Annuity Formula: Present and Future Value of an Annuity

Search Fintrev

Search our guides, reviews and glossary.

The annuity formula values a series of equal payments made at regular intervals. Four versions cover most problems: the present value (PV) of an ordinary annuity, the future value (FV) of an ordinary annuity, and the same two for an annuity due. Ordinary annuity payments come at the end of each period. Annuity due payments come at the start. The core formulas are:

• PV, ordinary annuity: PV = C × [1 - (1 + i)^-n] / i
• FV, ordinary annuity: FV = C × [(1 + i)^n - 1] / i
• PV, annuity due: PV = C × [1 - (1 + i)^-n] / i × (1 + i)
• FV, annuity due: FV = C × [(1 + i)^n - 1] / i × (1 + i)

Here C is the cash flow per period, i is the interest rate per period, and n is the number of payments. The annuity due versions are the ordinary formulas multiplied by one extra period of interest.

What the formula is actually doing

Every annuity formula is a shortcut for summing a list of individual compound interest calculations. Because money received sooner can be invested sooner, each payment in the series has a different value depending on when it arrives.

For present value, each payment is discounted back to today: the first by (1 + i)^1, the second by (1 + i)^2, and so on to the nth. Adding those discounted amounts gives a geometric series, and the bracketed term [1 - (1 + i)^-n] / i is that series in closed form, called the present value interest factor of an annuity (PVIFA). Present value of an ordinary annuity tables used in accounting and exam prep list this factor for combinations of i and n.

Future value works in the other direction. Each payment is compounded forward to the date of the final payment. In an ordinary annuity, the last payment earns no interest at all because it arrives on the valuation date, while the first payment earns n - 1 periods of interest. The bracketed term [(1 + i)^n - 1] / i is the future value interest factor of an annuity (FVIFA).

Ordinary annuity vs. annuity due

• Ordinary annuity: payments at the end of each period. Bond coupons paid every six months, quarterly dividends and most loan payments where interest accrues before the first installment fit this pattern.
• Annuity due: payments at the start of each period. Rent paid on the first of the month, insurance premiums paid in advance and lease payments due at signing fit this pattern.

Because each annuity due payment arrives one period earlier, it earns one more period of interest (for FV) or is discounted one fewer period (for PV). That is why the annuity due formulas multiply by (1 + i). In an annuity due, the first payment occurs at time 0, the moment the contract starts, and the future value is measured at the end of period n, one full period after the last deposit.

Worked examples with the four formulas

All four examples use the same hypothetical stream: five annual payments of $1,000 at 5% interest per year. Here (1.05)^5 = 1.27628, which feeds every calculation.

Example 1: Future value of an ordinary annuity

You deposit $1,000 at the end of each year for five years in an account earning 5% compounded annually.

FV = $1,000 × [(1.05)^5 - 1] / 0.05 = $1,000 × 5.52563 = $5,525.63

You put in $5,000, and the remaining $525.63 is interest.

Example 2: Present value of an ordinary annuity

How much would you need today, invested at 5%, to fund five year-end withdrawals of $1,000?

PV = $1,000 × [1 - (1.05)^-5] / 0.05 = $1,000 × 4.32948 = $4,329.48

That $4,329.48 earns enough interest along the way to cover all five payments and reach zero after the last one. The same logic lets you price a bond’s coupon stream or calculate the current value of an income stream you expect to receive.

Example 3: Both values for an annuity due

Now the same $1,000 payments arrive at the start of each year, as with rent paid in advance.

FV = $1,000 × 5.52563 × 1.05 = $5,801.91

PV = $1,000 × 4.32948 × 1.05 = $4,545.95

Both figures are exactly 5% higher than their ordinary annuity counterparts, because each payment gets one extra year of interest.

Matching the rate to the payment period

Most real annuities pay monthly, so i and n have to be restated per month. With a 6% annual rate compounded monthly, i = 0.06 / 12 = 0.005, and a 10-year plan has n = 120.

A hypothetical investor putting $500 into a fund at the end of each month for 10 years at that assumed rate would accumulate:

FV = $500 × [(1.005)^120 - 1] / 0.005 ≈ $500 × 163.88 ≈ $81,940

Contributions total $60,000, so roughly $21,940 is compounded growth. Real returns vary from month to month, so the formula gives the outcome for a constant assumed rate, not a forecast.

If the compounding frequency differs from the payment frequency (say, quarterly payments on an account that compounds monthly), convert the rate to an effective rate per payment period first: i = (1 + annual rate / m)^(m / p) - 1, where m is compounding periods per year and p is payments per year.

Rearranging the formula: payment, periods and rate

The same equation can be solved for whichever variable is unknown.

Periodic payment from a present value. This is the standard loan payment formula:

C = PV × i / [1 - (1 + i)^-n]

A hypothetical $20,000 loan at 6% annual interest repaid over 60 monthly payments (i = 0.005) gives C = $20,000 × 0.005 / [1 - (1.005)^-60] ≈ $386.66 per month.

Periodic payment from a future value target. To reach a savings goal: C = FV × i / [(1 + i)^n - 1].

Number of periods. For an ordinary annuity with a known PV, n = -ln(1 - PV × i / C) / ln(1 + i). For a known FV, n = ln(1 + FV × i / C) / ln(1 + i). If the PV version produces the log of a negative number, the payment is too small to ever cover the interest, so the balance never reaches zero.

Interest rate. There is no algebraic solution for i. Financial calculators, spreadsheet RATE functions and trial-and-error searches find it numerically. In spreadsheets, the PV, FV, PMT and NPER functions take a “type” argument: 0 for an ordinary annuity, 1 for an annuity due.

Variations on the basic annuity

Growing annuity

When payments rise by a constant rate g each period, as with a salary-linked contribution or an inflation-adjusted pension, the present value of an ordinary growing annuity is:

PV = C / (i - g) × [1 - ((1 + g) / (1 + i))^n]

C is the first payment. With a hypothetical $1,000 first payment growing 2% a year, a 5% discount rate and five payments, PV ≈ $4,497.47, compared with $4,329.48 for the level stream. The formula requires i ≠ g. When i = g, the PV is simply n × C / (1 + i). The future value of a growing annuity is the PV multiplied by (1 + i)^n.

Deferred annuity

If the first payment of an ordinary annuity arrives after d idle periods, value it as a normal annuity and then discount that value back d more periods:

PV = C × [1 - (1 + i)^-n] / i × (1 + i)^-d

Deferred annuity insurance contracts, which pay income or a lump sum at a future date, use this timing structure. An immediate annuity starts paying right away.

Perpetuity and continuous annuity

As n grows without limit, (1 + i)^-n approaches zero and the ordinary annuity PV collapses to PV = C / i. That is the perpetuity formula behind preferred stock valuation and the terminal value step in business valuation models. When payments flow continuously at a rate C per year with continuous compounding at rate r, PV = C × (1 - e^(-rn)) / r and the accumulated value is FV = C × (e^(rn) - 1) / r.

Life annuities

The formulas above assume a fixed number of payments. A single life annuity pays only while the annuitant is alive, so actuaries weight each discounted payment by the probability of survival to that date: PV = sum of C × (1 + i)^-t × (probability of surviving to time t). Insurers’ pricing also reflects their own mortality tables and expenses, so the certain-annuity formula gives an upper reference point rather than a contract’s quoted price.

When an annuity formula does not apply

The shortcut works only when payments are equal (or grow at a constant rate), evenly spaced and discounted at a single rate. A stream with irregular amounts, such as a business’s projected cash flows, has to be discounted one cash flow at a time, as in a net present value calculation. Mixed problems combine both: a bond’s price is the PV of its coupon annuity plus the PV of its single face-value payment at maturity. On the CFA Level 1 exam and in accounting exams, the annuity due factor is often found by taking the ordinary annuity factor for n - 1 periods and adding 1.

FAQs

The Bottom Line

Students, savers and anyone pricing loans, bonds or income streams can use four annuity formulas to value equal payments made at regular intervals. The key steps are matching the rate and number of periods to the payment schedule and knowing whether payments come at the start or end of each period. When payments are irregular, discount each cash flow separately, and remember the results assume a constant rate.