What Is the Present Value of an Annuity? Formula and Examples

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The present value of an annuity is the amount a series of equal, regularly spaced future payments is worth today, after each payment is discounted at a chosen rate of return. It answers one question: how much money would you need now, invested at that rate, to reproduce the same stream of payments? The present value of an annuity formula does that discounting in one step instead of payment by payment.

The Present Value of an Annuity Formula

For an ordinary annuity, where each payment arrives at the end of a period, the formula is:

PV = PMT × [1 − (1 + r)^−n] / r

where:

• PV = present value of the payment stream
• PMT = the dollar amount of each payment
• r = the discount rate (interest rate) per period
• n = the number of payment periods

The bracketed part, [1 − (1 + r)^−n] / r, is called the present value interest factor of an annuity (PVIFA), or annuity discount factor. It depends only on the rate and the number of periods, so the PV of any level annuity is simply the payment multiplied by that factor. Printed annuity tables list PVIFA values by rate and term for that reason. The single-payment version, the PV factor for one cash flow received n periods out, is (1 + r)^−n. The annuity factor is the sum of those single-payment factors for periods 1 through n.

Why the formula works

The formula is a shortcut for adding up each discounted payment. With v = 1 / (1 + r), the discount factor for one period, the PV of n end-of-period payments is PMT × (v + v² + … + vⁿ). Multiplying by (1 + r) shifts every term down one power, giving PMT × (1 + v + … + vⁿ⁻¹). Subtracting the first sum from the second cancels the middle terms: r × PV = PMT × (1 − vⁿ), so PV = PMT × (1 − vⁿ) / r. This is the sum of a finite geometric series, and it shows why the formula only works when every payment is equal and every period is the same length.

Match the rate to the payment period

The rate r and the count n must use the same period. If payments are monthly and you have an annual rate, divide the annual rate by 12 and multiply the number of years by 12. A 10-year stream of monthly payments at a 6% annual rate uses r = 0.005 and n = 120. Plugging in 6% and 10 would value ten annual payments, a different cash flow altogether. (Dividing a nominal annual rate by 12 assumes monthly compounding; if your rate is an effective annual rate, convert it with (1 + annual rate)^(1/12) − 1 instead.)

Worked Example: Discounting Payment by Payment

Hypothetical example. Suppose you will receive $1,000 at the end of each year for five years, and your discount rate is 5%.

Discounting each payment separately:

With the formula: (1.05)^5 = 1.27628, so (1.05)^−5 = 0.78353. Then (1 − 0.78353) / 0.05 = 4.32948. Multiply by $1,000 and PV = $4,329.48.

Both methods agree. The five $1,000 installments are worth only about $4,329 today at 5%, less than $5,000 in hand, because each later payment is discounted for longer.

Example: Lump Sum vs. Annuity Payments

The most common practical use is choosing between a lump sum now and a stream of payments later.

Hypothetical example. Someone is offered either $650,000 today or an ordinary annuity paying $50,000 a year for 25 years. They use a 6% discount rate.

1. Compute (1.06)^25 = 4.29187, so (1.06)^−25 = 0.23300.
2. Compute the annuity factor: (1 − 0.23300) / 0.06 = 12.7834.
3. Multiply by the payment: $50,000 × 12.7834 = $639,168.

At 6%, the payments are worth about $639,168 today, roughly $10,832 less than the $650,000 lump sum. On a time-adjusted basis, the lump sum comes out ahead. The annuity pays $1.25 million in total, but that nominal total is not the right comparison: most of those dollars arrive years from now.

The same comparison applies to a pension lump-sum offer (often called the commuted value), a lottery prize offered as cash or installments, a structured legal settlement or an annuity contract’s payout options. A pension plan computing a commuted value discounts its future benefit obligations under its own assumptions.

How the Discount Rate Changes the Answer

The discount rate is the assumption that moves the result most. Holding the $50,000-a-year, 25-year annuity fixed:

At 4%, the annuity beats the $650,000 lump sum by a wide margin. At 8%, it falls far short. A higher rate discounts future payments more heavily, and because distant payments are discounted by large powers of (1 + r), long annuities are especially sensitive to the rate. Testing a range of rates is more informative than one point estimate.

Choosing a discount rate

The discount rate should reflect your opportunity cost: the return you could reasonably earn on the money elsewhere at comparable risk. If you could earn 5% on a corporate bond, 5% is a sensible rate for comparing against an annuity of similar risk.

The usual floor is the risk-free rate, typically proxied by U.S. Treasury yields, since Treasuries are generally treated as the closest thing to a risk-free investment. Riskier payment streams deserve higher rates. In corporate settings, a company may discount project cash flows at its weighted average cost of capital. For U.S. gift and estate tax valuations of annuities, the IRS prescribes a specific rate (the Section 7520 rate) rather than letting the taxpayer choose.

Annuity Due: Payments at the Start of Each Period

An annuity due pays at the beginning of each period instead of the end. Rent is the classic example: you pay at the start of the month you occupy the property. Because every payment arrives one period earlier, each is discounted one period less, and the whole stream is worth more.

The present value of an annuity due (PVAD) formula multiplies the ordinary annuity result by (1 + r):

PV(due) = PMT × [1 − (1 + r)^−n] / r × (1 + r)

Applied to the earlier example, if the $50,000 payments began immediately:

$639,168 × 1.06 = $677,518

That is about $27,518 more than the $650,000 lump sum, so under this timing the annuity becomes the better choice. A one-period shift in timing reverses the decision.

An equivalent way to see it: an annuity due of n payments is one undiscounted payment today plus an ordinary annuity of n − 1 payments. Either method gives the same value.

Variations on the Basic Formula

Deferred annuity: payments starting in the future

If payments do not begin for several periods, value the stream as an ordinary annuity at the point just before payments start, then discount that amount back to today:

PV = PMT × [1 − (1 + r)^−n] / r × (1 + r)^−k

Here k is the number of periods of deferral before the ordinary annuity begins, so the first payment arrives at the end of period k + 1. Hypothetical example: if the $50,000, 25-year annuity at 6% were deferred so that the first payment arrives at the end of year 6 (k = 5), the value today would be $639,168 ÷ (1.06)^5 = $639,168 ÷ 1.33823, or about $477,600.

Growing annuity

When payments rise at a constant rate g each period, such as a benefit with a fixed annual increase, the present value of a growing annuity is:

PV = PMT / (r − g) × [1 − ((1 + g) / (1 + r))^n]

PMT is the first payment, received at the end of period 1. Hypothetical example: a first payment of $10,000 growing 2% a year for 10 years, discounted at 5%, gives $10,000 / 0.03 × [1 − (1.02 / 1.05)^10] = about $83,893. A level $10,000 annuity over the same term and rate is worth about $77,217. The formula requires r and g to differ; if r equals g, the PV is simply n × PMT / (1 + r).

Perpetuity

If the payments never end, the term (1 + r)^−n shrinks toward zero as n grows, and the formula collapses to PV = PMT / r. This is the value of a level perpetuity, and it is a useful upper bound: no finite annuity at the same rate can be worth more than PMT / r.

How the Present Value of an Annuity Is Used

• Comparing annuities with different terms. Two payout schedules with different amounts or lengths can be compared on equal footing by converting each to a present value at the same rate.
• Loans and mortgages. A loan’s principal is the present value of its payments at the loan rate. Rearranging the formula gives the payment for a given loan: PMT = PV × r / [1 − (1 + r)^−n]. Fixed-rate mortgages paid at the end of each month are ordinary annuities.
• Bond valuation. A bond’s coupons form an ordinary annuity. The bond’s price is the PV of the coupon annuity plus the PV of the face value repaid at maturity.
• Valuation and capital budgeting. Level cash flows inside a discounted cash flow model or a net present value calculation can be valued with the annuity factor instead of discounting each year separately.

Calculating it in a spreadsheet

Excel and Google Sheets use =PV(rate, nper, pmt, [fv], [type]). For the main example, =PV(0.06, 25, −50000) returns 639,168. Setting type to 1 treats payments as an annuity due and returns 677,518. The sign convention matters: entering the payment as a negative cash outflow returns a positive PV, and vice versa.

Limitations

• One constant rate. It applies a single discount rate across every period. If rates or your opportunity cost change over time, a single-rate PV is an approximation.
• Equal, regular payments. The closed-form formula holds only for level payments at evenly spaced intervals (or constant growth, in the growing version). Irregular amounts require discounting each cash flow separately.
• Certainty of payment. It treats every payment as guaranteed. A commercial annuity contract depends on the insurer’s ability to pay, and a life annuity ends at death, so the number of payments is not fixed. Insurers and actuaries value life annuities with mortality assumptions layered onto this math; the basic formula does not capture that.
• Inflation. Fixed nominal payments lose purchasing power over time. Using a nominal discount rate already reflects expected inflation, but if you want to think in real terms, you need a real rate and real payments, consistently.
• Taxes and fees. A lump sum and a stream of payments may be taxed differently, and annuity products can carry fees or surrender charges. The PV formula compares pre-tax, pre-fee cash flows unless you adjust them first.

Distinctions Readers Often Confuse

Present value vs. future value of an annuity. PV discounts payments back to today. Future value (FV) compounds them forward to the date of the last payment, using FV = PMT × [(1 + r)^n − 1] / r for an ordinary annuity. The two are linked: FV = PV × (1 + r)^n. FV helps estimate what regular savings will grow to; PV helps decide what a payment stream is worth now.

Discount rate vs. the annuity’s interest rate. The rate credited inside an annuity contract, or the interest rate on a loan, is a term of that product. The discount rate in a PV calculation is your own required return or opportunity cost. They can be the same number, but they answer different questions. Discounting a contract’s payments at its own credited rate tells you little about whether it beats your alternatives.

PV of an annuity vs. NPV. The PV of an annuity values only the payment stream. NPV subtracts the upfront cost from the present value of all expected cash flows. If you pay $600,000 for an annuity worth $639,168 at your discount rate, the NPV is $39,168.

PV vs. IRR. PV takes a rate and produces a value. The internal rate of return works the other way: given a price and the payments, it solves for the rate at which PV equals the price. For the lump-sum example, the IRR is the rate at which $50,000 for 25 years is worth exactly $650,000, slightly below 6%.

The financial product vs. the math. “Annuity” names both an insurance contract and any level series of payments. The PV formula applies to loan payments, bond coupons and rent just as it does to an insurance product.

The Bottom Line

Anyone weighing a lump sum against installments, such as a pension commuted value, lottery prize or settlement, can use this calculation to compare offers in today's dollars. The answer depends heavily on the discount rate and on payment timing, so test a range of rates that reflect your opportunity cost. Adjust separately for taxes, fees, inflation and the risk that payments stop.