How the maths works

Every calculator on this site shows its working. These guides explain the ideas behind that working: what the actuarial symbols mean, where the numbers come from, and how each step is calculated. Each guide uses real data, the US Social Security Administration’s 2023 period life table, so its examples match what the calculators do.

Why retirement maths is different

Most money calculators need one kind of maths: interest. Money grows, or is discounted, at some rate. Retirement adds a second kind: how long the money is needed for, which nobody knows in advance. Actuaries handle that uncertainty with probabilities from a life table rather than with a single guess.

The difference matters. A man aged 65 in the SSA table has a life expectancy of 18.1 years, but that is an average. His chance of reaching 90 is 24%, and for a woman of the same age it is 35%. A plan built around “life expectancy” leaves out everyone who lives longer than average, which at retirement ages is more than half of people. The guides below show how to put the whole range of outcomes into the numbers.

The three building blocks

Nearly every calculation on the site is built from three pieces.

1. Mortality: who is alive when. A life table gives the chance of dying within a year at each age, $q_{x}$ (Death probability qₓ)Death probability qₓThe chance that someone aged exactly x dies before reaching x + 1, read from the life table.How it works: Life tables. Multiplying one-year survival chances together gives the chance of being alive any number of years from now, ${}_{t}p_{x}$ (Survival probability ₜpₓ)Survival probability ₜpₓThe chance that someone aged x is still alive t years from now.How it works: Survival probability: see survival probability. Adding those chances up over every future year gives life expectancy, $\mathring{e}_{x}$ (Complete life expectancy e̊ₓ)Complete life expectancy e̊ₓThe average number of years still to live for someone aged x, counting part years. The ring marks the complete version.. For a couple, two people’s survival combines into the chance that both, or at least one, are alive: joint and last survivor.

2. Interest: what money is worth when. A payment due later is worth less today, because money today can earn interest in the meantime. The discount factor $v$ (Discount factor v)Discount factor vWhat 1 paid one period from now is worth today: 1/(1 + i).How it works: Present value = 1/(1 + $i$) turns future amounts into today’s: see present value. A series of regular payments for a fixed term has a closed formula, $a_{\angl{n}}$ (Annuity-immediate aₙ)Annuity-immediate aₙThe present value of 1 paid at the end of each period for n periods. The angle around n marks a fixed term.How it works: Annuity formulas: see annuity formulas.

3. The two together. Weight each discounted payment by the chance of being alive to receive it and you have the value of an income for life, the life annuity $\ddot{a}_{x}$ (Life annuity-due äₓ)Life annuity-due äₓThe present value of 1 a year paid at the start of each year for as long as someone aged x lives.. It is how pensions, Social Security and income annuities are valued: see life annuities.

Behind the life table sits a striking regularity: adult death rates rise by roughly the same percentage each year of age. Mortality laws explains the Gompertz and Makeham formulas that describe it, and how a smoker adjustment works.

Reading actuarial notation

Actuarial symbols look dense, but they follow a few rules.

• The main letter says what is measured: $p$ a survival probability, $q$ a death probability, $e$ an expectation of life, $a$ the value of an annuity, $s$ an accumulated value.
• The subscript on the right is the age, or ages: $q_{65}$ is the death probability at 65, and ${}_{t}p_{xy}$ involves two people aged $x$ and $y$.
• The subscript on the left is a time span. ${}_{10}p_{65}$ is the chance a 65-year-old survives 10 years. A vertical bar there, as in ${}_{n|}\ddot{a}_{x}$, means it starts only after that time.
• Marks above the letter change the version. Two dots, $\ddot{a}$, mean payments at the start of each period. A ring, $\mathring{e}$, means the complete expectation, counting part years. A bar over two ages means the last survivor.
• An angle around $n$ means a fixed term, $a_{\angl{n}}$, rather than for life.

Every symbol used anywhere on the site is defined in the glossary, and you can tap any symbol in the text to see its definition.

The assumptions behind every calculation

The calculators and guides share the same assumptions, stated once here and again wherever they matter:

• A period life table. The SSA 2023 period table applies one year’s death rates at every age. It does not allow for future changes in death rates. Over most of the last century US death rates fell, so period tables have tended to understate how long people went on to live.
• Uniform distribution of deaths. Between birthdays, deaths are assumed to be spread evenly through the year. This fills in survival to part years and reproduces the SSA’s published life expectancies.
• Independent lifetimes. For a couple, one person’s death is assumed not to change the other’s chances. Real couples’ lifetimes are somewhat linked.
• Population averages. The table splits by sex only. Health, income and smoking move individual chances away from it; the smoker option is a rough multiplier, not a smoker table.
• Nominal or real, always stated. Amounts in future dollars are discounted at nominal rates and amounts in today’s dollars at real rates. Each calculator says which it uses.

How to use these guides

If you are new to the subject, read the guides in order: life tables, survival probability, life expectancy, then present value and annuity formulas, and finally life annuities. Each one is self-contained, with the idea in plain English first, then the notation and formula, a worked example from the real table, and a small interactive tool to try your own numbers.

If you came from a calculator, follow the links from its “Show the working” panel: each step links to the guide that explains it. The retirement income calculators put the same ideas to work on your own figures.

These guides explain how the calculations work. They are educational illustrations, not financial advice.

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