When Prospective and Retrospective Reserves Agree

When Should the Two Reserve Methods Agree?

A prospective reserve values future cash flows from a valuation date. A retrospective reserve accumulates past cash flows to that date. They describe the same policy value from different directions, but they agree only when they use the same contract cash flows, assumptions, and timing conventions. A discrepancy can reflect inconsistent calculations rather than a difference in the underlying liability.

The central check is a balance equation: the reserve already held, together with the net cash flow at the start of the year, accumulates with interest to the expected value of claims and reserves at year-end. Omitting the opening reserve changes that equation and can produce an incorrect result even when all other inputs match.

Definition: Let $V_k$ be the reserve at time $k$, immediately before the beginning-of-year premium and expense. Let $P_k$ be the premium received then, $e_k$ the expense paid then, $B_{k+1}$ the benefit paid at the end of the year on death, $i$ the effective annual interest rate, $q_k$ the probability of death during the year, and $p_k=1-q_k$ the probability of survival. If a survivor has reserve $V_{k+1}$ at time $k+1$, the expected balance equation is $$ V_k+P_k-e_k=v\left(q_kB_{k+1}+p_kV_{k+1}\right), \qquad v=\frac{1}{1+i}. $$ Equivalently, $$ p_kV_{k+1}=(V_k+P_k-e_k)(1+i)-q_kB_{k+1}. $$

This equation uses a particular timing convention: the reserve is measured before the start-of-year premium and expense; those cash flows occur at time $k$; and death claims and survivor reserves are valued at time $k+1$. If a contract pays a benefit at another time, or charges an expense at another time, its amount must be valued at that actual time. The displayed equation is not a timing-free formula.

Why the Balance Equation Gives Equivalence

For a life aged $x+k$, write $q_k={}_1q_{x+k}$ and $p_k={}_1p_{x+k}$. On a fixed actuarial basis, the prospective reserve at time $k$ is the expected present value of future outgo less the expected present value of future income. Future outgo includes benefits and expenses; future income includes premiums. The premium and expense at time $k$ are included because the reserve is defined immediately before those cash flows.

Separate the cash flows during the next year from those after time $k+1$. At the end of the year, a death claim $B_{k+1}$ is paid with probability $q_k$. With probability $p_k$, the policy survives and has reserve $V_{k+1}$. Discounting those two possible year-end amounts to time $k$, while accounting for the premium and expense at time $k$, gives the balance equation.

V_k=P_k-e_k+v\left(q_kB_{k+1}+p_kV_{k+1}\right).

Rearranging gives the recursive form in the definition. Retrospectively, the same equation says that the opening reserve plus the net cash flow is accumulated through the year; death claims are then removed, and the remaining expected fund is assigned to survivors. Thus the two methods follow the same recursion when they use the same cash flows and assumptions.

For an issue-date policy with $V_0=0$, matching the same initial reserve and applying the same annual recursion at each duration gives matching reserves at subsequent dates. More generally, equality requires the two methods to start from the same $V_0$. This is why the opening reserve cannot be dropped merely because the calculation is retrospective.

Fix the valuation date.
Specify whether the reserve is measured before or after the premium, expense, or other cash flow due on that date.

List the contract cash flows.
Identify premium amounts and timing, expenses, death benefits, maturity benefits, and any changes by duration.

Use one consistent basis.
Use compatible interest and mortality assumptions in both calculations, and use the same benefit and premium patterns.

Carry forward the opening reserve.
Include $V_k$ when accumulating the year’s net cash flow. Compare the resulting balance with the prospective value at the same date.

Assumptions and Conventions That Must Match

“Same basis” means more than using the same interest rate. The future cash flows being valued and the past cash flows being accumulated must describe the same policy and be measured at compatible dates.

Item	What must be consistent	Why it matters
Interest	Use the same effective accumulation and discount basis over corresponding periods.	A different rate changes both the accumulation of past amounts and the present value of future amounts.
Mortality	Use compatible probabilities of death and survival for each duration.	Mortality affects expected claims and the number of survivors over which a retrospective fund is allocated.
Benefits	Use the same amounts, contingencies, and payment dates, including maturity or changing benefits.	A benefit paid at year-end is not valued like the same benefit paid at the beginning of the year.
Premiums	Match amounts, duration, payment frequency, and whether a premium due at the valuation date is included.	Counting or omitting a premium changes the prospective liability and the retrospective accumulation.
Expenses	Match amounts, timing, and whether expenses are included in the reserve basis.	An expense is an outflow; ignoring it or placing it at the wrong date changes the balance.
Valuation date	Compare reserves at the same duration and on the same side of any cash flow.	A reserve just before a premium need not have the same value as one just after it.

The mortality probabilities in a retrospective calculation also need interpretation. On a deterministic actuarial basis, expected death and survival proportions are used. With an observed cohort, actual claim counts and actual survivors may differ from those expected proportions. A retrospective amount based on actual experience need not equal a prospective reserve calculated using the original assumed basis. Agreement is a result under consistent assumptions and experience, not a claim that outcomes always match expectations.

Worked Example: Matching Reserves for a Two-Year Term Policy

A hypothetical two-year term policy pays $100$ at the end of the year of death. The annual effective interest rate is $25\%$. The one-year probabilities are $q_0=0.20$, $p_0=0.80$, $q_1=0.25$, and $p_1=0.75$. A level premium $P$ is paid at times 0 and 1 while the insured is alive. There are no expenses. Find the net premium and compare the prospective and retrospective reserves at time 1, immediately before the second premium.

Method. First use the prospective equivalence principle at issue to find $P$. Then value the time-1 reserve in both directions, using the same interest, mortality, benefit, and premium timing.

The discount factor is $v=1/1.25=0.8$. The expected present value of benefits at issue is

$$ 0.20(100)(0.8)+0.80(0.25)(100)(0.8^2) =16+12.8=28.8. $$

The premium at time 0 is certain, and the premium at time 1 is paid only if the insured survives the first year. Therefore the expected present value of premiums is

$$ P(1+0.80(0.8))=1.64P. $$

Equating the two present values gives $1.64P=28.8$, so $P=28.8/1.64=17.5609756\ldots$.

At time 1, immediately before the second premium, the prospective value of the remaining death benefit is $0.25(100)(0.8)=20$. The premium due at time 1 is future income at this valuation instant, so

V_1^{\mathrm{pros}}=20-P=20-17.5609756\ldots=2.4390244\ldots.

For the retrospective calculation, the time-0 premium accumulates to $1.25P=21.9512195\ldots$ at time 1. Per original policy, expected first-year claims cost $0.20(100)=20$. The remaining amount per original policy is $21.9512195\ldots-20=1.9512195\ldots$. Divide by the $0.80$ expected survivors to obtain the amount per survivor:

V_1^{\mathrm{ret}}= \frac{1.25P-0.20(100)}{0.80} =\frac{21.9512195\ldots-20}{0.80} =2.4390244\ldots.

The reserves agree. As a check, the time-0 balance is $P=v(0.20(100)+0.80V_1)=0.8(20+0.80(2.4390244\ldots))=17.5609756\ldots$, the premium found above. The common mistake avoided here is omitting the premium due at time 1 from the prospective value or counting it as already accumulated in the retrospective fund.

The Opening Reserve Is Part of the Fund

The issue-date condition $V_0=0$ is common for a policy valued at issue under a suitable premium basis. It is not a general condition at every duration. If a policy already has a reserve, that amount remains part of the fund available to support future claims. The retrospective balance must therefore accumulate $V_k+P_k-e_k$, not just the current net contribution $P_k-e_k$.

Worked Example: Carrying a Nonzero Opening Reserve

At the start of a year, immediately before premium and expense, a hypothetical policy has reserve $V_k=40$. The premium is $12$, the expense is $2$, the annual effective interest rate is $5\%$, the death probability is $q_k=0.10$, and the death benefit paid at year-end is $30$. Find the reserve for a survivor at year-end, $V_{k+1}$.

Method. Apply the expected balance equation, including the opening reserve. The survival probability is $p_k=1-0.10=0.90$. The amount available at the start of the year is $40+12-2=50$, which accumulates to $50(1.05)=52.5$. Expected death claims per policy in force at the start of the year are $0.10(30)=3$. Thus the amount assigned to survivors is $52.5-3=49.5$, and

V_{k+1}=\frac{49.5}{0.90}=55.

Check the discounted balance: $v(0.10(30)+0.90(55))=(1/1.05)(3+49.5)=50$, which equals $V_k+P_k-e_k$. If the opening reserve were incorrectly omitted, the accumulated amount would be $(12-2)(1.05)=10.5$, giving a different and invalid balance for this policy. The formula using only the current contribution applies only when the opening reserve is zero.

Expenses and the Side of the Valuation Date

An expense belongs in both methods if it is part of the reserve basis. If it is incurred at the start of the year, subtract it with the start-of-year premium before accumulating the net amount. If it is payable later, its value must reflect that later timing. The same principle applies to premiums: a premium due immediately after the valuation date cannot be treated as already held in the reserve at that date.

Worked Example: Matching a Reserve with an Initial Expense

A hypothetical one-year term policy has no opening reserve. At time 0, it collects a gross premium $P=12$ and incurs an acquisition expense of $e_0=2$. The effective annual interest rate is $10\%$. The death probability is $q_0=0.20$, and the death benefit of $55$ is paid at year-end. There are no other benefits or expenses. Find the end-of-year reserve for a survivor and check the issue-date prospective reserve.

Method. Treat the premium and acquisition expense as time-0 cash flows, then accumulate their net amount to year-end. The net contribution is $12-2=10$, and its accumulated value is $10(1.10)=11$. Expected claims per policy at issue are $0.20(55)=11$. Therefore the expected fund remaining for survivors is $11-11=0$, so $V_1=0/0.80=0$.

At issue, the present value of the death benefit is $0.20(55)/1.10=10$. The premium less the time-0 expense is also $12-2=10$, so the prospective reserve is $10-10=0$. The two calculations agree. The check would fail if the expense were omitted from one method or incorrectly moved to year-end in the other.

Common Mistakes and Practical Checks

A reserve comparison is most reliable when the cash-flow timeline is written before any arithmetic. In particular, state whether the policy is in force at the valuation date, which premium is due then, and whether the reserve is just before or just after that payment.

Dropping the reserve already held. The general accumulated amount starts with $V_k+P_k-e_k$. Replacing it with $P_k-e_k$ is valid only when $V_k=0$.
Mixing bases. Using one mortality basis for expected claims and another for the survivor count generally breaks the balance.
Changing a cash-flow pattern. A level premium, a varying premium, or a benefit that changes by duration must be represented identically in both methods.
Ignoring expenses in one calculation. If expenses are included in the reserve basis, they are outgo and must be included at their specified times.
Comparing different instants. A reserve before a premium and a reserve after it are different quantities; label the instant explicitly.
Treating expected experience as actual experience. A realized cohort can have different deaths, interest, or expenses from the valuation assumptions, so its accumulated fund can differ from the prospective value on the assumed basis.

For exam-style work aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus, the balance equation provides a compact audit. Substitute the opening reserve, net current cash flow, interest accumulation, expected claims, and survivor reserve. If both methods use the same conventions, the two sides must agree.

Check Your Understanding

For each problem, identify the applicable timing and basis before calculating.

A one-year hypothetical term policy has $V_0=0$, $i=5\%$, $q_0=0.10$, death benefit $80$ payable at year-end, and a time-0 expense of $1$. A single time-0 premium funds the policy, and $V_1=0$. Find the premium. (Answer: $8.62$)
At the start of a year, $V_k=25$. A premium of $6$ and expense of $1$ are paid then; $i=4\%$, $q_k=0.20$, and the year-end death benefit is $20$. Find $V_{k+1}$ for a survivor. (Answer: $34$)
A two-year term policy pays $50$ at the end of the year of death. The effective annual interest rate is $5\%$, with $q_0=0.20$, $p_0=0.80$, $q_1=0.25$, and $p_1=0.75$. A level premium is due at times 0 and 1 while alive; there are no expenses. Find the net premium and the reserve at time 1 immediately before the second premium. (Answer: net premium $10.55$; reserve $1.35$)
A one-year term premium of $10$ was priced using $i=0$, $q_0=0.10$, and a death benefit of $100$. There are no expenses. Suppose actual first-year mortality is $0.15$, while the premium and benefit are unchanged. Using actual expected cohort experience and $i=0$, find the retrospective amount per survivor at year-end; the prospective terminal reserve for a survivor is zero. (Answer: $-5.88$)

What You'll Learn