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Chapter 6: The Time Value of Money

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Financial Management of Health Care Organizations An Introduction 4th edition, Zelman

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Chapter 6: The Time Value of Money

  1. 1. The Time Value of Money Chapter 6
  2. 2. Learning Objectives • Explain why a dollar today is worth more than a dollar in the future • Define the terms future value • Calculate the future value of an amount and an annuity
  3. 3. The Time Value of Money • Consume Today or Tomorrow? • TVM is based on the belief that people prefer to consume goods today rather than wait to consume the same goods tomorrow •An apple we can have today is more valuable to us than an apple we can have in one year. •Money has a time value because buying an apple today is more important than buying an apple in one year.
  4. 4. The Time Value of Money • Consume Today or Tomorrow? • A dollar someone has today can be spent for consumption or loaned to earn interest • A dollar loaned earns interest that increases wealth and the ability to consume • The rate of interest determines the trade-off between consumption today and saving (investing)
  5. 5. The Time Value of Money • Future Value versus Present Value • Cash-flows are evaluated based on future value or present value • Future value measures what cash-flows are worth after a certain amount of time has passed • Present value measures what future cash-flows are worth before a certain amount of time has passed
  6. 6. Comparison of Future Value and Present Value
  7. 7. The Time Value of Money • Future Value versus Present Value • Compounding is the process of increasing cash-flows to a future value • Discounting is the process of reducing future cash-flows to a present value
  8. 8. Future Value • What a dollar invested today will be worth in the future depends on  Length of the investment period Method to calculate interest  Interest rate  2 types of methods to calculate interest Simple method-calculated only on the original principal each year Compound interest-calculated on both the original principal and on any accumulated interest earned up to that point. Future value implies the compound method Interest rate Simple interest is paid on the original principal amount only Compound interest consists of both simple interest and interest-on-interest
  9. 9. Present Value of an Amount to be Received in the Future • Taking future values back to the present is called discounting Present Value=Future Value x Present Value Factor PV = Present value (initial investment amount) i = the interest rate n= no. of time periods of the investment
  10. 10. Example 1 • Suppose we place $10000 in a savings account that pays 10% interest compounded annually. How will our savings grow? Value at the end of year 1 = present value X (1+i) i = interest rate • = $10000 ퟏ + ퟎ. ퟏ ퟒ • = $10000 (1.4641) • = $14,641
  11. 11. Future Value: Example 2 • If we place $1,000 in a savings account paying 5% interest compounded annually, how much will our account accrue to in 10 years? • Future value = present value X (ퟏ + 풓)풏 • 푭푽풏 = $1,000 (ퟏ + ퟎ. ퟎퟓ)ퟏퟎ • = $1,000 (1.62889) • = $1,628.89 13 Refer Table B-1
  12. 12. Example 3: • What is the future value of $10,000 with an interest rate of 16 percent and one annual period of compounding? With an annual interest rate of 16 percent and two semiannual periods of compounding? With an annual interest rate of 16 percent and four quarterly periods of compounding? Annually: 푭푽 = 푷푽 ∗ (ퟏ + 풊)풏 푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ+. ퟏퟔ)ퟏ 푭푽 = $ퟏퟎ, ퟎퟎퟎ ∗ (ퟏ. ퟏퟔ)ퟏ 푭푽 = $ퟏퟏ, ퟔퟎퟎ
  13. 13. Semi-annually: 푭푽 = 푷푽 푿 ퟏ + 풊 풎 풏푿풎 • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + .ퟏퟔ ퟐ ퟏ푿ퟐ • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟖ ퟐ • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟔퟒ • 푭푽 = $ퟏퟏ, ퟔퟔퟒ • Quarterly: • 푭푽 = 푷푽 푿 ퟏ + 풊 풎 풏푿풎 • 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ + .ퟏퟔ ퟒ ퟏ푿ퟒ 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟎퟒ ퟒ 푭푽 = $ퟏퟎ, ퟎퟎퟎ 푿 ퟏ. ퟏퟔퟗퟗ 푭푽 = $ퟏퟏ, ퟔퟗퟗ
  14. 14. Present Value and Discounting • Present Value concepts • A present value calculation takes end-of-the-period cash flows and reverses the effect of compounding to determine the equivalent beginning-of-the-period cash flows •Present Value Equation •P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 푿 푷푽푭풊,풏 •P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ ퟏ (ퟏ+풓)풏] •This is discounting and the interest rate i is called the discount rate. •Present value (PV) is often referred to as the discounted value of future cash-flows.
  15. 15. Present Value and Discounting • Time and the discount rate affect present value •The greater the amount of time before a cash flow is to occur, the smaller the present value of the cash-flow. •The higher the discount rate, the smaller the present value of a future cash-flow.
  16. 16. Example 4 What is the value of $500 to be received 10 years from today if our discount rate is 6%. P풓풆풔풆풏풕 푽풂풍풖풆 = 푭푽풏 [ ퟏ (ퟏ+풓)풏] FV = $500, n = 10, r = 6% or 0.06 = $500 [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] = $500 (0.558) = $279.20 18 Refer Table B -3
  17. 17. Example 5 What is the present value of an investment that yields $1,000 to be received in 7 years and $1,000 to be received in 10 years if the discount rate is 6 percent? ퟏ 풑풓풆풔풆풏풕 풗풂풍풖풆 = 푭푽풏 [ (ퟏ+풓)풏] + 푭푽풏 [ ퟏ (ퟏ+풓)풏] PV= $ퟏ, ퟎퟎퟎ [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟕ] + $1,000 [ ퟏ (ퟏ+ퟎ.ퟎퟔ)ퟏퟎ] • = $1,000 (0.665) + $1,000 (0.558) • = $665 + $558 • = $1223
  18. 18. Future and Present Values of Annuities • Annuity- A series of equal payment made or received at regular time intervals • Ordinary Annuity- A series of equal annuity payments made or received at the end of each period • Future value of an annuity-What an equal series of payments will be worth at some future date • Future Value Factor of an Annuity (FVFA)-A factor that when multiplied by a stream of equal payments equals the future value of that stream • Future Value of an Annuity Table- Table of factors that shows the future value of equal flows at the end of each period, given a particular interest rate
  19. 19. Example 6 Carlos Menendez is planning to invest $3,500 every year for the next six years in an investment paying 12 percent annually. What will be the amount he will have at the end of the six years? (Round to the nearest dollar.) A) $21,000 B) $28,403 C) $24,670 D) $26,124
  20. 20. Example 7 You plan to save $1,250 at the end of each of the next three years to pay for a vacation. If you can invest it at 7 percent, how much will you have at the end of three years? (Round to the nearest dollar.) A) $3,750 B) $3,918 C) $4,019 D) $4,589
  21. 21. Example 8: Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on any investment, what is the amount that she will have to invest every year at the end of each year for the next five years? (Round to the nearest dollar.) A) $5,000 B) $4,261 C) $4,640 D) $4,445
  22. 22. Example 9 Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of four years. She can invest a certain amount at the beginning of each of the next four years in a bank account that will pay her 6.8 percent annually. How much will she have to invest annually to reach her target? (Round to the nearest dollar.) A) $3,000 B) $2,980 C) $2,538 D) $2,711
  23. 23. Annuities Continued • Present Value of an Annuity- What the series of payments in the future is worth today • Present Value Factor of an Annuity (PVFA)- A factor that when multiplied by a stream of equal payments equals the present value of that stream • Present Value of an Annuity Table- Table of factors that shows the value today of equal flows at the end of each future period, given a particular interest rate
  24. 24. How to calculate present value of an annuity (6.1) 30 o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 PVA CF PVFA PVFA 1   (1 ) 1    1 0                     i i CF i CF n ퟏ−풑풓풆풔풆풏풕 풗풂풍풖풆 풇풂풄풕풐풓 풓 o 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 ퟏ−(ퟏ+풓)−풏 풓
  25. 25. Level Cash Flows: Annuities and • A contract will pay $2,000 at the end of each year for three years and the appropriate discount rate is 8%. What is a fair price for the contract? $5,154.19 Perpetuities • Present Value of an Annuity Example (1 1/(1 0.08)    0.08 PVA $2000 3  3      Refer Table B-4 31
  26. 26. Example 1 What is the value of 10-year $1,000 annuity discounted back to the present at 5 percent?
  27. 27. 푷푽풐풇 풂풏 풂풏풏풖풊풕풚 = 푷푴푻 ퟏ − (ퟏ + 풓)−풏 풓 푷푽 = $ퟏ, ퟎퟎퟎ ퟏ − (ퟏ + ퟎ. ퟎퟓ)−ퟏퟎ ퟎ. ퟎퟓ • = $1,000 (7.722) • = $7,722
  28. 28. Example 2 • Transit Insurance Company has made an investment in another company that will guarantee it a cash flow of $37,250 each year for the next five years. If the company uses a discount rate of 15 percent on its investments, what is the present value of this investment? (Round to the nearest dollar.) A) $101,766 B) $124,868 C) $251,154 D) $186,250 Annual payment = PMT = $37,250 No. of payments = n = 5 Required rate of return = 15% Present value of investment = PVA5
  29. 29.          5 1 1 (1 ) n 1 1 (1.15) $37,250 $37,250 3.3522 0.15 n i PVA PMT i                     $124,867.78
  30. 30. Example 3 Herm Mueller has invested in a fund that will provide him a cash flow of $11,700 for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value of this cash flow stream? (Round to the nearest dollar.) A) $234,000 B) $132,455 C) $110,721 D) $167,884
  31. 31.          20 1 1 (1 ) 1 1 n (1.085) $11,700 $11,700 9.4633 0.085 n i PVA PMT i                     $110,721.04
  32. 32. Example 4 Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period, starting at the end of this year. If the firm uses a 9 percent discount rate, what is the present value of this annuity? (Round to the nearest dollar.) A) $23,250 B) $29,000 C) $25,000 D) $20,000
  33. 33.          10 1 1 (1 ) 1 1 n (1.09) $3,895.50 $3,895.50 6.4177 0.09 n i PVA PMT i                     $24,999.99
  34. 34. Ordinary Annuity versus Annuity Due • Present Value of Annuity Due •Cash flows are discounted for one period less than in an ordinary annuity. • Future Value of Annuity Due •Cash flows are earn compound interest for one period more than in an ordinary annuity. 40 PVA PVA (1 ) (6.4) 1 Due 1 Due i    FVA FVA (1 i )   
  35. 35. Ordinary Annuity versus Annuity Due 41
  36. 36. Perpetuities • A stream of equal cash flows that goes on forever • Preferred stock and some bonds are perpetuities • Equation for the present value of a perpetuity can be derived from the present value of an annuity equation 42 PVP CF esent value factor for an annuity 1   1 1 0 ( i) ( . )   Pr 0 CF i ( ) i CF i CF 6 3  1                  
  37. 37. Valuing Perpetuity Example 1 • Suppose you decide to endow a chair in finance. The goal of the endowment is to provide $100,000 of financial support per year forever. If the endowment earns a rate of 8%, how much money will you have to donate to provide the desired level of support? 43 $1,250,000 $100,000 CF 0    0.08 i PVP
  38. 38. Example 2 • What is the value of a $500 perpetuity discounted back to the present at 8 percent?
  39. 39. 푷푽 = 푪푭 풊 푷푽 = $ퟓퟎퟎ ퟎ.ퟎퟖ = $6250
  40. 40. Example 3 Your father is 60 years old and wants to set up a cash flow stream that would be forever. He would like to receive $20,000 every year, beginning at the end of this year. If he could invest in account earning 9 percent, how much would he have to invest today to receive his perpetual cash flow? (Round to the nearest dollar.) A) $222,222 B) $200,000 C) $189,000 D) $235,200
  41. 41. Annual payment needed = PMT = $20,000 Investment rate of return = i = 9% Term of payment = Perpetuity Present value of investment needed = PV PMT $20,000 PV of Perpetuity   i 0.09  $222,222.22
  42. 42. Example 4 A lottery winner was given a perpetual payment of $11, 444. She could invest the cash flows at 7 percent. What is the present value of this perpetuity? (Round to the nearest dollar.) A) $112,344 B) $163,486 C) $191,708 D) $201,356
  43. 43. Annual payment needed = PMT = $11,444 Investment rate of return = i = 7% Term of payment = Perpetuity Present value of investment needed = PV PMT $11,444 PV of Perpetuity   i 0.07  $163,485.71
  44. 44. Cash Flows That Grow at a Constant Rate • Growing Annuity • equally-spaced cash flows that increase in size at a constant rate for a finite number of periods • Multiyear product or service contract with periodic cash flows that increase at a constant rate for a finite number of years • Growing Perpetuity • equally-spaced cash flows that increase in size at a constant rate forever • Common stock whose dividend is expected to increase at a constant rate forever
  45. 45. Cash Flows That Grow at a Constant Rate • Growing Annuity • Calculate the present value of growing annuity (only) when the growth rate is less than the discount rate. 51 CF   (6.5) 1 g  1 i  1 i - g PVA n 1    n            
  46. 46. Growing Annuity: Example • A coffee shop will operate for fifty more years. Cash flow was $300,000 last year and increases by 2.5% each year. The discount rate for similar firms is 15%. Estimate the value of the firm. 1 $300,000 (1 0.025) $307,500     $307,500 CF $2,460,000 0.9968   $2,452,128 1.025 1.15  1 0.15 0.025 50 0             PVA
  47. 47. Cash Flows That Grow at a Constant Rate • Growing Perpetuity • Use Equation 6.6 to calculate the present value of growing perpetuity (only) when the growth rate is less than discount rate. • It is derived from equation 6.5 when the number of periods approaches infinity 53 CF PVP 1   (6.6) i - g 0 
  48. 48. Cash Flows That Grow at a Constant Rate • Growing perpetuity example • A firm’s cash flow was $450,000 last year. You expect the cash flow to increase by 5% per year forever. If you use a discount rate of 18%, what is the value of the firm? 54 1 $450,000 (1 0.05) $307,500     $472,500 $3,634,615 $472,500 0.13 0.18 0.05 0     CF PVP
  49. 49. The Effective Annual Interest Rate • Describing interest rates • The most common way to quote interest rates is in terms of annual percentage rate (APR). It does not incorporate the effects of compounding. • The most appropriate way to quote interest rates is in terms of effective annual rate (EAR). It incorporates the effects of compounding. 55
  50. 50. The Effective Annual Interest Rate • Calculate Annual Percentage Rate (APR) APR = (periodic rate) x m m is the # of periods in a year • APR does not account for the number of compounding periods or adjust the annualized interest rate for the time value of money • APR is not a precise measure of the rates involved in borrowing and investing 56
  51. 51. Future and Present Value Calculations and Excel Functions for Special Situations • The more frequent the compounding for any given interest level and time period, the higher the future value. • In the Excel RATE and NPER functions, the Payment box or loan payment box must be a negative value to represent cash outflows
  52. 52. Summary • Future values determine the value of dollar payments in the future • Present value indicates the current value of future dollars • Formulas are used to calculate both future and present values • All calculations can be made using tables or spreadsheets

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