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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? Can invest the $1,000 today let it grow This is a fundamental building block of finance**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%? • PV = 100 / 1.15 = 86.96**FV Example**• What is the value in one year of $100, invested today at 15%? • FV = 100 * (1.15)1 = $115**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? • We don’t know • We cannot simply compare cash flows that occur at different times**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?**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**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 1: Draw out the cash flows**-$275,000 $300,000**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? • – 275,000 + (300,000/1.07) • – 275,000 + 280,373.83 • NPV = $5,373.83 • Positive NPV → Build the coffee shop**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**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 • 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**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 = • (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**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**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**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**$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**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 =**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 =**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**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**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**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**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**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: 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**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%**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**First, which project is more valuable? Why?**Insight Example**First, which project is more valuable? Why? B, gets the cash faster**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**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%: 100/0.1 = $1,000 …**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**$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**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%