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# Discounted Cash Flow Valuation

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1. Discounted Cash Flow Valuation

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

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. PV Example • What is the value today of \$100 in one year, if r = 15%?

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 thensells for \$98. What is your rate of return? PV = FV =

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. 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 C0 will be negative, and C1 will be positive • For Loans C0 will be positive, and C1 will be negative

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. 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. 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. Step 1: Draw out the cash flows

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. Step 3: Find NPV • The NPV of the project is?

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. The Discount Rate • Should take account of two things: • Time value of money • 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. 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. Calculations • Need to recalculate the NPV

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. Expectation Calculation • The expected value is the weighted average of X’s possible values, where the probability of any outcome is p • E(X) = p1X1 + p2X2 + …. psXs • 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. Risky Coffee Shop 2 • Now the Starbucks offer depends on the state of the economy

23. Calculations • Discount Rate = 12% • Expected Future Cash Flow = • NPV = • Do we still build the coffee shop?

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. 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. 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. Four Related Questions • How much must you deposit today to have \$1 million in 25 years? (r=12%) • If a \$58,823.31 investment yields \$1 million in 25 years, what is the rate of interest? • How many years will it take \$58,823.31 to grow to \$1 million if r=12%? • What will \$58,823.31 grow to after 25 years if r=12%?

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? \$10 0 1 2 3 4 5

29. 0 1 2 3 4 5 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%? \$20,000 PV

30. Historical Example From Fibonacci’s Liber Abaci, written in the year 1202: “A certain man gave 1 denariat 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

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. Compounding Periods • We have been assuming that compounding and discounting occurs annually, this does not need to be the case

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. 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?

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. 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. \$70.93 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

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 =

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 =

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.

43. Insight Example 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?

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. 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

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. C C C 0 1 2 3 Perpetuity • A stream of cash flows that lasts forever • PV: = C/r • What is PV if C=\$100 and r=10%: …

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?