Integer-Age Life-Table Inputs for Fractional Ages

What Integer-Age Life-Table Entries Tell You

A life table reports mortality information at specified ages. For adjacent integer ages, its entries describe what happens over the full year between those ages. This is useful for annual survival calculations, but it leaves an important question open: what happens within that year? A table of integer-age entries alone does not specify the exact survival probability over a fraction of a year.

This distinction matters whenever an age or a time interval is fractional. Before calculating, identify which information comes directly from the table and which would require an additional within-year mortality assumption. The calculations here are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: In a life table, $\ell_x$ is the number of lives surviving to exact integer age $x$, and $d_x$ is the number dying between exact ages $x$ and $x+1$. The one-year probability of death for a life aged $x$ is $q_x$, and the one-year probability of survival is $p_x=1-q_x$. When $\ell_x>0$, $$ d_x=\ell_x-\ell_{x+1}, \qquad q_x=\frac{d_x}{\ell_x}, \qquad p_x=\frac{\ell_{x+1}}{\ell_x}. $$

The entries may be counts for a hypothetical group or scaled values based on an arbitrary life-table radix. Their absolute scale does not affect probabilities: multiplying all the $\ell$ and $d$ entries by the same positive constant leaves $q_x$ unchanged.

Reading the Interval Between Adjacent Ages

The index on $q_x$ identifies the age at the start of the year. Thus $q_x$ is the probability that a life alive at age $x$ dies before reaching age $x+1$; it is not a probability for someone already aged $x+1$. The adjacent entries $\ell_x$ and $\ell_{x+1}$ determine the same annual information as $d_x$ and $q_x$, provided the table is internally consistent.

The relationship follows by counting the lives at age $x$. Each either survives to $x+1$ or dies during the interval, so the number at age $x$ is partitioned into $\ell_{x+1}$ survivors and $d_x$ deaths. Dividing that partition by $\ell_x$ gives the corresponding probability relationship.

\ell_x=\ell_{x+1}+d_x \quad\Longrightarrow\quad 1=\frac{\ell_{x+1}}{\ell_x}+\frac{d_x}{\ell_x} =p_x+q_x.

For an interval covering several complete years, the same reasoning gives an exact result from the integer-age entries. If $n$ is a positive integer and $\ell_x>0$, then the probability of surviving from age $x$ to age $x+n$ is $\ell_{x+n}/\ell_x$. The probability of dying during that interval is one minus that survival probability. These results use only the life table at the integer endpoints.

{}_np_x=\frac{\ell_{x+n}}{\ell_x}, \qquad {}_nq_x=1-{}_np_x =\frac{\ell_x-\ell_{x+n}}{\ell_x}.

Worked Example: Checking One Year of Table Entries

A hypothetical table has $\ell_{52}=10{,}000$ and $d_{52}=800$. Find $\ell_{53}$, $q_{52}$, and $p_{52}$, and check the result using the adjacent-age relationship.

Method. Use the one-year partition of the lives aged 52: those alive at age 53 plus those who die during the year must total $\ell_{52}$.

First calculate the number surviving to age 53: $$ \ell_{53}=\ell_{52}-d_{52}=10{,}000-800=9{,}200. $$

Next calculate the one-year death probability: $$ q_{52}=\frac{d_{52}}{\ell_{52}}=\frac{800}{10{,}000}=0.08. $$

The one-year survival probability is $$ p_{52}=\frac{\ell_{53}}{\ell_{52}}=\frac{9{,}200}{10{,}000}=0.92. $$

Check. The probabilities sum to one: $0.08+0.92=1$. The counts also reconcile: $9{,}200+800=10{,}000$. The mistake to avoid is treating $d_{52}$ as a probability; it is a number of deaths, so it must be divided by $\ell_{52}$ to obtain $q_{52}$.

Several Complete Years and Consistency Checks

To calculate survival over multiple complete years, the endpoint survivor counts give the probability directly. Equivalently, if every annual entry is available, the one-year survival probabilities multiply. Both methods should agree when the entries are consistent. Comparing them can expose an indexing or arithmetic error.

Worked Example: Combining Two Annual Intervals

A hypothetical table gives $\ell_{30}=50{,}000$, $\ell_{31}=49{,}000$, and $\ell_{32}=47{,}530$. Find $q_{30}$, $q_{31}$, the probability of surviving from age 30 to age 32, and the probability of dying during those two complete years.

Method. Calculate each annual death probability from the decrease in survivor counts, then check the two-year result using both endpoint counts and the annual survival probabilities.

For age 30, the number dying is $$ d_{30}=50{,}000-49{,}000=1{,}000, $$ so $$ q_{30}=\frac{1{,}000}{50{,}000}=0.02, \qquad p_{30}=1-0.02=0.98. $$

For age 31, the number dying is $$ d_{31}=49{,}000-47{,}530=1{,}470, $$ so $$ q_{31}=\frac{1{,}470}{49{,}000}=0.03, \qquad p_{31}=1-0.03=0.97. $$

The probability of surviving from 30 to 32, using the endpoint entries, is $$ {}_2p_{30}=\frac{47{,}530}{50{,}000}=0.9506. $$ The annual-probability calculation confirms it: $$ p_{30}p_{31}=0.98(0.97)=0.9506. $$ Therefore the probability of dying before age 32, conditional on being alive at age 30, is $$ {}_2q_{30}=1-0.9506=0.0494. $$

Check. The product of annual survival probabilities matches the endpoint ratio, and the two-year death and survival probabilities sum to one. This calculation concerns the full interval from age 30 to age 32; it does not identify how deaths are distributed within either year.

Key distinction: Integer-age entries determine probabilities between integer ages. They do not, by themselves, determine $\ell_{x+t}$ or a survival or death probability over a partial year when $0<t<1$. A specified within-year mortality assumption, or fractional-age information in the table, is needed for an exact fractional-age calculation.

What Can Be Said at a Fractional Age?

Suppose a life is known to be alive at integer age $x$, and the table gives $\ell_x$ and $\ell_{x+1}$. The table determines the probability of surviving the entire year, $p_x=\ell_{x+1}/\ell_x$. Without further information, it does not determine the probability of surviving just part of that year.

A limited conclusion is still possible if survival is nonincreasing with age. For $0<t<1$, survival from age $x$ to $x+t$ must be at least as likely as survival all the way to $x+1$, and cannot exceed certainty. Hence the fractional-year survival probability lies between $p_x$ and 1, while the probability of dying before $x+t$ lies between 0 and $q_x$. These bounds do not supply an exact value.

p_x\leq P(\text{survive from }x\text{ to }x+t\mid\text{alive at }x)\leq 1, $$ $$ 0\leq P(\text{die before }x+t\mid\text{alive at }x)\leq q_x.

Worked Example: Identifying What the Table Cannot Determine

For a hypothetical life table, $\ell_{67}=12{,}500$ and $d_{67}=500$. A question asks for the probability that a life aged 67 survives to age 67.6. Find the annual mortality information available, give the valid bounds for the requested probability, and state whether an exact answer follows from these entries alone.

Method. First calculate the one-year probabilities for ages 67 to 68. Then use monotonicity to bound survival to the fractional age; do not assume a particular pattern of deaths within the year.

The number surviving to age 68 is $$ \ell_{68}=12{,}500-500=12{,}000. $$ Thus $$ q_{67}=\frac{500}{12{,}500}=0.04, \qquad p_{67}=\frac{12{,}000}{12{,}500}=0.96. $$

Survival from age 67 to 67.6 cannot be less likely than surviving the entire year to age 68, and cannot be more likely than surviving from age 67 with certainty. Therefore the requested probability is bounded by $$ 0.96\leq P(\text{survive to }67.6\mid\text{alive at }67)\leq 1. $$ An exact probability cannot be calculated from the given entries alone. The entries specify that 4% die over the whole year, but not when within that year those deaths occur.

Check. The bounds use the annual survival probability as the lower bound and certainty as the upper bound. Multiplying $q_{67}$ by $0.6$ would impose an extra within-year rule not stated in the problem; it is not a consequence of these life-table entries.

Common Mistakes and a Reliable Workflow

A frequent exam error is to treat an annual probability as if it automatically applied proportionally to a fraction of a year. The table entry $q_x$ is defined for the complete interval from age $x$ to $x+1$. Applying it to a shorter interval requires more than the integer-age entry.

Another error is to use the wrong age index. The value $\ell_{x+1}$ is the number alive at the end of the year that starts at age $x$; $q_x$ is the death probability over that same year. When starting at age $x+1$, use the entries indexed by $x+1$, not $x$.

A disciplined calculation can be organized as follows:

Mark the interval.
Identify the starting age and ending age, and decide whether the interval covers a complete number of years.

Match table entries to the interval.
For the year from $x$ to $x+1$, use $\ell_x$, $\ell_{x+1}$, $d_x$, and $q_x$.

Calculate and reconcile.
Use the relevant count ratios, and check that $d_x=\ell_x-\ell_{x+1}$ and $p_x+q_x=1$.

Check for fractional ages.
If the interval ends or begins at a noninteger age, determine whether the problem supplies a within-year assumption or fractional-age entry before calculating an exact probability.

The key exam skill is not only calculating probabilities, but also recognizing the limits of the supplied information. Integer-age table entries are sufficient for annual and multi-year integer-age probabilities. A fractional-age result needs an additional basis; when none is supplied, state that the exact value is not determined rather than silently selecting an interpolation rule.

Check Your Understanding

For each problem, distinguish annual table information from information that would require a fractional-age assumption.

A hypothetical table has $\ell_{44}=8{,}000$ and $\ell_{45}=7{,}760$. Find $d_{44}$, $q_{44}$, and $p_{44}$. (Answer: $d_{44}=240,\ q_{44}=0.0300,\ p_{44}=0.9700$)
A table gives $\ell_{25}=20{,}000$, $\ell_{26}=19{,}600$, and $\ell_{27}=19{,}208$. Find the probability of surviving from age 25 to age 27 and verify it using the one-year survival probabilities. (Answer: $0.9604$; $p_{25}=0.98,\ p_{26}=0.98,\ 0.98^2=0.9604$)
A hypothetical table has $\ell_{61}=5{,}000$ and $d_{61}=150$. Give the bounds on the probability of surviving from age 61 to age 61.4, and state whether the exact probability is determined. (Answer: $0.9700\leq P(\text{survive to }61.4\mid\text{alive at }61)\leq1$; no exact value is determined)
At age 38, a table gives $\ell_{38}=12{,}000$ and $q_{38}=0.025$. Find $d_{38}$ and $\ell_{39}$, then explain whether these entries determine survival from age 38.2 to age 38.8. (Answer: $d_{38}=300,\ \ell_{39}=11{,}700$; the exact fractional-interval survival is not determined without additional within-year information)
A table gives $\ell_{70}=10{,}000$, $d_{70}=400$, and $\ell_{71}=9{,}600$. Check consistency, then give the probability of surviving to age 70.5 as bounds only. (Answer: consistent since $10{,}000-400=9{,}600$; $0.9600\leq P(\text{survive to }70.5\mid\text{alive at }70)\leq1$)

What You'll Learn