Annuity
A series of equal payments made at regular intervals, such as monthly contributions or yearly withdrawals.
PV = PMT × (1 − (1 + r)^−n) ÷ r ; FV = PMT × ((1 + r)^n − 1) ÷ r
Explanation
The word “annuity” has two meanings. In everyday use it is an insurance product that pays you income, often for life. In time-value-of-money math, which is how the app uses it, an annuity is simply a stream of equal payments at regular intervals: $500 every month, $40,000 every year, and so on.
Annuity formulas save you from valuing each payment one by one. The future value formula tells you what a series of deposits grows to. The present value formula tells you what a series of future payments is worth today, or equivalently, how big a lump sum you need to fund those payments.
If payments happen at the start of each period instead of the end (an “annuity due”), each payment gets one extra period of growth, so multiply the result by (1 + r).
In scenarios, recurring contributions are treated as annuities when projecting balances, and level spending in retirement can be valued as an annuity to estimate how much you need saved. Real plans usually adjust spending for inflation, which the app handles by working in real (inflation-adjusted) returns.
Example
How much do you need to fund $40,000 a year for 25 years, at 5%? Payments come at the end of each year.
- (1.05)^−25 = 0.2953
- 1 − 0.2953 = 0.7047
- 0.7047 ÷ 0.05 = 14.094
- PV = $40,000 × 14.094 = $563,758
If each payment comes at the start of the year instead, multiply by 1.05: $563,758 × 1.05 = $591,946.
Note that the 25 payments total $1,000,000. The lump sum is much smaller because the money not yet withdrawn keeps earning 5% while it waits.