Annuity Due: Definition, Calculation, Formula, and Examples

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An annuity due is a series of equal payments made at the start of each period, not at the end. Rent is the textbook case: you pay on the first of the month for the month ahead. The annuity due formula is the ordinary annuity formula multiplied by (1 + i). That one extra period of interest means an annuity due is always worth more, in present and future value terms, than an otherwise identical ordinary annuity whenever the interest rate is above zero.

How an annuity due works

Every annuity has four parts: a payment amount (C), a number of payments (n), an interest rate per period (i) and a timing convention. An annuity due fixes the timing: each payment lands at the opening of the period it covers.

A time diagram shows the difference. For a four-year annuity with annual payments:

Time:              0     1     2     3     4
Annuity due:       C     C     C     C
Ordinary annuity:        C     C     C     C

Both streams contain four payments. In the annuity due, the first payment arrives today (time 0) and the last arrives at the start of year 4 (time 3). Nothing is paid at time 4, because the final period was paid for in advance. In the ordinary annuity, each payment sits one period later.

Moving every payment one period earlier has two effects:

1. Each payment is discounted one period less when you calculate present value, so the present value rises.
2. Each payment earns one period more interest before the end date when you calculate future value, so the future value rises.

Both effects have the same size: a factor of (1 + i). That is why the formulas below are the ordinary annuity formulas with one extra multiplication.

Annuity due vs ordinary annuity

With an annuity due, you pay before you use what you paid for, such as the apartment or the insurance coverage. With an ordinary annuity, you pay after the period has passed: a mortgage payment covers interest that built up over the month just ended.

Annuity due formulas

The variables are:

• C = payment per period
• i = interest rate per period
• n = number of payments

The rate and the number of periods must match the payment frequency. For monthly payments at a 6% annual rate compounded monthly, i = 0.06 / 12 = 0.005, and a five-year stream has n = 60.

Present value of an annuity due

The present value tells you what the whole stream of future payments is worth today at the chosen discount rate.

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

The bracketed part is the ordinary annuity present value factor. Multiplying by (1 + i) shifts every payment one period closer.

An equivalent form makes the timing explicit:

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

The first payment is already in hand, so it counts at full value. The remaining n − 1 payments form an ordinary annuity starting one period from now.

Future value (accumulated value) of an annuity due

The future value tells you what the payments will have grown to by the end of the final period, assuming each one earns the rate i from the day it is made.

FV = C × [((1 + i)^n − 1) / i] × (1 + i)

It measures value at time n, one period after the last payment, so each deposit compounds for at least one full period and the first compounds for n periods. The growth comes from compound interest applied to every payment separately.

Actuarial notation using v and d

Actuarial notation writes the formulas more compactly with two symbols:

• v = 1 / (1 + i), the discount factor for one period
• d = i / (1 + i), the effective rate of discount

The present value of an annuity due of 1 per period for n periods is written ä with a subscript n (read “a-double-dot angle n”):

ä_n = (1 − v^n) / d

The accumulated value is s̈_n (“s-double-dot angle n”):

s̈_n = ((1 + i)^n − 1) / d

Ordinary annuity symbols have no double dots (a_n and s_n). Dividing by d instead of i is the same as multiplying the ordinary annuity factor by (1 + i). These identities link the two types:

• ä_n = (1 + i) × a_n
• ä_n = 1 + a_(n−1)
• s̈_n = s_(n+1) − 1

Solving for the payment

If you know the present value and need the payment, rearrange:

C = PV / {[(1 − (1 + i)^−n) / i] × (1 + i)}

For a savings target, use the future value version:

C = FV / {[((1 + i)^n − 1) / i] × (1 + i)}

Because each deposit earns one extra period of interest, the required payment is smaller than it would be for end-of-period deposits.

Solving for i has no closed-form answer. You find the rate by trial and error or with a financial calculator or spreadsheet that iterates to the solution.

Worked example

Hypothetical example. You are entitled to $1,000 a year for 10 years, with the first payment today, and you value the stream at an annual rate of 3%.

Present value

• Ordinary annuity factor: (1 − 1.03^−10) / 0.03 = 8.5302
• Annuity due factor: 8.5302 × 1.03 = 8.7861
• PV = $1,000 × 8.7861 = $8,786.11

Future value at the end of year 10

• Ordinary annuity factor: (1.03^10 − 1) / 0.03 = 11.4639
• Annuity due factor: 11.4639 × 1.03 = 11.8078
• FV = $1,000 × 11.8078 = $11,807.80

If the same ten payments came at the end of each year instead, the present value would be about $8,530.20 and the future value about $11,463.88. In both cases the gap is roughly 3%, the value of having each payment one year sooner.

Hypothetical savings example. You want $50,000 in 15 years, deposit at the start of each year, and assume 5% a year. The annuity due factor is [((1.05)^15 − 1) / 0.05] × 1.05 ≈ 21.5786 × 1.05 ≈ 22.6575. The required deposit is $50,000 / 22.6575 ≈ $2,206.78 a year. With end-of-year deposits, it would be about $2,317.11.

Calculating an annuity due on a calculator or spreadsheet

• Financial calculators have a payment timing setting, usually labeled BGN (beginning) and END. Switch to BGN, then enter N, I/Y, PMT and solve for PV or FV. Most calculators default to END, so a result that is off by a factor of (1 + i) usually means the mode wasn’t changed.
• Spreadsheets use PV(rate, nper, pmt, [fv], [type]) and FV(rate, nper, pmt, [pv], [type]). Set type to 1 for an annuity due and 0 (the default) for an ordinary annuity. For the example above, =PV(3%,10,-1000,0,1) returns 8,786.11. The payment is negative under spreadsheet sign conventions for cash flows.

Where annuities due appear

An annuity due arises whenever a recurring obligation is paid in advance of the period it covers:

• Rent. Landlords collect at the start of the month.
• Insurance premiums, paid at the beginning of each coverage period.
• Leases and subscriptions billed at the start of each term, such as many phone plans and equipment leases.
• Savings and retirement contributions made at the start of each month or year.
• Whole life annuity due. Insurers sell contracts that pay at the start of each monthly, quarterly or annual period for as long as the holder lives. In a straight life version, payments stop at death and the insurer keeps any remaining funds.

Which is better: annuity due or ordinary annuity?

It depends on which side of the payment you are on. For the recipient, an annuity due is better at any positive interest rate: money arrives a period earlier and can be invested, spent or used to pay down debt sooner. For the payer, an ordinary annuity is better, because paying at the end keeps the money working for that period. In practice the contract usually sets the timing. Where you control it, as with your own savings deposits, paying in at the start of each period lets each contribution compound for one extra period.

Annuity due vs immediate annuity

• In annuity math, an annuity-immediate is simply an ordinary annuity: payments at the end of each period. It is the opposite of an annuity due.
• In insurance, an immediate annuity is a product you buy with a lump-sum deposit that starts paying income right away instead of after a deferral period. Payments can be a stated amount, run for a stated number of years, or last for life.

The product name refers to when the income stream begins, not whether each payment falls at the start or end of its period. To value an immediate annuity contract, read its payment schedule and apply whichever formula matches.

Limitations of the annuity due formula

The formulas rest on assumptions that real cash flows don’t always meet:

• Level payments. The basic formulas assume equal payments. Payments that grow, such as rent with annual increases, need a growing annuity formula or a period-by-period calculation.
• Constant rate. One rate is applied across all periods, and small changes in i can move the result noticeably over long horizons.
• Matching periods. Payment frequency, compounding frequency and the rate must line up. Using an annual rate with monthly payments without converting gives a wrong answer.
• Certainty. The formulas treat every payment as guaranteed. Life-contingent annuities depend on how long the annuitant lives, which actuaries handle with mortality tables on top of these formulas.
• Nominal dollars. A fixed payment loses purchasing power as prices rise. Use a real (inflation-adjusted) rate to account for it.
• Taxes and fees. In the U.S., the taxable portion of annuity income payments is taxed as ordinary income, and contract fees reduce what the holder actually receives. Neither appears in the formula.

FAQs

The Bottom Line

Annuity due math matters to anyone valuing rent, leases, insurance premiums or savings deposits paid in advance. Recipients gain from the earlier timing at any positive rate, while payers give up a period's use of each payment. When you control the timing of your own savings, depositing at the start of each period lowers the payment needed to reach a target, as long as your calculator or spreadsheet is set to beginning mode.