Retrospective Reserves After a Single Premium

What a Single-Premium Retrospective Reserve Measures

A single-premium contract receives its premium at issue, so there are no later premium payments to add to the fund. By a valuation date, however, some policies in the original cohort may have paid death benefits. A retrospective reserve accounts for both sides: it accumulates the original premium, subtracts benefits paid before the valuation date with interest to that date, and then measures the remaining fund per policy still in force.

The last step is essential. The accumulated fund belongs to the surviving cohort, not to every policy originally issued. A reserve per surviving policy therefore requires dividing the cohort fund by the number of survivors, or equivalently by the probability of survival from issue to the valuation date.

Definition: Under a discrete annual model, the retrospective reserve at duration $k$ is the accumulated value at time $k$ of the original cohort's premiums less its accumulated benefits paid through time $k$, divided by the number or probability of policies surviving to time $k$. It is a reserve per surviving policy.

For the formulas and examples below, assume an effective annual interest rate $i$, with $v=1/(1+i)$, and annual mortality. The single premium $P_x$ is paid at time 0. A death benefit for a death during policy year $j+1$ is paid at the end of that year, time $j+1$. There are no expenses, lapses, or other cash flows. The calculation uses the expected cohort under the stated mortality model, unless actual life-table counts are supplied.

Formula for a Single Premium

Let $B_{j+1}$ be the benefit paid at the end of policy year $j+1$ if death occurs in that year. The probability, for one policy issued at age $x$, of dying in that year is ${}_jp_x q_{x+j}$: the policy must first survive $j$ years and then die within the next year. The probability of surviving to duration $k$ is ${}_kp_x$.

Theorem: For $k\geq 0$ with ${}_kp_x>0$, the discrete retrospective reserve per survivor is

{}_kV_x = \frac{ P_x(1+i)^k - \displaystyle\sum_{j=0}^{k-1} {}_jp_x q_{x+j} B_{j+1}(1+i)^{k-j-1} }{ {}_kp_x }.

The premium accumulates for $k$ years. A benefit paid at the end of year $j+1$ accumulates for only $k-j-1$ years to time $k$. In particular, a benefit paid exactly at time $k$ receives no additional accumulation. The formula at $k=0$ has an empty sum and gives ${}_0V_x=P_x$.

If the benefit is level at $B$ in every policy year, substitute $B_{j+1}=B$. The mortality probabilities still vary by year when the $q_{x+j}$ differ. Do not replace the year-specific death probabilities with a single probability unless the mortality assumptions justify doing so.

Why the Formula Divides by Survivors

The numerator is the accumulated fund for the original cohort on a per-policy-issued basis. The denominator converts that amount to a per-policy-surviving basis. This follows directly by counting expected policies and cash flows.

Start with the original cohort.
For $N$ policies issued, the cohort receives $NP_x$ at time 0. At time $k$, that amount has accumulated to $NP_x(1+i)^k$.

Subtract accumulated claims.
The expected number dying in year $j+1$ is $N{}_jp_x q_{x+j}$. Their benefits, accumulated from time $j+1$ to time $k$, total $N{}_jp_x q_{x+j}B_{j+1}(1+i)^{k-j-1}$.

Divide by the surviving cohort.
The expected number surviving to time $k$ is $N{}_kp_x$. Dividing the remaining fund by this count cancels $N$ and gives the stated formula per survivor.

This is a cohort accounting argument: the premium and claim amounts are first put on the same valuation date, then the remaining assets are allocated over the policies still in force. The survival division is not an optional adjustment; without it, the result is a fund per original policy issued, not a reserve per surviving policy.

Worked Calculations

Worked Example: A Level Benefit Over Two Years

A hypothetical contract receives a single premium of $P_x=\$1{,}000$. The effective annual interest rate is $i=5\%$. The annual mortality probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. A benefit of $\$1{,}200$ is paid at the end of the policy year of death. Find the retrospective reserve per survivor at duration 2.

Method. Calculate the two year-of-death probabilities, accumulate the premium and each year's benefits to time 2, then divide by the two-year survival probability. This avoids treating the second-year mortality rate as a probability measured from issue.

The probability of surviving one year is ${}_1p_x=1-0.10=0.90$. The probability of surviving two years is ${}_2p_x=0.90(1-0.20)=0.72$. Thus the probability of death in year 1 is $0.10$, and the probability of death in year 2 is $0.90(0.20)=0.18$.

The premium accumulated to time 2 is $1{,}000(1.05)^2=1{,}000(1.1025)=\$1{,}102.50$.

A year-1 benefit accumulates for one year: $1{,}200(1.05)=\$1{,}260$. Weighted by the probability of year-1 death, its expected accumulated amount per policy issued is $0.10(1{,}260)=\$126$. The year-2 benefit is paid at time 2, so its accumulated amount is $\$1{,}200$; weighted by the year-2 death probability, it contributes $0.18(1{,}200)=\$216$.

The remaining accumulated fund per policy issued is $1{,}102.50-126-216=\$760.50$. Therefore, ${}_2V_x=760.50/0.72=\$1{,}056.25$.

Check. Multiplying the reserve by the survival probability gives $1{,}056.25(0.72)=\$760.50$, the remaining accumulated fund per policy issued.

Worked Example: Benefits That Change by Policy Year

A hypothetical insurer receives a single premium of $\$5{,}000$ at issue. Interest is $i=10\%$ effective annually. The annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Benefits paid at the ends of years 1, 2, and 3 are respectively $\$1{,}000$, $\$2{,}000$, and $\$3{,}000$. Find the reserve per survivor at duration 3.

Method. Weight each year's benefit by the probability of death in that year, and accumulate it only from its payment date to duration 3. This prevents applying the same accumulation period to claims paid at different times.

The successive survival probabilities are ${}_1p_x=0.90$, ${}_2p_x=0.90(0.80)=0.72$, and ${}_3p_x=0.72(0.75)=0.54$. The death probabilities from issue are $0.10$ in year 1, $0.90(0.20)=0.18$ in year 2, and $0.72(0.25)=0.18$ in year 3.

The premium accumulates to $5{,}000(1.10)^3=5{,}000(1.331)=\$6{,}655$. The year-1 benefit accumulates two years, contributing $0.10[1{,}000(1.10)^2]=0.10(1{,}210)=\$121$ per policy issued. The year-2 benefit accumulates one year, contributing $0.18[2{,}000(1.10)]=0.18(2{,}200)=\$396$. The year-3 benefit contributes $0.18(3{,}000)=\$540$, since it is paid at the valuation date.

The accumulated claims total $121+396+540=\$1{,}057$. The remaining fund per policy issued is $6{,}655-1{,}057=\$5{,}598$. Divide by the probability of survival to duration 3: ${}_3V_x=5{,}598/0.54=\$10{,}366.67$, rounded to the nearest cent.

Check. The reserve multiplied by $0.54$ is $\$5{,}598$ to the nearest cent, matching the remaining fund per policy issued.

Worked Example: Calculating From Life-Table Counts

A hypothetical cohort has $\ell_x=2{,}000$, $\ell_{x+1}=1{,}800$, $\ell_{x+2}=1{,}440$, and $\ell_{x+3}=1{,}080$. Each policy paid a single premium of $\$700$ at issue. The effective annual interest rate is $5\%$, and each death receives $\$900$ at the end of its year of death. Find the duration-3 retrospective reserve per survivor.

Method. Work with total cohort amounts using the life-table counts, then divide by the number alive at duration 3. This is the count-based form of the same cohort calculation.

Deaths in years 1, 2, and 3 number $2{,}000-1{,}800=200$, $1{,}800-1{,}440=360$, and $1{,}440-1{,}080=360$, respectively. The original premiums total $2{,}000(700)=\$1{,}400{,}000$. Accumulated to time 3, this is $1{,}400{,}000(1.05)^3=1{,}400{,}000(1.157625)=\$1{,}620{,}675$.

Year-1 claims total $200(900)=\$180{,}000$ at time 1 and accumulate two years to $180{,}000(1.05)^2=\$198{,}450$. Year-2 claims total $360(900)=\$324{,}000$ at time 2 and accumulate one year to $\$340{,}200$. Year-3 claims total $360(900)=\$324{,}000$ at time 3. Total accumulated claims are $198{,}450+340{,}200+324{,}000=\$862{,}650$.

The remaining fund is $1{,}620{,}675-862{,}650=\$758{,}025$. There are $1{,}080$ survivors, so the reserve per survivor is $758{,}025/1{,}080=\$701.875$, or $\$701.88$ to the nearest cent.

Check. $1{,}080(701.875)=\$758{,}025$, exactly the calculated remaining cohort fund before rounding.

Common Exam Trap: Forgetting the Survival Denominator

Suppose the accumulated premium less accumulated claims is $\$760.50$ per policy issued and the probability of surviving to the valuation date is $0.72$. Reporting $\$760.50$ as the reserve is incorrect: that amount is still on a per-policy-issued basis. The correction is $760.50/0.72=\$1{,}056.25$ per survivor.

A related error is to use $q_{x+1}$ alone as the probability that a policy issued at age $x$ dies in year 2. That probability is ${}_1p_xq_{x+1}$, because a policy must survive the first year to be exposed to second-year mortality. Also check payment timing: a claim paid at time $k$ is not accumulated beyond time $k$.

Key takeaway: First accumulate the single premium and each past benefit to the same valuation date. Then subtract the cohort's accumulated benefits and divide by the probability or count of policies surviving to that date.

Check Your Understanding

For each problem, assume annual discrete mortality, an effective annual interest rate, benefits paid at the end of the year of death, and no expenses or other cash flows.

A single premium is $\$1{,}000$, $i=5\%$, and $q_x=0.20,\ q_{x+1}=0.25$. A level benefit of $\$1{,}000$ is paid on death. Find the reserve per survivor at duration 2. (Answer: $\$1{,}154.17$)
A policy receives a single premium of $\$1{,}500$. Let $i=4\%$, $q_x=0.10$, and $q_{x+1}=0.10$, with a $\$2{,}000$ death benefit in either year. Find the duration-2 reserve per survivor. (Answer: $\$1{,}523.95$)
A contract has single premium $\$900$, $i=0\%$, and annual death probabilities $0.20,\ 0.25,\ 0.20$. Benefits at the ends of years 1, 2, and 3 are $\$500,\ \$1{,}000,\ \$1{,}500$. Find the duration-3 reserve per survivor. (Answer: $\$875.00$)
A cohort has $\ell_x=2{,}000$, $\ell_{x+1}=1{,}800$, $\ell_{x+2}=1{,}440$, and $\ell_{x+3}=1{,}080$. Each policy paid $\$600$ at issue, $i=5\%$, and each death receives $\$700$ at the end of its year. Find the duration-3 reserve per survivor. (Answer: $\$665.00$)
A single-premium contract has $i=10\%$, $q_x=0.10$, and $q_{x+1}=0.20$. It pays a level $\$1{,}000$ death benefit at each year-end. What issue premium produces a duration-2 retrospective reserve of $\$1{,}500$ per survivor? (Answer: $\$1{,}132.23$)

What You'll Learn