Learning Objectives for Section 3.2 MAT 103 SPRING 2009 Learning Objectives for Section 3.2 Compound Interest After this lecture, you should be able to Compute compound interest. Compute the annual percentage yield of a compound interest investment.

MAT 103 SPRING 2009 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:

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 = _______

MAT 103 SPRING 2009 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.

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.

Graphical Illustration of Compound Interest MAT 103 SPRING 2009 Graphical Illustration of Compound 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. Dollars The red curve refers to the $1 at 6% being compounded monthly. Time (in years)

General Formula: Compound Interest MAT 103 SPRING 2009 General Formula: Compound Interest 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.

Alternate Formula: Future Amount with Compound Interest MAT 103 SPRING 2009 Alternate Formula: Future Amount with Compound Interest The formula for calculating the Future Amount with Compound Interest is An alternate formula: Let We now have,

MAT 103 SPRING 2009 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.

Changing the number of compounding periods per year MAT 103 SPRING 2009 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?

Effect of Increasing the Number of Compounding Periods MAT 103 SPRING 2009 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.

Computing the Inflation Rate MAT 103 SPRING 2009 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?

Computing the Inflation Rate (continued ) MAT 103 SPRING 2009 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?

MAT 103 SPRING 2009 Example Example 6: If $20,000 is invested at 4% compounded monthly, what is the amount after a) 5 years b) 8 years?

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

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?

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?

Annual Percentage Yield MAT 103 SPRING 2009 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: Amount at simple interest APY after one year = Amount at compound interest after one year This is also called the effective rate.

Annual Percentage Yield Example MAT 103 SPRING 2009 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