Whole-Life Insurance as an Infinite Death-Year Sum

Why Whole-Life Values Require a Death-Year Sum

A discrete whole-life policy pays a benefit at the end of the policy year in which the insured dies. Because the payment date depends on the year of death, its present value is random. To find the expected present value, consider one policy year at a time: the benefit is discounted to that year’s end and weighted by the probability that death occurs in that year.

Unlike an $n$-year term policy, a whole-life policy has no fixed final coverage year. Its expected present value is therefore written as an infinite sum, with one term for each possible policy year of death. The central calculation is not simply the conditional probability of dying during a year; it is the probability of surviving to that year and then dying in it.

Definition: Assume an insured is age $x$ at issue, the annual effective interest rate is $i$, and a level benefit $B$ is paid at the end of the policy year of death. Let $v=1/(1+i)$. The probability of surviving $k$ full years is ${}_kp_x$, and the conditional probability of dying during the next year, given survival to its start, is $q_{x+k}$. Thus the probability of death in policy year $k+1$ is ${}_kp_x q_{x+k}$, and the benefit’s expected present value is $B\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}$, when the sum converges.

Here $k=0,1,2,\ldots$. In particular, ${}_0p_x=1$, so the first term uses $q_x$, the probability of death during the first policy year. The exponent is $k+1$, not $k$, because a death in policy year $k+1$ produces a payment at the end of that year.

For a unit benefit, this expected present value is denoted $A_x$. For a level benefit $B$, linearity of expectation gives $B A_x$. In this tutorial, the key task is to form and evaluate $A_x$ as a death-year sum; it is not necessary to find a separate closed form for the series.

Building the Infinite Sum

The $k$th term is made from three parts: the probability of reaching the start of policy year $k+1$, the conditional probability of dying during that year, and the discount factor for a payment at its end.

P(\text{death in policy year }k+1) ={}_kp_xq_{x+k}

\operatorname{EPV} =\sum_{k=0}^{\infty} Bv^{k+1}{}_kp_xq_{x+k} =B A_x,\qquad A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}.

The conditional death probabilities $q_x,q_{x+1},\ldots$ are not themselves the probabilities of death in the corresponding policy years from issue. Survival to each year must be included. For example, the probability of death in policy year 3 is ${}_2p_xq_{x+2}$, not $q_{x+2}$ alone.

This sum is infinite because the possible year of death is not bounded by a fixed term. Under a standard model with $i\geq 0$ and a finite benefit, each present value is bounded above by $B$; the expectation is finite. If a model allows a positive probability of never dying, the policy pays nothing on that outcome, and the death-year probabilities in the sum need not add to one. More generally, the series expression applies when its expected value is well-defined and the sum converges.

Why the Formula Holds

Let $K$ be the number of complete policy years lived after issue before death. Thus $K=k$ means death occurs during policy year $k+1$, and the payment is made at time $k+1$. The present value is $Bv^{k+1}$ on that event. The events $K=0,K=1,\ldots$ are disjoint, so expectation can be calculated by summing the present value on each event times that event’s probability.

Partition by policy year.
Each possible finite year of death belongs to exactly one event: death in policy year $k+1$.

Find the event probability.
To die in that year, the insured must first survive $k$ years and then die during the next year, giving probability ${}_kp_xq_{x+k}$.

Discount the payment on that event.
The payment is $B$ at time $k+1$, so its present value is $Bv^{k+1}$.

Sum the event contributions.
Adding present value times probability over all possible death years gives $B\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}$.

This justification also shows why neither an undiscounted average benefit nor a sum of conditional death probabilities gives the insurance APV. Each possible payment must be weighted by its own unconditional probability from issue and discounted to its own payment date.

Worked Examples

Worked Example: Four Possible Death Years

A hypothetical discrete whole-life policy pays $10,000 at the end of the policy year of death. The annual effective interest rate is $25\%$. The conditional mortality rates for policy years 1 through 4 are $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, and $q_{x+3}=1$. Find the APV. Use the death-year sum and check that all possible outcomes have been included.

Method. Calculate each year’s unconditional death probability, then multiply the benefit by that probability and the appropriate discount factor. The rate $q_{x+3}=1$ means that anyone surviving the first three years dies in policy year 4.

Policy year 1: the death probability is $1(0.10)=0.10$. The present-value contribution is $10{,}000(0.10)/1.25=800$.

Policy year 2: survival through year 1 is $1-0.10=0.90$, so the death probability is $0.90(0.20)=0.18$. The contribution is $10{,}000(0.18)/1.25^2=1{,}800/1.5625=1{,}152$.

Policy year 3: survival through year 2 is $0.90(0.80)=0.72$. The death probability is $0.72(0.25)=0.18$, and the contribution is $1{,}800/1.25^3=1{,}800/1.953125=921.60$.

Policy year 4: the probability of reaching that year is $0.72(0.75)=0.54$, and death in that year is certain conditional on reaching it. The contribution is $10{,}000(0.54)/1.25^4=5{,}400/2.44140625=2{,}211.84$.

\operatorname{EPV}=800+1{,}152+921.60+2{,}211.84 =\$5{,}085.44.

Check. The death-year probabilities add to $0.10+0.18+0.18+0.54=1$, so the four terms cover all possible outcomes. The calculation avoids stopping after year 3 and omitting the remaining $0.54$ probability of payment.

Worked Example: Carrying the Sum Through the Last Possible Year

A hypothetical policy pays $12,000 at the end of the year of death. The annual effective interest rate is $5\%$. Conditional death probabilities in policy years 1 through 5 are $1/6$, $0.12$, $3/11$, $3/8$, and $1$, respectively. Find the whole-life APV, rounding the final amount to the nearest cent.

Method. Accumulate survival to the start of each year to obtain the unconditional death probabilities. This example also illustrates why a partial sum cannot be called a whole-life value if later death years remain possible.

The death probability in year 1 is $1/6$. Survival to year 2 is $5/6$, so the death probability in year 2 is $(5/6)(0.12)=0.10$. Survival to year 3 is $(5/6)(0.88)=11/15$; hence the year 3 death probability is $(11/15)(3/11)=0.20$.

Survival to year 4 is $11/15-0.20=8/15$, so the year 4 death probability is $(8/15)(3/8)=0.20$. Survival to year 5 is $8/15-0.20=1/3$, and the conditional death probability in year 5 is 1, giving a year 5 death probability of $1/3$. The five probabilities sum to 1.

The expected benefit amounts by year are therefore $2,000, $1,200, $2,400, $2,400, and $4,000. Discount each amount to issue:

\begin{aligned} \operatorname{EPV} &=\frac{2{,}000}{1.05}+\frac{1{,}200}{1.05^2} +\frac{2{,}400}{1.05^3}+\frac{2{,}400}{1.05^4} +\frac{4{,}000}{1.05^5}\\ &=1{,}904.7619+1{,}088.4354+2{,}073.2102+1{,}974.4859+3{,}134.1047\\ &\approx \$10{,}175.00. \end{aligned}

The displayed contributions are shown to four decimal places; the final amount is rounded to the nearest cent using unrounded contributions. The check is that the year 5 term is required: the first four death probabilities total $2/3$, leaving $1/3$ of the probability mass—and a possible benefit—to year 5.

Worked Example: Avoiding Conditional-Probability Weighting

A hypothetical discrete whole-life policy pays $5,000 at the end of the year of death. The annual effective interest rate is $10\%$. The conditional death probabilities in policy years 1 through 4 are $0.25$, $0.20$, $0.40$, and $1$. Find the APV to the nearest cent.

Method. Convert each conditional rate into a death probability from issue before discounting. The year 2 rate applies only to those who survive year 1.

The unconditional death probabilities are $0.25$ in year 1, $0.75(0.20)=0.15$ in year 2, $0.75(0.80)(0.40)=0.24$ in year 3, and $0.75(0.80)(0.60)(1)=0.36$ in year 4. They sum to 1.

The discounted contributions are:

\begin{aligned} \operatorname{EPV} &=\frac{5{,}000(0.25)}{1.10} +\frac{5{,}000(0.15)}{1.10^2} +\frac{5{,}000(0.24)}{1.10^3} +\frac{5{,}000(0.36)}{1.10^4}\\ &=1{,}136.363636+619.834711+901.577761+1{,}229.424221\\ &\approx \$3{,}887.20. \end{aligned}

Check. The four probabilities sum to 1 and each contribution is discounted for its own payment year. A common error would be to use $0.20$ as the unconditional year 2 death probability: that would give $5{,}000(0.20)/1.10^2=\$826.4463$, instead of the correct $5{,}000(0.75)(0.20)/1.10^2=\$619.8347$. The incorrect calculation counts some people who had already died in year 1.

Common Timing and Probability Traps

The death-year sum is compact, but two details are easy to misread under exam conditions: whether a probability is conditional or unconditional, and when the benefit is paid.

Issue	Incorrect approach	Correction
Conditional mortality	Use $q_{x+k}$ alone as the probability of death in year $k+1$ from issue.	Use ${}_kp_xq_{x+k}$; survival to that policy year is required.
Payment timing	Discount a year $k+1$ benefit by $v^k$.	Use $v^{k+1}$, because payment occurs at the end of policy year $k+1$.
Whole-life horizon	Stop after a few terms and treat the partial sum as the whole-life APV.	Include all possible later death years, or explicitly identify the result as a partial sum or approximation.
Probability check	Assume the conditional $q$ values should add to 1.	Check the unconditional death-year probabilities. They add to 1 only if death eventually occurs with probability 1.

The sum also makes several basic checks possible. Every term must be nonnegative for a nonnegative benefit and ordinary interest assumptions. Earlier payment years have smaller discount exponents. If the death-year probabilities sum to 1, the undiscounted expected benefit is $B$; with positive interest, discounting reduces the APV below $B$.

Key takeaway: For a level benefit paid at the end of the year of death, build one term for every possible policy year: benefit multiplied by that year’s discount factor and by the probability of surviving to the year and dying during it. Keep the survival factor and the end-of-year discount exponent explicit.

Check Your Understanding

For each hypothetical policy, form death-year probabilities before applying the discount factor.

A $8,000 whole-life benefit is paid at each year-end of death. The annual effective interest rate is $25\%$, and conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.25$, and $1$. Find the APV. (Answer: $4,556.80)
A $6,000 benefit is paid at the end of the year of death. At an annual effective interest rate of $10\%$, conditional death probabilities in years 1, 2, and 3 are $0.20$, $0.30$, and $1$. Find the APV to the nearest cent. (Answer: $4,805.41)
A $9,000 benefit is payable at the end of the year of death. The annual effective interest rate is $25\%$, and conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.50$. What is the present-value contribution from death in policy year 3? (Answer: $1,658.88)
A hypothetical policy pays $4,000 at the end of the death year. The annual effective interest rate is $25\%$; conditional death probabilities in years 1, 2, and 3 are $0.50$, $0.50$, and $1$. Find the whole-life APV. (Answer: $2,752.00)
A $5,000 benefit is paid at the end of the death year. The annual effective interest rate is $25\%$, and the unconditional probabilities of death in years 1 through 4 are $0.10$, $0.20$, $0.30$, and $0.40$. Find the APV. (Answer: $2,499.20)

Issue	Incorrect approach	Correction
Conditional mortality	Use \(q_{x+k}\) alone as the probability of death in year \(k+1\) from issue.	Use \({}_kp_xq_{x+k}\); survival to that policy year is required.
Payment timing	Discount a year \(k+1\) benefit by \(v^k\).	Use \(v^{k+1}\), because payment occurs at the end of policy year \(k+1\).
Whole-life horizon	Stop after a few terms and treat the partial sum as the whole-life APV.	Include all possible later death years, or explicitly identify the result as a partial sum or approximation.
Probability check	Assume the conditional \(q\) values should add to 1.	Check the unconditional death-year probabilities. They add to 1 only if death eventually occurs with probability 1.

What You'll Learn