What a life table is
A life table starts with an imaginary group of people, all the same age, and follows them until none is left. The group's starting size is arbitrary — 1,000, 10,000, 100,000 — because every answer is a ratio and the starting size cancels. That single idea is what makes the table usable: you are never asked to remember a probability, only to divide one number in a column by another.
Everything in this topic is one of five quantities, and three of them are differences or ratios of the first.
The functions
lx — the number of the original group still alive at age x. This is the only column that need actually be printed. The starting value, l at the youngest age in the table, is called the radix.
dx — the number who die between age x and age x+1:
qx — the probability that a life aged exactly x dies within the year. This is the rate of mortality, and it is a conditional probability: it is conditional on having reached age x at all.
px — the probability the same life survives the year. Since they must do one or the other:
tpx — the probability of surviving t whole years. This is the one that does the work in annuity and assurance calculations, and it is a single division:
It is worth seeing why that is a single division rather than a product of t separate survival probabilities. It is both:
Every numerator cancels the next denominator and only lx+t/lx survives. Once you have seen that cancellation you will stop multiplying chains of survival probabilities in an exam, which is where arithmetic slips come from.
The deferred probability, which is where people get stuck
t|qx is the probability that a life aged x survives t years and then dies within the following year. Two things have to happen, so it is a product:
The right-hand form is the one to use, and it is the reason the d column is printed at all: a deferred mortality probability is a single entry from the d column divided by the l value at the starting age. No multiplication, no chain.
Note carefully which denominator it is. qx+t divides by lx+t — it is a probability for someone who has already reached age x+t. t|qx divides by lx — it is a probability for someone standing at age x today, who does not yet know whether they will get there. Mixing these two denominators is the commonest error in this topic, and it is invisible in the answer because both numbers are small and plausible.
The curtate expectation of life
The expected number of complete years a life aged x will live is the sum of the survival probabilities:
"Curtate" means whole years only — a life who dies after ten years and eight months contributes ten. The derivation is worth one minute of your time: the probability of living at least one more year contributes 1 if it happens, the probability of living at least two more contributes another 1, and so on, so the expectation is the sum of those probabilities. That is the standard result that the expectation of a non-negative integer variable is the sum of its tail probabilities, and it is the whole proof.
The temporary version, ex:n, stops the sum after n terms and is the expected number of complete years lived in the next n. It cannot exceed n, which is a free check.
Where marks are lost
Dividing by the wrong l. Covered above and worth repeating, because it costs more marks in this topic than everything else combined. Before dividing, ask: at what age is the person standing when this probability is quoted? That age gives the denominator.
Confusing qx with the force of mortality. qx is a probability over a whole year and can never exceed 1. The force of mortality μx is an instantaneous rate and can be any positive number. They are close in value when mortality is light, which makes the confusion hard to spot.
Reading the table at the wrong age. "A life aged 60 dies in the third year" means death between ages 62 and 63, so the numerator is d62, not d63. Write the ages out rather than counting in your head.
Questions students ask
Does the radix matter?
No. Every quantity in this topic is a ratio of two l values, so multiplying the whole column by any constant changes nothing. Tables are printed with a round radix purely so the entries are convenient whole numbers.
What happens between integer ages?
The table says nothing, so an assumption has to be supplied — most often that deaths are spread uniformly across the year, or that the force of mortality is constant across it. Those assumptions are a syllabus topic of their own. The thing to carry forward is that they are assumptions, chosen and stated, not facts the table contains.
Why is it called a "rate" of mortality if it is a probability?
Historical usage again, and it is genuinely confusing because the force of mortality has a better claim to the word. Treat qx as a probability in every calculation and let the name be a name.
A worked example
Here is a short life table. The figures are invented for this example — they are not from any published mortality table, and nothing here should be used as a mortality assumption.
| Age x | l(x) | d(x) |
|---|---|---|
| 60 | 9,500 | 120 |
| 61 | 9,380 | 140 |
| 62 | 9,240 | 165 |
| 63 | 9,075 | 195 |
| 64 | 8,880 | — |
The d column is just the differences: 9,500 − 9,380 = 120, and so on.
The probability a life aged 60 dies within a year
The probability the same life survives the year
And 0.012632 + 0.987368 = 1, as it must.
The probability of surviving three years
One division. Doing it as p60 × p61 × p62 gives 0.987368 × 0.985075 × 0.982143, which is the same number, three rounding opportunities later.
The probability a life aged 60 survives two years and then dies in the third
Check it the long way: 2p60 × q62 = (9,240/9,500) × (165/9,240). The 9,240s cancel and 165/9,500 is left. The cancellation is the shortcut, and seeing it once is better than memorising the formula.
Note the contrast with q62 = 165/9,240 = 0.017857 — the probability for someone who has already reached 62. The two differ by about 3%, and using one where the other is wanted is the error this topic punishes hardest.
The probability of dying between ages 61 and 63
Equivalently d61 + d62 over l60, which is (140 + 165)/9,500. Same 305.
The expected number of complete years lived in the next four
= (9,380 + 9,240 + 9,075 + 8,880) / 9,500 = 36,575 / 9,500 = 3.8500
The maximum possible is 4, and mortality at these ages is light, so a figure a little under 4 is exactly what the table should give. An answer of 3.2 or 4.1 would be telling you that something had gone wrong before you looked for it.
Where this fits in CM1
This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CM1 topics explained
- Simple and compound interest, and the principle that rules out one of them
- Present value, the discount factor, and why d is not i
- Nominal rates payable p times a year, and the loan that costs more than it says
- Force of interest explained, and how it relates to i, v and d
- When the interest rate changes: steps, and a force that varies
- Continuous payment streams, and why they are worth more than they look
- Annuity functions in CM1: immediate, due, deferred and temporary
- Perpetuities and increasing annuities: payments that never stop, or never stay level
- Money rates, real rates and inflation: the exact link, not the subtraction
- Real yields from an inflation index, and index-linked payments
- Loan schedules: the prospective and retrospective methods
- Net premiums and the equivalence principle, worked from first principles
- Reserves in CM1: prospective, retrospective, and why they agree
- Thiele's differential equation, explained term by term
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.