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Chapter 5 The Time Value of Money

Chapter 5 The Time Value of Money. Laurence Booth, Sean Cleary and Pamela Peterson Drake. Outline of the chapter. 5.1 Time value of money. Simple interest. Simple interest is interest that is paid only on the principal amount. Interest = rate × principal amount of loan.

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Chapter 5 The Time Value of Money

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  1. Chapter 5The Time Value of Money Laurence Booth, Sean Cleary and Pamela Peterson Drake

  2. Outline of the chapter

  3. 5.1 Time value of money

  4. Simple interest Simple interest is interest that is paid only on the principal amount. Interest = rate × principal amount of loan

  5. Simple interest: example A 2-year loan of $1,000 at 6% simple interest • At the end of the first year, interest = 6% × $1,000 = $60 • At the end of the second year,interest = 6% × $1,000 = $60 and loan repayment of $1,000

  6. Compound interest Compound interest is interest paid on both the principal and any accumulated interest. Interest =

  7. Compounding Compounding is translating a present value into a future value, using compound interest. Future value interest factor is also referred to as the compound factor.

  8. Terminology and notation

  9. Compare: simple versus compound Suppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is: Simple interest? Compound interest?

  10. Simple interest

  11. Compound interest

  12. Interest on interest How much interest on interest? Interest on interest = FVcompound – FVsimple Interest on interest = $6,077.53 – 6,000.00 = $77.53

  13. Comparison

  14. Try it: Simple v. compound Suppose you are comparing two accounts: The Bank A account pays 5.5% simple interest. The Bank B account pays 5.4% compound interest. If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?

  15. Try it: Answer Bank A: $12,750.00 Bank B: $13,007.78

  16. A note about interest Because compound interest is so common, assume that interest is compounded unless otherwise indicated.

  17. Short-cuts Example: Consider $1,000 deposited for three years at 6% per year.

  18. The long way FV1 = $1,000.00 × (1.06) = $1,060.00 FV2 = $1,060.00 × (1.06) = $1,123.60 FV3 = $1,123.60 × (1.06) = $1,191.02 or FV3 = $1,000 × (1.06)3 = $1,191.02 or FV3 = $1,000 × 1.191016 = $1,191.02 Future value factor

  19. Short-cut: Calculator Known values: PV = 1,000 n = 3 i = 6% Solve for: FV

  20. Input three known values, solve for the one unknown Known: PV, i , n Unknown: FV

  21. Short-cut: spreadsheetMicrosoft Excel or Google Docs =FV(RATE,NPER,PMT,PV,TYPE) TYPE default is 0, end of period =FV(.06,3,0,-1000) or

  22. Problems Set 1

  23. Problem 1.1 Suppose you deposit $2,000 in an account that pays 3.5% interest annually. How much will be in the account at the end of three years? How much of the account balance is interest on interest?

  24. Problem 1.2 If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?

  25. Discounting

  26. Discounting Discounting is translating a future value into a present value. The discount factor is the inverse of the compound factor: To translate a future value into a present value, PV=

  27. Example Suppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?

  28. Example, continued Known values: FV = $100,000 n = 3 i = 4% Unknown: PV

  29. Example, continued PV = = PV = $100,000 × 0.8889964 PV = $88,899.64 Check: FV3 = $88,899.64 × (1 + 0.04)3 = $100,000

  30. Short-cut: Calculator

  31. Short-cut: spreadsheetMicrosoft Excel or Google Docs =PV(RATE,NPER,PMT,PV,TYPE) TYPE default: end of period =PV(.06,3,0,-1000) or

  32. Try it: Present value What is the today’s value of $10,000 promised ten years from now if the discount rate is 3.5%?

  33. Try it: Answer Given: FV = $10,000 N = 10 I = 3.5% Solve for PV PV = = $7,089.19

  34. Frequency of compounding If interest is compounded more than once per year, we need to make an adjustment in our calculation. The stated rate or nominal rate of interest is the annual percentage rate (APR). The rate per period depends on the frequency of compounding.

  35. Discrete compounding:Adjustments • Adjust the number of periods and the rate per period. • Suppose the nominal rate is 10% and compounding is quarterly: • The rate per period is 10%  4 = 2.5% • The number of periods is number of years × 4

  36. Continuous compounding:Adjustments • The compound factor is eAPR x n. • The discount factor is . • Suppose the nominal rate is 10%. • For five years, the continuous compounding factor is e0.10 x 5 = 1.6487 • The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 = 0.60653

  37. Try it: Frequency of compounding If you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?

  38. Try it: Answer Given: PV = $1,000 n = 5 × 2 = 10 i = 0.05  2 = 0.25 Solve for FV FV = $1,000 × (1 + 0.025)10= $1,280.08

  39. Problem Set 2

  40. Problem 2.1 Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

  41. Problem 2.2 Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

  42. Problem 2.3 Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

  43. 5.2 Annuities and Perpetuities

  44. What is an annuity? • An annuity is a periodic cash flow. • Same amount each period • Regular intervals of time • The different types depend on the timing of the first cash flow.

  45. Type of annuities

  46. Time lines: 4-payment annuity

  47. Key to valuing annuities The key to valuing annuities is to get the timing of the cash flows correct. When in doubt, draw a time line.

  48. Example: PV of an annuity What is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?

  49. Example: PV of an annuityThe long way

  50. Example: PV of an annuityIn table form PV = $4,000.00 × 2.67301 = $10,692.05

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