1 Learning Objectives for Section 3.2 After this lecture, you should be able to Compute compound interest. Compute the annual percentage yield of a compound.

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Learning Objectives for Section 3.2

After this lecture, you should be able to Compute compound interest. Compute the annual percentage yield of a compound

interest investment.

Compound Interest

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Compound Interest

Compound interest: Interest paid on interest reinvested. Compound interest is always greater than or equal to simple

interest in the same time period, given the same annual rate.

Annual nominal rates: How interest rates are generally quoted

Rate per compounding period: annual nominal rate

# of compounding periods per year

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Compounding Periods

The number of compounding periods per year (m):

If the interest is compounded annually, then m = _______

If the interest is compounded semiannually, then m = _______

If the interest is compounded quarterly, then m = _______

If the interest is compounded monthly, then m = _______

If the interest is compounded daily, then m = _______

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Example

Example 1: Suppose a principal of $1 was invested in an account paying 6% annual interest compounded monthly. How much would be in the account after one year?

See next slide.

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Solution

Solution: Using the Future Value with simple interest formula A = P (1 + rt) we obtain the following amount:

after one month:

after two months:

after three months:

After 12 months, the amount is: ________________________.

With simple interest, the amount after one year would be _______.

The difference becomes more noticeable after several years.

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Graphical Illustration ofCompound Interest

The growth of $1 at 6% interest compounded monthly compared to 6% simple interest over a 15-year period.

The blue curve refers to the $1 invested at 6% simple interest.

The red curve refers to the $1 at 6% being compounded monthly.

Time (in years)

Dol

lars

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The formula for calculating the Future Amount with Compound Interest is

Where

A is the future amount,

P is the principal,

r is the annual interest rate as a decimal,

m is the number of compounding periods in one year, and

t is the total number of years.

General Formula: Compound Interest

1mt

rA P

m

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The formula for calculating the Future Amount with Compound Interest is

An alternate formula: Let

We now have,

Alternate Formula: Future Amount with Compound Interest

1mt

rA P

m

1n

A P i

ri and n mt

m

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Example

Example 2a: Find the amount to which $1,500 will grow if compounded quarterly at 6.75% interest for 10 years.

Example 2b: Compare your answer from part a) to the amount you would have if the interest was figured using the simple interest formula.

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Changing the number of compounding periods per year

Example 3: To what amount will $1,500 grow if compounded daily at 6.75% interest for 10 years?

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Effect of Increasing the Number of Compounding Periods

If the number of compounding periods per year is increased while the principal, annual rate of interest and total number of years remain the same, the future amount of money will increase slightly.

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Computing the Inflation Rate

Example 4: Suppose a house that was worth $68,000 in 1987 is worth $104,000 in 2004. Assuming a constant rate of inflation from 1987 to 2004, what is the inflation rate?

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Computing the Inflation Rate(continued )

Example 5: If the inflation rate remains the same for the next 10 years, what will the house from Example 4 be worth in the year 2014?

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Example

Example 6: If $20,000 is invested at 4% compounded monthly, what is the amount after a) 5 years b) 8 years?

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Which is Better?

Example 7: Which is the better investment and why: 8% compounded quarterly or 8.3% compounded annually?

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Inflation

Example 8: If the inflation rate averages 4% per year compounded annually for the next 5 years, what will a car costing $17,000 now cost 5 years from now?

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Investing

Example 9: How long does it take for a $4,800 investment at 8% compounded monthly to be worth more than a $5,000 investment at 5% compounded monthly?

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Annual Percentage Yield

The simple interest rate that will produce the same amount as a given compound interest rate in 1 year is called the annual percentage yield (APY). To find the APY, proceed as follows:

(1 ) 1

1 1

1 1

m

m

m

rP APY P

m

rAPY

m

rAPY

m

This is also called the effective rate.

Amount at simple interest APY after one year = Amount at compound interest after one year

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Annual Percentage Yield Example

What is the annual percentage yield for money that is invested at

6% compounded monthly?

General formula:

Substitute values:

Effective rate is

1 1m

rAPY

m