Do My NumbersMethodology

How the annuity calculator values income

The annuity calculator has two engines behind one switch. The annuity-certain side is pure interest arithmetic. The life annuity side adds a survival weight to every payment, which turns a premium into a fair income: the income that uses the premium up exactly, on average, with nothing kept for costs or profit.

Annuity-certain: payments for a fixed term

i=(1+iyr)1/f−1i = \left(1 + i_{\text{yr}}\right)^{1/f} - 1
Interest rate i: The interest rate per period, used to move money between dates.
an=1−vnia_{\angl{n}} = \frac{1 - v^{n}}{i}
Annuity-immediate aₙ: The present value of 1 paid at the end of each period for n periods. The angle around n marks a fixed term.
a¨n=(1+i) an\ddot{a}_{\angl{n}} = (1 + i)\, a_{\angl{n}}
Annuity-due äₙ: The present value of 1 paid at the start of each period for n periods. The two dots mark payments in advance.
(Ga)n=1−(1+g1+i)ni−g(Ga)_{\angl{n}} = \frac{1 - \left(\frac{1+g}{1+i}\right)^{n}}{i - g}
Growing annuity: The present value of n payments that start at 1 and grow by g each period, paid at the end of each period.
a∞=1ia_{\infty} = \frac{1}{i}
Perpetuity: The present value of 1 paid at the end of every period forever: 1/i.

A yearly rate is converted to the equivalent rate per payment, so twelve monthly rates compound back to exactly the yearly rate. The payment, present value or future value then comes from multiplying or dividing by the right factor.

Life annuity: payments while someone is alive

a¨x(m)=1m∑j≥0vj/m(1+e)⌊j/m⌋ j/mpx\ddot{a}^{(m)}_{x} = \frac{1}{m} \sum_{j \ge 0} v^{j/m} (1+e)^{\lfloor j/m \rfloor}\, {}_{j/m}p_{x}
a¨x:n‾=a¨n+n∣a¨x\ddot{a}_{\overline{x:\angl{n}}} = \ddot{a}_{\angl{n}} + {}_{n|}\ddot{a}_{x}
Guaranteed life annuity: Payments for n years whether or not the person is alive, then for as long as they live.
a¨=a¨x+ρ(a¨y−a¨xy)\ddot{a} = \ddot{a}_{x} + \rho\left(\ddot{a}_{y} - \ddot{a}_{xy}\right)
Total annuity factor: The value of 1 a year of starting income, allowing for every feature chosen: guarantee, reversion and increases.
P=Premiuma¨P = \frac{\text{Premium}}{\ddot{a}}
Payment P: The amount of each payment.

Survival tpx{}_{t}p_{x} comes from the SSA table, with deaths spread evenly within each year so monthly payments can be weighted between birthdays. A spouse’s life is assumed independent of yours. For a CPI-linked annuity, ee is one expected inflation rate held for every year.

Starting values

  • Annuity-certain: $200,000 paid out monthly over 20 years at 5%, payments at the end of each month, no growth. A plain case whose factor is easy to check by hand.
  • Life annuity: a 65-year-old man with $100,000, monthly level income valued at 4.5%. The fair value moves a lot with the rate, so it is shown as an input to change, not a market quote.
  • CPI-linked option: expected inflation of 2.5% a year until changed.
  • Spouse share: 50% of the income continues to a surviving spouse unless another share is chosen.

The table behind the survival weights

  • SSA period life table (2023), Social Security Administration, Office of the Chief Actuary. Edition: Period Life Table, 2023, as used in the 2026 Trustees Report. Retrieved October 22, 2026.

Each file keeps its source address, edition and retrieval date, and is tested against figures the publisher prints. All of them are listed on the data sources page.

Missing from a fair value

  • Insurer expenses, commission, profit and capital costs, all of which real quotes include.
  • The insurer’s own life tables for annuity buyers, who tend to live longer than the population, and improvements in death rates after 2023.
  • Tax on annuity income, and the insurer’s credit risk.
  • Interest rates that change over time, and the fact that most life annuities can’t be cashed in.