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Chapter 3

Chapter 3. Compound Interest. START. EXIT. Chapter Outline. 3.1 Compound Interest: The Basics 3.2 Compounding Frequencies 3.3 Effective Interest Rates 3.4 Comparing Effective and Nominal Rates Chapter Summary Chapter Exercises. 3.1 Compound Interest: The Basics.

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Chapter 3

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  1. Chapter 3 Compound Interest START EXIT

  2. Chapter Outline 3.1 Compound Interest: The Basics 3.2 Compounding Frequencies 3.3 Effective Interest Rates 3.4 Comparing Effective and Nominal Rates Chapter Summary Chapter Exercises

  3. 3.1 Compound Interest: The Basics • Let’s consider a $5,000 loan for 5 years at 8% simple interest. We can easily calculate the simple interest: I = PRT I = $5,000 x 8% x 5 I = $2,000 • Maturity Value = $7,000 • How does the amount of interest grow over the loan’s term?

  4. 3.1 Compound Interest: The Basics • Now imagine that this is a deposit that you have made in a bank certificate. • At the end of the first year, you would have $5,400 in your account. By leaving it on deposit at the bank for the second year, you are in effect loaning the bank $5,400. • Yet, you are being paid interest only on your original $5,000. • That’s how simple interest works. No matter how long the loan continues, under simple interest the borrower pays (and lender receives) interest only on the original principal.

  5. 3.1 Compound Interest: The Basics • This doesn’t seem quite fair. It seems reasonable that you should receive interest on the entire amount of your account balance. • If the bank has the use of $5,400 of your money in year 2, then you have every reason to expect that it should pay you interest on the full amount, $5,400. • In other words, you want to receive interest on your accumulated interest. • That is precisely the point of compound interest. • With compound interest, interest is paid on both the original principal and on any interest that accumulates along the way. • When interest is paid on interest, we say that it compounds.

  6. 3.1 Compound Interest: The Basics • Suppose that we look at the account balance year by year, assuming that interest is credited to the account annually. • Year 1 I = PRT I = $5,000 x 8% x 1 = $400 • Year 2 I = $5,400 x 8% x 1 = $432 • Year 3 I = $5,832 x 8% x 1 = $466.56

  7. 3.1 Compound Interest: The Basics

  8. 3.1 Compound Interest: The Basics • Looking at things year by year is a good way to get a sense of how compound interest works, but a tedious and impractical way of doing an actual calculation. • Note that 8% interest for a single year on $1 amounts to exactly 8 cents. Therefore, in 1 year $1 turns into $1.08. • What happens to $5,000? • End of year 1 balance $5,000 x $1.08 = $5,400 • End of year 2 balance $5,000 x $1.08 x $1.08 = $5,832 • End of year 5 balance $5,000 x $1.08 x $1.08 x $1.08 x $1.08 x $1.08 = $7,346.64

  9. 3.1 Compound Interest: The Basics • This approach is far more efficient than building an entire table, yet the repeated multiplications are still tedious. • We can accomplish the same thing more efficiently by using exponents. • An exponent is a way of denoting repeated multiplication of the same number. • (1.08)(1.08)(1.08)(1.08)(1.08) = (1.08)5

  10. 3.1 Compound Interest: The Basics • Most calculators have a key for exponents. Different calculator models, though, may mark their exponent keys differently. • It may be labeled “^”, or “xy”, or “yx”. • Be careful – “ex” is NOT the key you’re looking for!!! • Enter 1.08 • Press the exponent key • Enter 5 • Press an equal sign (=) • The result should be 1.469328 • If we multiply this result by $5,000, we come up with $7,346.64.

  11. 3.1 Compound Interest: The Basics FORMULA 3.1.1 The Compound Interest Formula FV = PV(1 + i)n where FV represents the FUTURE VALUE (the ending amount) PV represents the PRESENT VALUE (the starting amount) i represents the INTEREST RATE (per time period) n represents the NUMBER OF TIME PERIODS

  12. 3.1 Compound Interest: The Basics Example 3.1.1 • Problem • Use the compound interest formula to find how much $5,000 will grow to in 50 years at 8% annual compound interest. • Solution FV = PV(1 + i)n FV = $5,000(1 + 0.08)50 FV = $5,000(46.9016125132) FV = $234,508.06

  13. 3.1 Compound Interest: The Basics • Notice that we used the exponent before multiplying, even though reading from left to right it might appear as though the multiplication should have come first.

  14. 3.1 Compound Interest: The Basics Example 3.1.2 • Problem • Suppose you invest $14,075 at 7.5% annually compounded interest. How much will this grow to over 20 years? • Solution FV = PV(1 + i)n FV = $14,075(1 + 0.075)20 FV = $14,075(4.247851100) FV = $59,788.50

  15. 3.1 Compound Interest: The Basics Example 3.1.3 • Problem • Suppose you invest $14,075 at 7.5% annually compounded interest. How much total interest will you earn over 20 years? • Solution I = FV – PV I = $59,788.50 -- $14,075 = $45,713.50

  16. 3.1 Compound Interest: The Basics Example 3.1.4 • Problem • Suppose that $2,000 is deposited at a compound interest rate of 6% annually. Find (a) the total account value after 12 years and (b) the total interest earned in those 12 years. • Solution (a) FV = PV(1 + i)n FV = $2,000(1 + 0.06)12 FV = $4,024.39 (b) Total Interest = $4,024.39 -- $2,000 = $2,024.39

  17. 3.1 Compound Interest: The Basics Example 3.1.5 • Problem • How much money should I deposit today into an account earning 7 3/8% annually compounded in order to have $2,000 in the account 5 years from now? • Solution FV = PV(1 + i)n $2,000 = PV(1 + 0.07375)5 $2,000 = PV(1.42730203237) PV = $1,401.25

  18. 3.1 Compound Interest: The Basics FORMULA 3.1.2 The Rule of 72 The time required for a sum of money to double at a compound interest rate of x% is approximately 72/x years. The interest rate should not be converted to a decimal.

  19. 3.1 Compound Interest: The Basics Example 3.1.6 • Problem • Jarron deposited $3,200 into a retirement account, which he expects to earn 7% annually compounded interest. If his expectations about the interest rate is correct, how much will his deposit grow to between now and when he retires 40 years from now? Use the Rule of 72 to obtain an approximate answer, then use the compound interest formula to find the exact value. • Solution • Using the Rule of 72, we know that his money should double approximately every 10 years, since 72/7 FV = PV(1 + i)n $2,000 = PV(1 + 0.07375)5 $2,000 = PV(1.42730203237) PV = $1,401.25

  20. 3.1 Compound Interest: The Basics Example 3.1.6 • Problem • Jarron deposited $3,200 into a retirement account, which he expects to earn 7% annually compounded interest. If his expectations about the interest rate is correct, how much will his deposit grow to between now and when he retires 40 years from now? Use the Rule of 72 to obtain an approximate answer, then use the compound interest formula to find the exact value. • Solution • Using the Rule of 72, we know that his money should double approximately every 10 years, since 72/7 ≈ 10.2857. So in 40 years, his account should experience approximately 40/10 = 4 doublings. • FV = PV(1 + i)n FV = $3,200(1 + 0.07)40 = $47,918.27

  21. 3.1 Compound Interest: The Basics FORMULA 3.1.3 The Rule of 72 (Alternate Form) The compound interest rate required for a sum of money to double in x years is approximately 72/x percent.

  22. 3.1 Compound Interest: The Basics Example 3.1.7 • Problem • What compound interest rate is required to double $50,000 in 5 years? • Solution • 72/5 = 14.4, so the interest rate would need to be approximately 14.4%.

  23. Section 3.1 Exercises

  24. Problem 1 • Suppose that you deposit $5,000 in an account that pays 3% annually compounded interest for 2 years. Calculate (by hand) the balance at the end of Year 2. CHECK YOUR ANSWER

  25. Solution 1 • Suppose that you deposit $5,000 in an account that pays 3% annually compounded interest for 2 years. Calculate (by hand) your total balance at the end of Year 2. • Year 1 I = PRT I = $5,000 x 3% x 1 = $150 • Year 2 I = $5,150 x 3% x 1 = $154.50 • $5,150 + $154.50 = $5,304.50 BACK TO GAME BOARD

  26. Problem 2 • Jolanda deposited $2,000 into a 5-year certificate of deposit paying 3% interest compounded annually. How much will her CD be worth at maturity? Use the compound interest formula for the future value (FV). CHECK YOUR ANSWER

  27. Solution 2 • Jolanda deposited $2,000 into a 5-year certificate of deposit paying 3% interest compounded annually. How much will her CD be worth at maturity? Use the compound interest formula for the future value (FV). • FV = PV(1 + i)n FV = $2,000(1 + 0.03)5 FV = $2,000(1.159274074) = $2,318.55 BACK TO GAME BOARD

  28. Problem 3 • You are planning a trip to a dream destination. How much would you need to deposit today into an account paying 4.5% annually compounded interest in order to have $2,500 in 2 years? CHECK YOUR ANSWER

  29. Solution 3 • You are planning a trip to a dream destination. How much would you need to deposit today into an account paying 4.5% annually compounded interest in order to have $2,500 in 2 years? • FV = PV(1 + i)n $2,500 = PV(1 + 0.045)2 $2,500 = PV(1.092025) PV = $2,289.32 BACK TO GAME BOARD

  30. Problem 4 • Kristen has just invested $5,000 at 4% annually compounded interest. How long will it take for this investment to double (grow to $10,000). CHECK YOUR ANSWER

  31. Solution 4 • Kristen has just invested $5,000 at 4% annually compounded interest. How long will it take for this investment to double (grow to $10,000). Use the Rule of 72. • It will take 72/4% ≈ 18 years for Kristen’s investment to double. • We can check our estimation: FV = PV(1 + i)n FV = $5,000(1 + 0.04)18 FV = $10,129.08 BACK TO GAME BOARD

  32. 3.2 Compounding Frequencies Example 3.2.1 • Problem • Find the future value of $2,500 at 6% interest compounded monthly for 7 years. • Solution • Since the interest is compounded monthly, the annual rate of 6% needs to be divided by 12 (since each month is 1/12 of a year) to make it a monthly rate, 6% ÷ 12 = 0.5%. • In addition, the term of 7 years must be expressed in months, so there are 7 x 12 = 84 compounding periods. • FV = PV(1 + i)n FV = $2,500(1 + 0.005)84 FV = $3,800.92

  33. 3.2 Compounding Frequencies Example 3.2.2 • Problem • Find the future value of $3,250 at 4.75% interest compounded daily for 4 years. • Solution • Daily interest rate = 4.75% ÷ 365 = 0.000130137 • Number of compounding periods = 4 x 365 = 1,460 • FV = PV(1 + i)n FV = $3,250(1 + 0.000130137)1460 FV = $3,930.01

  34. 3.2 Compounding Frequencies Example 3.2.3 • Problem • Find the future value of $85.75 at 8.37% interest compounded monthly for 15 years. • Solution • Monthly interest rate = 8.37% ÷ 12 = 0.6975% • Compounding periods = 15 x 12 = 180 • FV = PV(1 + i)n FV = $85.75(1 + 0.006975)180 FV = $85.75(3.494330151) = $299.64

  35. 3.2 Compounding Frequencies Example 3.2.4 • Problem • How much do I need to deposit today into a CD paying 6.06% compounded monthly in order to have $10,000 in the account in 3 years? • Solution • Monthly interest rate = 6.06% ÷ 12 • Compounding periods = 3 x 12 = 36 • FV = PV(1 + i)n $10,000 = PV(1 + 0.00505)36 $10,000 = PV(1.1988257) PV = $8,341.50

  36. 3.2 Compounding Frequencies Example 3.2.5 • Problem • Find the future value of $5,000 in 5 years at 8% interest compounded annually, semiannually, quarterly, monthly, biweekly, weekly, and daily. • Solution • See page 105 in your textbook.

  37. Section 3.2 Exercises

  38. Problem 1 • Sonya deposited $5,000 in a certificate of deposit (CD) paying $3% compounded monthly for 5 years. How much will she end up with in her account? CHECK YOUR ANSWER

  39. Solution 1 • Sonya deposited $5,000 in a certificate of deposit (CD) paying 3% compounded monthly for 5 years. How much will she end up with in her account? • Monthly interest rate = 3% ÷ 12 = 0.25% • Compounding periods = 5 x 12 = 60 • FV = PV(1 + i)n FV = $5,000(1 + 0.0025)60 FV = $5,000(1.161616782) = $5,808.08 BACK TO GAME BOARD

  40. Problem 2 • Raj wants to have $60,000 in an investment account when his daughter starts college 15 years from now. Assuming that the account pays 4.5% compounded daily, how much should Raj deposit now? CHECK YOUR ANSWER

  41. Solution 2 • Raj wants to have $60,000 in an investment account when his daughter starts college 15 years from now. Assuming that the account pays 4.5% compounded daily, how much should Raj deposit now? • Daily interest rate = 4.5% ÷ 365 • Compounding periods = 15 x 365 = 5,475 • FV = PV(1 + i)n $60,000 = PV(1 + 0.045/365)5475 $60,000 = PV(1.963951249) PV = $30,550.66 BACK TO GAME BOARD

  42. 3.3 Effective Interest Rates Example 3.3.1 • Problem • Which of the following banks is offering the best rate for a certificate of deposit?

  43. 3.3 Effective Interest Rates Example 3.3.1 Cont. • Solution • Both rates are compounded annually, so it’s clear that Maplehurst Savings and Loan is offering a slightly higher rate.

  44. 3.3 Effective Interest Rates Example 3.3.2 • Problem • Which of these two banks is offering the best CD rate?

  45. 3.3 Effective Interest Rates Example 3.3.2 Cont. • Solution • Both rates are the same, but Cato National compounds interest daily. Since more frequent compounding means more interest overall, we know that Cato National will end up paying more interest, so their rate is better.

  46. 3.3 Effective Interest Rates Example 3.3.3 • Problem • Leo has a life insurance policy with Trustworthy Mutual Life of Nebraska. The company credits interest to his policy’s cash value and offers Leo the choice of two different options shown below. • Which option would give Leo the most interest?

  47. 3.3 Effective Interest Rates Example 3.3.3 Cont. • Solution • Daily Dividends FV = $100(1 + 0.08/360)360 = $108.33 • Annual Advancement FV = $100(1 + 0.0833)1 = $108.33 • Therefore, there is no difference.

  48. 3.3 Effective Interest Rates Definition 3.3.1 • The annually compounded rate which produces the same results as a given interest rate and compounding is called the equivalent annual rate (EAR) or the effective interest rate. • The original interest rate is called the nominal rate. • So, for example, we could say that if the nominal rate is 8.00% compounded daily (banker’s rule), then the equivalent annual rate (or the effective rate) is 8.33%.

  49. 3.3 Effective Interest Rates FORMULA 3.3.1 Finding the Effective Interest Rate To find the effective rate for a given nominal rate and compounding frequency, simply find the FV of $100 in 1 year using the nominal rate and compounding. The effective interest rate (rounded to two decimal places) will be the same number as the amount of interest earned.

  50. 3.3 Effective Interest Rates Example 3.3.4 • Problem • Find the equivalent annual rate for 7.35% compounded quarterly. • Solution FV = $100(1 + 0.0735/4)4 = $107.56 The interest earned is $7.56, and so we conclude that the equivalent annual rate is 7.56%.

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