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Overview of the Present Value Concept, Investment Criteria and Free-cash flows. The fundamental of valuation. Today’s plan. Review the concept of the time value of money present value (PV) discount rate (r) discount factor (DF) net present value (NPV)

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## Overview of the Present Value Concept, Investment Criteria and Free-cash flows

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**Overview of the Present Value Concept, Investment Criteria**and Free-cash flows The fundamental of valuation FIN 819: Lecture 2**Today’s plan**• Review the concept of the time value of money • present value (PV) • discount rate (r) • discount factor (DF) • net present value (NPV) • Review of two rules for making investment decisions • The NPV rule • The rate of return rule • Review the formula for calculating the present value of • perpetuity with and without growth • annuity with and without growth • Review the concepts about interest compounding FIN 819: Lecture 2**Today’s plan (continue)**• Why do we always argue for the use of the NPV rule • Examination of two other investment criteria • Payback rule • IRR rule • Some specific questions in using NPV • Sunk costs, opportunity cost • Incremental cash flows and incidental cash flows • Working capital • Inflation, real interest rate and nominal interest rate FIN 819: Lecture 2**Today’s plan (continue)**• How to calculate cash flows in Finance • Depreciations are not actual cash flows • Three approaches to calculate cash flows from operations FIN 819: Lecture 2**Financial choices**• Which would you rather receive today? • TRL 1,000,000,000 ( one billion Turkish lira ) • USD 652.72 ( U.S. dollars ) • Both payments are absolutely guaranteed. • What do we do? FIN 819: Lecture 2**Financial choices**• We need to compare “apples to apples” - this means we need to get the TRL:USD exchange rate • From www.bloomberg.com we can see: • USD 1 = TRL 1,789,320 • Therefore TRL 1bn = USD 558 FIN 819: Lecture 2**Financial choices at different times**• Which would you rather receive? • $1000 today • $1200 in one year • Both payments have no risk, that is, • there is 100% probability that you will be paid • there is 0% probability that you won’t be paid FIN 819: Lecture 2**Financial choices at different times (2)**• Why is it hard to compare ? • $1000 today • $1200 in one year • This is not an “apples to apples” comparison. They have different units • $1000 today is different from $1000 in one year • Why? • A cash flow is time-dated money • It has a money unit such as USD or TRL • It has a date indicating when to receive money FIN 819: Lecture 2**Present value**• In order to have an “apple to apple” comparison, we convert future payments to the present values • this is like converting money in TRL to money in USD • Certainly, we can also convert the present value to the future value to compare payments we get today with payments we get in the future. • Although these two ways are theoretically the same, but the present value concept is more important and has more applications, as to be shown in stock and bond valuations. FIN 819: Lecture 2**Present value for the cash flow at period 1**C1 is the cash in period 1 PV is the present value of the cash flow in period 1 DF1 is called discount factor for the cash flow in period 1 r1 is the discount rate FIN 819: Lecture 2**Example 1**• What is the present value of $100 received in one year (next year) if the discount rate is 7%? • PV=100/(1.07)1 = $100 PV=? Year one FIN 819: Lecture 2**Present value for the cash flow at period t**• Replacing “1” with “t” allows the formula to be used for cash flows at any point in time FIN 819: Lecture 2**Example 2**• What is the present value of $100 received in year five if the discount rate is 7%? • PV=100/(1.07)5 = $100 PV=? Year 5 FIN 819: Lecture 2**Example 3**• What is the present value of $100 received in year 20 if the discount rate is 7%? • PV=100/(1.07)20 = $100 PV=? Year 20 FIN 819: Lecture 2**Example 4**You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? FIN 819: Lecture 2**Explanation of the discount factor**Discount Factor FIN 819: Lecture 2**Example for the discount factor**• Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%. What is the present value for each dollar received? • DF1=1.00/(1+0.2)=0.83 • DF2=1.00/(1+0.07)2=0.87 FIN 819: Lecture 2**Present value of multiple cash flows**• For a cash flow received in year one and a cash flow received in year two, different discount rates may be used. • The present value of these two cash flows is the sum of the present value of each cash flow, since two present value have the same unit: time zero USD. FIN 819: Lecture 2**Present Values of future cash flows**• PVs can be added together to evaluate multiple cash flows. FIN 819: Lecture 2**Example 5**• John is given the following set of cash flows and discount rates. What is the PV? • PV=100/(1.1)1 + 100/(1.09)2 = $100 $100 PV=? Year one Year two FIN 819: Lecture 2**Example 6**• John is given the following set of cash flows and discount rates. What is the PV? • PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 = $100 $200 $50 PV=? Yr 3 Yr 1 Yr 2 FIN 819: Lecture 2**Projects**• A “project” is a term that is used to describe the following activity: • spend some money today • receive cash flows in the future • A stylized way to draw project cash flows is as follows: Expected cash flows in year one (probably positive) Expected cash flows in year two (probably positive) Initial investment (negative cash flows) FIN 819: Lecture 2**Examples of projects**• An entrepreneur starts a company: • initial investment is negative cash outflow. • future net revenue is cash inflow . • An investor buys a share of IBM stock • cost is cash outflow; dividends are future cash inflows. • A lottery ticket: • investment cost: cash outflow of $1 • jackpot: cash inflow of $20,000,000 (with some very small probability…) • Thus projects can range from real investments, to financial investments, to gambles (the lottery ticket). FIN 819: Lecture 2**Firms or companies**• A firm or company can be regarded as a set of projects. • capital budgeting is about choosing the best projects in real asset investments. • How do we know one project is worth taking? FIN 819: Lecture 2**Net present value**• A net present value NPV is the sum of the initial investment (usually made at time zero) and the PV of expected future cash flows. FIN 819: Lecture 2**NPV rule**• If the NPV of a project is positive, the firm should go ahead to take this project. • This rule is often called the DCF approach, because we have to use the discount rate to calculate the PV of the future cash flows of a project FIN 819: Lecture 2**Example 7**• Given the data for project A, what is the NPV? • NPV=-50+50/(1.075)+10/(1.08)2 = $50 $10 -$50 Yr 1 Yr 2 Yr 0 FIN 819: Lecture 2**Example 8**Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value. FIN 819: Lecture 2**Present Values**FIN 819: Lecture 2**Example 9**• John got his MBA from SFSU. When he was interviewed by a big firm, the interviewer asked him the following question: • A project costs 10 m and produces future cash flows, as shown in the next slide, where cash flows depend on the state of the economy. • In a “boom economy” payoffs will be high • over the next three years, there is a 20% chance of a boom • • In a “normal economy” payoffs will be medium • over the next three years, there is a 50% chance of normal • In a “recession” payoffs will be low • over the next 3 years, there is a 30% chance of a recession • In all three states, the discount rate is 8% over all time horizons. • Tell me whether to take the project or not FIN 819: Lecture 2**Cash flows diagram in each state**• Boom economy • Normal economy • Recession $3 m $8 m $3 m -$10 m $7 m $2 m $1.5 m -$10 m $6 m $1 m $0.9 m -$10 m FIN 819: Lecture 2**Example 9 (continues)**• The interviewer then asked John: • Before you tell me the final decision, how do you calculate the NPV? • Should you calculate the NPV at each economy or take the average first and then calculate NPV • Can your conclusion be generalized to any situations? FIN 819: Lecture 2**Calculate the NPV at each economy**• In the boom economy, the NPV is • -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36 • In the average economy, the NPV is • -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613 • In the bust economy, the NPV is • -10+ 6/1.08 + 1/1.082 + 0.9/1.083 =-$2.87 The expected NPV is 0.2*2.36+0.5*(-.613)+0.3*(-2.87)=-$0.696 FIN 819: Lecture 2**Calculate the expected cash flows at each time**• At period 1, the expected cash flow is • C1=0.2*8+0.5*7+0.3*6=$6.9 • At period 2, the expected cash flow is • C2=0.2*3+0.5*2+0.3*1=$1.9 • At period 3, the expected cash flows is • C3=0.2*3+0.5*1.5+0.3*0.9=$1.62 • The NPV is • NPV=-10+6.9/1.08+1.9/1.082+1.62/1.083 • =-$0.696 FIN 819: Lecture 2**The rate of return rule for a one-period project with**negative C0 • Another way to decide whether a project (with one piece of cash flow occurring in the future) should be taken or not is to compare the rate of return and the discount rate. • If the rate of return of a project is larger than the discount rate (the cost of capital, or hurdle rate), the firm should go ahead to take this project. • The rate of return is defined as the ratio of the profit to the cost. FIN 819: Lecture 2**Example**• If you invest $30 today in one share of stock (no dividends), you will get $36 next year. What is the rate of return for your investment? • Profit=36-30=$6 • Rate of return = 6/30=20% FIN 819: Lecture 2**NPV rule and the rate of return rule?**• What is the relationship between these two rules? • If there is some relation between these two rules, can you show formally? FIN 819: Lecture 2**Perpetuities**• We are going to look at the PV of a perpetuity starting one year from now (please see the cash flow diagram below). • Definition: if a project makes a level, periodic payment into perpetuity, it is called a perpetuity. • Let’s suppose your friend promises to pay you $1 every year, starting next year. His future family will continue to pay you and your future family forever. The discount rate is assumed to be constant at 8.5%. How much is this promise worth? $C $C $C $C $C $C PV ??? Yr2 Yr3 Yr4 Yr5 Time=infinity Yr1 FIN 819: Lecture 2**Perpetuities (continue)**• Calculating the PV of the perpetuity could be hard FIN 819: Lecture 2**Perpetuities (continue)**• To calculate the PV of perpetuities, we can have some math exercise as follows: FIN 819: Lecture 2**Perpetuities (continue)**• Calculating the PV of the perpetuity could also be easy if you ask George FIN 819: Lecture 2**Calculate the PV of a perpetuity**• Consider the perpetuity of one dollar every period your friend promises to pay you. The interest rate or discount rate is 8.5%. • Then PV =1/0.085=$11.765, not a big gift. FIN 819: Lecture 2**Perpetuity (continue)**• What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ? $C $C $C $C $C $C Yr0 t+2 t+3 t+4 T+5 Time=t+inf t+1 FIN 819: Lecture 2**Perpetuity (continue)**• What is the PV of a perpetuity of paying $C every year, starting from year t +1, with a constant discount rate of r ? FIN 819: Lecture 2**Perpetuity (alternative method)**• What is the PV of a perpetuity that pays $C every year, starting in year t+1, at constant discount rate “r”? • Alternative method: we can think of PV of a perpetuity starting year t+1. The normal formula gives us the value AS OF year “t”. We then need to discount this value to account for periods “1 to t” • That is FIN 819: Lecture 2**Annuities**• Well, a project might not pay you forever. Instead, consider a project that promises to pay you $C every year, for the next “T” years. This is called an annuity. • Can you think of examples of annuities in the real world? $C $C $C $C $C $C PV ??? Yr2 Yr3 Yr4 Yr5 Time=T Yr1 FIN 819: Lecture 2**Value the annuity**• Think of it as the difference between two perpetuities • add the value of a perpetuity starting in yr 1 • subtract the value of perpetuity starting in yr T+1 FIN 819: Lecture 2**Example for annuities**• you win the million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won (in PV-terms) ? FIN 819: Lecture 2**My solution**• Using the formula for the annuity FIN 819: Lecture 2**Lottery example**• Paper reports: Today’s JACKPOT = $20mm !! • paid in 20 annual equal installments. • payment are tax-free. • odds of winning the lottery is 13mm:1 • Should you invest $1 for a ticket? • assume the risk-adjusted discount rate is 8% FIN 819: Lecture 2

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