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Time value of money Some important concepts Financial management: Lecture 3 Today’s agenda Review of what we have learned in the last lecture Discuss the concept of the time value of money • present value (PV) • discount rate (r) • net present value (NPV) Learn how to draw cash flows of projects Learn how to calculate the present value of annuities Learn how to calculate the present value of perpetuities Inflation, real interest rates and nominal interest rates, and their relationship Financial management: Lecture 3 What have we learned in the last lecture Three types of business organization and their characteristics Three types of financial managers The goal of corporation and agency problem The motivation for the study of the financial market The seven functions of a financial market The cost of capital Financial management: Lecture 3 Financial choices Which would you rather receive today? • TRL 1,000,000,000 ( one billion Turkish lira ) • USD 850 ( U.S. dollars ) Both payments are absolutely guaranteed. What do we do? Financial management: Lecture 3 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: Therefore TRL 1bn = USD 627 • USD 1 = TRL 1,594,500 Financial management: Lecture 3 Financial choices with time 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 Financial management: Lecture 3 Financial choices with time (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 $100 today is different from $100 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 Financial management: Lecture 3 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 way is more important and has more applications, as to be shown in stock and bond valuations. Financial management: Lecture 3 Present value (2) The formula for converting future cash flows or payments: Ct PV0 (1 rt )t PV0 = present value at time zero Ct = cash flow in the future (in year t) = discount rate for the cash flow in year t rt Ct i Financial management: Lecture 3 Example 1 What is the present value of $100 received in one year (next year) if the discount rate is 7%? • Step 1: draw the cash flow diagram • Step 2: think ! $100 • PV<?> $100 PV=? Step 3: PV=100/(1.07)1= Year one Financial management: Lecture 3 Example 2 What is the present value of $100 received in year 5 if the discount rate is 7%? • Step 1: draw the cash flow diagram • Step 2: think ! $100 • PV<?> $ 100 PV=? Step 3: PV=100/(1.07)5 = Financial management: Lecture 3 Year 5 Example 3 What is the present value of $100 received in year 20 if the discount rate is 7%? • Step 1: draw the cash flow diagram • Step 2: think ! PV<?> $ 100 PV=? • Step 3: PV=100/(1.07)20 = Financial management: Lecture 3 $100 Year 20 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 must 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. PVt 0 (C1, C2 ) PVt 0 (C1 ) PVt 0 (C2 ) C1 (1 r1 )1 C2 (1 r2 ) 2 Financial management: Lecture 3 Example 4 John is given the following set of cash flows and discount rates. What is the PV? $100 $100 C1 100 r1 10% C2 100 r2 9% • • • PV=? Year one Step 1: draw the cash flow diagram Step 2: think ! PV<?> $200 Step 3: PV=100/(1.1)1 + 100/(1.09)2 = Financial management: Lecture 3 Year two Example 5 John is given the following set of cash flows and discount rates. What is the PV? $100 $200 $50 C1 100 r1 0.1 C2 200 C3 50 r2 0.09 • • • PV=? Yr 1 Yr 2 Yr 3 r3 0.07 Step 1: draw the cash flow diagram Step 2: think ! PV<?> $350 Step 3: PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 = Financial management: Lecture 3 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 Expected cash flows Expected cash flows as follows: in year one (probably positive) in year two (probably positive) Initial investment (negative cash flows) Financial management: Lecture 3 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). Financial management: Lecture 3 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? Financial management: Lecture 3 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. NPV C0 PV (C1 CT ) T C0 Ct t 1(1 rt ) t Financial management: Lecture 3 NPV rule If NPV > 0, the manager should go ahead to take the project; otherwise, the manager should not. Financial management: Lecture 3 Example 6 Given the data for project A, what is the NPV? $50 $10 -$50 C0 50 C1 50 r1 7.5% C2 10 r2 8.0% Yr 1 • Step1: draw the cash flow graph • Step 2: think! NPV<?>10 • Step 3: NPV=-50+50/(1.075)+10/(1.08)2 = Yr 0 Financial management: Lecture 3 Yr 2 Example 1 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 • over the next three years, there is a 50% chance of normal • over the next 3 years, there is a 30% chance of a recession • In a “normal economy” payoffs will be medium In a “recession” payoffs will be low In all three states, the discount rate is 8% over all time horizons. Tell me whether to take the project or not Financial management: Lecture 3 Cash flows diagram in each state Boom economy Normal economy -$10 m $8 m $3 m $3 m $7 m $2 m $1.5 m $1 m $0.9 m -$10 m $6 m Recession -$10 m Financial management: Lecture 3 Example 1 (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? Financial management: Lecture 3 Calculate the NPV at each economy In the boom economy, the NPV is In the average economy, the NPV is In the bust economy, the NPV is • -10+ 8/1.08 + 3/1.082 + 3/1.083=$2.36 • -10+ 7/1.08 + 2/1.082 + 1.5/1.083=-$0.613 • -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 Financial management: Lecture 3 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 Financial management: Lecture 3 Perpetuities We are going to look at the PV of a perpetuity starting one year from now. 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 in one 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 ??? Yr1 Yr2 Yr3 Yr4 Yr5 Financial management: Lecture 3 Time=infinity Perpetuities (continue) Calculating the PV of the perpetuity could be hard PV C (1 r )1 C C (1 r ) 2 1 i 1(1 r ) i Financial management: Lecture 3 C (1 r ) Perpetuities (continue) To calculate the PV of perpetuities, we can have some math exercise as follows: 1 1 1 (1 r ) S 2 S 2 3 S S 1 /(1 r ) 1 S 1 1 1 /(1 r ) r Financial management: Lecture 3 Perpetuities (continue) Calculating the PV of the perpetuity could also be easy if you ask George C C C PV (1 r )1 (1 r ) 2 (1 r ) 1 C C C. C.S i r i 1(1 r ) i 1 i Financial management: Lecture 3 Calculate the PV of the 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. Financial management: Lecture 3 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 ? PV C (1 r ) t 1 C (1 r ) C Yr0 t+1 t 2 C t+2 C C C (1 r ) C t+3 t+4 T+5 Financial management: Lecture 3 C Time=t+inf 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 ? PV C (1 r )t 1 C (1 r )t 2 C (1 r ) 1 1 1 t 1 2 (1 r ) (1 r ) (1 r ) (1 r ) C C 1 C 1 C . t i t r (1 r ) i 1(1 r ) (1 r ) (1 r )t r Financial management: Lecture 3 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” Vt C That is r PV Vt (1 r ) t C (1 r )t r Financial management: Lecture 3 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 ??? Yr1 Yr2 Yr3 Yr4 Yr5 Financial management: Lecture 3 Time=T 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 1 C C 1 PV C r (1 r )T r r (1 r )T r Financial management: Lecture 3 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) ? Financial management: Lecture 3 My solution Using the formula for the annuity 1 1 PV 50,000 * 0.07 1.07 20 * 0.07 $529,700 .71 Financial management: Lecture 3 Example You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? Financial management: Lecture 3 Solution 1 1 Lease Cost 300 48 .005 .0051 .005 Cost $12,774.10 Financial management: Lecture 3 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% Financial management: Lecture 3 My solution Should you invest ? Step1: calculate the PV 1.0mm 1.0mm 1.0mm PV 2 (1.08) (1.08) (1.08) 20 $9.818 mm Step 2: get the expectation of the PV 1 1 E[ PV ] * 9.818 mm (1 )*0 13mm 13mm $0.76 $1 Pass up this this wonderful opportunity Financial management: Lecture 3 Mortgage-style loans Suppose you take a $20,000 3-yr car loan with “mortgage style payments” • • annual payments interest rate is 7.5% “Mortgage style” loans have two main features: • • They require the borrower to make the same payment every period (in this case, every year) The are fully amortizing (the loan is completely paid off by the end of the last period) Financial management: Lecture 3 Mortgage-style loans The best way to deal with mortgage-style loans is to make a “loan amortization schedule” The schedule tells both the borrower and lender exactly: • • • what the loan balance is each period (in this case year) how much interest is due each year ? ( 7.5% ) what the total payment is each period (year) Can you use what you have learned to figure out this schedule? Financial management: Lecture 3 My solution year Beginning balance Interest payment Principle payment Total payment Ending balance 0 1 $20,000 $1,500 $6,191 $7,691 $13,809 2 13,809 1,036 6,655 7,691 7,154 3 7,154 537 7,154 7,691 Financial management: Lecture 3 0 Future value The formula for converting the present value to future value: FVt i PVt 0 (1 rt i )i PVt 0 = present value at time zero FVt i = future value in year i rt i = discount rate during the i years Ct i Financial management: Lecture 3 Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%. Financial management: Lecture 3 Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? To answer, determine $24 is worth in the year 2003, compounded at 8%. FV $24 (1 .08) $75.979 trillion 374 FYI - The value of Manhattan Island land is well below this figure. Financial management: Lecture 3 Inflation What is inflation? What is the real interest rate? What is the nominal interest rate? Financial management: Lecture 3 Inflation rule Be consistent in how you handle inflation!! Use nominal interest rates to discount nominal cash flows. Use real interest rates to discount real cash flows. You will get the same results, whether you use nominal or real figures Financial management: Lecture 3 Example You own a lease that will cost you $8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease? 1 real interest rate = 1+ nominal interest rate 1+inflation rate Financial management: Lecture 3 Inflation Example - nominal figures Year 1 2 3 4 Cash Flow 8000 8000x1.03 = 8240 8000x1.03 2 = 8487.20 8000x1.03 3 = 8741.82 PV @ 10% 8000 1.10 7272.73 8240 6809.92 1.102 8487.20 6376.56 1.103 8741.82 5970.78 1.104 $26,429.99 Financial management: Lecture 3 Inflation Example - real figures Year 1 2 3 4 Cash Flow 8000 1.03 = 7766.99 8240 = 7766.99 1.032 8487.20 = 7766.99 1.033 8741.82 = 7766.99 4 1.03 [email protected]% 7766.99 1.068 7272.73 7766.99 6809.92 1.0682 7766.99 6376.56 1.0683 7766.99 4 5970.78 1.068 = $26,429.99 Financial management: Lecture 3