Whole-Life Insurance From Age-Specific Mortality Rates

From Annual Mortality Rates to a Whole-Life Value

A sequence of annual mortality rates describes conditional probabilities: each rate applies only to a life that is still alive at the start of the corresponding year. A whole-life insurance value, however, depends on the probability of death in each policy year measured from the original issue date. The key step is therefore to combine each year’s mortality rate with the probability of surviving all earlier policy years.

This tutorial considers a discrete whole-life policy with a level benefit paid at the end of the policy year of death. The insured is age $x$ at issue, and the annual effective interest rate is $i$. The calculation is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: The annual mortality rate $q_{x+k}$ is the probability that a life aged $x+k$ at the start of a year dies before reaching age $x+k+1$. The corresponding one-year survival probability is $p_{x+k}=1-q_{x+k}$. Thus, $q_{x+k}$ is conditional on being alive at the start of that year; it is not, by itself, the probability of death in policy year $k+1$.

For example, $q_{x+1}$ applies during the second policy year. To die in that year, the insured must first survive the first policy year and then die during the second. That sequence of events is why the second-year unconditional death probability is $p_xq_{x+1}$, not simply $q_{x+1}$.

Unconditional Death Probabilities and the APV

Let $K_x$ be the curtate future lifetime: $K_x=k$ means death occurs in policy year $k+1$. For a death in that year, the life must survive the first $k$ years and then die during the next year. Applying conditional probabilities successively gives

P(K_x=k) ={}_kp_xq_{x+k} =\left(\prod_{j=0}^{k-1}p_{x+j}\right)q_{x+k}, \qquad k=0,1,2,\ldots

For $k=0$, the product over earlier years is an empty product, defined to equal 1. Hence $P(K_x=0)=q_x$. For $k=1$, the probability is $p_xq_{x+1}$. Each result is an unconditional probability from the issue date.

Theorem: For a discrete whole-life insurance paying a level benefit of 1 at the end of the policy year of death, its actuarial present value is

A_x =\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k} =\sum_{k=0}^{\infty}v^{k+1} \left(\prod_{j=0}^{k-1}(1-q_{x+j})\right)q_{x+k}, \qquad v=\frac{1}{1+i}.

This formula assumes annual mortality rates are specified for all relevant future policy years, the benefit is paid at the end of the year of death, and the effective annual interest rate is $i$. For a level benefit $B$, the APV is $BA_x$. If a sequence of rates stops before future mortality is determined, that sequence alone is not enough to calculate the exact whole-life value.

Why the Formula Holds

The possible death years divide the outcomes into mutually exclusive events: death in the first policy year, death in the second, and so on. For death in policy year $k+1$, the probability is ${}_kp_xq_{x+k}$, and the benefit of 1 is paid at time $k+1$, with present value $v^{k+1}$. The expected present value is the sum of each possible present value multiplied by its probability:

E[\text{present value}] =\sum_{k=0}^{\infty} \bigl(v^{k+1}\bigr)P(K_x=k) =\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}.

This argument is a death-year partition: it pairs the probability of each event with the present value paid if that event occurs. It also explains two essential details. The mortality probability must include survival through earlier years, and discounting must use the year in which the benefit is paid.

A Reliable Calculation Procedure

A table makes it easier to keep conditional rates, unconditional death probabilities, and discount factors separate. Proceed year by year, carrying forward survival from the issue date.

Record the conditional rate.
Use the rate for the age at the start of that policy year. Do not treat it as an unconditional probability from issue.

Calculate survival to the year.
Multiply the one-year survival probabilities for all earlier years. For the first policy year, survival to the year is 1.

Calculate death probability and present value.
Multiply survival to the year by its mortality rate, then multiply that death probability by the discount factor for the payment year.

Sum the contributions and scale the benefit.
Add the discounted contributions for a unit benefit. Multiply by the face amount if the benefit is not 1.

As a basic check, the unconditional death probabilities should sum to 1 if the supplied sequence specifies certain death by its final year. Otherwise, their sum is the probability of death over the years represented by the sequence, not necessarily the probability of eventual death.

Worked Calculations

Worked Example: Four Policy Years With Different Mortality Rates

A hypothetical policy is issued to a life aged 40. The annual mortality rates are $q_{40}=0.10$, $q_{41}=0.20$, $q_{42}=0.25$, and $q_{43}=1$. The last rate completes the supplied sequence. A benefit of $10,000 is paid at the end of the year of death, and $i=5\%$. Calculate the APV. Give the unit APV to eight decimal places and the policy APV to the nearest cent.

Method. First form the unconditional probability of death in each policy year by multiplying the mortality rate by survival to that year.

Policy year 1: $q_{40}=0.10$, so the unconditional death probability is $0.10$. Survival after year 1 is $1-0.10=0.90$.

Policy year 2: the death probability is $0.90(0.20)=0.18$. Survival after year 2 is $0.90(0.80)=0.72$.

Policy year 3: the death probability is $0.72(0.25)=0.18$. Survival after year 3 is $0.72(0.75)=0.54$.

Policy year 4: the death probability is $0.54(1)=0.54$. The probabilities check: $0.10+0.18+0.18+0.54=1.00$.

The discount factors are $v=1/1.05$, $v^2=1/1.05^2$, $v^3=1/1.05^3$, and $v^4=1/1.05^4$. The unit APV is

\begin{aligned} A_{40} &=0.10v+0.18v^2+0.18v^3+0.54v^4\\ &=\frac{0.10}{1.05}+\frac{0.18}{1.05^2} +\frac{0.18}{1.05^3}+\frac{0.54}{1.05^4}\\ &=0.09523809524+0.16326530612 +0.15549076773+0.44425933640\\ &=0.85825350549. \end{aligned}

Rounded to eight decimal places, the unit APV is $0.85825351$. The APV for $10,000 is $10{,}000(0.85825350549)=\$8{,}582.5350549$, or $8,582.54 to the nearest cent. The result is less than the face amount, as expected for a positive interest rate and a benefit paid in the future.

Common mistake avoided: The later-year rates are conditional. Using $0.20$, $0.25$, and $1$ directly as death probabilities would ignore the chance of dying earlier and would fail the probability check.

Worked Example: A Larger Benefit and Policy-Year Discounting

A hypothetical whole-life policy is issued at age 52. The rates are $q_{52}=0.15$, $q_{53}=0.10$, $q_{54}=0.20$, and $q_{55}=1$. It pays $25,000 at the end of the year of death, and $i=4\%$. Find the APV to the nearest cent.

Method. Calculate survival to the start of each year before applying that year’s rate. The unconditional death probabilities are as follows:

Policy year 1: $0.15$.
Policy year 2: $(1-0.15)(0.10)=0.85(0.10)=0.085$.
Policy year 3: $0.85(1-0.10)(0.20)=0.85(0.90)(0.20)=0.153$.
Policy year 4: $0.85(0.90)(1-0.20)(1)=0.612$.

The probabilities sum to $0.15+0.085+0.153+0.612=1$. The timing of the benefit determines the discount powers: the four death-year payments are discounted by $v,v^2,v^3,v^4$, respectively, where $v=1/1.04$.

\begin{aligned} A_{52} &=0.15v+0.085v^2+0.153v^3+0.612v^4\\ &=\frac{0.15}{1.04}+\frac{0.085}{1.04^2} +\frac{0.153}{1.04^3}+\frac{0.612}{1.04^4}\\ &\approx 0.1442307692+0.0785872781 +0.1360164429+0.5231401655\\ &\approx 0.8819746557. \end{aligned}

The policy APV is $25{,}000(0.8819746557)=\$22{,}049.3664$, or $22,049.37 to the nearest cent.

Common mistake avoided: The probability for year 3 is $0.153$, not $0.20$, because the life must survive both earlier years. The year-3 contribution is also discounted by $v^3$, not $v^2$.

Worked Example: Checking a High-Probability Final Year

A hypothetical policy is issued at age 60 with rates $q_{60}=0.08$, $q_{61}=0.12$, $q_{62}=0.18$, and $q_{63}=1$. The benefit is $50,000, payable at the end of the year of death, and $i=10\%$. Calculate the whole-life APV to the nearest cent.

Method. The successive unconditional death probabilities are $0.08$, $0.92(0.12)=0.1104$, $0.92(0.88)(0.18)=0.145728$, and $0.92(0.88)(0.82)(1)=0.663872$. Their sum is $0.08+0.1104+0.145728+0.663872=1$.

The last probability is large because it is the probability of surviving the first three years; the final conditional mortality rate is 1. With $v=1/1.10$, the unit APV is

\begin{aligned} A_{60} &=0.08v+0.1104v^2+0.145728v^3+0.663872v^4\\ &=\frac{0.08}{1.10}+\frac{0.1104}{1.10^2} +\frac{0.145728}{1.10^3}+\frac{0.663872}{1.10^4}\\ &=0.07272727273+0.09123966942 +0.10948760331+0.45343350866\\ &=0.72688805412. \end{aligned}

Thus the policy APV is $50{,}000(0.72688805412)=\$36{,}344.4027$, or $36,344.40 to the nearest cent. The fourth-year term is discounted four years even though its probability is the remaining survival probability after year 3.

Common mistake avoided: A large death probability in the final year does not mean the benefit is paid immediately. Its present value still uses the discount factor for the fourth policy year.

Common Errors and Checks

Age-specific rates are easy to misapply when moving quickly through a sequence. Keep the conditional rate, survival to the year, unconditional death probability, and discount power distinct. The following checks help detect indexing and arithmetic errors before the final APV is reported.

Check	What to verify
Conditional versus unconditional	For policy year $k+1$, multiply $q_{x+k}$ by survival through the preceding $k$ years.
Policy-year indexing	A death in policy year $k+1$ is paid at time $k+1$, so its discount factor is $v^{k+1}$.
Probability total	If the supplied sequence ends with certain death, unconditional death probabilities should add to 1.
Completeness of data	If future rates are not specified, do not assume the listed rates determine the whole-life value.
Benefit amount	Calculate the unit APV first, then multiply by the level benefit.

A tempting but incorrect calculation is to use $\sum_k v^{k+1}q_{x+k}$. This treats every age-specific rate as if the life were certain to reach the corresponding year. The correction is to include survival to each year:

v^{k+1}q_{x+k} \quad\longrightarrow\quad v^{k+1}{}_kp_xq_{x+k}.

A second common error is to use $v^k$ instead of $v^{k+1}$. The first policy year’s benefit is paid at the end of year 1, so its discount factor is $v$, not 1. Check the first term of the sum: it must be $v q_x$.

Key takeaway: For each policy year, multiply the conditional mortality rate by survival to the start of that year to obtain the unconditional death probability. Multiply that probability by the discount factor for the end of that year, then sum across all possible death years.

Check Your Understanding

Work each problem by forming unconditional death probabilities before discounting. Each scenario is hypothetical.

A policy issued at age 30 has $q_{30}=0.20$, $q_{31}=0.25$, and $q_{32}=1$. It pays $1,000 at the end of the year of death, with $i=5\%$. Find the APV to the nearest cent. (Answer: $890.18)
At age 45, the rates are $q_{45}=0.05$, $q_{46}=0.10$, $q_{47}=0.15$, and $q_{48}=1$. For a unit benefit and $i=4\%$, find the whole-life APV to four decimal places. (Answer: $0.8712$)
A policy issued at age 70 has $q_{70}=0.30$, $q_{71}=0.20$, $q_{72}=0.10$, and $q_{73}=1$. With a unit benefit and $i=3\%$, state the unconditional death probabilities by policy year and the APV to four decimal places. (Answer: probabilities $0.3000, 0.1400, 0.0560, 0.5040$; APV $0.9223$)
A hypothetical policy pays $30,000 at the end of the year of death. Its rates are $q_{60}=0.12$, $q_{61}=0.18$, and $q_{62}=1$, with $i=6\%$. Find the APV to the nearest cent. (Answer: $25,801.57)
For a life aged 70, suppose $q_{70}=q_{71}=q_{72}=0.08$ and $q_{73}=1$. At $i=5\%$, find (a) the unconditional probability of death in policy year 4 and (b) that year’s contribution to the unit-benefit APV, to four decimal places. (Answer: (a) $0.7787$; (b) $0.6406$)

Check	What to verify
Conditional versus unconditional	For policy year \(k+1\), multiply \(q_{x+k}\) by survival through the preceding \(k\) years.
Policy-year indexing	A death in policy year \(k+1\) is paid at time \(k+1\), so its discount factor is \(v^{k+1}\).
Probability total	If the supplied sequence ends with certain death, unconditional death probabilities should add to 1.
Completeness of data	If future rates are not specified, do not assume the listed rates determine the whole-life value.
Benefit amount	Calculate the unit APV first, then multiply by the level benefit.

What You'll Learn