Future Value of Annuity Formula: Ordinary, Due & Growing Annuity

What Is an Annuity?

An annuity is a series of equal payments made at regular intervals over a fixed period of time. The concept is central to personal finance, corporate finance, actuarial science, and insurance. Whenever you make monthly mortgage payments, contribute to a 401(k), pay a car lease, or receive pension checks, you are dealing with an annuity.

Annuities are classified along two primary dimensions:

By Payment Timing

• Ordinary Annuity (Annuity in Arrears) — Payments occur at the end of each period. This is the most common type. Examples include mortgage payments, bond coupon payments, and most savings account deposits.
• Annuity Due (Annuity in Advance) — Payments occur at the beginning of each period. Examples include rent payments, insurance premiums, and lease payments.

By Payment Pattern

• Level (Fixed) Annuity — All payments are identical in amount. This is the "standard" annuity assumed in most formulas.
• Growing Annuity — Payments increase at a constant rate each period (e.g., retirement contributions that rise 3% per year with salary growth).
• Perpetuity — A special case where payments continue forever (n → ∞). Examples include preferred stock dividends and certain endowments.

Understanding the differences between these types is essential before applying any formula. Throughout this guide, we use i for the interest rate per period and n for the total number of periods, unless stated otherwise. For a broader overview of time value of money concepts, see our Future Value Formula guide.

Future Value of Ordinary Annuity Formula

The future value of an ordinary annuity calculates the total accumulated value of a series of equal payments, each made at the end of a period, after all payments have been made and interest has compounded.

Where:

• FV_OA — Future value of the ordinary annuity
• PMT — Equal payment amount per period
• i — Interest rate per compounding period (annual rate ÷ compounding frequency)
• n — Total number of payment periods

Step-by-Step Derivation

To understand why this formula works, consider making a payment of PMT at the end of each period for n periods at interest rate i per period. Each payment compounds for a different number of remaining periods:

1. Payment 1 (end of period 1) — compounds for (n − 1) periods: PMT × (1 + i)ⁿ⁻¹
2. Payment 2 (end of period 2) — compounds for (n − 2) periods: PMT × (1 + i)ⁿ⁻²
3. Payment 3 (end of period 3) — compounds for (n − 3) periods: PMT × (1 + i)ⁿ⁻³
4. ...
5. Payment n−1 — compounds for 1 period: PMT × (1 + i)¹
6. Payment n (final) — no compounding time remains: PMT × (1 + i)⁰ = PMT

The total future value is the sum:

This is a geometric series with first term a = 1, common ratio r = (1 + i), and n terms. The closed-form sum of a geometric series is (rⁿ − 1) / (r − 1). Substituting:

Worked Example

Scenario: You deposit $500 at the end of each month into an investment account earning 6% annual interest, compounded monthly, for 20 years.

1. Identify variables: PMT = $500, annual rate = 6%, compounding = monthly
2. Calculate i: i = 0.06 / 12 = 0.005 per month
3. Calculate n: n = 20 × 12 = 240 months
4. Apply the formula: FV = $500 × [((1.005)²⁴⁰ − 1) / 0.005]
5. Compute (1.005)²⁴⁰: = 3.3102
6. Solve: FV = $500 × [(3.3102 − 1) / 0.005] = $500 × 462.041 = $231,020.26

Your total contributions are $500 × 240 = $120,000. The remaining $111,020.26 is compound interest — nearly as much as you invested! Try this scenario on our Annuity FV Calculator.

Future Value of Annuity Due Formula

An annuity due is identical to an ordinary annuity except that each payment is made at the beginning of the period rather than the end. Because each payment has one extra period to earn interest, the future value of an annuity due is always higher.

This is simply the ordinary annuity formula multiplied by an extra factor of (1 + i).

Why the Extra (1 + i) Factor?

In an annuity due, every payment arrives one period earlier compared to an ordinary annuity. This means each payment compounds for exactly one additional period. Multiplying the entire ordinary annuity FV by (1 + i) accounts for this uniform one-period shift:

Ordinary Annuity vs Annuity Due: Side-by-Side

Worked Example

Scenario: Using the same inputs as the ordinary annuity example above — $500/month at 6% for 20 years — but now payments are made at the beginning of each month.

1. Ordinary annuity FV: $231,020.26 (computed previously)
2. Multiply by (1 + i): FV_AD = $231,020.26 × 1.005 = $232,175.36

The annuity due yields $1,155.10 more than the ordinary annuity — simply because each payment earns one extra month of interest. Over longer time horizons or at higher rates, this difference grows substantially.

Future Value of Growing Annuity Formula

A growing annuity is a series of payments that increase at a constant rate (g) each period. This is far more realistic for many real-world scenarios — retirement contributions that grow with salary raises, rental income that increases with inflation, or business revenues expected to grow at a steady rate.

When g ≠ i (Standard Case)

Where:

• PMT — First payment amount (at end of period 1)
• i — Interest rate per period
• g — Constant growth rate of payments per period
• n — Total number of periods

Each subsequent payment grows: the payment in period k is PMT × (1 + g)^k−1. Payment k compounds for (n − k) periods. Summing the resulting geometric series and simplifying yields the closed-form formula above.

When g = i (Special Case)

When the growth rate equals the interest rate, the standard formula produces a 0/0 indeterminate form. Applying L'Hôpital's rule (or direct summation) yields:

This special case arises occasionally in theoretical problems and should be handled separately in any spreadsheet model.

Worked Example

Scenario: You contribute $6,000 per year to a retirement account. Contributions grow by 3% annually to match salary increases. The expected annual return is 7%, and you invest for 30 years.

1. Identify variables: PMT = $6,000 (first year), g = 0.03, i = 0.07, n = 30
2. Apply the formula: FV = $6,000 / (0.07 − 0.03) × [(1.07)³⁰ − (1.03)³⁰]
3. Calculate (1.07)³⁰: = 7.6123
4. Calculate (1.03)³⁰: = 2.4273
5. Solve: FV = $6,000 / 0.04 × (7.6123 − 2.4273) = $150,000 × 5.1850 = $777,749

Present Value of Annuity Formulas

While future value tells you what a stream of payments will accumulate to, present value tells you what that stream is worth today. PV of annuity formulas are essential for loan pricing, bond valuation, lease analysis, and deciding between a lump sum and periodic payments (e.g., lottery winnings).

PV of Ordinary Annuity

This formula answers: "What single lump sum today is equivalent to receiving PMT at the end of each period for n periods, given discount rate i?"

PV of Annuity Due

As with FV, the annuity due PV is the ordinary annuity PV multiplied by (1 + i), because payments arrive one period sooner.

FV vs PV: The Relationship

The future value and present value of the same annuity are connected by the standard compound interest factor:

You can verify: take the PV ordinary annuity formula, multiply by (1 + i)ⁿ, and you get the FV ordinary annuity formula. They are two views of the same cash-flow stream — one looking forward from today, the other looking back from the future.

For interactive PV calculations, use our Present Value Calculator, or explore the full TVM framework on our Time Value of Money Calculator.

FVIFA and PVIFA Tables

Before spreadsheets and calculators, finance professionals relied on interest factor tables to perform annuity calculations quickly. While these tables are less necessary today, understanding them deepens your grasp of annuity mechanics and they still appear frequently in finance textbooks and CFA exams.

FVIFA — Future Value Interest Factor of an Annuity

FVIFA represents the future value of $1 per period for n periods at rate i:

To find the FV of any annuity: FV = PMT × FVIFA(i, n)

PVIFA — Present Value Interest Factor of an Annuity

PVIFA represents the present value of $1 per period for n periods at rate i:

To find the PV of any annuity: PV = PMT × PVIFA(i, n)

Sample FVIFA Table

How to read: At 7% for 20 periods, FVIFA = 40.9955. If you deposit $200 per period, FV = $200 × 40.9955 = $8,199.10.

Sample PVIFA Table

How to read: At 5% for 30 periods, PVIFA = 15.3725. If you receive $1,000 per period, PV = $1,000 × 15.3725 = $15,372.50.

Annuity Formula Examples

Below are three detailed examples covering different real-world scenarios. Each walks through variable identification, formula selection, and full step-by-step computation.

Example 1: Monthly Retirement Savings (Ordinary Annuity)

Scenario: You contribute $400/month to your 401(k) plan. Your portfolio earns an average return of 8% per year, compounded monthly. How much will you have after 25 years?

1. Identify variables: PMT = $400, annual rate = 8%, m = 12 (monthly), years = 25
2. Calculate i: i = 0.08 / 12 = 0.006667 per month
3. Calculate n: n = 25 × 12 = 300 months
4. Apply the formula: FV = $400 × [((1.006667)³⁰⁰ − 1) / 0.006667]
5. Compute (1.006667)³⁰⁰: = 7.2446
6. Solve FVIFA: (7.2446 − 1) / 0.006667 = 936.684
7. Final answer: FV = $400 × 936.684 = $374,673.63

Your total contributions: $400 × 300 = $120,000. Interest earned: $254,673.63 — more than double your out-of-pocket investment! This is the power of consistent investing with compound interest over time.

Example 2: Lease Payments (Annuity Due)

Scenario: A company makes $2,000 quarterly lease payments at the beginning of each quarter. The implicit interest rate is 6% annually, compounded quarterly. What is the FV of all payments after 5 years?

1. Identify variables: PMT = $2,000, annual rate = 6%, m = 4 (quarterly), years = 5, annuity due
2. Calculate i: i = 0.06 / 4 = 0.015 per quarter
3. Calculate n: n = 5 × 4 = 20 quarters
4. Ordinary annuity FV: FV_OA = $2,000 × [((1.015)²⁰ − 1) / 0.015]
5. Compute (1.015)²⁰: = 1.3469
6. Solve FVIFA: (1.3469 − 1) / 0.015 = 23.1237
7. Ordinary FV: $2,000 × 23.1237 = $46,247.37
8. Annuity due FV: $46,247.37 × 1.015 = $46,941.08

The beginning-of-period timing adds $693.71 compared to the ordinary annuity. Total lease payments: $40,000. Total interest accumulated: $6,941.08.

Example 3: Growing Salary Contributions

Scenario: A new graduate starts saving $3,000/year in a Roth IRA. She plans to increase contributions by 5% each year as her salary grows. Expected return is 9% annually for 35 years.

1. Identify variables: PMT = $3,000, g = 0.05, i = 0.09, n = 35
2. Verify g ≠ i: 0.05 ≠ 0.09 ✔
3. Apply the formula: FV = $3,000 / (0.09 − 0.05) × [(1.09)³⁵ − (1.05)³⁵]
4. Calculate (1.09)³⁵: = 20.4140
5. Calculate (1.05)³⁵: = 5.5160
6. Solve: FV = $3,000 / 0.04 × (20.4140 − 5.5160) = $75,000 × 14.8980 = $1,117,350

With growing contributions, she becomes a millionaire! Her total out-of-pocket contributions over 35 years amount to $3,000 × [(1.05)³⁵ − 1] / 0.05 = $271,024. The remaining $846,326 is investment growth. Compare this to flat $3,000/year at 9% for 35 years = $814,736 — the 5% contribution growth adds over $302,000.

Annuity Formulas in Excel

Microsoft Excel and Google Sheets provide built-in functions that make annuity calculations effortless. The two key functions are FV() for future value and PV() for present value.

The FV Function

• rate — Interest rate per period (e.g., 0.06/12 for 6% monthly)
• nper — Total number of payment periods (e.g., 10*12 = 120)
• pmt — Payment per period (enter as negative for deposits)
• pv — Present value / initial lump sum (negative for investment; 0 if annuity-only)
• type — 0 = ordinary annuity (default); 1 = annuity due

Excel FV Examples

Ordinary annuity: $200/month at 6% for 10 years, no initial deposit:

Annuity due: Same inputs, but payments at beginning of month:

Combined lump sum + annuity: $10,000 initial + $200/month at 6% for 10 years:

The PV Function

This is the mirror of FV — it calculates the present value of a series of future payments.

Excel PV Examples

PV of ordinary annuity: What is $500/month for 15 years worth today at 5%?

Loan payment calculation with PMT: What monthly payment for a $300,000 mortgage at 6.5% for 30 years?

For more advanced Excel tips including data tables and scenario analysis, see our FV Formula Guide.

Frequently Asked Questions

• What is the future value of an ordinary annuity formula?

  The future value of an ordinary annuity is FV = PMT × [((1 + i)ⁿ − 1) / i], where PMT is the equal payment per period, i is the interest rate per compounding period, and n is the total number of periods. "Ordinary" means payments are made at the end of each period. The bracketed expression [((1 + i)ⁿ − 1) / i] is called the Future Value Interest Factor of an Annuity (FVIFA).

• What is the difference between an ordinary annuity and an annuity due?

  An ordinary annuity makes payments at the end of each period (e.g., mortgage payments, bond coupons), while an annuity due makes payments at the beginning (e.g., rent, insurance premiums). Because annuity due payments arrive one period earlier, each payment earns an extra period of interest. The annuity due FV equals the ordinary annuity FV multiplied by (1 + i). For a 6% monthly rate over 20 years, annuity due adds roughly 0.5% more to your total accumulation.

• What is the growing annuity formula?

  The growing annuity FV formula is FV = PMT / (i − g) × [(1 + i)ⁿ − (1 + g)ⁿ], where g is the constant growth rate of payments. This models scenarios where contributions increase over time, like retirement savings that grow with annual salary raises. When the interest rate equals the growth rate (i = g), use the special case: FV = PMT × n × (1 + i)ⁿ⁻¹.

• What is FVIFA and how is it used?

  FVIFA (Future Value Interest Factor of an Annuity) = ((1 + i)ⁿ − 1) / i. It represents the future value of receiving $1 at the end of each period for n periods at interest rate i. To calculate the FV of any annuity, simply multiply: FV = PMT × FVIFA. Before calculators, FVIFA was looked up in pre-computed tables. Similarly, PVIFA = (1 − (1 + i)⁻ⁿ) / i is used for present value of annuity calculations.

• How do I calculate the present value of an annuity?

  For an ordinary annuity: PV = PMT × [(1 − (1 + i)⁻ⁿ) / i]. For an annuity due, multiply by (1 + i). Present value tells you the lump-sum equivalent today of a stream of future payments. This is widely used for loan pricing (how much can you borrow given a monthly payment?), comparing settlement options, and valuing pension benefits. Use our Present Value Calculator for instant results.

• How do I use the FV function in Excel for annuities?

  Use =FV(rate, nper, pmt, [pv], [type]). For an ordinary annuity of $200/month at 6% for 10 years: =FV(0.06/12, 120, -200). For annuity due, add type=1: =FV(0.06/12, 120, -200, 0, 1). PMT is negative because it's cash you pay out. For present value, use =PV(rate, nper, pmt, [fv], [type]). Google Sheets uses identical syntax.

• Can I combine a lump sum and annuity in one calculation?

  Yes! The combined formula is: FV = PV × (1 + i)ⁿ + PMT × [((1 + i)ⁿ − 1) / i]. The first term grows your initial lump sum, and the second accumulates your periodic payments. In Excel, include both: =FV(rate, nper, pmt, pv) where both pmt and pv are entered as negative numbers. Our Future Value Calculator handles this automatically.

• What is a perpetuity and how does it differ from an annuity?

  A perpetuity is an annuity that continues forever (n → ∞). The present value of a perpetuity simplifies to PV = PMT / i, since the (1 + i)⁻ⁿ term approaches zero. A growing perpetuity has PV = PMT / (i − g). Perpetuities have no future value in the traditional sense (it would be infinite). Real-world examples include preferred stock dividends and certain endowment funds. The TVM Calculator can handle very long annuity periods as an approximation.

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