Common mistake 1 with Life table

Reading e_0 as a forecast for a specific person. It is a cohort average under fixed mortality, with no allowance for future improvement.

Common mistake 2 with Life table

Using a Gompertz fit calibrated on middle age to describe mortality at very young or very old ages, where the law fits poorly.

Common mistake 3 with Life table

Confusing the force of mortality mu(x) (an instantaneous rate, can exceed 1) with the probability q_x (bounded by 1).

Common mistake 4 with Life table

Comparing life expectancies computed from a period table with those from a cohort table. They answer different questions and the cohort figure is routinely years higher.

EconStatsCraft › Life table

Life table

Builds a full single-decrement life table from a mortality law - Gompertz, Makeham or a constant force - and reports l_x, d_x, q_x, curtate and complete life expectancy, the median and modal age at death, and the survival and death-density curves.

The intuition

A life table follows a notional cohort - conventionally 100,000 newborns - and records how many are still alive at each age. Everything else in life contingencies is arithmetic on that one column. The mortality rate qx is the probability of dying within the year given you reached age x, and it is a CONDITIONAL probability, which is why it can rise towards one while the number of deaths at that age falls to almost nothing.

When to use it

Any life-contingent calculation starts here: annuity and assurance values, reserves, pension liabilities, or simply answering how long a cohort at a given age is expected to live.

Where the estimator comes from

From the survival function S(x)=P(T>x): the one-year mortality rate is q_x=[S(x)-S(x+1)]/S(x), and p_x=1-q_x. The cohort recursion is \ell_{x+1}=\ell_x p_x with deaths d_x=\ell_x-\ell_{x+1}. Person-years lived are L_x=\int_0^1 \ell_{x+t}\,dt\approx\tfrac12(\ell_x+\ell_{x+1}) under a uniform distribution of deaths, total future person-years are T_x=\sum_{k\ge x}L_k, and life expectancy is simply e_x=T_x/\ell_x. The central death rate m_x=d_x/L_x is the quantity actually observed in population data, and q_x\approx m_x/(1+\tfrac12 m_x) converts between them.

The model

{}_{}p_x = e^{-\int_0^1 \mu(x+s)\,ds},\qquad \mu(x)=A + B c^{x}\ \text{(Makeham)},\qquad e_x = \sum_{k\ge 1} {}_{k}p_x

What it assumes

A single decrement: the only way to leave the table is death. No lapse, no transfer, no new entrants.
The chosen law describes mortality over the whole age range. Makeham adds a constant term for accident-type deaths that do not rise with age; Gompertz has only the exponential ageing term.
Deaths are spread uniformly over each year of age when converting the curtate expectation to the complete one (the +0.5).
The population is stationary enough that a period table means something. A period life table applies this year's age-specific rates to a hypothetical cohort, so it describes nobody who actually lives - a real cohort meets improving rates as it ages.

Statistical properties

qx is a probability and is bounded by one; mx is a rate per person-year and is not. Confusing them understates mortality at old ages, where the two diverge sharply.
Life expectancy at birth is not the age most people die at. It is an average dragged down by infant deaths, which is why e0 can be 40 in a population where most adults reach 60.
Period tables apply one calendar year's rates to a hypothetical cohort. Cohort tables follow real people. For pricing an annuity the cohort table is correct, and using a period table understates longevity and therefore the liability.
Mortality is roughly exponential in age over the adult range - Gompertz - so log qx is close to linear from about 30 to 90. That is the single most useful diagnostic plot for a table.
The uniform-distribution-of-deaths assumption behind Lx is harmless in mid-life and poor in the first year, where deaths cluster in the first days. Infant mortality uses a separation factor instead.
Selection matters: people who have just passed a medical are healthier than the general population of the same age, so insurers use select-and-ultimate tables where qx depends on time since underwriting as well as age.

How to read the output

q_x is the probability a life exact age x dies before x+1; p_x = 1 - q_x is survival. Under Gompertz-Makeham q_x roughly doubles every 8 to 9 years in adulthood.
e_x is the average number of further years lived. The curtate version counts whole years only; the complete version adds about half a year.
The modal adult age at death is where d_x peaks - it is higher than the median because the death distribution is left-skewed.
Raising c steepens the ageing term and pulls every expectation down; raising the Makeham A shifts mortality up at all ages equally.

A worked reading

A Makeham table (A = 0.0002, B = 3.5e-05, c = 1.10) on a radix of 100,000 closing at age 121 gives complete e₀ = 76.36 years and e₆₅ = 16.42. So a 65-year-old is expected to reach 81.42 - five years past the figure quoted at birth. That is not a contradiction: e₀ averages in everyone who dies young, and having survived to 65 you have escaped all of them. Note also that the median age at death is 79 and the modal adult age at death is 82, both above e₀: the distribution of age at death is left-skewed, so the mean is the lowest of the three summaries and the one least like a typical life. For an annuity provider only e₆₅ matters, and pricing a pension off e₀ would understate the liability by years of payments per policy.

Symptom → diagnosis → what to do

If you see this	It means	Do this
qx above 1	A rate has been used as a probability	Convert with qx = mx/(1 + mx/2)
log qx not close to linear in adulthood	Data error, or a genuine shock such as an epidemic	Plot it before doing anything else
Life expectancy rising with age	Correct and often surprising	Having survived to 65, you are expected to pass the e0 figure
A jagged qx at old ages	Very few exposed lives	Graduate the rates before using them
Annuity priced from a period table looks cheap	Longevity improvement ignored	Use a projected cohort basis
Table stops at 100 with survivors remaining	Truncation	Close the table with a limiting age or an exponential tail

Common mistakes

Reading e_0 as a forecast for a specific person. It is a cohort average under fixed mortality, with no allowance for future improvement.
Using a Gompertz fit calibrated on middle age to describe mortality at very young or very old ages, where the law fits poorly.
Confusing the force of mortality mu(x) (an instantaneous rate, can exceed 1) with the probability q_x (bounded by 1).
Comparing life expectancies computed from a period table with those from a cohort table. They answer different questions and the cohort figure is routinely years higher.

Where to read this properly

Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks (2020), ch. 3.
Bowers et al., Actuarial Mathematics (1997), ch. 3.
Preston, Heuveline and Guillot, Demography (2001), ch. 3, for the period versus cohort distinction.

Related Actuarial

Life annuity · Life insurance pricing · Chain ladder reserving · Buhlmann credibility · Survival analysis · Mortality graduation · Aggregate loss model · Ruin probability · Multi-state Markov model · Loss severity fitting

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Life table

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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