# Discounted cash flow valuation Student | Coordinator - Omega, The Operations Interest Group | IIM Kozhikode em Indian Institute of Management, Kozhikode
12 de Sep de 2016                                                                              1 de 78

### Discounted cash flow valuation

• 1. Discounted Cash Flow Valuation
• 2. 2 BASIC PRINCIPAL  Would you rather have \$1,000 today or \$1,000 in 30 years? Why?
• 3. Present and Future Value  Present Value: value of a future payment today  Future Value: value that an investment will grow to in the future  We find these by discounting or compounding at the discount rate Also know as the hurdle rate or the opportunity cost of capital or the interest rate 3
• 4. 4 One Period Discounting  PV = Future Value / (1+ Discount Rate) V0 = C1 / (1+r)  Alternatively  PV = Future Value * Discount Factor V0 = C1 * (1/ (1+r)) Discount factor is 1/ (1+r)
• 5. 5 PV Example  What is the value today of \$100 in one year, if r = 15%?
• 6. 6 FV Example  What is the value in one year of \$100, invested today at 15%?
• 7. Discount Rate Example  Your stock costs \$100 today, pays \$5 in dividends at the end of the period, and then sells for \$98. What is your rate of return?  PV =  FV = 7
• 8. 8 NPV  NPV = PV of all expected cash flows Represents the value generated by the project To compute we need: expected cash flows & the discount rate  Positive NPV investments generate value  Negative NPV investments destroy value
• 9. 9 Net Present Value (NPV)  NPV = PV (Costs) + PV (Benefit) Costs: are negative cash flows Benefits: are positive cash flows  One period example NPV = C0+ C1/ (1+r) For Investments C0will be negative, and C1will be positive For Loans C0will be positive, and C1will be negative
• 10. 10 Net Present Value Example  Suppose you can buy an investment that promises to pay \$10,000 in one year for \$9,500. Should you invest?
• 11. 11 Net Present Value  Since we cannot compare cash flow we need to calculate the NPV of the investment If the discount rate is 5%, then NPV is?  At what price are we indifferent?
• 12. 12 Coffee Shop Example  If you build a coffee shop on campus, you can sell it to Starbucks in one year for \$300,000  Costs of building a coffee shop is \$275,000  Should you build the coffee shop?
• 13. 13 Step 1: Draw out the cash flows Today Year 1
• 14. 14 Step 2: Find the Discount Rate  Assume that the Starbucks offer is guaranteed  US T-Bills are risk-free and currently pay 7% interest This is known as rf  Thus, the appropriate discount rate is 7% Why?
• 15. 15 Step 3: Find NPV  The NPV of the project is?
• 16. 16 If we are unsure about future?  What is the appropriate discount rate if we are unsure about the Starbucks offer rd = rf rd > rf rd < rf
• 17. 17 The Discount Rate  Should take account of two things: 1. Time value of money 2. Riskiness of cash flow  The appropriate discount rate is the opportunity cost of capital  This is the return that is offer on comparable investments opportunities
• 18. 18 Risky Coffee Shop  Assume that the risk of the coffee shop is equivalent to an investment in the stock market which is currently paying 12%  Should we still build the coffee shop?
• 19. 19 Calculations  Need to recalculate the NPV
• 20. 20 Future Cash Flows  Since future cash flows are not certain, we need to form an expectation (best guess) Need to identify the factors that affect cash flows (ex. Weather, Business Cycle, etc). Determine the various scenarios for this factor (ex. rainy or sunny; boom or recession) Estimate cash flows under the various scenarios (sensitivity analysis) Assign probabilities to each scenario
• 21. 21 Expectation Calculation  The expected value is the weighted average of X’s possible values, where the probability of any outcome is p  E(X) = p1 X1 + p2 X2 + …. ps Xs E(X) – Expected Value of X Xi − Outcome of X in state i pi – Probability of state i s – Number of possible states  Note that = p1 + p2 +….+ ps = 1
• 22. 22 Risky Coffee Shop 2  Now the Starbucks offer depends on the state of the economy Recession Normal Boom Value 300,000 400,000 700,000 Probability 0.25 0.5 0.25
• 23. 23 Calculations  Discount Rate = 12%  Expected Future Cash Flow =  NPV =  Do we still build the coffee shop?
• 24. 24 Valuing a Project Summary  Step 1: Forecast cash flows  Step 2: Draw out the cash flows  Step 3: Determine the opportunity cost of capital  Step 4: Discount future cash flows  Step 5: Apply the NPV rule
• 25. 25 Reminder  Important to set up problem correctly  Keep track of • Magnitude and timing of the cash flows • TIMELINES  You cannot compare cash flows @ t=3 and @ t=2 if they are not in present value terms!!
• 26. 26 General Formula PV0 = FVN/(1 + r)N OR FVN = PVo*(1 + r)N  Given any three, you can solve for the fourth Present value (PV) Future value (FV) Time period Discount rate
• 27. 27 Four Related Questions 1. How much must you deposit today to have \$1 million in 25 years? (r=12%) 2. If a \$58,823.31 investment yields \$1 million in 25 years, what is the rate of interest? 3. How many years will it take \$58,823.31 to grow to \$1 million if r=12%? 4. What will \$58,823.31 grow to after 25 years if r=12%?
• 28. 28 FV Example  Suppose a stock is currently worth \$10, and is expected to grow at 40% per year for the next five years.  What is the stock worth in five years? 0 1 2 3 4 5 \$10
• 29. 29 PV Example  How much would an investor have to set aside today in order to have \$20,000 five years from now if the current rate is 15%? 0 1 2 3 4 5 \$20,000PV
• 30. Historical Example  From Fibonacci’s Liber Abaci, written in the year 1202: “A certain man gave 1 denari at interest so that in 5 years he must receive double the denari, and in another 5, he must have double 2 of the denari and thus forever. How many denari from this 1denaro must he have in 100 years?”  What is rate of return? Hint: what does the investor earn every 5 years 30
• 31. 31 Simple vs. Compound Interest  Simple Interest: Interest accumulates only on the principal  Compound Interest: Interest accumulated on the principal as well as the interest already earned  What will \$100 grow to after 5 periods at 35%? • Simple interest  FV2 = (PV0 * (r) + PV0 *(r)) + PV0 = PV0 (1 + 2r) = • Compounded interest  FV2 = PV0 (1+r) (1+r)= PV0 (1+r)2 =
• 32. 32 Compounding Periods We have been assuming that compounding and discounting occurs annually, this does not need to be the case
• 33. 33 Non-Annual Compounding  Cash flows are usually compounded over periods shorter than a year  The relationship between PV & FV when interest is not compounded annually FVN = PV * ( 1+ r / M) M*N PV = FVN / ( 1+ r / M) M*N  M is number of compounding periods per year  N is the number of years
• 34. 34 Compounding Examples  What is the FV of \$500 in 5 years, if the discount rate is 12%, compounded monthly?  What is the PV of \$500 received in 5 years, if the discount rate is 12% compounded monthly?
• 35. Another Example  An investment for \$50,000 earns a rate of return of 1% each month for a year. How much money will you have at the end of the year? 35
• 36. 36 Interest Rates  The 12% is the Stated Annual Interest Rate (also known as the Annual Percentage Rate) This is the rate that people generally talk about  Ex. Car Loans, Mortgages, Credit Cards  However, this is not the rate people earn or pay  The Effective Annual Rate is what people actually earn or pay over the year The more frequent the compounding the higher the Effective Annual Rate
• 37. 37 Compounding Example 2  If you invest \$50 for 3 years at 12% compounded semi-annually, your investment will grow to:
• 38. Compounding Example 2: Alt.  If you invest \$50 for 3 years at 12% compounded semi-annually, your investment will grow to:  Calculate the EAR: EAR = (1 + R/m)m – 1  So, investing at compounded annually is the same as investing at 12% compounded semi-annually 38 \$70.93
• 39. 39 EAR Example  Find the Effective Annual Rate (EAR) of an 18% loan that is compounded weekly.
• 40. Credit Card  A bank quotes you a credit card with an interest rate of 14%, compounded daily. If you charge \$15,000 at the beginning of the year, how much will you have to repay at the end of the year?  EAR = 40
• 41. Credit Card  A bank quotes you a credit card with an interest rate of 14%, compounded daily. If you charge \$15,000 at the beginning of the year, how much will you have to repay at the end of the year?  EAR = 41
• 42. 42 Present Value Of a Cash Flow Stream  Discount each cash flow back to the present using the appropriate discount rate and then sum the present values. PV C r C r C r C r C r N N N t t t t N = + + + + + + + + += ∑ 1 1 2 2 2 3 3 3 1 1 1 1 1 1 ( ) ( ) ( ) ... ( ) ( ) =
• 43. 43 Insight Example r = 10% Year Project A Project B 1 100 300 2 400 400 3 300 100 PV Which project is more valuable? Why?
• 44. Various Cash Flows  A project has cash flows of \$15,000, \$10,000, and \$5,000 in 1, 2, and 3 years, respectively. If the interest rate is 15%, would you buy the project if it costs \$25,000? 44
• 45. 45 Example (Given)  Consider an investment that pays \$200 one year from now, with cash flows increasing by \$200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows?  If the issuer offers this investment for \$1,500, should you purchase it?
• 46. 46 Multiple Cash Flows (Given) 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93 Don’t buy
• 47. Various Cash Flow (Given)  A project has the following cash flows in periods 1 through 4: –\$200, +\$200, –\$200, +\$200. If the prevailing interest rate is 3%, would you accept this project if you were offered an up-front payment of \$10 to do so?  PV = –\$200/1.03 + \$200/1.032 – \$200/1.033 + \$200/1.034  PV = –\$10.99.  NPV = \$10 – \$10.99 = –\$0.99.  You would not take this project 47
• 48. 48 Common Cash Flows Streams  Perpetuity, Growing Perpetuity A stream of cash flows that lasts forever  Annuity, Growing Annuity A stream of cash flows that lasts for a fixed number of periods  NOTE: All of the following formulas assume the first payment is next year, and payments occur annually
• 49. 49 Perpetuity  A stream of cash flows that lasts forever  PV: = C/r  What is PV if C=\$100 and r=10%: …0 1 C 2 C 3 C + + + + + + = 32 )1()1()1( r C r C r C PV
• 50. Perpetuity Example  What is the PV of a perpetuity paying \$30 each month, if the annual interest rate is a constant effective 12.68% per year? 50
• 51. Perpetuity Example 2  What is the prevailing interest rate if a perpetual bond were to pay \$100,000 per year beginning next year and costs \$1,000,000 today? 51
• 52. 52 Growing Perpetuities  Annual payments grow at a constant rate, g PV= C1/(1+r) + C1(1+g)/(1+r)2 + C1(1+g)2 (1+r)3 +…  PV = C1/(r-g)  What is PV if C1=\$100, r=10%, and g=2%? … 0 1 2 3 C1 C2(1+g) C3(1+g)2
• 53. Growing Perpetuity Example  What is the interest rate on a perpetual bond that pays \$100,000 per year with payments that grow with the inflation rate (2%) per year, assuming the bond costs \$1,000,000 today? 53
• 54. 54 Growing Perpetuity: Example (Given)  The expected dividend next year is \$1.30, and dividends are expected to grow at 5% forever.  If the discount rate is 10%, what is the value of this promised dividend stream? 0 … 1 \$1.30 2 \$1.30×(1.05) = \$1.37 3 \$1.30 ×(1.05)2 = \$1.43 PV = 1.30 / (0.10 – 0.05) = \$26
• 55. 55 Example An investment in a growing perpetuity costs \$5,000 and is expected to pay \$200 next year. If the interest is 10%, what is the growth rate of the annual payment?
• 56. 56 Annuity A constant stream of cash flows with a fixed maturity 0 1 C 2 C 3 C T r C r C r C r C PV )1()1()1()1( 32 + + + + + + + =        + −= T rr C PV )1( 1 1 T C 
• 57. 57 Annuity Formula T r r C r C PV )1( + −=  Simply subtracting off the PV of the rest of the perpetuity’s cash flows 0 1 C 2 C 3 C T C T+1 C T+2 C T+3 C
• 58. 58 Annuity Example 1  Compute the present value of a 3 year ordinary annuity with payments of \$100 at r=10%  Answer:
• 59. 59 Alternative: Use a Financial Calculator  Texas Instruments BA-II Plus, basic  N = number of periods  I/Y = periodic interest rate  P/Y must equal 1 for the I/Y to be the periodic rate  Interest is entered as a percent, not a decimal  PV = present value  PMT = payments received periodically  FV = future value  Remember to clear the registers (CLR TVM) after each problem  Other calculators are similar in format
• 60. 60 Annuity Example 2  You agree to lease a car for 4 years at \$300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? Work through on your financial calculators
• 61. 61 Annuity Example 3  What is the value today of a 10-year annuity that pays \$600 every other year? Assume that the stated annual discount rate is 10%. What do the payments look like? What is the discount rate?
• 62. 62 Annuity Example 3  What is the value today of a 10-year annuity that pays \$600 every other year? Assume that the stated annual discount rate is 10%. What do the payments look like? 0 2 4 6 8 10 PV \$600\$600 \$600\$600\$600
• 63. 63 Annuity Example 3  What is the value today of a 10-year annuity that pays \$600 every other year? Assume that the stated annual discount rate is 10%. What is the discount rate?
• 64. 64 Annuity Example 4  What is the present value of a four payment annuity of \$100 per year that makes its first payment two years from today if the discount rate is 9%? What do the payments look like? 0 1 2 3 4 5
• 65. 65 Annuity Example 5  What is the value today of a 10-pymt annuity that pays \$300 a year if the annuity’s first cash flow is at the end of year 6. The interest rate is 15% for years 1-5 and 10% thereafter?
• 66. 66 Annuity Example 5  What is the value today of a 10-pymt annuity that pays \$300 a year (at year-end) if the annuity’s first cash flow is at the end of year 6. The interest rate is 15% for years 1-5 and 10% thereafter?  Steps: 1. Get value of annuity at t= 5 (year end) 2. Bring value in step 1 to t=0
• 67. Annuity Example 6  You win the \$20 million Powerball. The lottery commission offers you \$20 million dollars today or a nine payment annuity of \$2,750,000, with the first payment being today. Which is more valuable is your discount rate is 5.5%? 67
• 68. Alt: Annuity Example 6  You win the \$20 million Powerball. The lottery commission offers you \$20 million dollars today or a nine payment annuity of \$2,750,000, with the first payment being today. Which is more valuable if your discount rate is 5.5%? 68
• 69. 69 Delayed first payment: Perpetuity  What is the present value of a growing perpetuity, that pays \$100 per year, growing at 6%, when the discount rate is 10%, if the first payment is in 12 years?
• 70. 70 Growing Annuity A growing stream of cash flows with a fixed maturity 0 1 C T T r gC r gC r C PV )1( )1( )1( )1( )1( 1 2 + +× ++ + +× + + = −                + + − − = T r g gr C PV )1( 1 1  2 C×(1+g) 3 C ×(1+g)2 T C×(1+g)T-1
• 71. 71 Growing Annuity: Example A defined-benefit retirement plan offers to pay \$20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%? 0 1 \$20,000  2 \$20,000×(1.03) 40 \$20,000×(1.03)39
• 72. 72 Growing Annuity: Example (Given) You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be \$8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? PV = (8,500/(.12-.07)) * [ 1- {1.07/1.12}5 ] = \$34,706.26 0 1 2 3 4 5
• 73. 73 Growing Perpetuity Example  What is the value today a perpetuity that makes payments every other year, If the first payment is \$100, the discount rate is 12%, and the growth rate is 7%? r: g: Price:
• 74. 74 Valuation Formulas               + + − − = T r g gr C PV )1( 1 1 1      + −= T rr C PV )1( 1 1 gr C PV − = 1 r C PV = n n r FV PV )1( + = n n rPVFV )1(* +=
• 75. 75 Remember  That when you use one of these formula’s or the calculator the assumptions are that:  PV is right now  The first payment is next year
• 76. 76 What Is a Firm Worth?  Conceptually, a firm should be worth the present value of the firm’s cash flows.  The tricky part is determining the size, timing, and risk of those cash flows.
• 77. 77 Quick Quiz 1. How is the future value of a single cash flow computed? 2. How is the present value of a series of cash flows computed. 3. What is the Net Present Value of an investment? 4. What is an EAR, and how is it computed? 5. What is a perpetuity? An annuity?
• 78. Why We Care  The Time Value of Money is the basis for all of finance  People will assume that you have this down cold 78