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CHAPTER 2 Time Value of Money

CHAPTER 2 Time Value of Money. Future value Present value Annuities Rates of return Amortization. Time lines. 0. 1. 2. 3. I%. CF 0. CF 1. CF 2. CF 3. Show the timing of cash flows.

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CHAPTER 2 Time Value of Money

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  1. CHAPTER 2Time Value of Money Future value Present value Annuities Rates of return Amortization

  2. Time lines 0 1 2 3 I% CF0 CF1 CF2 CF3 • Show the timing of cash flows. • Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

  3. $100 lump sum due in 2 years 0 1 2 I% 100 3 year $100 ordinary annuity 0 1 2 3 I% 100 100 100 Drawing time lines

  4. 0 2 3 1 I% -50 100 75 50 Drawing time lines Uneven cash flow stream (mixed stream)

  5. Today Future Compounding and Discounting Single Sums We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interestwe could have earned if we had received the $1 sooner.

  6. Today Future ? Today Future ? Compounding VS Discounting • Translate $1 today into its equivalent in the future(compounding). • Translate $1 in the future into its equivalent today(discounting).

  7. Compound Interest and Future Value

  8. Compound Interest and Future Value • 1) Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? PV = -100 FV = 106 Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .06, 1 ) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)1 = $106

  9. Compound Interest and Future Value • 2) Future Value - single sums • If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .06, 5 ) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82

  10. Compound Interest and Future Value • 3) Future Value - single sums • If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .015, 20 ) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68

  11. Compound Interest and Future Value • 4) Future Value - single sums • If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n ) FV = 100 (FVIF .005, 60 ) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = $134.89

  12. Compound Interest and Future Value • 5) Future Value - continuous compounding • What is the FV of $1,000 earning 8% with continuous compounding, after 100 years? Mathematical Solution: FV = PV (e in) FV = 1000 (e .08x100) = 1000 (e 8) FV = $2,980,957.99

  13. 0 1 2 3 10% 100 FV = ? What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? • Finding the FV of a cash flow or series of cash flows is called compounding. • FV can be solved by using the step-by-step (formula/table), financial calculator, and spreadsheet methods (excel).

  14. Solving for FV:The step-by-step and formula methods • After 1 year: • FV1 = PV (1 + I) = $100 (1.10) = $110.00 • After 2 years: • FV2 = PV (1 + I)2 = $100 (1.10)2 =$121.00 • After 3 years: • FV3 = PV (1 + I)3 = $100 (1.10)3 =$133.10 • After N years (general case): • FVN = PV (1 + I)N

  15. Solving for FV:The calculator method • Solves the general FV equation. • Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.) 3 10 -100 0 INPUTS N I/YR PV PMT FV OUTPUT 133.10

  16. Present Value

  17. Present Value • 1) Present Value - single sums • If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF .06, 1 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)1 = $94.34

  18. Present Value • 2) Present Value - single sums • If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF .06, 5 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73

  19. Present Value • 3) Present Value - single sums • What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? Mathematical Solution: PV = FV (PVIF i, n ) PV = 100 (PVIF .07, 15 ) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.07)15 = $362.45

  20. Present Value • 4) Present Value - single sums • If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Mathematical Solution: PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19

  21. Present Value • 5) Present Value - single sums • Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i)n 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months

  22. What is the present value (PV) of $100 due in 3 years, if I/YR = 10%? • Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). • The PV shows the value of cash flows in terms of today’s purchasing power. 0 1 2 3 10% PV = ? 100

  23. Solving for PV:The formula method • Solve the general FV equation for PV: • PV = FVN / (1 + I)N • PV = FV3 / (1 + I)3 = $100 / (1.10)3 = $75.13

  24. Solving for PV:The calculator method • Solves the general FV equation for PV. • Exactly like solving for FV, except we have different input information and are solving for a different variable. 3 10 0 100 INPUTS N I/YR PV PMT FV OUTPUT -75.13

  25. Solving for I:What interest rate would cause $100 to grow to $125.97 in 3 years? • Solves the general FV equation for I. • Hard to solve without a financial calculator or spreadsheet. 3 -100 0 125.97 INPUTS N I/YR PV PMT FV OUTPUT 8

  26. Solving for N:If sales grow at 20% per year, how long before sales double? • Solves the general FV equation for N. • Hard to solve without a financial calculator or spreadsheet. 20 -1 0 2 INPUTS N I/YR PV PMT FV OUTPUT 3.8

  27. Hint for single sum problems • In every single sum present value and future value problem, there are four variables: FV, PV, i and n. • When doing problems, you will be given three variables and you will solve for the fourth variable. • Keeping this in mind makes solving time value problems much easier!

  28. 0 1 2 3 4 Annuities • Annuity: a sequence of equal cash flows, occurring at the end of each period.

  29. Examples of Annuities: • If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

  30. Annuities • 1) Future Value – annuity • If you invest $1,000 each year at 8%, how much would you have after 3 years? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 FV = $3,246.40

  31. Future Value - annuityIf you invest $1,000 each year at 8%, how much would you have after 3 years? Mathematical Solution: FV = PMT (FVIFA i, n) FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or) FV = PMT (1 + i)n - 1 i FV = 1,000 (1.08)3 - 1 = $3246.40 .08

  32. Annuities • 1) Present Value - annuity • What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = -1,000 PV = $2,577.10

  33. Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Mathematical Solution: PV = PMT (PVIFA i, n) PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i)n i 1 PV = 1000 1 - (1.08 )3 = $2,577.10 .08

  34. Ordinary Annuity (end) 0 1 2 3 i% PMT PMT PMT Annuity Due (beginning) 0 1 2 3 i% PMT PMT PMT What is the difference between an ordinary annuity and an annuity due?

  35. Solving for FV:3-year ordinary annuity of $100 at 10% • $100 payments occur at the end of each period, but there is no PV. 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 331

  36. Solving for PV:3-year ordinary annuity of $100 at 10% • $100 payments still occur at the end of each period, but now there is no FV. 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -248.69

  37. Solving for FV:3-year annuity due of $100 at 10% • Now, $100 payments occur at the beginning of each period. • FVAdue= FVAord(1+I) = $331(1.10) = $364.10. • Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity: BEGIN 3 10 0 -100 INPUTS N I/YR PV PMT FV OUTPUT 364.10

  38. Solving for PV:3-year annuity due of $100 at 10% • Again, $100 payments occur at the beginning of each period. • PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. • Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity: BEGIN 3 10 100 0 INPUTS N I/YR PV PMT FV OUTPUT -273.55

  39. What is the present value of a 5-year $100 ordinary annuity at 10%? • Be sure your financial calculator is set back to END mode and solve for PV: • N = 5, I/YR = 10, PMT = 100, FV = 0. • PV = $379.08

  40. What if it were a 10-year annuity? A 25-year annuity? A perpetuity? • 10-year annuity • N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46. • 25-year annuity • N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70. • Perpetuity • PV = PMT / I = $100/0.1 = $1,000.

  41. Other Cash Flow Patterns • Perpetuities • Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. • You can think of a perpetuity as an annuity that goes on forever.

  42. Present Value of a Perpetuity • When we find the PV of an annuity, we think of the following relationship: PV = PMT (PVIFA i, n )

  43. 1 n 1 - (1 + i) i Mathematically, (PVIFA i, n ) = We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?

  44. 1 n 1 - (1 + i) i 1 i When n gets very large, this becomes zero. So we’re left with PVIFA =

  45. PMT PV = i Present Value of a Perpetuity • So, the PV of a perpetuity is very simple to find:

  46. PMT $10,000 i .08 = $125,000 PV = = What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

  47. The Power of Compound Interest A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%. How much money will she have when she is 65 years old?

  48. 45 12 0 -1095 INPUTS N I/YR PV PMT FV OUTPUT 1,487,262 Solving for FV:If she begins saving today, how much will she have when she is 65? • If she sticks to her plan, she will have $1,487,261.89 when she is 65. • N = 45, I/YR = 12, PMT = -1095, PV = 0; solve for FV = $1,487,262.

  49. Solving for FV:If you don’t start saving until you are 40 years old, how much will you have at 65? • If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20. • Lesson: It pays to start saving early. 25 12 0 -1095 INPUTS N I/YR PV PMT FV OUTPUT 146,001

  50. Solving for PMT:How much must the 40-year old deposit annually to catch the 20-year old? • To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT. 25 12 0 1,487,262 INPUTS N I/YR PV PMT FV OUTPUT -11,154.42

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