Life Table¶

Definition¶

A life table specifies survivor numbers $l_x$ at ages $x$ for a hypothetical cohort with a chosen initial size, the radix. Annual deaths are $d_x=l_x-l_{x+1}$ $l_{x + 1}$ . Within the mortality model represented by the table, for $l_x>0$ $0$ ,

{}_tp_x=\frac{l_{x+t}}{l_x},\qquad q_x=\frac{d_x}{l_x}.

Fractional ages require an interpolation assumption if only integer-age entries are supplied.

Interpretation¶

The table expresses a survival model through cohort counts. Its numbers may be modelled or estimated; the displayed ratios do not remove sampling or projection uncertainty.

Cohort diagram: 1000 lives at age 60, of whom 20 die within the year, leaving 980 at age 61 and 900 at age 65, with q_60 = 0.02 and 5p60 = 0.9 written as ratios.

The figure follows the cohort in the Examples. Of $l_{60}=1000$ lives, $d_{60}=20$ die before age 61, so $q_{60}=20/1000=0.02$ $0.02$ ; after five years $l_{65}=900$ remain, so ${}_5p_{60}=900/1000=0.9$ $0.9$ . Every probability is a ratio of table entries, which is why rescaling the radix leaves it unchanged.

Key Results¶

${}_{u|t}q_x=(l_{x+u}-l_{x+u+t})/l_x$ $l_{x + u + t}) / l_{x}$ .
Scaling all entries by the same positive constant leaves survival probabilities unchanged.
A population table and an insured-life table can differ through selection. Choose the table appropriate to the covered population.
Use table ratios only when a table is given or a corresponding table model has been specified.

Examples¶

For $l_{60}=1000$ and $l_{61}=980$ $980$ , $q_{60}=20/1000=0.02$ $0.02$ . If $l_{65}=900$ $900$ , then ${}_5p_{60}=0.9$ $0.9$ .

Prerequisites¶

Actuarial Survival and Death Probabilities

Sources¶

Life Insurance · Lecture 2

Aliases: mortality table, survivorship table, cohort life table, actuarial table Keywords: radix, l x survivors, d x = l x - l x+1, q x = d x / l x, t p x = l x+t / l x

Last updated: 17 October 2026 21:53