Annuities Formula

The Formula

Future value of ordinary annuity:
FV = PMT \cdot \frac{(1 + i)^n - 1}{i}
Present value of ordinary annuity:
PV = PMT \cdot \frac{1 - (1 + i)^{-n}}{i}
where PMT = payment per period, i = interest rate per period, n = total number of payments.

When to use: Imagine depositing \$100 every month into a savings account. Each deposit earns interest for a different amount of time—the first deposit earns interest for the full term, the last deposit barely earns any. An annuity formula adds up all these differently-growing deposits in one clean expression, instead of computing compound interest on each payment separately.

Quick Example

Save \$200/month at 6% annual rate (0.5% monthly) for 20 years:
FV = 200 \cdot \frac{(1.005)^{240} - 1}{0.005} = 200 \cdot 462.04 = \$92{,}408
You deposited \48,000 total but earned \44,408 in interest.

Notation

PMT = payment amount per period, i = periodic interest rate (annual rate \div periods per year), n = total number of periods, FV = future value, PV = present value.

What This Formula Means

A series of equal payments made at regular intervals over a fixed period of time. The future value and present value formulas calculate the total worth of these payment streams.

Imagine depositing \$100 every month into a savings account. Each deposit earns interest for a different amount of time—the first deposit earns interest for the full term, the last deposit barely earns any. An annuity formula adds up all these differently-growing deposits in one clean expression, instead of computing compound interest on each payment separately.

Formal View

FV = PMT \cdot \frac{(1+i)^n - 1}{i} = PMT \cdot \sum_{k=0}^{n-1}(1+i)^k; PV = PMT \cdot \frac{1 - (1+i)^{-n}}{i}

Worked Examples

Example 1

easy
You deposit \200 at the end of each month into an account earning 6\% annual interest compounded monthly. How much will you have after 1$ year?

Solution

  1. 1
    This is an ordinary annuity. The future value formula is FV = P \cdot \frac{(1+r)^n - 1}{r}.
  2. 2
    Monthly rate: r = \frac{0.06}{12} = 0.005. Number of payments: n = 12.
  3. 3
    FV = 200 \cdot \frac{(1.005)^{12} - 1}{0.005} = 200 \cdot \frac{1.06168 - 1}{0.005} = 200 \cdot \frac{0.06168}{0.005}.
  4. 4
    FV = 200 \cdot 12.336 = \2{,}467.20$.

Answer

\$2{,}467.20
An ordinary annuity involves equal payments made at the end of each period. The future value formula accounts for the fact that earlier payments earn more interest than later ones. The total deposits are \2{,}400, so \67.20 is earned in interest.

Example 2

medium
What monthly payment is needed to accumulate \50{,}000 in 10 years if the account earns 4.8\%$ annual interest compounded monthly?

Common Mistakes

  • Using the annual interest rate instead of the periodic rate: for monthly payments at 6% annual, i = 0.005 (not 0.06), and n = 240 months (not 20 years).
  • Confusing ordinary annuity (payments at END of period) with annuity due (payments at BEGINNING of period). An annuity due has one extra compounding period, so multiply the ordinary annuity result by (1 + i).
  • Forgetting that loan payments (like mortgages) use the present value formula—you're solving for PMT given PV, not the future value formula.

Why This Formula Matters

Annuities model mortgages, car loans, retirement savings, pension payouts, and any financial plan involving regular payments. Knowing the formulas lets you calculate monthly payments, total interest paid, or how much to save for retirement.

Frequently Asked Questions

What is the Annuities formula?

A series of equal payments made at regular intervals over a fixed period of time. The future value and present value formulas calculate the total worth of these payment streams.

How do you use the Annuities formula?

Imagine depositing \$100 every month into a savings account. Each deposit earns interest for a different amount of time—the first deposit earns interest for the full term, the last deposit barely earns any. An annuity formula adds up all these differently-growing deposits in one clean expression, instead of computing compound interest on each payment separately.

What do the symbols mean in the Annuities formula?

PMT = payment amount per period, i = periodic interest rate (annual rate \div periods per year), n = total number of periods, FV = future value, PV = present value.

Why is the Annuities formula important in Math?

Annuities model mortgages, car loans, retirement savings, pension payouts, and any financial plan involving regular payments. Knowing the formulas lets you calculate monthly payments, total interest paid, or how much to save for retirement.

What do students get wrong about Annuities?

The interest rate i in the formula is the PERIODIC rate, not the annual rate. For monthly payments at 6% annual, use i = 0.06/12 = 0.005, and n is the total number of months, not years.

What should I learn before the Annuities formula?

Before studying the Annuities formula, you should understand: compound interest.

← Back to Annuities See All Examples Practice Problems