Annuities and savings Updated November 2026

How long savings will last

A balance PV paying out PMT per period at rate r lasts n = -ln(1 - PV x r / PMT) / ln(1 + r) periods. If PMT is no more than PV x r, the interest covers the withdrawal and the money lasts forever. In Excel use =NPER(rate, -withdrawal, balance).

Syntax

n = -ln(1 - PV x r / PMT) / ln(1 + r)

Part	What it is
`PV`	Starting balance
`PMT`	Withdrawal per period, taken at the end of the period
`r`	Interest rate per period earned on the balance

Examples

100,000 earning 6% a year, withdrawing 1,000 per month
n = -ln(1 - 100,000 x 0.005 / 1,000) / ln(1.005)

Result: n = 138.98 months (about 11.6 years)

Excel: =NPER(6%/12, -1000, 100000)

200,000 at 0.4% per month, withdrawing 2,000 per month
n = -ln(1 - 200,000 x 0.004 / 2,000) / ln(1.004)

Result: n = 127.96 months (10.7 years)

Reverse: monthly income from 500,000 over 25 years at 6%
PMT = 500,000 x 0.005 / (1 - 1.005^-300)

Result: PMT = 3,221.51 per month

Excel: =PMT(6%/12, 300, -500000)

Common mistakes

Dividing the balance by the withdrawal (100,000 / 1,000 = 100 months) ignores the interest still being earned, so it understates how long the money lasts.
Taking the log of a negative number: if PMT is less than or equal to PV x r the balance never runs out and the formula does not apply.
Ignoring inflation: a fixed withdrawal buys less each year, so a sustainable plan usually increases the withdrawal over time.

Practice How long savings will last

These drills check your answer on live data, free, no account needed.

How long a balance lastsA retiree has $110,000 invested at 3.25% APR, compounded monthly, and withdraws $500 at the end of each month. How many monthly withdrawals can be made before the money runs out? Give N to 2 decimals, as a calculator shows it.
Monthly income from savingsYou retire with $200,000 earning 3.25% APR, compounded monthly. What equal withdrawal at the end of each month uses up the balance in exactly 35 years?

Questions

What withdrawal can I take forever?

At most the interest: PV x r per period. 100,000 at 6% supports 500 per month indefinitely; anything above that draws down the principal.

What is the 4% rule?

A retirement rule of thumb: withdraw 4% of the starting balance in the first year, then raise it with inflation. Historically a diversified portfolio has usually lasted 30 years at that rate, but it is not guaranteed.

Present value of an annuityHow to find the present value of equal payments with PV = PMT x (1 - (1 + r)^-n) / r. Worked examples, Excel PV and why this is the loan formula in disguise.
Monthly deposit for a savings goalThe monthly deposit needed to reach a target is PMT = FV x r / ((1 + r)^n - 1). Examples for 10,000 in 5 years and 50,000 in 10 years, Excel PMT and errors.
PerpetuityA perpetuity pays a fixed amount forever and is worth PV = C / r. The growing perpetuity formula C1 / (r - g), examples for preferred stock and dividends.
Real vs nominal returnThe real return strips inflation out of a nominal return: real = (1 + nominal) / (1 + inflation) - 1. Examples, the quick approximation and its errors.