, the retrospective and prospective amounts need not agree. A difference is not automatically an arithmetic error; it may reveal a different funding basis.

Worked Example: Reconcile a Two-Year Term Policy

A hypothetical two-year term policy pays $100 at the end of a year if death occurs during that year. The annual effective interest rate is $i=0.10$. The probabilities of death in years 1 and 2, conditional on being alive at the start of each year, are $q_0=0.20$ and $q_1=0.2475$, respectively. Annual premiums of $P$ are payable at issue and at time 1 if the policyholder is alive. Find the reserve at time 1 immediately after the second premium, using both methods. Assume no expenses.

Method. First verify the equivalence premium, then calculate the future liability per survivor and the accumulated past net cash flows per policy issued. This checks both the funding basis and survival conditioning.

Let $v=1/1.10=10/11$, and let ${}_1p_0=1-q_0=0.80$. The expected present value of benefits at issue is

$$ 100(0.20)v+100(0.80)(0.2475)v^2 =18.1818+16.3636 =34.5455. $$

The expected present value of premiums per policy issued is $P$ at issue plus $0.80Pv$ at time 1. Thus

P(1+0.80v)=P(1.7272727)=34.5455, \qquad P=20.

Immediately after the time-1 premium, there are no future premiums. The only future outgo is the year-2 death benefit. Conditional on being alive at time 1, its expected present value is

{}_1V_0^{\text{pros}} =100q_1v =100(0.2475)\left(\frac{10}{11}\right) =22.50.

For the retrospective calculation, accumulate the issue premium of $20 for one year: $20(1.10)=22$ per policy issued. The expected year-1 death benefits are $0.20(100)=20$ per policy issued, paid at time 1. The time-1 premium is received by the $0.80$ who survive, giving $0.80(20)=16$ per policy issued. The accumulated net amount per policy issued is therefore

$$ 22-20+16=18. $$

Divide by the probability $0.80$ of being in force at time 1:

{}_1V_0^{\text{ret}} =\frac{18}{0.80} =22.50.

The two values agree. The check is that the past cohort amount of $18 equals $0.80$ times the prospective reserve of $22.50. A common mistake is to omit the $20 of year-1 death benefits because those policies did not survive. Doing so gives $(22+16)/0.80=47.50$, which is not the retrospective reserve: those past claims are part of the original cohort’s cash-flow history.

Align the Valuation Instant and Premium Treatment

“Reserve at time $t$” is incomplete unless the premium timing convention is clear. A reserve immediately before a premium includes that premium as a future inflow. A reserve immediately after the premium does not. The past-cash-flow calculation changes at the same instant: before payment, the premium is absent from accumulated past inflows; after payment, it is included for policies that remain in force.

Worked Example: Before Versus After a Premium

Use the policy in the previous example, with $i=0.10$, $q_0=0.20$, $q_1=0.2475$, benefit $100, and premium $20 at time 1 if alive. Calculate the reserve immediately before and immediately after that premium. State the timing mistake that would create a $20 discrepancy.

Immediately before the premium: The future benefit has expected present value $100(0.2475)/1.10=22.50$, conditional on survival to time 1. The premium of $20 is due immediately, so its value at that instant is $20. The prospective reserve is

{}_1V_0^{\text{before}} =22.50-20=2.50.

Retrospectively, the issue premium has accumulated to $22, and the year-1 claims amount to $20 per policy issued. The net past amount is $22-20=2$. Dividing by the $0.80$ survival probability gives $2/0.80=2.50$, the same value.

Immediately after the premium: The $20 premium is no longer a future inflow. The prospective reserve is therefore $22.50. Retrospectively, the premium contributes $0.80(20)=16$ per policy issued, making the cohort net amount $22-20+16=18$. Dividing by $0.80$ gives $18/0.80=22.50$.

The difference between the before- and after-premium reserves is $20, exactly the premium due at that instant. Comparing a before-premium prospective value with an after-premium retrospective value would falsely suggest that the methods disagree. The correction is to use one valuation instant in both calculations.

Trace Differences to Their Assumptions

A reserve comparison is most efficient when each source of a possible mismatch is checked in a fixed order. Recalculate only after confirming that the two methods describe the same contract and valuation time.

Fix the valuation instant.
Record whether the reserve is before or after any premium, benefit, or expense payment at duration $t$.

Match the assumptions.
Check the effective interest rate, mortality probabilities, and any changes in benefits, premiums, or expenses. A different $q$, $i$, or expense basis generally changes the answer.

Rebuild the cash-flow schedule.
Identify which policies pay or receive each amount and when it occurs. In particular, include premiums only for policies in force when they are due.

Check the cohort denominator.
Accumulate past cash flows per policy issued, including benefits paid on policies that terminate, then divide by the probability of surviving to the valuation time.

Check the funding basis.
Agreement follows when the initial balance and premium basis are consistent with the same projected cash flows. If the premium was set under another basis, a genuine difference can remain.

For example, suppose a spreadsheet uses a different year-2 death probability from the retrospective calculation. That changes future expected benefits but does not change the already accumulated year-1 cohort cash flows. The discrepancy may therefore be caused by a mortality assumption rather than a reserve-method error.

Worked Example: Diagnose a Mortality Assumption Difference

A spreadsheet values the time-1 reserve immediately after the premium for the two-year policy described earlier, but uses $q_1=0.30$ rather than $q_1=0.2475$. It keeps $i=0.10$, the $100 benefit, and the $20 premium. The retrospective calculation uses the original year-1 data and gives $22.50. Find the spreadsheet’s prospective value under its stated mortality assumption and diagnose the disagreement.

Method. At time 1 immediately after premium payment, no future premium remains. The prospective value is the present value of the year-2 death benefit conditional on being alive at time 1:

{}_1V_0^{\text{pros, spreadsheet}} =100(0.30)\left(\frac{1}{1.10}\right) =\frac{30}{1.10} =27.2727.

The retrospective value based on the original cohort data is still $22.50$. The difference is $27.2727-22.50=4.7727$, or $4.77 to the nearest cent. The disagreement is explained by the mortality mismatch: the spreadsheet assumes a higher probability of a year-2 claim.

To verify the diagnosis, restore $q_1=0.2475$. The prospective value becomes $100(0.2475)/1.10=22.50$, matching the retrospective value. If the spreadsheet instead intended a reserve immediately before the time-1 premium while using $q_1=0.30$, it would also subtract the $20 premium and report $27.2727-20=7.2727$. That is a different valuation instant, not a correction to the after-premium value.

The common mistake is to label any difference an arithmetic error without checking the inputs. Here the stated spreadsheet calculation is internally consistent; its mortality assumption does not match the retrospective basis.

Common Diagnostic Traps

Omitting claims on terminated policies. The retrospective numerator is based on the original cohort, not just survivors. Past death benefits must be included even though those policies are no longer in force.
Dividing at the wrong stage. First accumulate the cohort’s expected net cash flows per policy issued. Then divide by ${}_tp_x$. Do not divide only past premiums by survival while leaving claims out.
Mixing before- and after-payment reserves. A premium received at time $t$ is a future inflow just before payment and a past inflow just after payment.
Using different assumptions without noticing. Matching interest rates alone is insufficient; mortality, benefits, premiums, expenses, and payment timing must also match.
Assuming equivalence pricing when it was not used. A premium calculated under another mortality or interest basis may create an initial surplus or deficit. The two reserve methods can then differ even when each is correctly calculated on its own basis.

Key takeaway: To explain a reserve disagreement, compare like with like: use the same valuation instant, cash-flow schedule, assumptions, and funding basis. In a retrospective calculation, include all past cohort cash flows and divide the resulting net amount by the probability of being in force at the valuation time.

Check Your Understanding

For each problem, keep the valuation instant and the policies included in the cohort explicit.

A hypothetical one-year contract has $i=0.05$, death probability $q_0=0.10$, a $200 benefit at the end of the year of death, and a $30 premium at issue. Compute the retrospective amount at time 1 per policy in force, including the claim cash flows across the issued cohort. (Answer: $12.78)
At time 1 just before a premium, the expected present value of future benefits conditional on survival is $48.08. A premium of $40 is due immediately. What is the prospective reserve just before payment, and what is it immediately after payment if there are no other changes? (Answer: $8.08 before; $48.08 after)
A reserve is taken immediately after a time-1 premium. The expected year-2 death benefit is $300, $i=0.05$, and the year-2 death probability is $0.08$. A spreadsheet instead uses $0.12$. Find the spreadsheet value and its excess over the value using the stated mortality assumption. (Answer: $34.29; excess $11.43)
A cohort has ${}_1p_x=0.75$. The issue premium is $30 per policy and accumulates to time 1 at $i=0.10$. A $10 renewal premium is paid at time 1 by survivors. Year-1 deaths have a $80 benefit, paid at time 1. Calculate the retrospective reserve immediately after the renewal premium. (Answer: $27.33)
For the data in Question 4, a calculation omits the year-1 death benefits, then divides the accumulated premiums by $0.75$. What value does it produce, and why is it not the retrospective reserve? (Answer: $54.00; it omits $20 of expected claims per policy issued before dividing by survival.)

What You'll Learn