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Time Value of Money

Time Value of Money. Present value of future cash flows or payments. This week’s plan. Some administrative issues Understand the concept of the time value of money Learn how to compare: Cash flows or payments you get today Cash flows or payments you get in the future

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Time Value of Money

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  1. Time Value of Money Present value of future cash flows or payments Fin 351: lectures 3-4

  2. This week’s plan • Some administrative issues • Understand the concept of the time value of money • Learn how to compare: • Cash flows or payments you get today • Cash flows or payments you get in the future • • Understand the following terms: • present value (PV) • discount rate (r) • net present value (NPV) • annuity • perpetuity Fin 351: lectures 3-4

  3. Today’s plan (2) • 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 Fin 351: lectures 3-4

  4. 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 351: lectures 3-4

  5. 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 1,637,600 • Therefore TRL 1bn = USD 610.64 Fin 351: lectures 3-4

  6. 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 Fin 351: lectures 3-4

  7. 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 Fin 351: lectures 3-4

  8. 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. Fin 351: lectures 3-4

  9. Present value (2) • The formula for converting future cash flows or payments: = present value at time zero = cash flow in the future (in year i) = discount rate for the cash flow in year i Fin 351: lectures 3-4

  10. 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 ! PV<?> $100 • Step 3: PV=100/(1.07)1 = $100 PV=? Year one Fin 351: lectures 3-4

  11. 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 ! PV<?> $ • Step 3: PV=100/(1.07)5 = $100 PV=? Year 5 Fin 351: lectures 3-4

  12. 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<?> $ • Step 3: PV=100/(1.07)20 = $100 Year 20 PV=? Fin 351: lectures 3-4

  13. 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. Fin 351: lectures 3-4

  14. Example 4 • John is given the following set of cash flows and discount rates. What is the PV? • Step 1: draw the cash flow diagram • Step 2: think ! PV<?> $200 • Step 3: PV=100/(1.1)1 + 100/(1.09)2 = $100 $100 PV=? Year one Year two Fin 351: lectures 3-4

  15. Example 5 • John is given the following set of cash flows and discount rates. What is the PV? • 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 = $100 $200 $50 PV=? Yr 1 Yr 2 Yr 3 Fin 351: lectures 3-4

  16. 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 351: lectures 3-4

  17. 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 351: lectures 3-4

  18. 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 351: lectures 3-4

  19. 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 351: lectures 3-4

  20. NPV rule • If NPV > 0, the manager should go ahead to take the project; otherwise, the manager should not. Fin 351: lectures 3-4

  21. Example 6 • Given the data for project A, what is the NPV? • Step1: draw the cash flow graph • Step 2: think! NPV<?>10 • Step 3: NPV=-50+50/(1.075)+10/(1.08)2 = $50 $10 -$50 Yr 1 Yr 2 Yr 0 Fin 351: lectures 3-4

  22. Example 7 • 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 351: lectures 3-4

  23. 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 351: lectures 3-4

  24. Example 7 (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 351: lectures 3-4

  25. 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 351: lectures 3-4

  26. 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 351: lectures 3-4

  27. 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? $1 $1 $1 $1 $1 $1 PV ??? Yr2 Yr3 Yr4 Yr5 Time=infinity Yr1 Fin 351: lectures 3-4

  28. Perpetuities (continue) • Calculating the PV of the perpetuity could be hard Fin 351: lectures 3-4

  29. Perpetuities (continue) • To calculate the PV of perpetuities, we can have some math exercise as follows: Fin 351: lectures 3-4

  30. Perpetuities (continue) • Calculating the PV of the perpetuity could also be easy if you ask George Fin 351: lectures 3-4

  31. 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. Fin 351: lectures 3-4

  32. 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 ? $1 $1 $1 $1 $1 $1 Yr0 t+2 t+3 t+4 T+5 Time=t+inf t+1 Fin 351: lectures 3-4

  33. 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 351: lectures 3-4

  34. 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 351: lectures 3-4

  35. 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? $1 $1 $1 $1 $1 $1 PV ??? Yr2 Yr3 Yr4 Yr5 Time=T Yr1 Fin 351: lectures 3-4

  36. 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 351: lectures 3-4

  37. 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 351: lectures 3-4

  38. My solution • Using the formula for the annuity Fin 351: lectures 3-4

  39. 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? Fin 351: lectures 3-4

  40. Solution Fin 351: lectures 3-4

  41. 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 351: lectures 3-4

  42. My solution • Should you invest ? • Step1: calculate the PV • Step 2: get the expectation of the PV • Pass up this this wonderful opportunity Fin 351: lectures 3-4

  43. 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) Fin 351: lectures 3-4

  44. 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? Fin 351: lectures 3-4

  45. My solution Ending balance Total payment Interest payment Principle payment year Beginning balance 0 $20,000 $1,500 $6,191 $7,691 $13,809 1 7,154 13,809 1,036 6,655 7,691 2 7,154 7,691 0 7,154 537 3 Fin 351: lectures 3-4

  46. Future value • The formula for converting the present value to future value: = present value at time zero = future value in year i = discount rate during the i years Fin 351: lectures 3-4

  47. Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%. Fin 351: lectures 3-4

  48. 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%. FYI - The value of Manhattan Island land is well below this figure. Fin 351: lectures 3-4

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