Retrospective Reserve Checks and Bounds

Why Check a Retrospective Reserve?

A retrospective reserve is built from the past: premiums and other funds accumulated to the valuation date, less claims already paid. That makes it possible to check a calculation in two complementary ways: reconcile the fund directly from its cash flows, and compare the resulting fund per survivor with what the reserve represents. A result can satisfy an equation and still be implausible if the wrong number of survivors, a mistimed payment, or an inconsistent mortality assumption was used.

The checks below use a simple annual model. They are designed to identify errors, not to replace careful specification of the policy and the valuation date. In particular, premiums paid at the beginning of a year and claims paid at its end must not be treated as if they occur at the same time.

Definition: Let $n_k$ be the number of policies in force immediately before the premium due at time $k$, and let $F_k$ be the aggregate fund at that time. Suppose each in-force policy pays an annual premium $P$ immediately after time $k$; interest is earned at effective annual rate $i$; $d_k$ claims of amount $B$ are paid at the end of the year; and there are no expenses. Then $$ F_{k+1}=(F_k+n_kP)(1+i)-d_kB. $$ If $n_{k+1}=n_k-d_k>0$, the retrospective reserve per policy in force immediately before the next premium is ${}_{k+1}V_x=F_{k+1}/n_{k+1}$, with the subscript understood to identify the policy duration and valuation basis.

This balance assumes that the premium is paid at the beginning of the year, earns a full year of interest, and that the claim is paid at year-end. If premiums are paid at year-end, claims are paid earlier, or the fund earns interest over different periods, the cash-flow equation must be adjusted. A reserve check based on the wrong timing convention can produce a neat but incorrect answer.

Reconcile the Fund Before Dividing by Survivors

A useful first check is to calculate the aggregate fund independently in two ways: apply the annual fund equation successively, then list the contributions and claims accumulated to the same valuation date. Only after the aggregate fund reconciles should it be divided by the number of policies still in force. The reserve is per surviving policy, not per original policy.

Worked Example: Two-Year Cohort Cash-Flow Reconciliation

A hypothetical cohort begins with 1,000 policies. The annual premium is $1,200 per policy, paid at the beginning of each year. The death benefit is $10,000, paid at the end of the year of death. The effective annual interest rate is 5%. There are 100 deaths in year 1 and 90 deaths in year 2. There are no expenses, and the fund at time 0 is zero. Find the retrospective reserve per policy at time 2, immediately before the third premium would be due.

Method. Reconcile the fund year by year, then verify the result by accumulating each cash flow to time 2. The number in force at the start of year 2 is $1{,}000-100=900$; at time 2 it is $900-90=810$.

At time 0, the 1,000 premiums produce $1,200,000. After one year of interest, the fund is $1,200,000 × 1.05 = $1,260,000. The 100 year-1 claims total $1,000,000, so:

F_1=1{,}260{,}000-1{,}000{,}000=260{,}000.

At the beginning of year 2, 900 premiums add $900(1{,}200)=\$1{,}080{,}000$. Accumulating that combined amount for the year and subtracting the year-2 claims gives:

F_2=(260{,}000+1{,}080{,}000)(1.05)-90(10{,}000) =1{,}407{,}000-900{,}000 =507{,}000.

The independent time-2 reconciliation is: the first-year premiums accumulate for two years, the second-year premiums for one year, and the two sets of claims are accumulated from their payment dates to time 2.

1{,}200{,}000(1.05)^2+1{,}080{,}000(1.05) -1{,}000{,}000(1.05)-900{,}000 =1{,}323{,}000+1{,}134{,}000-1{,}050{,}000-900{,}000 =507{,}000.

There are 810 policies in force at time 2, so the reserve is $507{,}000/810=\$625.9259\ldots$, or $625.93 per policy to the nearest cent. The direct fund calculation and the accumulated-cash-flow calculation agree.

Check and common mistake. Dividing $507,000 by the original 1,000 policies would give $507.00, but that is not a reserve per policy in force at time 2. The denominator must be 810 survivors.

When a Prospective Comparison Is Valid

A prospective reserve is the present value of future benefits less the present value of future premiums, conditional on a policy being in force at the valuation date. Comparing it with a retrospective reserve is a strong check only when both calculations use the same benefit, premium, interest, mortality, and payment-timing assumptions. For a realized cohort, the observed number of deaths must also be consistent with the mortality counts used in the reconciliation; otherwise, an experience difference can explain a difference in values.

The reason for equivalence is a cash-flow identity. If a net premium is set by the equivalence principle, the present value at issue of premiums equals the present value at issue of benefits. Moving the past cash flows forward to the valuation date leaves the fund accumulated from past premiums less past claims. The same balance can be written as the present value of the remaining benefits less the remaining premiums for the policies still in force. Both descriptions value the same policy cash flows on the same basis.

Worked Example: Prospective and Retrospective Values on One Basis

A hypothetical two-year term policy pays $10,000 at the end of the year of death. The first-year death probability is $q_0=0.10$, and, conditional on surviving the first year, the second-year death probability is $q_1=0.20$. Premiums are level, paid at the beginning of each year while the policy is in force. Interest is 5% effective annually, and there are no expenses. Find the equivalence premium and compare the time-1 reserve immediately before the second premium under both methods. For the cohort calculation, use 1,000 policies, with deaths matching the assumed probabilities in each year.

Method. First equate the issue-date expected present values to find the net premium. Then calculate the time-1 prospective value and independently reconcile the time-1 cohort fund.

Write $v=1/1.05=20/21$. The expected present value of benefits per policy is the first-year death benefit discounted one year plus the second-year death benefit discounted two years, allowing for first-year survival:

10{,}000\left(0.10v+0.90(0.20)v^2\right) =10{,}000\left(\frac{38}{147}\right).

The expected present value of premiums per dollar of annual premium is $1+0.90v=13/7$: every policy pays at time 0, and only first-year survivors pay at time 1. Thus:

P=\frac{10{,}000(38/147)}{13/7} =\frac{380{,}000}{273} =\$1{,}391.94

to the nearest cent. At time 1, a surviving policy has a 20% chance of a $10,000 claim at time 2. The second premium of $P$ is due immediately at time 1. Therefore, the prospective reserve immediately before that premium is:

{}_{1}V_x=10{,}000(0.20)v-P =\frac{2{,}000}{1.05}-\frac{380{,}000}{273} =\$512.82

to the nearest cent. For the retrospective calculation, 100 of the 1,000 policies die in year 1, leaving 900. The time-1 fund is:

F_1=1{,}000P(1.05)-100(10{,}000) =1{,}461{,}538.46-1{,}000{,}000 =461{,}538.46.

Dividing by the 900 surviving policies gives $F_1/900=\$512.82$ per policy, matching the prospective result. The agreement checks the premium, timing, survivor count, and valuation basis together.

Check and common mistake. The time-1 premium is due immediately and is included in the prospective value as a subtraction of $P$. Omitting it would give a value of $1,904.76, not $512.82, and would fail to match the retrospective fund.

Signs, Magnitudes, and Adjacent Durations

When the cohort fund is divided by survivors, the annual balance gives a direct relation between reserves at adjacent durations. Let $q_k=d_k/n_k$ and $p_k=1-q_k=n_{k+1}/n_k$. Under the annual timing assumptions already stated, divide the fund equation by $n_{k+1}$:

{}_{k+1}V_x =\frac{({}_{k}V_x+P)(1+i)-q_kB}{p_k}.

This relation explains how to interpret changes in the reserve. The amount $({}_{k}V_x+P)(1+i)$ is the accumulated reserve and premium per policy at the start of the year; $q_kB$ is the expected claim cost per starting policy during the year. The remaining fund is shared among the fraction $p_k$ still in force. The relation is a check on an adjacent-duration calculation, not a claim that reserves must always rise or must always be positive.

Bound and sign checks: If $p_k>0$, then ${}_{k+1}V_x\geq 0$ exactly when $q_kB\leq({}_{k}V_x+P)(1+i)$. Also, ${}_{k+1}V_x\leq{}_{k}V_x$ exactly when $q_k(B-{}_{k}V_x)\geq({}_{k}V_x+P)(1+i)-{}_{k}V_x$. These tests follow by multiplying the adjacent-duration equation by the positive quantity $p_k$ and rearranging. They depend on the stated timing and benefit assumptions.

Worked Example: Checking a Decline Between Adjacent Reserves

At the start of a year, a hypothetical cohort has 800 policies and a retrospective reserve of $450 per policy. Each in-force policy pays a $300 premium immediately. The effective annual interest rate is 4%. During the year, 64 policies die and each receives a $5,000 benefit at year-end. There are no expenses. Calculate the next reserve and assess its sign and direction of change.

Method. Apply the cohort fund balance, divide by the number of survivors, and use the adjacent-duration inequalities as independent plausibility checks. The starting aggregate fund is $800(450)=\$360{,}000$, and the premiums add $800(300)=\$240{,}000$. Thus the amount accumulated before claims is:

(360{,}000+240{,}000)(1.04)=\$624{,}000.

The claims total $64(5{,}000)=\$320{,}000$, leaving $F_{k+1}=624{,}000-320{,}000=\$304{,}000$. There are $800-64=736$ survivors, so:

{}_{k+1}V_x=\frac{304{,}000}{736} =\$413.0434\ldots =\$413.04

to the nearest cent. The reserve is positive because $q_k=64/800=0.08$, and $q_kB=\$400$ is less than $(450+300)(1.04)=\$780$. It is also below the starting $450. For the decline check, $q_k(B-{}_{k}V_x)=0.08(5{,}000-450)=\$364$, while $({}_{k}V_x+P)(1+i)-{}_{k}V_x=780-450=\$330$. Since $364\geq330$, a decline is consistent with the stated cash flows.

Check and common mistake. The $320,000 of claims is subtracted after interest is credited because the benefits are paid at year-end. Subtracting the claims before accumulating the premiums and reserve would apply interest to cash that has already been paid out and would not match the contract timing.

Practical Reserve-Checking Sequence

Fix the valuation instant.
State whether the reserve is before or after the premium, and identify when claims and interest are applied.

Reconcile the aggregate fund.
Track the number in force, premiums collected, interest earned, and claims paid. Where useful, independently accumulate each cash flow to the valuation date.

Use the correct denominator.
Divide the fund by policies in force at the valuation instant, not by the original cohort unless everyone remains in force.

Check direction and comparability.
Use the adjacent-duration balance to test sign and direction. Compare with a prospective value only if the assumptions and timing are aligned.

A negative retrospective fund is not automatically an arithmetic error: it can indicate that accumulated premiums and interest were insufficient to cover claims under the observed experience. Likewise, a reserve need not increase at every duration. The benefit pattern, premium pattern, mortality, and valuation timing all affect the balance. The useful question is whether the result follows from those inputs and reconciles to the cohort cash flows.

Check Your Understanding

Use the stated annual timing: premiums at the beginning of the year, claims at year-end, and the reserve measured immediately before the next premium.

A hypothetical cohort has 500 policies, a $400 annual premium, a $3,000 death benefit, and a 5% effective annual interest rate. There are 50 deaths during the year and the starting fund is zero. Find the next reserve per survivor. (Answer: $133.33)
At a valuation date, the aggregate fund is $120,000 for 400 policies in force. Each pays a $250 premium immediately; interest is 3%; 20 claims of $4,000 are paid at year-end. Find the next reserve per survivor. (Answer: $385.79)
A one-year hypothetical term policy has death probability 0.08, a $12,000 end-of-year benefit, and 4% effective annual interest. A single premium is paid at issue. Find the net equivalence premium and the reserve for a survivor at the end of the year. (Answer: Premium $923.08; end-of-year reserve $0.00)
For one year, ${}_{k}V_x=\$200$, $P=\$300$, $i=0.04$, $q_k=0.10$, and ${}_{k+1}V_x=\$400$. Under the stated timing and a death benefit $B$, find the implied benefit amount. (Answer: $1,600)
A hypothetical cohort has ${}_{k}V_x=\$300$, $P=\$200$, $i=0.05$, $q_k=0.10$, and $B=\$4{,}000$. Calculate ${}_{k+1}V_x$ and state whether it rises or falls. (Answer: $138.89; it falls)

What You'll Learn