Retrospective Reserves for Level-Premium Contracts

Why Level-Premium Retrospective Reserves Need Careful Weighting

A level premium is the same dollar amount at each scheduled payment date, but it is not paid by every policy in the original group at every date: payment depends on the policyholder being alive. Similarly, death benefits are paid only to policies that end in death during a specified year. A retrospective reserve must account for both facts before it can describe the accumulated value associated with a policy that has survived to the valuation date.

The calculation has two stages. First, accumulate the premiums and benefits for the original cohort, weighting each cash flow by the probability that it occurs. Then divide the accumulated net amount by the probability that a policy from that cohort is still in force at the valuation date. This gives a reserve per surviving policy. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a discrete contract issued at age $x$, with annual effective interest rate $i$, annual level premium $P$ payable at the beginnings of years while the insured is alive, and death benefit $B$ payable at the end of the year of death. For an integer duration $t$, the retrospective reserve per policy in force is the accumulated value of past premiums less the accumulated value of past death benefits, for the original cohort, divided by ${}_tp_x$, the probability of surviving from age $x$ to age $x+t$. This setup assumes premiums and benefits are the only cash flows being considered, and that the same interest and mortality basis is used throughout.

The Level-Premium Retrospective Formula

Use the following timing convention: the premium for policy year $k+1$ is paid at time $k$, provided the insured is alive then. A death during that policy year generates a benefit at time $k+1$. Thus, by time $t$, a premium paid at time $k$ has accumulated for $t-k$ years, while a benefit paid at time $k+1$ has accumulated for $t-k-1$ years.

The probability of being alive to pay the premium at time $k$ is ${}_kp_x$. The probability of dying in policy year $k+1$ is ${}_kp_xq_{x+k}$. The latter equals ${}_kp_x-{}_{k+1}p_x$. These weights apply to the original cohort; the final division by ${}_tp_x$ conditions the accumulated net amount on survival to time $t$.

{}_tV_x^{(R)} = \frac{ P\displaystyle\sum_{k=0}^{t-1}{}_kp_x(1+i)^{t-k} - B\displaystyle\sum_{k=0}^{t-1}{}_kp_xq_{x+k}(1+i)^{t-k-1} }{{}_tp_x}.

Here ${}_tV_x^{(R)}$ denotes the retrospective reserve at duration $t$. For $t=0$, there are no past premiums or benefits, so the retrospective reserve is zero. For $t>0$, the premium sum includes payment at issue, $k=0$, if the contract is in force at issue. It does not include a premium at time $t$: that payment would be due at the valuation date, not part of the past cash flows being accumulated.

Why the Formula Works

Imagine a large group of identical policies issued at age $x$. For each original policy, the expected accumulated premiums through time $t$ equal the premium amount times the probability of being alive at each payment date, times the accumulation factor from that date to $t$. The expected accumulated benefits are found in the same way, using each year’s probability of death and the accumulation period after payment.

These are expected amounts per original policy. In the cohort, the proportion expected to remain in force at time $t$ is ${}_tp_x$. Dividing the cohort’s expected accumulated net amount by that proportion gives the amount per policy that has survived. This justifies both the survival weights inside the sums and the survival probability in the denominator.

List past premium dates.
For each $k=0,\ldots,t-1$, include $P$ multiplied by ${}_kp_x$.

Accumulate premiums to time $t$.
Multiply the term for time $k$ by $(1+i)^{t-k}$.

List past death-benefit dates.
For each $k=0,\ldots,t-1$, include $B$ multiplied by ${}_kp_xq_{x+k}$.

Accumulate benefits and condition on survival.
Multiply the benefit term by $(1+i)^{t-k-1}$, subtract the accumulated benefits from premiums, and divide by ${}_tp_x$.

Worked Calculations

The examples use annual premiums at the beginning of each policy year and death benefits at the end of the year of death. All mortality probabilities are conditional on survival to the start of the stated year. Unless otherwise specified, no expenses or other contract cash flows are included.

Worked Example: Net Premium and a One-Year Retrospective Reserve

A hypothetical two-year term contract pays a death benefit of $1,000 at the end of the year of death. The annual effective interest rate is $i=0.05$. The annual death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. Find the level annual net premium by the equivalence principle, then find the retrospective reserve at duration 1.

Step 1: Find the survival probabilities. We have ${}_0p_x=1$, ${}_1p_x=1-0.10=0.90$, and ${}_2p_x=0.90(1-0.20)=0.72$.

Step 2: Calculate the present value of premiums per unit premium. Premiums are due at times 0 and 1 if the insured is alive, so the annuity factor is $1+{}_1p_xv=1+0.90/1.05=1+6/7=13/7$.

Step 3: Calculate the expected present value of the benefit. Death in the first year has probability $0.10$; death in the second year has unconditional probability $0.90(0.20)=0.18$. Discounting each end-of-year payment gives $1000(0.10/1.05+0.18/1.05^2)=1000(2/21+8/49)=\$38{,}000/147$.

Equating the expected present values of premiums and benefits gives $P(13/7)=38{,}000/147$, so $P=38{,}000/273=\$139.194139\ldots$. The net premium is $\$139.19$ to the nearest cent; retain the unrounded value in the reserve calculation.

Step 4: Accumulate the time-0 premium and first-year death benefits to time 1. The premium accumulation per original policy is $P(1.05)=\$146.153846\ldots$. The expected benefit accumulation is $1000(0.10)=\$100$.

Step 5: Divide the accumulated net amount by survival to time 1.

{}_1V_x^{(R)} = \frac{(38{,}000/273)(1.05)-100}{0.90} = \frac{2000}{39} = \$51.28.

The result is rounded to the nearest cent. The method is the cohort accumulation formula with the premium chosen on a net equivalence basis. A useful check is that the reserve is positive: after one year, the premium accumulated for the surviving cohort exceeds the first-year death benefit amount allocated across the original cohort.

Common mistake avoided: The first-year benefit is not accumulated for a full year to time 1; it is paid at time 1, so its accumulation factor is 1. Multiplying the $100 expected benefit by 1.05 would give the wrong timing.

Worked Example: A Contractual Level Premium With Changing Mortality

A hypothetical contract pays $1,000 at the end of the year of death. Its annual premium is $200, payable at the beginning of each year while the insured is alive. The annual effective interest rate is $i=0.05$, with $q_x=0.10$ and $q_{x+1}=0.20$. Calculate the retrospective reserve at duration 2.

Step 1: Find the survival and death probabilities. The probabilities of being alive at times 0, 1, and 2 are $1$, $0.90$, and $0.90(0.80)=0.72$. The probabilities of death in years 1 and 2, measured from issue, are $0.10$ and $0.90(0.20)=0.18$.

Step 2: Accumulate the premium contributions to time 2. The time-0 premium accumulates for two years: $200(1.05)^2=\$220.50$. The time-1 premium is paid by $0.90$ of the original cohort and accumulates for one year: $200(0.90)(1.05)=\$189.00$. Total accumulated premiums per original policy are $\$409.50$.

Step 3: Accumulate the benefit contributions to time 2. First-year deaths receive a benefit at time 1, which accumulates one year: $1000(0.10)(1.05)=\$105.00$. Second-year deaths receive a benefit at time 2, so there is no further accumulation: $1000(0.18)=\$180.00$. Total accumulated benefits per original policy are $\$285.00$.

Step 4: Convert to a reserve per surviving policy. The accumulated net amount per original policy is $\$409.50-\$285.00=\$124.50$. Divide by the probability of surviving to time 2:

{}_2V_x^{(R)} = \frac{124.50}{0.72} = \$172.92.

The reserve is rounded to the nearest cent. This calculation uses the stated contractual premium; it does not assume that $200 is a net premium. As a check, multiplying the reserve by the survival probability gives $0.72(\$172.916666\ldots)=\$124.50$, the accumulated cohort net amount.

Common mistake avoided: Reporting $\$124.50$ as the reserve would leave the result on an original-cohort basis. The requested amount is per policy in force, so it must be divided by $0.72$.

Worked Example: Three-Year Accumulation With Unequal Death Probabilities

A hypothetical policy has a level annual premium of $180, payable at the beginning of each year while the insured is alive, and a death benefit of $1,500 payable at the end of the year of death. The annual effective interest rate is $i=0.04$, and the annual death probabilities are $q_x=0.05$ and $q_{x+1}=0.15$. Find the retrospective reserve at duration 2.

Step 1: Establish the relevant probabilities. Survival to time 1 is $0.95$. Survival to time 2 is $0.95(0.85)=0.8075$. The unconditional probabilities of death in years 1 and 2 are $0.05$ and $0.95(0.15)=0.1425$.

Step 2: Accumulate premiums. The time-0 premium accumulates for two years: $180(1.04)^2=180(1.0816)=\$194.688$. The time-1 premium is paid by $0.95$ of the original cohort and accumulates for one year: $180(0.95)(1.04)=\$177.84$. Their total is $\$372.528$.

Step 3: Accumulate benefits. The first-year expected benefit contribution at time 1 is $1500(0.05)=\$75$, which accumulates to $75(1.04)=\$78$. The second-year expected benefit contribution is $1500(0.1425)=\$213.75$, paid at time 2. Total accumulated benefits are $\$78+\$213.75=\$291.75$.

Step 4: Divide by survival to time 2. The cohort net amount is $\$372.528-\$291.75=\$80.778$. Therefore,

{}_2V_x^{(R)} = \frac{80.778}{0.8075} = \$100.034055\ldots \approx \$100.03.

The answer is rounded to the nearest cent. A check is $0.8075(\$100.034055\ldots)=\$80.778$. The method weights each premium by survival to its payment date and each benefit by the corresponding year-of-death probability.

Common mistake avoided: The year-2 premium contribution is not weighted by $0.95$ alone. It is paid at time 1 by policies alive then, which is indeed $0.95$, while the denominator is survival through time 2, $0.8075$. These probabilities refer to different dates and serve different purposes.

Common Timing and Conditioning Errors

The most reliable way to avoid indexing errors is to write the payment date, probability weight, and accumulation period beside every term before combining them. The general pattern is summarized below.

Cash flow	Cohort probability weight	Accumulation to time $t$
Premium at time $k$	${}_kp_x$	$(1+i)^{t-k}$
Benefit at time $k+1$ for death in year $k+1$	${}_kp_xq_{x+k}$	$(1+i)^{t-k-1}$
Convert cohort total to per surviving policy	Survival to time $t$	Divide by ${}_tp_x$

For example, in the second worked example, incorrectly accumulating the year-2 benefit for an extra year would use $\$189$ instead of $\$180$. That would give a cohort net amount of $\$409.50-(\$105+\$189)=\$115.50$, and an incorrect reserve of $\$115.50/0.72=\$160.42$, rather than $\$172.92$. The benefit is paid at time 2, the valuation date, so its accumulation factor is 1.

A second frequent error is to divide by the probability of survival to the start of the final policy year instead of survival to the valuation date. The denominator is determined by the reserve’s duration: at time $t$, use ${}_tp_x$. This matters because the reserve is stated per policy still in force at that exact date.

Key takeaway: A level premium does not remove the need for year-by-year weighting. Accumulate each premium from its payment date and each death benefit from its payment date, weight by the probability of that cash flow, subtract the cohort totals, and divide by survival to the valuation date.

Check Your Understanding

For each problem, use annual premiums at the beginnings of policy years, death benefits at year-end, and the retrospective reserve formula. Round dollar answers to the nearest cent.

A hypothetical two-year contract has $i=0$, $q_x=0.10$, $q_{x+1}=0.20$, annual premium $P=\$100$, and death benefit $B=\$500$. Find the reserve at duration 1. (Answer: $\$55.56$)
A contract has $i=0.04$, $q_x=0.10$, $q_{x+1}=0.20$, annual premium $P=\$120$, and death benefit $B=\$800$. Find the reserve at duration 2. (Answer: $\$20.71$)
A contract has $i=0.03$, $q_x=0.05$, $q_{x+1}=0.10$, annual premium $P=\$150$, and death benefit $B=\$2{,}000$. Find the reserve at duration 2. (Answer: $\$15.10$)
A contract has $i=0.05$, annual death probabilities $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$, annual premium $P=\$250$, and death benefit $B=\$1{,}000$. Find the reserve at duration 3. (Answer: $\$457.81$)

Cash flow	Cohort probability weight	Accumulation to time \(t\)
Premium at time \(k\)	\({}_kp_x\)	\((1+i)^{t-k}\)
Benefit at time \(k+1\) for death in year \(k+1\)	\({}_kp_xq_{x+k}\)	\((1+i)^{t-k-1}\)
Convert cohort total to per surviving policy	Survival to time \(t\)	Divide by \({}_tp_x\)

What You'll Learn