Accumulated Death Benefits Before Valuation

Why Past Death Benefits Must Be Accumulated

A retrospective calculation looks back at the cash flows associated with a group of policies and brings those cash flows forward to a valuation date. When the cash flow is a death benefit, two adjustments are essential: a benefit paid earlier earns interest up to the valuation date, and the probability of that payment depends on surviving to the policy year in which death occurs.

This tutorial considers annual policies with death benefits paid at the end of the policy year of death. The quantity of interest is the accumulated death-benefit outgo for the cohort, divided by the probability of being alive at the valuation date. It is a cohort measure: it allocates the accumulated claims from policies that ended in death across the policies from the cohort that remain in force.

Definition: Suppose a policy is issued at age $x$, the valuation date is the end of policy year $n$, and $B_k$ is the death benefit paid at the end of policy year $k$, for $k=1,\ldots,n$. Under annual effective interest rate $i$, the benefit paid at time $k$ accumulates to $B_k(1+i)^{n-k}$ at time $n$. The accumulated death-benefit outgo per survivor at time $n$ is the expected accumulated death benefits per policy issued, divided by the probability ${}_np_x$ that a policy is still in force at time $n$.

Probability Weights and the General Calculation

Let $q_{x+k-1}$ be the one-year probability of death during policy year $k$, conditional on being alive at its start. The unconditional probability, measured from issue, of death during that year is ${}_{k-1}p_x q_{x+k-1}$. The factor ${}_{k-1}p_x$ matters: a policy cannot have a death in year $k$ unless it survives the first $k-1$ years.

The year-$k$ benefit is paid at time $k$. It therefore earns interest for $n-k$ years before the valuation date—not $k$ years, and not $n-k+1$ years. Combining the death probability and accumulation factor gives the standard expression.

{}_nR_x^{(\mathrm{death})} = \frac{\displaystyle\sum_{k=1}^{n} {}_{k-1}p_x\,q_{x+k-1}\,B_k(1+i)^{n-k}} {{}_np_x}.

Here ${}_nR_x^{(\mathrm{death})}$ denotes the accumulated death-benefit outgo per survivor; it is introduced as a label for this calculation. The denominator is positive when there are survivors at the valuation date. If ${}_np_x=0$, a per-survivor value at that date is undefined.

Life-table form: If $\ell_x$ is the number alive at age $x$ in a life table, then ${}_{k-1}p_x q_{x+k-1}=(\ell_{x+k-1}-\ell_{x+k})/\ell_x$, and ${}_np_x=\ell_{x+n}/\ell_x$. Thus $$ {}_nR_x^{(\mathrm{death})} = \frac{\displaystyle\sum_{k=1}^{n} (\ell_{x+k-1}-\ell_{x+k})B_k(1+i)^{n-k}} {\ell_{x+n}}. $$

Why the Formula Works

The formula follows by summing the accumulated claim amounts across mutually exclusive policy-year death events, then expressing that expected cohort outgo per survivor. The steps also clarify why the probabilities in the numerator are not conditional on survival all the way to time $n$.

Separate the possible death years.
Let $D_k$ be the event that death occurs during policy year $k$. These events are mutually exclusive, and $P(D_k)={}_{k-1}p_xq_{x+k-1}$.

Value each claim at time $n$.
On $D_k$, the claim is $B_k$ at time $k$, so its accumulated value at time $n$ is $B_k(1+i)^{n-k}$.

Take the expected outgo per policy issued.
Summing the value on each event multiplied by its probability gives $\sum_{k=1}^{n}{}_{k-1}p_xq_{x+k-1}B_k(1+i)^{n-k}$.

Express outgo per survivor.
The expected number of survivors per policy issued is ${}_np_x$. Dividing expected cohort claim outgo by this survivor proportion gives the stated per-survivor quantity.

This is a ratio of cohort expectations, not the expected past claim of an individual known to have survived to time $n$. A survivor cannot also have died in one of the earlier policy years. In a retrospective cohort calculation, however, claims paid on behalf of deaths in the original cohort are part of the group’s accumulated outgo, while survivors provide the denominator.

Worked Example: Using Life-Table Counts

A hypothetical annual policy pays a level death benefit of $8,000 at the end of the policy year of death. The life table has $\ell_x=1{,}000$, $\ell_{x+1}=960$, and $\ell_{x+2}=900$. The effective annual interest rate is $i=0.05$. Find accumulated death-benefit outgo per survivor at time 2.

Method. Use life-table deaths in each year, accumulate each benefit to time 2, add the cohort outgo, and divide by the number alive at time 2.

There are $1{,}000-960=40$ deaths in year 1 and $960-900=60$ deaths in year 2. A year-1 claim accumulates for one year, so its time-2 value is $8{,}000(1.05)=\$8{,}400$. A year-2 claim is already paid at time 2, so its value remains $8,000.

The accumulated cohort outgo is $40(8{,}400)+60(8{,}000)=336{,}000+480{,}000=\$816{,}000$. There are 900 survivors, so $\$816{,}000/900=\$906.666\ldots$.

Answer: The accumulated death-benefit outgo per survivor is $906.67, rounded to the nearest cent. As a check using probabilities, the numerator per original policy is $0.04(8{,}400)+0.06(8{,}000)=\$816$, and $816/0.90=\$906.666\ldots$. The method avoids confusing the 40 and 60 deaths with percentages of the surviving 900.

Worked Example: Varying Benefits Over Three Years

A hypothetical policy pays $10,000, $20,000, or $30,000 at the end of the year of death in policy years 1, 2, or 3, respectively. Assume $i=0.05$, $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find accumulated death-benefit outgo per survivor at time 3.

Method. First calculate the unconditional probability of death in each year and the probability of surviving to time 3. Then accumulate each benefit to time 3 and weight it by its death probability.

The year-1 death probability is $0.10$. The year-2 probability is $0.90(0.20)=0.18$. Survival through two years is $0.90(0.80)=0.72$, so the year-3 death probability is $0.72(0.25)=0.18$. Survival to time 3 is $0.72(0.75)=0.54$.

The year-1 benefit accumulates for two years: $10{,}000(1.05)^2=\$11{,}025$, giving weighted outgo $0.10(11{,}025)=\$1{,}102.50$. The year-2 benefit accumulates for one year: $20{,}000(1.05)=\$21{,}000$, giving $0.18(21{,}000)=\$3{,}780$. The year-3 benefit accumulates for zero years, giving $0.18(30{,}000)=\$5{,}400$.

The numerator is $1{,}102.50+3{,}780+5{,}400=\$10{,}282.50$. Divide by the survival probability: $10{,}282.50/0.54=\$19{,}041.666\ldots$.

Answer: The accumulated outgo per survivor is $19,041.67, rounded to the nearest cent. The check is $0.54(19{,}041.666\ldots)=\$10{,}282.50$, the expected accumulated outgo per policy issued. This method avoids using each conditional $q$ as though it were an unconditional death probability.

Solving for an Unknown Benefit

The same expression can be rearranged when the accumulated outgo per survivor is given and one policy-year benefit is unknown. Multiply the target per-survivor amount by ${}_np_x$ to recover the expected accumulated outgo per policy issued. Then subtract the known weighted claims and solve for the unknown benefit.

Worked Example: Finding the First-Year Benefit

A hypothetical two-year policy pays an unknown benefit $B_1$ at the end of year 1 and $8,000 at the end of year 2. Assume $i=0.04$, $q_x=0.10$, and $q_{x+1}=0.20$. The accumulated death-benefit outgo per survivor at time 2 is $2,500. Find $B_1$.

Method. Use the year-specific death probabilities and multiply the target by the probability of surviving to time 2.

The year-1 death probability is $0.10$. The year-2 death probability is $0.90(0.20)=0.18$, and survival to time 2 is $0.90(0.80)=0.72$. The year-1 benefit accumulates one year, so its weighted value per policy issued is $0.10(1.04)B_1=0.104B_1$. The year-2 weighted value is $0.18(8{,}000)=\$1{,}440$.

The target per-survivor amount implies expected accumulated outgo per policy issued of $2{,}500(0.72)=\$1{,}800$. Therefore, $0.104B_1+1{,}440=1{,}800$, so $0.104B_1=360$, and $B_1=360/0.104=\$3{,}461.538\ldots$.

Answer: $B_1=$$3,461.54, rounded to the nearest cent. Substitution gives $0.104(3{,}461.538\ldots)+1{,}440=360+1{,}440=\$1{,}800$, and $1{,}800/0.72=\$2{,}500$. This method avoids dividing the target by survival before setting up the weighted cohort equation.

Common Mistakes and When the Formula Applies

The calculation depends on the stated timing and mortality model. It assumes annual death benefits paid at policy-year ends, a constant effective annual interest rate $i$ over the accumulation period, and one-year mortality probabilities appropriate to the successive ages. If a benefit is paid immediately at death or at another time within the year, its accumulation period changes. If the interest rate varies by year, each benefit must be accumulated using the product of the applicable annual accumulation factors rather than a single power of $1+i$.

A frequent error is to use $q_{x+k-1}$ alone as the probability of death in year $k$. That probability is conditional on survival to the start of year $k$; the unconditional probability from issue is ${}_{k-1}p_xq_{x+k-1}$. For the three-year example, incorrectly using $0.20$ and $0.25$ as the year-2 and year-3 probabilities would produce a numerator of $0.10(11{,}025)+0.20(21{,}000)+0.25(30{,}000)=\$12{,}802.50$, and a result of $12{,}802.50/0.54=\$23{,}708.33$. The correct year-2 and year-3 probabilities are $0.18$ and $0.18$, so the correct result is $19,041.67.

Wrong accumulation duration: A claim paid at time $k$ accumulates for $n-k$ years. In particular, the final-year claim has no accumulation before time $n$.
Wrong denominator: Divide by ${}_np_x$, the probability of surviving to the valuation date, not by the probability of dying before it.
Mixing conditional and unconditional probabilities: Include survival to the start of each policy year when weighting deaths from the issue date.
Mixing units: If life-table counts are used, the accumulated dollar claims among deaths and the number of survivors must refer to the same cohort. Their ratio is dollars per survivor.

Careful separation of timing, probability, and cohort denominator is useful in retrospective reserve work. The calculation isolates the accumulated claim component; premiums and other cash flows require their own timing and weighting before they can be combined.

Check Your Understanding

For each problem, assume annual effective interest and benefits paid at the end of the policy year of death. Round money to the nearest cent.

A two-year policy pays $5,000 on death in either year. If $i=0.05$, $q_x=0.10$, and $q_{x+1}=0.20$, find accumulated death-benefit outgo per survivor at time 2. (Answer: $1,979.17)
A life table has $\ell_x=2{,}000$, $\ell_{x+1}=1{,}900$, and $\ell_{x+2}=1{,}700$. A benefit of $6,000 is paid in either year, and $i=0.03$. Find accumulated outgo per survivor at time 2. (Answer: $1,069.41)
A three-year policy pays $12,000, $15,000, and $20,000 in years 1, 2, and 3, respectively. Given $i=0.04$, $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$, find accumulated outgo per survivor at time 3. (Answer: $8,115.44)
A two-year policy pays $B_1$ in year 1 and $5,000 in year 2. Given $i=0.05$, $q_x=0.10$, and $q_{x+1}=0.20$, accumulated outgo per survivor at time 2 is $2,000. Find $B_1$. (Answer: $5,142.86)
A life table has $\ell_x=5{,}000$, $\ell_{x+1}=4{,}850$, and $\ell_{x+2}=4{,}600$. Benefits are $4,000 in year 1 and $7,000 in year 2, with $i=0.02$. Find accumulated outgo per survivor at time 2. (Answer: $513.48)

What You'll Learn