How this instrument works
In annuity terminology, 'immediate' is a specific word: it means the income starts right away, inside the first payment period, rather than after years of deferral. This instrument answers the question an insurer answers when pricing that kind of contract — given Monthly income payment, $ for Payment period, years, discounted at Annual discount rate, %, what lump sum handed over today is exactly equivalent? The output, Present value (cost today), is that lump sum: small enough that it could not simply be divided into equal slices, but large enough that, growing at the stated rate while it is drawn down, it stretches to cover every payment with nothing left over at the end.
The formula treats each future monthly payment as its own small future value and discounts it back separately, then sums all of those discounted payments in closed form. Because the payments recur every month rather than once, the annual rate is first converted to a monthly rate by dividing by 1,200 (percent to decimal, then divided by 12), and the term is converted to a payment count by multiplying years by 12 — so a 20-year stream is priced across 240 discounted monthly payments, not 20 annual ones. That monthly convention matches how retail immediate annuity quotes are usually built and how the golden figures on this page were checked.
This is fixed-term-certain pricing, not life-contingent pricing. A real immediate annuity sold by an insurance company usually pools mortality risk across many buyers and keeps paying for as long as the annuitant lives, however long that turns out to be — a promise this arithmetic cannot make, since it only prices a stream that is known in advance to stop after a fixed count of payments. It also excludes commissions, administrative loads, and the profit margin an insurer builds into a quoted price, all of which push a real quote above this pure time-value-of-money number.
- Enter the income you want each month into Monthly income payment, $.
- Set Annual discount rate, % to the yearly rate you want applied to that stream.
- Enter Payment period, years for how many years the income is guaranteed to run.
- Read Present value (cost today) — the lump sum equivalent to that entire stream, priced today.
- Raise or lower Annual discount rate, % alone to see how sensitive the price is to the assumed rate.
Worked example — pricing $2,000 a month for 20 years
Set Monthly income payment, $ to 2,000, Annual discount rate, % to 4, and Payment period, years to 20. That is a monthly rate of 4 ÷ 1,200, or about 0.3333%, applied across 240 monthly payments. Present value (cost today) reads $330,043.72 — the sum a retiree would need to hand over today, at that assumed rate, to fund $2,000 arriving every month for the next two decades.
Two thousand dollars a month for 240 months, added up with no discounting at all, would be $480,000; the true price comes in well under that because money owed years from now is worth less today than money owed next month. Cut the term to 10 years at the same payment and rate and the price falls to $197,540.35, less than half — most of the value in a 20-year stream sits in its later payments, and cutting the term in half removes far more than half the price. Doubling the monthly payment to $4,000 while holding the rate and term fixed exactly doubles the price to $660,087.43, since the formula scales linearly with the payment amount alone.
Questions
What does 'immediate' mean in an immediate annuity?
It means the income begins right away, inside the first payment period after purchase, rather than after a waiting stretch. That distinguishes it from a deferred annuity, where premiums are paid in and left to grow for years before any income phase starts. This calculator only prices the immediate case: payments begin at once and run for Payment period, years with nothing accumulating beforehand.
How does this differ from a calculator that finds the payment from a balance?
This one runs the arithmetic in the other direction. A payout-style calculator starts from a balance you already hold and solves for how large a level withdrawal it can sustain; this one starts from the income you want and solves for what lump sum buying that income costs today. Same underlying formula, rearranged — one solves for the payment, this one solves for the price.
Why does raising Annual discount rate, % lower Present value (cost today)?
A higher rate means a dollar sitting idle today grows faster, so fewer dollars are needed up front to eventually cover the same monthly payments. Every payment gets divided by a larger compounding factor before being added into the total, and payments further in the future shrink the most, since that factor is raised to a higher power the later a payment falls in the schedule.
Does this match what an insurance company would actually quote me?
Not exactly, and it is not meant to. A real quote for a life-contingent immediate annuity prices in mortality pooling — the insurer collects from buyers who die early to keep paying buyers who live long — plus administrative costs and profit margin. This figure is the pure fixed-term-certain price with none of that layered on, useful as a floor to compare a real quote against, not as a substitute for one.
Why is Payment period, years multiplied by 12 instead of used directly?
Because the income is monthly, not annual, so the formula needs a monthly payment count and a matching monthly rate to discount each one correctly. Payment period, years times 12 gives that payment count, and Annual discount rate, % divided by 1200 gives the matching monthly rate — mixing an annual rate with a monthly count, or the reverse, would misprice every payment in the stream.
References
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.