Annuities Formula

Annuities are a series of equal payments made at regular intervals over a fixed period of time.

The Formula

Future value of ordinary annuity:
FV=PMT⋅(1+i)n−1iFV = PMT \cdot \frac{(1 + i)^n - 1}{i}
Present value of ordinary annuity:
PV=PMT⋅1−(1+i)−niPV = PMT \cdot \frac{1 - (1 + i)^{-n}}{i}
where PMTPMT = payment per period, ii = interest rate per period, nn = 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⋅(1.005)240−10.005=200⋅462.04=$92,408FV = 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

PMTPMT = payment amount per period, ii = periodic interest rate (annual rate ÷\div periods per year), nn = total number of periods, FVFV = future value, PVPV = 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⋅(1+i)n−1i=PMT⋅∑k=0n−1(1+i)kFV = PMT \cdot \frac{(1+i)^n - 1}{i} = PMT \cdot \sum_{k=0}^{n-1}(1+i)^k; PV=PMT⋅1−(1+i)−niPV = PMT \cdot \frac{1 - (1+i)^{-n}}{i}

Worked Examples

Example 1

easy
You deposit $200\$200 at the end of each month into an account earning 6%6\% annual interest compounded monthly. How much will you have after 11 year?

Answer

$2,467.20\$2{,}467.20

First step

1
This is an ordinary annuity. The future value formula is FV=P⋅(1+r)n−1rFV = P \cdot \frac{(1+r)^n - 1}{r}.

Full solution

  1. 2
    Monthly rate: r=0.0612=0.005r = \frac{0.06}{12} = 0.005. Number of payments: n=12n = 12.
  2. 3
    FV=200⋅(1.005)12−10.005=200⋅1.06168−10.005=200⋅0.061680.005FV = 200 \cdot \frac{(1.005)^{12} - 1}{0.005} = 200 \cdot \frac{1.06168 - 1}{0.005} = 200 \cdot \frac{0.06168}{0.005}.
  3. 4
    FV=200⋅12.336=$2,467.20FV = 200 \cdot 12.336 = \$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\$2{,}400, so $67.20\$67.20 is earned in interest.

Example 2

medium
What monthly payment is needed to accumulate $50,000\$50{,}000 in 1010 years if the account earns 4.8%4.8\% annual interest compounded monthly?

Example 3

medium
You deposit \$150 at the end of each month for 5 years into an account earning 4.8% annual interest compounded monthly. Find the future value.

Common Mistakes

  • Using the annual rate as ii - the periodic rate ii is the annual rate divided by payments per year (e.g. 6%6\% monthly gives i=0.005i=0.005).
  • Confusing nn with years - nn is the TOTAL number of payments (months ×\times years for monthly), not the number of years.
  • Mixing up future-value and present-value forms - use (1+i)n−1i\frac{(1+i)^n-1}{i} to find what deposits grow to, and 1−(1+i)−ni\frac{1-(1+i)^{-n}}{i} to find what a future stream is worth now.

Why This Formula Matters

It is how real mortgages, car loans, retirement savings, and pensions are actually computed; without it you would have to run a separate compound-interest calculation on every single payment, which is intractable for 360 monthly payments. Recognizing it by "Is the same amount paid repeatedly at fixed intervals, rather than once?" — rather than by familiar numbers — is what lets a student tell it apart from compound interest (single deposit) and present/future value (single amount) and net present value (npv) in a mixed problem set.

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?

PMTPMT = payment amount per period, ii = periodic interest rate (annual rate ÷\div periods per year), nn = total number of periods, FVFV = future value, PVPV = present value.

Why is the Annuities formula important in Math?

It is how real mortgages, car loans, retirement savings, and pensions are actually computed; without it you would have to run a separate compound-interest calculation on every single payment, which is intractable for 360 monthly payments. Recognizing it by "Is the same amount paid repeatedly at fixed intervals, rather than once?" — rather than by familiar numbers — is what lets a student tell it apart from compound interest (single deposit) and present/future value (single amount) and net present value (npv) in a mixed problem set.

What do students get wrong about Annuities?

The procedure for annuities is the easy part; the trap is using the annual rate as ii. Asking "Is the same amount paid repeatedly at fixed intervals, rather than once?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Annuities formula?

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

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