Simple And Compound Interest

8-1: Simple Interest and Compound Interest Unit 8: Discrete Functions – Financial Applications Financial institutions borrow and lend money. The amount borrowed or lent is called . Interest is the paid for using someone else’s money. The principal plus interest is called the . The bank pays you interest for using your money to lend to others. You pay the bank interest when you borrow the bank’s money. (ex. Mortgage, line of credit, credit card, etc.) We will study two types of interest calculations, simple interest and compound interest. Simple versus Compound Interest Which would be the wisest way to invest a principal of $1000: at 9% simple interest for 10 years, or at 7% compounded monthly for 10 years? Consider the graphs below in selecting an answer. Which graph represents each scenario above?

Simple interest

Amount ($) (future value)

Compound interest

(A)

SIMPLE INTEREST

Time (years)

Simple Interest: the interest earned or paid only on the original sum of money invested or borrowed. Principal: the amount of money borrowed or lent (invested). Ex. 1: Mary buys a $1000 regular (simple) interest Canada Savings Bond. It has a ten year term and earns 5% interest annually. Complete the following table: Year Principal Interest Earned Accumulated Amount at End of for year Interest Year 1 1000 50 1050 1000  0.05  50 2 1000 100 1100 1000  0.05  50 3 1000 150 1150 1000  0.05  50 4 1000 5 1000

Use the following variables to answer the questions below: P = Principal (original investment) r = Interest Rate (in decimal form, not percent) t = time (in years) I = Interest (in dollars) A = Amount a) What type of sequence could you use to represent the amount of her investment for successive years? Explain.

b) Write an equation for the interest earned. c) Write an equation for the Final Amount.

Ex. 2 Richard invests $715 at an annual rate of 6.2% for 10 months: a) How much interest does he earn? b) How much will his investment be worth at the end of the year?

Ex.3: Daniel had a credit card balance of $550 that was 31 days overdue. The annual interest rate on the card was 23.9%. How much interest did he have to pay?

Ex.4: Five years ago, Natalie lent Sasha money. Sasha repaid her a total of $2100, including simple interest charged at 10%. How much did Natalie originally lend Sasha?

Ex.5:

How many days will $800 have to be invested at 7% annually to earn $13.50 in interest?

(B)

COMPOUND INTEREST

Compound Interest means the interest is at regular intervals. The interest is added to the principal to earn interest for the next interval of time (or compounding period). Ex. Kirsten invests $1000 in a compound interest Canada Savings Bond (CSB). It has a ten year term and earns 5% interest annually. Year

Principal

1

1000

2

1050

3

1102.50

4

1157.63

Interest Earned

Amount at End of Year

1000  0.05  1  50 1050  0.05  1  52.50

1000 + 50 = 1050

1102 .50  0.05  1  55.13

1102.50+55.13 = 1157.63

1050 + 52.50 = 1102.50

5 Think of the above table in a different way: Year

Principal

1

1000

2

Amount at End of Year

Amount as a Power

1000(1.05)

1000(1.05)1

1000(1.05)1

1000(1.05)1(1.05)

1000(1.05)2

3

1000(1.05)2

1000(1.05)2(1.05)

1000(1.05)3

4

1000(1.05)3

5 a) What type of sequence could you use to represent the amount of Jessica’s investment for successive years? Explain.

b) Write an equation for the Final Amount. The amount from compound interest is represented by an exponential function with a constant ratio, (1+i). It is calculated with the formula:

A  P (1  i ) n

A= P= N= i = interest rate per compounding period (effective rate) = r (r = annual interest rate as a decimal) N n = total # of compounding periods = N x t

(t in years)

Compounding period Types of Compounding Periods, N Semi- Annually

2 times a year, N =

Quarterly

4 times a year, N =

Monthly

12 times a year, N =

Weekly:

52 times a year, N =

Daily

365 times a year, N =

Ex. 1:

Luke invests $10000 in a compound interest account for five years. He considers 3 different compounding periods: Option A: 6%/a compounded semi-annually Option B: 6%/a compounded quarterly Option C: 6%/a compounded monthly a) For each option, what is the amount at the end of 5 years? Which option is best? b) How does changing the compounding period affect the amount of the investment? Why?

Ex. 2: Jasmine borrows $3500 at an interest rate of 5.25% /a compounded semi-annually for her tuition. She plans to pay back the loan in 4 years. How much will she owe after 4years? How much interest will she pay for the loan?

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