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## Discounted Cash Flow Valuation

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**BASIC PRINCIPAL**• Would you rather have $1,000 today or $1,000 in 30 years? • Why?**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**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)**PV Example**• What is the value today of $100 in one year, if r = 15%?**FV Example**• What is the value in one year of $100, invested today at 15%?**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 =**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**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**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?**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?**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?**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?**Step 3: Find NPV**• The NPV of the project is?**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**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**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?**Calculations**• Need to recalculate the NPV**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**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**Risky Coffee Shop 2**• Now the Starbucks offer depends on the state of the economy**Calculations**• Discount Rate = 12% • Expected Future Cash Flow = • NPV = • Do we still build the coffee shop?**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**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!!**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**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%?**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**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**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**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 =**Compounding Periods**• We have been assuming that compounding and discounting occurs annually, this does not need to be the case**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**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?**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?**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**Compounding Example 2**• If you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to:**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**EAR Example**• Find the Effective Annual Rate (EAR) of an 18% loan that is compounded weekly.**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 =**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 =**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.**Insight Example**Which project is more valuable? Why?**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?**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?**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**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**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**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%: …**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?