A one time investment of $67,000invested at 7% per year, compounded annually, will grow to $1 million in 40 years. Most graduates don’t have $67,000 available to invest. >Thus, others use regular investments and take advantage of compound interest.>A series of equal investments at regular time periods is called an ANNUITY

7.1 ANNUITIES

GETTING STARTED EX. 1 $100 is deposited at the end of

each month at 6% interest per year compounded monthly.MONTH STARTING

BALANCEINTEREST EARNED

DEPOSIT ENDING BALANCE

1 $0.00 $0.00 $100.00 $100.00

2 $100.00 $0.50 $100.00 $200.50

3 $200.50 $1.00 $100.00 $301.50

4 $301.50 $1.51 $100.00 $403.01

𝑖=6%12

=0.0612

×0.0612

×0.0612

×0.0612

GETTING STARTED CONT’D Determine i, the interest rate per

compounding period as a decimal, and n, the number of compounding periods for each annuity.

TIME OF PAYMENT

LENGTH OF

ANNUITY

INTEREST RATE PER

YEAR

FREQUENCY OF

COMPOUNDING

End of each year 7 years 3% Annually

End of every 6 months

(semi-annual)

12 years 9% Semi-annually

End of each quarter

8 years 2.4% Quarterly

End of each month

5 years 18% Monthly

i = 0.03

= 0.045

= 0.006

i = = 0.015

n = 1x7=7

n = 2x12=24

n = 4x8=32

n = 12x5=60

ANNUITIES How much money should you invest annually

at 7% per year, compounded annually, to have $1 000 000 by the time you retire?

TVM Solver – www.grunderware.com

FUTURE VALUE OF AN ORDINARY SIMPLE ANNUITY

Emily works part time and is saving for a car for college. She deposits $400 at the end of each month into an account that pays 3.6% interest per year, compounded monthly. How much will Emily have saved at the end of six months?

FUTURE VALUE OF AN ORDINARY SIMPLE ANNUITY

Use Compound Interest Formula to calculate future value of each deposit.

1st deposit: earns interest for 5 months. 2nd deposit: earns interest for 4 months. 3rd deposit: earns interest for 3 months. Etc…

FV = PV(1 + i)n

Amount Invested

Growth Factor per compounding period

# of compounding periods

FUTURE VALUE OF AN ORDINARY SIMPLE ANNUITY Recall, simple interest was 3.6% per

year, but it is compounded monthly, so we need to figure out percentage per month:

FIRST DEPOSITFV1 = $400(1.003)5

=$406.0361

SECOND DEPOSITFV2 = $400(1.003)4

=$404.8216

THIRD DEPOSITFV3 = $400(1.003)3

=$403.6108

FOURTH DEPOSITFV4 = $400(1.003)2

=$402.4036

FIFTH DEPOSITFV5 = $400(1.003)1

=$401.20

SIXTH DEPOSITFV6 = $400(1.003)0

=$400

Emily’s first deposit is made at the end of the 1st month, so it earns interest for 5 months. Her second deposit earns interest

for four months, and so on.

PRESENT VALUE OF AN ANNUITY Joe has just purchased his first car. His

bank has given him car loan payments of $229.19 per month for the first year of the car loan at 10.5% per year, compounded monthly.

a) What is the actual cost of the car if Joe were to pay for it in cash today (hint:PV)?

Rearrange the compound interest formula:FV: $229.19; PV: ?; i = , n = 1 (1st month)

PAYMENT FOR AN ANNUITYFV: $229.19; PV: ?; i = , n=1 Rearrange compound interest formula for PV:

FV(1 + i)-n = PV(1 + i)n(1 + i)-n

FV(1 + i)-n = PVPV = FV(1 + i)-n =229.19(1 + 0.00875)-1

=227.20 – Thus, the present value of the first payment is $227.20.

PAYMENT FOR AN ANNUITY Present Value for 1st month of car: $227.20,

but he will pay $229.19. Present Value for 2nd month of car:

PV = FV(1 + i)-n =229.19(1 + 0.00875)-2

=225.23Thus the present value of the car for the 2nd month is only $225.23, whereas he will pay $229.19.The amount is getting less because there are less months to pay interest on every time.

7.1 HOMEWORK p. 409 #1