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A Mathematical View of Our World

A Mathematical View of Our World. 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer. Chapter 13. Consumer Mathematics: Buying and Saving. Section 13.1 Simple and Compound Interest. Goals Study simple interest Calculate interest Calculate future value Study compound interest

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A Mathematical View of Our World

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  1. A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

  2. Chapter 13 Consumer Mathematics: Buying and Saving

  3. Section 13.1Simple and Compound Interest • Goals • Study simple interest • Calculate interest • Calculate future value • Study compound interest • Calculate future value • Compare interest rates • Calculate effective annual rate

  4. 13.1 Initial Problem • Suppose you discover you are the only direct descendant of a man who loaned the Continental Congress $1000 in 1777 and was never repaid. • Using an interest rate of 6% and a compounding period of 3 months, how much should you demand from the government? • The solution will be given at the end of the section.

  5. Simple Interest • If P represents the principal, r the annual interest rate expressed as a decimal, and t the time in years, then the amount of simple interest is:

  6. Example 1 • Find the interest on a loan of $100 at 6% simple interest for time periods of: • 1 year • 2 years • 2.5 years

  7. Example 1, cont’d • Solution: We have P = 100 and r = 0.06. • For t = 1 year, the calculation is:

  8. Example 1, cont’d • Solution, cont’d: We have P = 100 and r = 0.06. • For t = 2 years, the calculation is: • For t = 2.5 years, the calculation is:

  9. Future Value • For a simple interest loan, the future value of the loan is the principal plus the interest. • If P represents the principal, I the interest, r the annual interest rate, and t the time in years, then the future value is:

  10. Example 2 • Find the future value of a loan of $400 at 7% simple interest for 3 years.

  11. Example 2, cont’d • Solution: Use the future value formula with P = 400, r = 0.07, and t = 3.

  12. Example 3 • In 2004, Regular Canada Savings Bonds paid 1.25% simple interest on the face value of bonds held for 1 year. • If the bond is cashed early, the investor receives the face value plus interest for every full month. • Suppose a bond was purchased for $8000 on November 1, 2004.

  13. Example 3, cont’d • What was the value of the bond if it was redeemed on November 1, 2005? • What was the value of the bond if it was redeemed on July 10, 2004?

  14. Example 3, cont’d • Solution: If the bond was redeemed on November 1, 2005, it had been held for 1 year. • The future value of the bond after 1 year is:

  15. Example 3, cont’d • Solution: If the bond was redeemed on July 10, 2004, it had been held for 7 full months. • The future value of the bond after 7/12 of a year is:

  16. Example 4 • What is the simple interest on a $500 loan at 12% from June 6 through October 12 in a non-leap year?

  17. Example 4, cont’d • Solution: The time must be converted to years. • (30 - 6) + 31 + 31 + 30 + 12 = 128 days • A non-leap year has 365 days. • The interest will be:

  18. Question: What is the simple interest on a $2000 loan at 8% from March 19th through August 15th in a leap year? a. $65.32 b. $653.15 c. $651.37 d. $65.14

  19. Ordinary Interest • Ordinary interest simplifies calculations by using 2 conventions: • Each month is assumed to have 30 days. • Each year is assumed to have 360 days.

  20. Example 5 • A homeowner owes $190,000 on a 4.8% home loan with an interest-only option. • An interest-only option allows the borrower to pay only the ordinary interest, not the principal, for the first year. • What is the monthly payment for the first year?

  21. Example 5, cont’d • Solution: Use the simple interest formula, measuring time according to ordinary interest conventions. • The monthly payments are:

  22. Compound Interest • Reinvesting the interest, called compounding, makes the balance grow faster. • To calculate compound interest, you need the same information as for simple interest plus you need to know how often the interest is compounded.

  23. Example 6 • Suppose a principal of $1000 is invested at 6% interest per year and the interest is compounded annually. • Find the balance in the account after 3 years.

  24. Example 6, cont’d • Solution: We must calculate the interest at the end of each year and then add that interest to the principal. • After 1 year: • The interest is: • The new balance is $1060.00 • We could also have used the future value formula.

  25. Example 6, cont’d • Solution, cont’d: • After 2 years the new balance is: • After 3 years the new balance is:

  26. Example 6, cont’d • Solution, cont’d: The interest earned each year increases because of the increasing principal.

  27. Example 6, cont’d • Solution, cont’d: The following table shows the pattern in the calculations for subsequent years.

  28. Compound Interest, cont’d • If P represents the principal, r the annual interest rate expressed as a decimal, m the number of equal compounding periods per year, and t the time in years, then the future value of the account is:

  29. Example 7 • Find the future value of each account at the end of 3 years if the initial balance is $2457 and the account earns: • 4.5% simple interest. • 4.5% compounded annually. • 4.5% compounded every 4 months. • 4.5% compounded monthly. • 4.5% compounded daily.

  30. Example 7, cont’d • Solution: We have P = 2457 and t = 3. • We have r = 0.045 with simple interest. • We have r = 0.045 compounded annually.

  31. Example 7, cont’d • Solution, cont’d: We have r = 0.045 • Compounded every 4 months: • Compounded monthly: • Compounded daily:

  32. Example 7, cont’d • Solution, cont’d: The results are summarized below.

  33. Example 8 • Find the future value of each account at the end of 100 years if the initial balance is $1000 and the account earns: • 7.5% simple interest. • 7.5% compounded annually.

  34. Example 8, cont’d • Solution: We have P = 1000, t = 100, and r = 0.075. • With simple interest, the future value is: • With annually compounded interest, the future value is:

  35. Question: If you loan your friend $100 at 3% interest compounded daily, how much will she owe you at the end of 1 year? a. $103.05 b. $103.00 c. $103.33 d. $103.02

  36. Interest, cont’d • Simple interest exhibits arithmetic growth. • The same amount is added each year. • Compound interest exhibits exponential growth. • The same amount is multiplied each year.

  37. Example 9 • How much money should be invested at 4% interest compounded monthly in order to have $25,000 eighteen years later?

  38. Example 9, cont’d • Solution: We know F = 25,000, r = 0.04, m = 12 and t = 18. • Solve for P, the necessary principal.

  39. Effective Annual Rate • The effective annual rate (EAR) or annual percentage yield (APY) is the simple interest that would give the same result in 1 year. • The stated rate is called the nominal rate. • This provides a basis for comparing different savings plans. • APY is used only for savings accounts. • EAR is used in any context.

  40. Example 10 • Find the effective annual rate by computing what happens to $100 over 1 year at 12% annual interest compounded every 3 months.

  41. Example 10, cont’d • Solution: The balance at the end of 1 year is: • Since the account increased by $12.55 in 1 year, the EAR is 12.55%.

  42. Effective Annual Rate, cont’d • If r represents the annual interest rate expressed as a decimal and m is the number of equal compounding periods per year, then the effective annual rate is: • Note: The same formula is used for APY.

  43. Effective Annual Rate, cont’d

  44. Example 11 • A bank offers a savings account with an interest rate of 0.25% compounded daily, with a minimum deposit of $100. • The same bank offers an 18-month CD with an interest rate of 2.13% compounded monthly, with deposits less than $10,000. • Find the effective annual rate for each option.

  45. Example 11, cont’d • Solution, cont’d: The effective annual rate for the savings account is: • The EAR for the account is about 0.2503%.

  46. Example 11, cont’d • Solution: The effective annual rate for the CD is: • The EAR for the CD is about 2.1509%.

  47. 13.1 Initial Problem Solution • Suppose you discover you are the only direct descendant of a man who loaned the Continental Congress $1000 in 1777 and was never repaid. • Using an interest rate of 6% and a compounding period of 3 months, how much should you demand from the government?

  48. Initial Problem Solution, cont’d • We have P = $1000, r = 0.06, m = 4 and t = 223. • The value of your ancestor’s loan is:

  49. Section 13.2Loans • Goals • Study amortized loans • Use an amortization table • Use the amortization formula • Study rent-to-own

  50. 13.2 Initial Problem • Home mortgage rates have decreased and Howard plans to refinance his home. • He will refinance $85,000 at either 5.25% for 15 years or 5.875% for 30 years. • In each case, what is his monthly payment and how much interest will he pay? • The solution will be given at the end of the section.

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