Infer Level Premiums from Reserve Recursion Data

Why Infer a Premium from Reserves?

A reserve sequence records how much an insurer holds at successive policy durations. When the annual premium is not stated, the reserve recursion can recover the premium implied by one year of data. If several consecutive reserve intervals are available, each gives an independent check: a genuine level premium must be consistent with all of them under the stated mortality, interest, benefit, and payment-timing assumptions.

The calculation is only meaningful after the reserve convention is fixed. This tutorial uses reserves measured at the start of each policy year, immediately before that year's premium is paid. The standard case has a premium paid at the start of the year while the insured is alive, interest earned through the year, and a death benefit paid at year-end if death occurs during the year. The formulas change if cash-flow timing changes.

Definition: Let $V_t$ be the reserve at duration $t$, immediately before the premium due at that time. Let $P$ be the level annual premium, $B$ the death benefit payable at the end of the year of death, and $i$ the effective annual interest rate. For the year from duration $t$ to $t+1$, let $q_t={}_1q_{x+t}$ and $p_t={}_1p_{x+t}=1-q_t$. In the standard premium-at-start convention, the one-year reserve recursion is $$ p_t V_{t+1}=(V_t+P)(1+i)-q_tB. $$

The formula assumes annual discrete benefits and premiums, a fixed benefit $B$, and a reserve defined consistently at both durations. It also assumes that the stated $p_t$ and $q_t$ apply to the policy year in question. It is not the appropriate recursion for a continuous benefit, a premium payable only at year-end, a different reserve timing, or a policy with additional cash flows unless those features are included in the equation.

Solving for the Premium

Rearranging the recursion isolates the premium implied by one reserve interval:

P_t=\frac{p_tV_{t+1}+q_tB}{1+i}-V_t.

The subscript on $P_t$ emphasizes that an individual interval may imply its own value. For a level premium, every interval used must yield the same $P_t$, apart from any stated rounding in the data. The expression $p_tV_{t+1}+q_tB$ is the expected end-of-year amount that must be financed from the start-of-year fund, after the premium has been added.

Fix the cash-flow timing.
Check whether the reserve is measured before or after premium payment and when benefits and premiums are paid.

Use the mortality for that policy year.
Set $p_t=1-q_t$; do not substitute a probability from another duration.

Calculate the implied premium.
Substitute the two reserves, benefit, mortality, and interest into the rearranged recursion.

Check other intervals.
A level premium is supported only if the other intervals imply the same amount under the same conventions.

Why the Recursion Holds

At the start of the year, the insurer has reserve $V_t$ and receives premium $P$, giving $V_t+P$. After interest, the fund is $(V_t+P)(1+i)$. On death during the year, probability $q_t$, the insurer pays $B$, leaving no continuing policy reserve for that claim. On survival, probability $p_t$, the fund must provide the year-end reserve $V_{t+1}$. Equating the fund before the death benefit to the expected required amounts gives

(V_t+P)(1+i)=q_tB+p_tV_{t+1}.

Solving this equality for $P$ gives the premium formula above. This is an expected-value reserve recursion: the survival and death outcomes are weighted by their probabilities. It does not say that the insurer holds $V_{t+1}$ for a policy that has just ended in a death claim.

Worked Example: Detecting Inconsistent Reserve Intervals

A hypothetical annual-pay whole life contract has a death benefit of $100,000 payable at the end of the year of death. The effective annual interest rate is 5%. At duration 0, $V_0=\$4,000$, and at duration 1, $V_1=\$5,000$. The first-year death probability is 0.02. At duration 2, $V_2=\$6,000$, and the second-year death probability is 0.03. Determine the premium implied by each interval and whether one level premium fits both.

Method. Apply the premium-at-start recursion separately to each year. The death probabilities are $q_0=0.02$ and $q_1=0.03$, so the corresponding survival probabilities are $p_0=0.98$ and $p_1=0.97$.

Year 0 to 1. The expected year-end amount is

p_0V_1+q_0B =0.98(5{,}000)+0.02(100{,}000) =4{,}900+2{,}000 =6{,}900.

Discount this amount one year and subtract the initial reserve:

P_0=\frac{6{,}900}{1.05}-4{,}000 =6{,}571.428571\ldots-4{,}000 =\$2{,}571.43.

Year 1 to 2. The expected year-end amount is

p_1V_2+q_1B =0.97(6{,}000)+0.03(100{,}000) =5{,}820+3{,}000 =8{,}820.

Therefore,

P_1=\frac{8{,}820}{1.05}-5{,}000 =8{,}400-5{,}000 =\$3{,}400.00.

The two intervals imply different premiums: $2,571.43 and $3,400.00. Thus the supplied values are inconsistent with one level premium under the stated assumptions. The calculation avoids the common mistake of using $V_{t+1}$ without weighting it by survival probability.

Checking a Consistent Sequence

When reserve data come from a contract with a level premium, successive intervals should reproduce the same premium. This is more than a final arithmetic check: it tests whether the mortality rates, benefit amount, interest rate, and reserve timing can jointly explain the observed sequence.

Worked Example: Verifying One Level Premium Across Two Years

A hypothetical policy pays $10,000 at the end of the year of death. The effective annual interest rate is 25%, and the death probability is 0.20 in each of the first two years. Reserves immediately before premiums are $V_0=\$1,000$, $V_1=\$3,000$, and $V_2=\$6,125$. Find the premium implied by each year and check whether it is level.

The survival probability in both years is $p=1-0.20=0.80$. For the first interval,

pV_1+qB =0.80(3{,}000)+0.20(10{,}000) =2{,}400+2{,}000 =4{,}400,

P_0=\frac{4{,}400}{1.25}-1{,}000 =3{,}520-1{,}000 =\$2{,}520.

For the second interval,

pV_2+qB =0.80(6{,}125)+0.20(10{,}000) =4{,}900+2{,}000 =6{,}900.

Then

P_1=\frac{6{,}900}{1.25}-3{,}000 =5{,}520-3{,}000 =\$2{,}520.

Both intervals imply a premium of $2,520, so the reserve sequence is consistent with a level premium under the assumptions. As a direct check, in year 0 the end-of-year fund after the premium is $(1{,}000+2{,}520)(1.25)=4{,}400$; subtracting the expected death payment of $2,000 leaves $2,400, which equals $0.80(3{,}000)$. The same check for year 1 gives $(3{,}000+2{,}520)(1.25)-2{,}000=4{,}900=0.80(6{,}125)$.

When the Premium Is Paid at Year-End

A different timing convention requires a different equation. Suppose the reserve is still measured at the start of the year, but the premium is paid at the end of the year only if the insured survives. The fund at the start accumulates to $V_t(1+i)$. It pays $B$ on death, while on survival it must provide both the premium $P$ and the next reserve $V_{t+1}$. The recursion is

V_t(1+i)=q_tB+p_t(P+V_{t+1}), \qquad P=\frac{(1+i)V_t-q_tB}{p_t}-V_{t+1}.

This formula should not be used for a premium payable at the beginning of the year. In particular, the year-end premium is conditional on survival and is not added to $V_t$ before interest is earned.

Worked Example: Inferring a Survival-Contingent Year-End Premium

A hypothetical contract pays a death benefit of $100,000 at year-end on death. A premium is paid at year-end only if the insured survives. The reserve at the start of the year is $V_t=\$4,000$, the reserve at the next year-start is $V_{t+1}=\$1,000$, the effective annual interest rate is 5%, and $q_t=0.02$. Infer the premium.

Here $p_t=0.98$. The amount available at year-end before claims is $4{,}000(1.05)=\$4{,}200$, while the expected death benefit is $0.02(100{,}000)=\$2{,}000$. The recursion therefore gives

P=\frac{4{,}200-2{,}000}{0.98}-1{,}000 =\frac{2{,}200}{0.98}-1{,}000 =2{,}244.897959\ldots-1{,}000 =\$1{,}244.90.

Check by substituting the unrounded premium: $0.02(100{,}000)+0.98(1{,}244.897959\ldots+1{,}000)=2{,}000+2{,}200=\$4{,}200$, equal to $V_t(1+i)$. The common mistake avoided here is to add the premium to the initial reserve and accumulate it for the year, even though this premium is paid only at year-end by survivors.

Common Errors and Interpretation

An inferred premium is conditional on the entire model specification. A discrepancy between intervals can indicate inconsistent data, but it can also signal a mismatch in assumptions—for example, an incorrect benefit, a mortality rate assigned to the wrong year, or reserves quoted on a different timing basis. Do not average the interval premiums and call the result a level premium unless the problem explicitly asks for an estimate from noisy or rounded observations.

Omitting survival weighting. In the premium-at-start recursion, the next reserve contributes $p_tV_{t+1}$, not $V_{t+1}$, because it is held only for survivors.
Using the wrong mortality year. The transition from $V_t$ to $V_{t+1}$ uses the mortality for ages during that specific policy year.
Changing timing silently. A premium at the start of the year earns interest for the year; a premium paid at year-end does not.
Ignoring consistency checks. One interval identifies an implied premium; it cannot establish that the premium is level across multiple years.
Rounding too early. Keep intermediate values unrounded when comparing intervals, then round monetary answers to cents if requested.

These reserve-based calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The central exam skill is to translate the stated timing into the correct one-year equation before substituting values.

Check Your Understanding

For each problem, use the stated reserve convention and round the premium to the nearest cent.

A hypothetical policy pays $20,000 at year-end on death. With premium at the start of the year, $i=0.04$, $q_t=0.10$, $V_t=\$2,000$, and $V_{t+1}=\$3,000$, find the implied premium. (Answer: $2,519.23)
A $50,000 death benefit is payable at year-end, and the premium is paid at the start of the year. Given $i=0.05$, $q_t=0.02$, $V_t=\$1,000$, and $V_{t+1}=\$2,000$, find the implied premium. (Answer: $1,819.05)
A policy has $B=\$10,000$, $i=0.10$, $V_0=\$500$, $V_1=\$1,000$, and $V_2=\$2,000$. The first-year death probability is 0.10 and the second-year death probability is 0.20. Premiums are due at the start of each year. Find both implied premiums and state whether they are level. (Answer: Year 0 implies $1,227.27; year 1 implies $2,272.73; they are not level.)
A $10,000 benefit is payable at year-end on death, and a premium is payable at year-end only on survival. Given $i=0.04$, $q_t=0.10$, $V_t=\$3,000$, and $V_{t+1}=\$1,000$, find the implied premium. (Answer: $1,355.56)
For a $20,000 death benefit with premiums at the start of the year, $i=0.05$, $V_0=\$1,000$, $V_1=\$3,000$, and $V_2=\$4,000$, the first-year death probability is 0.10 and the second-year probability is 0.05. Find the premium implied by each interval and determine whether one level premium fits. (Answer: Year 0 implies $3,476.19; year 1 implies $1,571.43; they are not level.)

What You'll Learn