Forward Calculation of Whole-Life Values by Age

Using a Later-Age Value to Calculate Earlier Values

A whole-life insurance value at one age depends on both the probability of death during the next year and the value of the insurance if the life survives that year. This relationship makes it possible to calculate a sequence of values from a supplied value at a later age. The essential discipline is to keep the age indices, probabilities, and discounting aligned.

The calculation is conditional on being alive at the age represented by the subscript. For example, $A_{x+1}$ is the expected present value of the future whole-life benefit for a life aged $x+1$, conditional on being alive at that age. It is not the value of a benefit paid at age $x+1$.

Definition: For a unit whole-life insurance on a life aged $x$, let $A_x$ be the expected present value of a benefit of 1, payable at the end of the year of death. Let $i$ be the effective annual interest rate, $v=1/(1+i)$, $q_x={}_1q_x$ the probability that a life aged $x$ dies within one year, and $p_x={}_1p_x=1-q_x$ the probability of surviving one year. Under this annual discrete-benefit model, $$ A_x=v(q_x+p_xA_{x+1}). $$

To use this relation repeatedly, begin with a supplied value at a later age, then calculate the value one age earlier. Repeat until reaching the age of interest. The input at each step is the value at the next age, not the value at the same age or two ages later.

Why the One-Year Recursion Holds

Condition on the two possible outcomes during the first year. If death occurs, the unit benefit is paid at the end of that year, so its present value at age $x$ is $v$. If the life survives, the future insurance has value $A_{x+1}$ at age $x+1$; its value at age $x$ is $vA_{x+1}$. Weighting those outcomes by their probabilities gives the recursion.

\begin{aligned} A_x &=q_x(v)+p_x(vA_{x+1})\\ &=v(q_x+p_xA_{x+1}). \end{aligned}

This argument relies on the stated benefit timing: death benefits are paid at the end of the year of death. It also assumes the same annual effective interest rate applies over the year, and that $A_{x+1}$ is calculated under mortality and benefit assumptions consistent with those used at age $x$. A continuous benefit payable at the moment of death, or a different payment timing, would require a different relationship.

A Reliable Calculation Procedure

For several ages, organize the work in a fixed order. At each age, pair $q_x$ with $p_x$, use the supplied value at age $x+1$, form the probability-weighted amount, and discount that amount once. The next calculation uses the value just obtained.

Identify the boundary value.
Record the supplied whole-life value at the latest age, such as $A_{x+n}$. It is the starting value for the age-by-age calculation.

Check the one-year probabilities.
At each age, verify that $p_x+q_x=1$, and use the mortality probabilities for that specific age.

Calculate one age earlier.
Substitute $A_{x+1}$ into $A_x=v(q_x+p_xA_{x+1})$, using the same $v$ at each age if the interest rate is level.

Carry the result to the next step.
Use the calculated $A_x$ as the next input when moving from age $x$ to age $x-1$. Keep unrounded values until the final answer.

The recursion is often described as propagating values from a terminal age to earlier ages. The numerical sequence moves toward younger ages: a value at age $x+1$ produces a value at age $x$, then a value at age $x-1$, and so on. The formula itself always links an age to the immediately following age.

Worked Example: One Step from a Supplied Value

A hypothetical insurer uses a 4% effective annual interest rate. For a unit whole-life insurance with benefit payable at the end of the year of death, suppose $q_{70}=0.08$, $p_{70}=0.92$, and the supplied next-age value is $A_{71}=0.6153846$. Calculate $A_{70}$ to four decimal places. Name the method and identify the common error this calculation must avoid.

Method. Apply the one-year whole-life recursion, using $v=1/(1+i)$. The rate is $i=0.04$, so $v=1/1.04$. The probabilities are consistent because $0.08+0.92=1$.

Substitute the inputs:

\begin{aligned} A_{70} &=\frac{q_{70}+p_{70}A_{71}}{1.04}\\ &=\frac{0.08+0.92(0.6153846)}{1.04}\\ &=\frac{0.08+0.566153832}{1.04}\\ &=\frac{0.646153832}{1.04}\\ &=0.621301761538\ldots \end{aligned}

Therefore, $A_{70}=0.6213$ to four decimal places. The unrounded calculation uses the supplied value $0.6153846$, so the intermediate product is $0.566153832$.

Common error avoided. Omitting discounting would give $0.08+0.92(0.6153846)=0.646153832$, which is not the value at age 70. Both the death payment and the surviving policy value are one year in the future, so the probability-weighted amount must be multiplied by $v$, or divided by $1.04$.

Propagating Values Across Several Ages

For a sequence, the order of calculation matters: first obtain the value one age before the boundary, then use it to obtain the next earlier value. The mortality probability changes with age, while a level interest assumption keeps the discount factor fixed. The following example shows how to keep those age-specific inputs organized.

Worked Example: Two Steps from a Terminal-Age Value

A hypothetical unit whole-life insurance pays at the end of the year of death. Use a 5% effective annual interest rate. Given $A_{63}=0.50$, $q_{62}=0.10$, and $q_{61}=0.08$, calculate $A_{61}$ to four decimal places. State each intermediate value and check that the probabilities at each age sum to 1.

Method. Work from age 63 toward younger ages. The discount factor is $v=1/1.05$. At age 62, $p_{62}=1-0.10=0.90$; at age 61, $p_{61}=1-0.08=0.92$.

First calculate $A_{62}$ from $A_{63}$:

\begin{aligned} A_{62} &=\frac{q_{62}+p_{62}A_{63}}{1.05}\\ &=\frac{0.10+0.90(0.50)}{1.05}\\ &=\frac{0.10+0.45}{1.05}\\ &=\frac{0.55}{1.05}\\ &=0.5238095238\ldots \end{aligned}

Now use the unrounded $A_{62}$ to calculate $A_{61}$:

\begin{aligned} A_{61} &=\frac{q_{61}+p_{61}A_{62}}{1.05}\\ &=\frac{0.08+0.92(0.5238095238\ldots)}{1.05}\\ &=\frac{0.08+0.4819047619\ldots}{1.05}\\ &=\frac{0.5619047619\ldots}{1.05}\\ &=0.5351473923\ldots \end{aligned}

Thus $A_{61}=0.5351$ to four decimal places. The probability checks are $0.10+0.90=1$ at age 62 and $0.08+0.92=1$ at age 61. The common mistake here is to insert $A_{63}$ directly into the age-61 equation. The recursion uses the next age, so the correct input for $A_{61}$ is the newly calculated $A_{62}$.

Worked Example: Three Ages with a Different Interest Rate

For a hypothetical unit whole-life insurance with end-of-year death benefits, use an effective annual interest rate of 3%. A supplied value is $A_{80}=0.4000$; the one-year death probabilities are $q_{79}=0.10$ and $q_{78}=0.05$. Calculate $A_{78}$ to four decimal places, showing the value at age 79 first.

Method. Set $p_{79}=1-0.10=0.90$ and $p_{78}=1-0.05=0.95$. At both ages the discount divisor is $1.03$. Begin with the age directly before the supplied boundary:

\begin{aligned} A_{79} &=\frac{0.10+0.90(0.4000)}{1.03}\\ &=\frac{0.10+0.3600}{1.03}\\ &=\frac{0.4600}{1.03}\\ &=0.4466019417\ldots \end{aligned}

Then use that result at age 78:

\begin{aligned} A_{78} &=\frac{0.05+0.95(0.4466019417\ldots)}{1.03}\\ &=\frac{0.05+0.4242718447\ldots}{1.03}\\ &=\frac{0.4742718447\ldots}{1.03}\\ &=0.4604581016\ldots \end{aligned}

Therefore, $A_{78}=0.4605$ to four decimal places. The probabilities check: $0.10+0.90=1$ and $0.05+0.95=1$. Both calculated values lie between 0 and 1, as expected for the expected present value of a discounted unit benefit under these assumptions. The common mistake avoided is discounting only the death term: the survival term $p_xA_{x+1}$ also represents a value one year in the future and must be discounted.

Checks, Interpretation, and Common Traps

A computed sequence is easier to audit when each line is interpreted as a one-year conditional calculation. The term $q_x$ weights the death benefit of 1, while $p_x A_{x+1}$ weights the value after survival. Discounting applies to the combined expected value because both outcomes are valued one year ahead.

Age indexing: In the equation for $A_x$, use $q_x$, $p_x$, and $A_{x+1}$. Do not skip an age or use the mortality probability from the next age.
Discounting: With effective annual rate $i$, use $v=1/(1+i)$. The recursion is not $q_x+p_xA_{x+1}$ unless the interest rate is zero.
Survival probability: Use $p_x=1-q_x$ for a one-year calculation. If a problem supplies both probabilities, check their sum before proceeding.
Boundary value: Treat the supplied $A_{x+n}$ as the whole-life value conditional on reaching age $x+n$. It is not itself the value at the original, younger age.
Rounding: Carry sufficient precision through the sequence. Rounding an intermediate value too early can change the last reported digit.

This calculation is useful whenever a set of whole-life values must be built from a known value at a later age. It is also a direct way to check whether a proposed series of values behaves consistently with the stated interest and mortality inputs. The recursion does not by itself determine a missing mortality probability or interest rate; here, those inputs are supplied and used to calculate earlier-age values. This topic is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: Starting from a supplied later-age value, calculate one age earlier at a time using $A_x=v(q_x+p_xA_{x+1})$. The next-age index, the age-specific survival and death probabilities, and the discount factor must all be correct at every step.

Check Your Understanding

For each problem, assume a unit whole-life benefit payable at the end of the year of death and use the stated effective annual interest rate. Round final values to four decimal places.

A hypothetical policy has $i=0.04$, $q_{50}=0.12$, and $A_{51}=0.5000$. Calculate $A_{50}$. (Answer: 0.5385)
At a 5% effective annual rate, a hypothetical whole-life insurance has $q_{62}=0.10$ and $A_{63}=0.6000$. Calculate $A_{62}$. (Answer: 0.6095)
At $i=0.03$, suppose $q_{41}=0.06$, $q_{40}=0.04$, and $A_{42}=0.5000$. Calculate $A_{40}$, showing the age-41 value as an intermediate step. (Answer: 0.5184)
A hypothetical insurer uses $i=0.02$, with $q_{61}=0.10$, $q_{60}=0.05$, and $A_{62}=0.7000$. Calculate $A_{60}$ in two steps. (Answer: 0.7156)

What You'll Learn