C. Discrete Functions · Financial Applications: Interest, Present Value and Annuities

The Future Value of an Ordinary Simple Annuity

C3.5 C3.7About 70 minutes

Learning goals

I can use FV = R[(1 + i)ⁿ − 1] ÷ i to find the future value (amount) of an ordinary simple annuity.
I can find the total deposits and the interest earned by an annuity.
I can solve multi-step problems where an annuity is followed by a lump sum growing at compound interest.
I can explain why starting to save early makes such a large difference.

Words to know

future value of an annuity (FV)
The sum of all the payments and the interest they earn, valued at the end of the term.

regular payment (R)
The equal amount paid or deposited each interval. Sometimes written PMT.

total deposits
The number of payments times the payment: nR.

interest earned
Future value minus total deposits: FV − nR.

From series to formula

In Lesson 7 you added up payments one by one, then used the geometric series formula. That always works, so let's do it once in general and get a formula we can use every time.

For n payments of R at the end of each period, at rate i per period:

FV = R + R(1 + i) + R(1 + i)² + … + R(1 + i)ⁿ⁻¹

This geometric series has a = R, r = 1 + i and n terms, so

FV = R[(1 + i)ⁿ − 1] ÷ i

This formula is only for ordinary simple annuities: end-of-period payments, with the payment period equal to the compounding period.

Find i and n exactly as for compound interest: i = annual rate ÷ periods per year, and n = years × periods per year (which is also the number of payments).

Worked example

Example 1: Monthly savings

Kiran saves $150 at the end of every month for 5 years in an account paying 6% compounded monthly. (a) How much will she have? (b) How much of it is interest?

R = 150, i = 0.06 ÷ 12 = 0.005, n = 5 × 12 = 60.
FV = 150[(1.005)⁶⁰ − 1] ÷ 0.005.
Calculator: 1.005 ^ 60 ≈ 1.348850; subtract 1: 0.348850; divide by 0.005: 69.7700; multiply by 150.
(a) FV ≈ $10 465.50.
(b) Total deposits = 60 × 150 = $9 000. Interest = $10 465.50 − $9 000 ≈ $1 465.50.

She will have about $10 465.50, including about $1 465.50 of interest.

Worked example

Example 2: Quarterly deposits

A family deposits $500 at the end of every 3 months for 10 years into an education savings account paying 4.8% compounded quarterly. Find the future value and the interest.

i = 0.048 ÷ 4 = 0.012 and n = 10 × 4 = 40.
FV = 500[(1.012)⁴⁰ − 1] ÷ 0.012 ≈ $25 477.65.
Deposits: 40 × 500 = $20 000. Interest ≈ $5 477.65.

About $25 477.65, including $5 477.65 of interest.

Worked example

Example 3: Start early or start later?

Two friends each save $200 at the end of every month until age 65 at 6% compounded monthly. Ana starts at 25 (480 deposits). Ben starts at 35 (360 deposits). Compare their totals.

Ana: FV = 200[(1.005)⁴⁸⁰ − 1] ÷ 0.005 ≈ $398 298.15. Her deposits: 480 × 200 = $96 000.
Ben: FV = 200[(1.005)³⁶⁰ − 1] ÷ 0.005 ≈ $200 903.01. His deposits: 360 × 200 = $72 000.
Ana deposited only $24 000 more but ends with about $197 395.14 more. Her extra early deposits had 30 to 40 years to compound.

Ana ≈ $398 298.15, Ben ≈ $200 903.01.

Both lines curve upward (exponential growth), but the early saver's curve has ten extra years to bend. At 65 Ana has about twice as much as Ben.

Worked example

Example 4: An annuity, then a lump sum

Noor deposits $100 at the end of every month for 3 years at 3.6% compounded monthly. She then makes no more deposits but leaves the money in the account for 2 more years at the same rate. How much does she have at the end?

Step 1 (annuity): i = 0.003, n = 36.
FV₁ = 100[(1.003)³⁶ − 1] ÷ 0.003 ≈ $3 795.59.
Step 2 (lump sum): the whole balance now grows as a single amount for 24 more months.
A = 3 795.59(1.003)²⁴ ≈ $4 078.51.

About $4 078.51.

Calculator order

Work from the inside out: (1 + i) ^ n, then − 1, then ÷ i, then × R. Or type it in one line with brackets: 150 × ((1.005 ^ 60 − 1) ÷ 0.005). A common mistake is to divide only the 1 by i.

Just for fun

Saving $5 a week sounds tiny. But if you put it away at the end of every week from age 16 to 66, in an account paying 5% compounded weekly, the math says you would end up with more than $57 000. That's a lot of fancy coffees you didn't buy.

Hands-on activity

Your own savings plan

You need: a calculator or a spreadsheet, and paper.

Choose an amount you could realistically save at the end of every month (for example, from a part-time job): $20, $50 or $100.
Choose a rate of 3% compounded monthly (a cautious savings rate for planning).
Find the future value after 1, 2, 3 and 5 years. For each, find the total deposits and the interest.
Repeat at 5% compounded monthly.
Make a small table and write two sentences: what changes the total most in the short term, and in the long term?

What to look for (model for $50 a month): at 3% the future values are about $608.32 (1 year), $1 235.14 (2 years), $1 881.03 (3 years) and $3 232.34 (5 years). At 5% they are about $613.94, $1 259.30, $1 937.67 and $3 400.30. In the short term the deposits matter most; over long terms the interest rate (and time) matter more and more.

Your turn

Find the future value of $150 at the end of every month for 5 years at 6% compounded monthly.

Find the future value of $2 000 at the end of every year for 10 years at 4% compounded annually.

How much interest is earned in the last question?

Find the future value of $300 at the end of every quarter for 5 years at 7% compounded quarterly.

Find the future value of $75 at the end of every month for 2 years at 3% compounded monthly.

Find the future value of $40 at the end of every month for 4 years at 4.2% compounded monthly.

More practice: warm-up

For $80 at the end of every month for 3 years at 6% compounded monthly, state R, i and n.

For $1 000 at the end of every half-year for 4 years at 5% compounded semi-annually, state R, i and n.

What are the total deposits for $60 a month for 2 years?

Find the future value of $1 000 at the end of one year at 6% compounded annually (n = 1). Explain the answer.

Which formula is the future value of an ordinary simple annuity: R[(1 + i)ⁿ − 1] ÷ i or R(1 + i)ⁿ?

More practice: on your own

A family deposits $500 at the end of every quarter for 10 years at 4.8% compounded quarterly. Find the future value and the interest.

$100 at the end of every month for 3 years at 3.6% compounded monthly, then left for 2 more years. Find the final amount.

$2 500 at the end of every year for 15 years at 5% compounded annually. Find the future value.

$100 at the end of every month for 10 years at 6% compounded monthly. What percent of the final amount is interest?

Two friends save $200 a month until 65 at 6% compounded monthly; one starts at 25, the other at 35. How much more does the early saver have?

Challenge

Jo saves $100 at the end of every month for 10 years at 6% compounded monthly, then stops and leaves the money for 20 more years. Max saves nothing for 10 years, then $100 at the end of every month for 20 years at the same rate. Who has more after 30 years?

Show that the future value of an annuity is the sum of the future values of each payment, by checking that 1000[(1.05)⁴ − 1] ÷ 0.05 equals the total of $1 157.63, $1 102.50, $1 050.00 and $1 000.00.

How much will you have if you save $5 at the end of every week from age 16 to age 66 (50 years) at 5% compounded weekly? (Use 52 weeks a year.)

Check your understanding

Quiz

Find the future value of $100 at the end of each year for 3 years at 10% compounded annually.

For $250 at the end of every month for 4 years at 4.8% compounded monthly, i and n are:

An annuity has future value $9 800 after 48 deposits of $180. How much interest was earned?

Which calculator entry is correct for FV = 75[(1.003)³⁶ − 1] ÷ 0.003?

Why does starting to save 10 years earlier make such a big difference?

The future value formula R[(1 + i)ⁿ − 1] ÷ i comes from:

For parents and teachers

Big idea: This lesson completes the connection in C3.5 (annuities and geometric series) and begins C3.7: solving problems involving the amount (future value) of an ordinary simple annuity.

How to help: Before calculating, ask your teen to say R, i and n out loud and to estimate ("60 deposits of $150 is $9 000, so the answer should be a bit more than that"). This catches most calculator errors.

Common difficulties:

Using the annuity formula for a single deposit, or A = P(1 + i)ⁿ for regular deposits.
Bracket errors on the calculator.
In two-step problems, forgetting to grow the annuity's total as a lump sum afterwards.
Using the formula when payments and compounding do not match (not a simple annuity).