C. Discrete Functions · Financial Applications: Interest, Present Value and AnnuitiesWhat Is an Annuity? Annuities and Geometric Series
C3.5About 70 minutes
Learning goals- I can explain what an annuity is, and what makes one an ordinary simple annuity.
- I can draw a timeline that shows each payment and how long it earns interest.
- I can show that the future value of an ordinary simple annuity is the sum of a geometric series.
- I can describe how the balance of an annuity grows, using a recursive formula, a table and a graph.
Words to know
annuity
A series of equal payments (or deposits) made at equal time intervals.
payment interval (payment period)
The time between payments, for example one month.
term
The total time from the start of the first payment period to the end of the last one.
ordinary annuity
An annuity whose payments are made at the end of each payment interval.
simple annuity
An annuity whose payment interval is the same as the compounding period (for example, monthly payments with monthly compounding).
geometric series
The sum of the terms of a geometric sequence: a + ar + ar² + … + arⁿ⁻¹.
One deposit is nice. Many deposits are a plan.
So far every problem has had one amount of money moving through time. Real saving and borrowing usually involve many equal payments: putting $50 into savings every month, paying $400 a month for a car, or receiving the same scholarship payment each term. A series of equal payments at equal intervals is called an annuity.
The big idea of this lesson: if you move every payment to the same date and add them up, you get a geometric series, the same kind of sum you learned in Sequences and Series.
What kind of annuity?
In this course every annuity is an ordinary simple annuity:
- ordinary: each payment is made at the end of its interval (the first payment comes one interval after the start);
- simple: the payment interval matches the compounding period (monthly payments with monthly compounding, yearly payments with yearly compounding, and so on).
Other kinds exist. If payments are made at the beginning of each interval it is an annuity due. If the payments are monthly but interest is compounded semi-annually, it is a general annuity. Those are studied in later courses; we will not use them here.
Worked exampleExample 1: Building a future value one payment at a time
You deposit $1 000 at the end of each year for 4 years into an account paying 5% compounded annually. How much is in the account right after the last deposit? How much of that is interest?
- Draw a timeline (above). Move each payment to the end of year 4.
- Payment 4 (made at year 4): 1000. Payment 3: 1000(1.05). Payment 2: 1000(1.05)². Payment 1: 1000(1.05)³.
- Future value: FV = 1000 + 1000(1.05) + 1000(1.05)² + 1000(1.05)³.
- This is a geometric series with first term a = 1000, common ratio r = 1.05 and n = 4 terms.
- Use Sₙ = a(rⁿ − 1) ÷ (r − 1): S₄ = 1000(1.05⁴ − 1) ÷ (1.05 − 1) = 1000(0.21550625) ÷ 0.05 = $4 310.13 (rounded to the cent).
- Interest = 4310.13 − 4 × 1000 = $310.13.
The account holds $4 310.13, of which $310.13 is interest.
The balance, step by step
Another way to see an annuity is to follow the balance through time. At the end of each year, the old balance earns interest, and then the new payment is added:
Bₖ = 1.05Bₖ₋₁ + 1000, with B₀ = 0.
This is a recursive formula, like the ones in Sequences and Series. A spreadsheet does it easily: put 0 in B2, then in B3 type **=B21.05+1000* and copy down.
| Year | Balance at start | Interest (5%) | Deposit | Balance at end |
|---|
| 1 | $0.00 | $0.00 | $1 000.00 | $1 000.00 |
| 2 | $1 000.00 | $50.00 | $1 000.00 | $2 050.00 |
| 3 | $2 050.00 | $102.50 | $1 000.00 | $3 152.50 |
| 4 | $3 152.50 | $157.63 | $1 000.00 | $4 310.13 |
Why the growth is exponential
For $1 000 a year at 5%, the sum of the geometric series is
FV(n) = 1000(1.05ⁿ − 1) ÷ 0.05 = 20 000(1.05)ⁿ − 20 000.
This is the exponential function y = 1.05ⁿ, stretched vertically by a factor of 20 000 and translated down 20 000: the same kind of transformation you used in Exponential Functions. So the future value of an annuity grows exponentially as the number of payments increases.
Worked exampleExample 2: A short monthly annuity
Dev deposits $50 at the end of each month for 6 months into an account paying 6% compounded monthly. Write the future value as a series and find it.
- Monthly payments and monthly compounding: an ordinary simple annuity. i = 0.06 ÷ 12 = 0.005, n = 6.
- FV = 50 + 50(1.005) + 50(1.005)² + 50(1.005)³ + 50(1.005)⁴ + 50(1.005)⁵.
- Geometric series with a = 50, r = 1.005, n = 6: S₆ = 50(1.005⁶ − 1) ÷ 0.005 ≈ $303.78.
Worked exampleExample 3: Is it an ordinary simple annuity?
Decide for each situation.
(a) $200 deposited at the end of every month; interest 3% compounded monthly.
(b) $200 deposited at the beginning of every month; interest 3% compounded monthly.
(c) $500 paid at the end of every month; interest 4% compounded semi-annually.
(d) One deposit of $5 000; interest 4% compounded annually.
- (a) Payments at the end, payment interval = compounding period. Yes, ordinary simple annuity.
- (b) Payments at the beginning: an annuity due, not ordinary.
- (c) Monthly payments but semi-annual compounding: the periods do not match, so it is not simple (it is a general annuity).
- (d) One deposit is not an annuity at all. Use A = P(1 + i)ⁿ.
Only (a) is an ordinary simple annuity.
Just for funThe word annuity comes from the Latin word annus, meaning "year", because early annuities paid once a year. So a monthly annuity is, word for word, a "yearly thing that happens every month".
Hands-on activityA timeline on the floor
You need: masking tape or string, 5 sticky notes (or scraps of paper), a pen and a calculator.
- Make a line on the floor with tape and mark 0, 1, 2, 3 and 4 along it (one big step apart). These are the ends of years.
- Write "$1 000" on four sticky notes and place one at each of 1, 2, 3 and 4, not at 0. (Ordinary annuity: payments at the end of each year.)
- Walk each note to the 4 mark. For every step it moves, multiply its value by 1.05 and write the new value on the note.
- Add the four notes. Compare with Example 1.
- Now try 3% instead of 5%. Before calculating, predict: more or less than $4 310.13?
What to look for: at 5% the notes read $1 157.63, $1 102.50, $1 050.00 and $1 000.00, and the total is $4 310.13. At 3% the total is 1000(1.03⁴ − 1) ÷ 0.03 ≈ $4 183.63, which is less, as predicted.
Your turn
1Is it an ordinary simple annuity? $75 deposited at the end of every month, interest 4.8% compounded monthly.
Yes: payments at the end, and the payment period (month) matches the compounding period.
2Is it an ordinary simple annuity? $1 000 paid at the end of each year, interest 6% compounded quarterly.
No: yearly payments but quarterly compounding (a general annuity).
3$200 is deposited at the end of each year for 5 years at 3% compounded annually. Write the future value as a series.
FV = 200 + 200(1.03) + 200(1.03)² + 200(1.03)³ + 200(1.03)⁴.
4Find the sum of the series in the last question.
a = 200, r = 1.03, n = 5: S₅ = 200(1.03⁵ − 1) ÷ 0.03 ≈ $1 061.83.
5In the annuity of Example 1, how many years of interest does the second payment earn?
It is made at the end of year 2 and the term ends at year 4, so it earns 2 years of interest.
6$75 is deposited at the end of each month for 1 year at 4.8% compounded monthly. Find the future value using a geometric series.
i = 0.004, n = 12: 75(1.004¹² − 1) ÷ 0.004 ≈ $920.07.
7Write a recursive formula for the balance in the last question.
B₀ = 0, Bₖ = 1.004Bₖ₋₁ + 75.
More practice: warm-up
A series of equal payments made at equal time intervals.
2In an ordinary annuity, when are payments made?
At the end of each payment interval.
3What makes an annuity "simple"?
The payment interval is the same as the compounding period.
4In an ordinary annuity, how much interest does the last payment earn by the end of the term?
None: it is made on the last day of the term.
5Find the sum 100 + 100(1.1) + 100(1.1)².
More practice: on your own
1$250 is deposited at the end of every quarter for 2 years at 8% compounded quarterly. Find the future value.
i = 0.02, n = 8: 250(1.02⁸ − 1) ÷ 0.02 ≈ $2 145.74.
2How much of the last answer is interest?
$2 145.74 − 8 × $250 ≈ $145.74.
3$1 200 is deposited at the end of each year for 6 years at 4.5% compounded annually. Find the future value.
1200(1.045⁶ − 1) ÷ 0.045 ≈ $8 060.27.
4Explain why the yearly interest in the balance table keeps increasing even though the deposit is always $1 000.
Interest is 5% of the balance at the start of the year, and the balance grows every year (deposits plus earlier interest), so 5% of it gets larger.
5Show that FV = 1000(1.05ⁿ − 1) ÷ 0.05 can be written as 20 000(1.05)ⁿ − 20 000, and describe the transformation of y = 1.05ⁿ.
1000 ÷ 0.05 = 20 000, so FV = 20 000(1.05ⁿ − 1) = 20 000(1.05)ⁿ − 20 000. It is a vertical stretch by a factor of 20 000 and a translation 20 000 down.
Challenge
1Prove that the future value of n payments of R at the end of each period, at rate i per period, is R[(1 + i)ⁿ − 1] ÷ i.
FV = R + R(1 + i) + … + R(1 + i)ⁿ⁻¹ is geometric with a = R, r = 1 + i and n terms. Sₙ = a(rⁿ − 1) ÷ (r − 1) = R[(1 + i)ⁿ − 1] ÷ [(1 + i) − 1] = R[(1 + i)ⁿ − 1] ÷ i.
2Sam deposits $400 at the end of every quarter for 3 years at 5% compounded quarterly. Alex deposits $1 600 at the end of every year for 3 years at 5% compounded annually. Both deposit $4 800 in total. Who has more at the end, and why?
Sam: 400(1.0125¹² − 1) ÷ 0.0125 ≈ $5 144.14. Alex: 1600(1.05³ − 1) ÷ 0.05 = $5 044.00. Sam has more, because his money is deposited earlier on average and compounded more often.
3A $1 000-per-year annuity at 5% compounded annually has a balance of $5 525.63 at the end of year 5. Use the recursive formula to find the balance at the end of year 6 and year 7.
B₆ = 1.05(5525.63) + 1000 ≈ $6 801.91. B₇ = 1.05(6 801.91) + 1000 ≈ $8 142.01.
Quiz
1. An ordinary simple annuity has payments made:
2. $500 is deposited at the end of each year for 3 years at 4% compounded annually. Which series gives the future value?
3. In the future value series of an ordinary simple annuity, the common ratio is:
4. Which situation is an ordinary simple annuity?
5. As the number of payments increases, the future value of an annuity grows:
6. Find 200 + 200(1.1) + 200(1.1)².
Big idea: Expectation C3.5 asks students to explain what an annuity is and to connect an ordinary simple annuity to geometric series and exponential growth. This lesson builds the future value of an annuity from separate payments; Lesson 8 turns it into a formula.
How to help: The floor-timeline activity is worth doing even for a strong student. Ask, "How many periods of interest does this payment earn?" for each payment.
Common difficulties:
- Giving the first payment interest for the whole term. In an ordinary annuity it is made at the end of the first period, so it earns n − 1 periods of interest.
- Treating annuities with mismatched periods (monthly payments, yearly compounding) as simple annuities.
- Using the geometric series formula with r = i instead of r = 1 + i.