Proving Prospective-Retrospective Reserve Equivalence

Why Two Reserve Methods Can Give the Same Value

A reserve can be viewed from two directions. Prospectively, it is the expected value of benefits still to be paid, less the expected value of premiums still to be received. Retrospectively, it is the accumulated value of premiums received so far, less benefits paid so far, allocated across the policies still in force. The two calculations look different because one examines the future and the other the past.

Under the equivalence principle, a net premium is chosen so that the expected present value of premiums at issue equals the expected present value of benefits. This balance links the past and future: once past cash flows have been accumulated to a valuation time, their expected amount per surviving policy must match the conditional expected value of the future net outgo. The timing convention matters. Throughout this tutorial, a reserve at time $t$ is measured immediately after any premium due at time $t$; benefits due at time $t$ have already been paid.

Definition: At time $t$, the prospective reserve is the expected present value at time $t$ of future benefits, conditional on the policy being in force at $t$, minus the corresponding expected present value of future premiums. The retrospective reserve is the accumulated value at time $t$ of past premiums minus past benefits for the original group, divided by the probability that a policy from that group is still in force at $t$. The equivalence principle requires equality of expected present values at issue, on the same valuation basis.

For example, if premiums are payable at the beginnings of policy years while the policy is in force, the premium at time $t$ is part of the past cash flows for a reserve measured immediately after it is paid. It is not a future premium in the prospective calculation. If a different timing convention is used, both definitions and the cash-flow balance must be adjusted consistently.

The Equivalence Argument

Let $p_t$ be the probability, at issue, that the policy is still in force immediately after time $t$. Let $V_t$ denote the prospective reserve at that time, conditional on being in force. Write $A_t$ for the accumulated net past cash flow per original policy: past premiums minus past benefits, with each cash flow accumulated to time $t$ at the assumed interest rate. In the retrospective calculation, cash flows from policies that have already terminated remain part of the original group’s past cash-flow total; the total is then allocated over the $p_t$ fraction expected to remain in force.

The argument needs two ingredients: the equivalence principle at issue and a consistent division of cash flows into those paid by time $t$ and those due after time $t$. Accumulation to time $t$ converts past cash flows to a common date; conditional expected present values describe future cash flows per policy in force at that date.

Write the issue-date balance.
Under equivalence, expected present value of all premiums minus expected present value of all benefits is zero.

Separate past from future.
Past net premiums and benefits, accumulated to $t$, have issue-date value $v^t E[A_t]$. Future net premiums minus benefits have value $-v^t p_t V_t$, because the policy must survive to $t$ before its conditional future cash flows apply.

Use the zero balance.
The sum of these two issue-date values is zero. Since $v^t$ is positive, this gives $E[A_t]=p_tV_t$.

Allocate over survivors.
Divide by $p_t$, the probability of being in force, to obtain the expected accumulated past net cash flow per policy still in force.

0=v^t E[A_t]-v^t p_t V_t \quad\Longrightarrow\quad V_t=\frac{E[A_t]}{p_t}.

This is the short derivation of reserve equivalence. The first term is the issue-date expected value of the past cash-flow balance; the second is the issue-date expected value of the future net outgo, conditional on reaching time $t$. Equivalence sets their difference to zero. The common factor $v^t$ cancels, leaving the accumulated past balance per survivor equal to the prospective reserve.

Theorem: If the premium is determined by the equivalence principle, the retrospective and prospective reserves agree at a valuation time when both use the same benefits, premiums, interest and survival assumptions, and the same cash-flow timing. The theorem is an accounting identity under those assumptions; changing the basis or timing in only one calculation breaks the argument.

Worked Calculations

In each example, show the two views separately. A useful check is to track the survival probability used to move between an original group and the policies that remain in force.

Worked Example: Two-Year Term Insurance With Unequal Premiums

A hypothetical two-year term policy pays a benefit $B$ at the end of the year of death. Premiums are \$300 at time 0 and \$100 at time 1 if the insured is alive. The conditional death probabilities are 0.20 in year 1 and 0.50 in year 2. The effective annual interest rate is $i=0.25$, so $v=1/1.25=0.80$. Find $B$ by equivalence and the reserve at time 1 immediately after the renewal premium. Use the method of equating expected present values, then verify the reserve retrospectively.

Equivalence calculation. The time-0 premium is certain. The time-1 premium is paid with probability $0.80$, and is discounted one year. Thus

\operatorname{EPV}(\text{premiums}) =300+(0.80)(100)(0.80)=364.

The first-year death benefit is paid with probability $0.20$ at time 1. The second-year death benefit is paid with probability $(0.80)(0.50)=0.40$ at time 2. Therefore

\operatorname{EPV}(\text{benefits}) =B\big[(0.20)(0.80)+(0.80)(0.50)(0.80)^2\big] =B(0.160+0.256)=0.416B.

Equating the two expected present values gives $0.416B=364$, so $B=364/0.416=\$875$.

Prospective reserve at time 1. Immediately after the renewal premium, the only possible future claim is a year-2 death benefit. Conditional on being alive at time 1, its probability is 0.50 and it is paid one year later. No further premium is due. Hence

{}_1V_x=(0.50)(875)(0.80)=\$350.

Retrospective check. For each original policy, the time-0 premium accumulates to $300(1.25)=\$375$ by time 1. The time-1 premium contributes an expected $(0.80)(100)=\$80$ to the original group’s accumulated cash flow. First-year claims subtract $(0.20)(875)=\$175$. The expected accumulated past net cash flow is therefore $375+80-175=\$280$. Since $p_1=0.80$, the retrospective reserve is $280/0.80=\$350$, matching the prospective value. All amounts here are exact to the cent. The calculation avoids the mistake of discounting a time-1 premium as though it were paid at time 0, or of omitting the first-year claim from the past balance.

Worked Example: Endowment-Style Benefit and a Renewal Premium

A hypothetical two-year contract pays \$1,500 at time 1 if death occurs in year 1. If the insured survives year 1, it pays \$1,500 at time 2, regardless of what happens during year 2. Level premiums $P$ are payable at times 0 and 1 while the insured is in force. The conditional death probability is 0.20 in each year, and $i=0$. Find the net premium and the reserve at time 1 after the renewal premium. Apply equivalence first, then compare the prospective and retrospective reserves.

The probability of a first-year claim is 0.20. The time-2 payment occurs whenever the insured survives year 1, an event of probability 0.80; the contract specifies that this payment is not conditional on survival through year 2. With zero interest, the expected present value of benefits is

$$ (0.20)(1{,}500)+(0.80)(1{,}500)=300+1{,}200=1{,}500. $$

The time-0 premium is certain, and the time-1 premium is payable with probability 0.80. Thus the expected present value of premiums is $P+0.80P=1.80P$. Equivalence gives $P=1{,}500/1.80=\$833.33$ to the nearest cent (the unrounded value is \$833.333\ldots).

Prospective reserve. For a policy in force at time 1 after its renewal premium, the future time-2 payment of \$1,500 is certain under the contract wording, and no further premium is payable. Therefore ${}_1V_x=\$1,500$.

Retrospective check. With $i=0$, the expected accumulated premiums received from each original policy are $P+0.80P=1.80P=\$1,500$. The first-year claims paid from the original group total an expected $(0.20)(1{,}500)=\$300$. The expected accumulated past net cash flow is $1{,}500-300=\$1,200$. Divide by the probability $0.80$ of remaining in force at time 1: $1{,}200/0.80=\$1,500$. The two methods agree. A common error is to treat the time-2 payment as payable only if the insured survives year 2; that would contradict the stated benefit condition and change the expected value.

Worked Example: Three-Year Term Insurance at Positive Interest

A hypothetical three-year term policy pays \$10,000 at the end of the year of death. Conditional death probabilities in years 1, 2 and 3 are 0.10, 0.20 and 0.25. A level premium $P$ is paid at times 0, 1 and 2 while the policy is in force. The effective annual interest rate is $i=0.10$, so $v=10/11$. Find $P$, then calculate the reserve at time 1 immediately after the renewal premium by both methods. Round final dollar amounts to the nearest cent.

The probabilities of being in force at times 1 and 2 are $0.90$ and $(0.90)(0.80)=0.72$. Unconditional probabilities of death in years 1, 2 and 3 are $0.10$, $(0.90)(0.20)=0.18$, and $(0.72)(0.25)=0.18$. The expected present value of benefits is

10{,}000\big[(0.10)v+(0.18)v^2+(0.18)v^3\big] =10{,}000\left(\frac{1}{11}+\frac{18}{121}+\frac{180}{1331}\right) =10{,}000\left(\frac{499}{1331}\right) =\$3{,}749.8122\ldots .

The premium expected present value is $P[1+0.90v+0.72v^2]$. The bracketed factor is $1+9/11+72/121=292/121$. Equivalence therefore gives

P=\frac{10{,}000(499/1331)}{292/121} =\frac{4{,}990{,}000}{3212} =\$1{,}553.5492\ldots,

so the premium is \$1,553.55 to the nearest cent.

Prospective reserve at time 1. Conditional on being in force then, the year-2 death probability is 0.20, and the year-3 death probability is $(0.80)(0.25)=0.20$. A premium is due at time 2 only if the policy survives year 2. Consequently,

{}_1V_x =10{,}000\big[(0.20)v+(0.20)v^2\big] -P(0.80)v =10{,}000\left(\frac{2}{11}+\frac{20}{121}\right) -P\left(\frac{8}{11}\right).

The benefit value is $10{,}000(42/121)=\$3,471.0744\ldots$. The future premium value is $(8/11)(1{,}553.5492\ldots)=\$1,129.8540\ldots$. Thus the prospective reserve is $\$2,341.22$ to the nearest cent.

Retrospective check. At time 1, the time-0 premium has accumulated to $1.10P$. The time-1 premium is paid with probability 0.90 and has no time to accumulate, so its expected contribution for the original group is $0.90P$. First-year claims subtract an expected $0.10(10{,}000)=\$1,000$. Therefore

E[A_1]=1.10P+0.90P-1{,}000 =2P-1{,}000 =\$2{,}107.0984\ldots .

Dividing by $p_1=0.90$ gives ${}_1V_x=\$2{,}341.22$ to the nearest cent, as in the prospective calculation. The method avoids treating the renewal premium as certain across the original cohort: only 90% of that cohort is expected to pay it.

Common Traps and a Practical Check

Reserve equivalence is a statement about matched cash flows and assumptions, not a license to mix conventions. Before accepting a calculation, check which premiums and benefits belong to each side of the valuation date and whether each is conditional on survival. Then confirm that the same interest and mortality assumptions were used in both methods.

Wrong valuation instant. A reserve immediately before a premium includes that premium among future premiums; a reserve immediately after it does not. State the instant explicitly.
Forgetting survival conditioning. A future premium payable only while in force is weighted by the probability of reaching its payment date. A future benefit conditional on being in force at $t$ uses conditional probabilities from $t$, not unconditional probabilities from issue.
Dividing by the wrong cohort size. The retrospective amount is an expected accumulated balance per original policy until it is divided by $p_t$. Dividing by one, or by a later survival probability, does not produce the value per policy in force at $t$.
Using different bases. The derivation starts with zero expected present value of premiums minus benefits. If the premium was not set by equivalence on the basis used for the reserve, this zero-balance step is unavailable.
Omitting past claims. The retrospective accumulated balance subtracts benefits already paid, accumulated to the valuation time. A past claim is not a future outgo in the prospective calculation.

A compact exam check is to write the issue-date balance, split it at the valuation time, and make the survival factor visible. If the calculation does not reduce to $E[A_t]=p_tV_t$, look for a missing cash flow, a timing mismatch, or a survival probability applied at the wrong point. This treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, state the timing convention and use the given assumptions consistently. Dollar answers are rounded to the nearest cent where needed.

A two-year term policy pays \$1,200 at the end of the year of death. Conditional death probabilities are 0.25 in year 1 and 0.40 in year 2; $i=0$. Level premiums are payable at times 0 and 1 while in force. Find the net premium. (Answer: \$377.14)
A two-year term policy pays \$2,000 at the end of the year of death. Conditional death probabilities are 0.10 and 0.30, and $i=0$. Level premiums are paid at times 0 and 1 while in force. Find the net premium and the reserve at time 1 immediately after the renewal premium. (Answer: premium \$389.47; reserve \$600.00)
A three-year term policy pays \$5,000 at the end of the year of death. Conditional death probabilities are 0.20, 0.25 and 0.40; $i=0.05$. Level premiums are paid at times 0, 1 and 2 while in force. Find the net premium and the prospective reserve at time 1 immediately after the renewal premium. (Answer: premium \$1,255.79; reserve \$1,696.74)
A two-year contract pays \$1,000 at time 1 on first-year death, and pays \$1,000 at time 2 if the insured survived year 1, regardless of the year-2 outcome. Conditional death probabilities are 0.20 in year 1 and 0.25 in year 2; $i=0$. Level premiums are paid at times 0 and 1 while in force. Find the net premium and both reserve values at time 1 immediately after the renewal premium. (Answer: premium \$555.56; prospective and retrospective reserves \$1,000.00)

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