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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? Can invest the \$1,000 today let it grow This is a fundamental building block of finance

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%? • PV = 100 / 1.15 = 86.96

6. FV Example • What is the value in one year of \$100, invested today at 15%? • FV = 100 * (1.15)1 = \$115

7. 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

8. 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

9. 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? • We don’t know • We cannot simply compare cash flows that occur at different times

10. 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? • NPV = -9,500 + 10,000/1.05 • NPV = -9,500 + 9,523.81 • NPV = 23.81 • At what price are we indifferent?

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? • NPV = -9,500 + 10,000/1.05 • NPV = -9,500 + 9,523.81 • NPV = 23.81 • At what price are we indifferent? \$9,523.81 • NPV would be 0

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 -\$275,000 \$300,000

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? • – 275,000 + (300,000/1.07) • – 275,000 + 280,373.83 • NPV = \$5,373.83 • Positive NPV → Build the coffee shop

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. 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

18. 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

19. 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?

20. Calculations • Need to recalculate the NPV • NPV = – 275,000 + (300,000/1.12) • NPV = – 275,000 + 267,857.14 • NPV = -7,142.86 • Negative NPV → Do NOT build the coffee shop

21. 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

22. 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

23. Risky Coffee Shop 2 • Now the Starbucks offer depends on the state of the economy

24. Calculations • Discount Rate = 12% • Expected Future Cash Flow = • (0.25*300) + (0.50*400) + (0.25*700) = 450,000 • NPV = • -275,000 + 450,000/1.12 • -275,000 + 401,786 = 126,786 • Do we still build the coffee shop? • Build the coffee shop, Positive NPV

25. 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

26. 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!!

27. 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

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? • \$53.78 = \$10×(1.40)5 \$53.78 \$10 14 19.6 27.44 38.42 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. \$20,000 9,943.53 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/(1+0.15)5 = 9,943.53

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 + 5r) = • Compounded interest • FV2 = PV0 (1+r) (1+r) * …= PV0 (1+r)5 =

32. 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 + 5r) = \$275 • Compounded interest • FV2 = PV0 (1+r) (1+r) * …= PV0 (1+r)5 =

33. 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 • FV5 = (PV0*(r) + PV0*(r)+…) + PV0 = PV0 (1 + 5r) = \$275 • Compounded interest • FV5 = PV0 (1+r) (1+r) * …= PV0 (1+r)5 = \$448.40

34. Less than Annual Compounding • Cash flows are usually discounted/ compounded over periods shorter than a year • Then the rates generally talked about are Stated Annual Interest Rate • Also known as the Annual Percentage Rate • These are simple interest rates, and DO NOT REFLECT WHAT PEOPLE ACTUALLY EARN • Effective Annual Rate: what people actually earn • Increase as compounding frequency increases

35. Compounding Periods • The relationship between PV & FV when interest is accumulated is less than a year: • FVN = PV * ( 1+ r / M) M*N • PV = FVN / ( 1+ r / M) M*N • r is the State Annual Rate • N is the number of years • M is number of compounding periods per year • We divide r by M to determine the rate earned each period • We multiple N by M to determine the number of periods

36. Compounding Examples • What is the FV of \$500 in 5 years, if the discount rate is 12%, compounded monthly? • FV = 500 * ( 1+ 0.12 / 12) 12*5 = 908.35 • What is the PV of \$500 received in 5 years, if the discount rate is 12% compounded monthly? • PV = 500 / ( 1+ 0.12 / 12) 12*5 = 275.22

37. Compounding Example 2 • If you invest \$50 for 3 years at 12% compounded semi-annually, your investment will grow to: \$70.93 FV = 50 * (1+(0.12/2))2*3 = \$70.93

38. 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

39. Compounding Example 2: Alt. \$70.93 12.36% • 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 • EAR = (1 + 0.12 / 2)2 – 1 = 12.36% • FV = 50 * (1+0.1236)3 = \$70.93 • So, investing at compounded annually is the same as investing at 12% compounded semi-annually

40. EAR Example • Find the Effective Annual Rate (EAR) of an 18% loan that is compounded weekly. • EAR = (1 + 0.18 / 52)52 – 1 = 19.68%

41. 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.

42. Insight Example First, which project is more valuable? Why?

43. Insight Example First, which project is more valuable? Why? B, gets the cash faster

44. 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?

45. Multiple Cash Flows (Given) 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93

46. 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

47. 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%: 100/0.1 = \$1,000 …

48. C1 C2(1+g) C3(1+g)2 … 0 1 2 3 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%? • PV = 100 / (0.10 – 0.02) =1,250

49. \$1.30 1 0 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? \$1.30 ×(1.05)2 = \$1.43 \$1.30×(1.05) = \$1.37 … 2 3 PV = 1.30 / (0.10 – 0.05) = \$26

50. 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? 5,000 = 200/ (0.10 – g) 5,000 * (0.10 – g) = 200 0.10 – g = 200 / 5,000 0.10 – (200 / 5,000) = g = 0.06 = 6%