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Chapter. 5. Discounted Cash Flow Valuation. Key Concepts and Skills. Be able to compute the present value of multiple cash flows Be able to compute loan payments Be able to find the interest rate on a loan Understand how loans are amortized or paid off

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  1. Chapter 5 Discounted Cash Flow Valuation

  2. Key Concepts and Skills • Be able to compute the present value of multiple cash flows • Be able to compute loan payments • Be able to find the interest rate on a loan • Understand how loans are amortized or paid off • Understand how interest rates are quoted

  3. Uneven Cash Flows • You are considering an investment that will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? • 1 N; 10 I/Y; 1000 FV; PV?  -909.09 • 2 N; 10 I/Y; 2000 FV; PV?  -1652.89 • 3 N; 10 I/Y; 3000 FV; PV?  -2253.94 • PV = 909.09 + 1652.89 + 2253.94 = 4815.93

  4. Multiple Uneven Cash Flows – TI/LW • Another way to use the financial calculator for uneven cash flows is to use the cash flow keys • Texas Instruments BA-II Plus • Clear the cash flow keys by pressing CF and then 2nd CLR Work • Press CF and enter the cash flows beginning with year 0. • You have to press the “Enter” key for each cash flow • Use the down arrow key to move to the next cash flow • The “F” is the number of times a given cash flow occurs in consecutive years • Use the NPV key to compute the present value by [ENTER]ing the interest rate for I, pressing the down arrow, and then computing NPV

  5. Uneven Cash Flows • Use CF button to enter CF’s • 2ND , CLR WORK clears the CF register • Trick: 1st CF = CF0 Enter CF 0 ↓ C01 1000 ENTER ↓ ↓ C02 2000 ENTER ↓ ↓ C03 3000 ENTER ↓ ↓ NPV, 10, ENTER, ↓ CPT NPV  $4815.93

  6. Decisions, Decisions • Your broker has an investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return, should you take the investment? • Use the CFj keys to compute the value of the investment • No  You’re paying $100; it’s worth $91 • What return would you actually make? • 2ND, CLR WORK • CFo -100 ENTER ↓, C01 40 ENTER ↓ ↓, C02 75 ENTER, IRR, CPT

  7. Quick Quiz – Part 1 • Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7% • What is the value of the cash flows today? • Here’s where the F’s come in handy… • 2ND, CLR WORK • CFo 0 ↓, C01 ENTER ↓ ↓, C02 200 ENTER ↓, F02 2 ENTER ↓, C03 300 ENTER ↓, F03 2 ENTER, NPV 7 ENTER ↓ CPT

  8. Annuities and Perpetuities Defined • Annuity – finite series of equal payments that occur at regular intervals • If the first payment occurs at the end of the period, it is called an ordinary annuity • If the first payment occurs at the beginning of the period, it is called an annuity due • Perpetuity – infinite series of equal payments

  9. Annuities and Perpetuities – Basic Formulas • Perpetuity: PV = CF / r • Annuities:

  10. Annuities and the Calculator • PMT key = annuity • Ordinary annuity versus annuity due • You can switch your calculator between the two types by using the <GOLD> BEG/END • If you see “BEGIN” in the display of your calculator, you have it set for an annuity due • Most problems are ordinary annuities

  11. Future Value of an Annuity • The FV of annuity = amount received + the interest earned from time received until the future date

  12. 0 5% 1 2 3 100 315.25 Future Value of an Annuity • If you deposit $100 at the end of each year for three years in a savings account that pays 5% interest per year, how much will you have at the end of three years? 100 100.00 = 100 (1.05)0 105.00 = 100 (1.05)1 110.25 = 100 (1.05)2

  13. Future Value of an Annuity • Financial calculator solution: • Inputs: 3 N; 5 I/Y; -100 PMT CPT FV  315.25 • To solve the same problem, but for the present value instead of the future value, change the final input from FV to PV

  14. Annuities Due • If the three $100 payments at the beginning of each year, the annuity = an annuity due. • How will your answer be different?

  15. 100 100 105.00 = 100 (1.05)1 0 5% 1 2 3 110.25 = 100 (1.05)2 115.7625 = 100 (1.05)3 100 331.0125 Future Value of an Annuity Due • $100 at the start of each year • Each payment = one year earlier earn interest for an additional year (period).

  16. Future Value of an Annuity Due • Financial calculator solution: • Switch to the beginning-of-period mode  <color> BEG/END • Inputs: 3 N; 5 I/Y; -100 PMT FV?  331.0125

  17. Present Value of an Annuity • If you require a 5% return, how much would you pay today for a three-year annuity with payments of $100 at the end of each year?

  18. 100 100 0 5% 1 2 3 100 272.325 Present Value of an Annuity

  19. Present Value of an Annuity Due • Payments at the beginning of each year • Payments all come one year sooner • Each payment would be discounted for one less year • Present value of annuity due will exceed the value of the ordinary annuity by one year’s interest on the present value of the ordinary annuity

  20. 0 5% 1 2 3 100 285.941 Present Value of Annuity Due 100 95.24 90.70 100

  21. Present Value of Annuity Due • Financial calculator solution: • Switch to the beginning-of-period mode  2ND BGN 2ND SET • Inputs: Inputs: 3 N; 5 I/Y; -100 PMT CPT PV  285.94 • Then switch back to the END mode • 2ND BGN 2ND SET

  22. More Annuities • $1 million Lotto winner!!!! • $40,000 per year for 25 years starting today • $500,000 lump sum today • Which is better? @8%? @6%? http://www.txlottery.org/faq/morequestions.cfm

  23. Lotto • $1 million Lotto winner!!!! • $40,000 per year for 25 years starting today • $500,000 lump sum today • Pre-CVO, private firms would “loan” you the lump sum • You repay it with your $40K check every year • Rate on $300K loan? • Rate on $500K “loan”?

  24. Future Values for Annuities • Suppose you begin saving for your retirement by depositing $2000 per year in an IRA. If the interest rate is 12.3%, how much will you have in 35 years? • 35 N • 12.3 I/Y • -2000 PMT • CPT FV 926,533

  25. Future Values for Annuities • Suppose you begin saving for your retirement by depositing $2000 per year in an IRA. If the interest rate is 12.3%, how much will you have in 40 years? • FV?  1,667,635 • Last slide, saving for 35 years  FV = 926,533 • $ 1,667,635 – 926,533 = $741,102 from 5 extra years of saving $2000 each year.

  26. Annuity Due • You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? • 2ND BGN 2ND SET • 3 N • -10,000 PMT • 8 I/Y • CPT FV  35,061.12

  27. Perpetuity • Perpetuity formula: PV = C / r • Preferred Stock is a perpetuity. Buyer of PS is promised a fixed cash dividend every period forever. • If PS pays a $5 yearly dividend and investors require a 12% return, what is the price?

  28. Finding the Payment • Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year. If you take a 4 year loan, what is your monthly payment? • What’s different? • Two methods…

  29. Finding the Payment • Interest rate = 8% per year, but monthly pmt • Int rate = 8%/12 = 0.6667% per month • 1 P/YR • 48 N; 20,000 PV; 0.6667 I/Y; • CPT PMT  488.26 • Or…

  30. Finding the Payment • Monthly PMT  2ND P/Y , 12 ENTER • N = # of PMT’s; I/Y still = annual rate • 2ND QUIT • 48 N; 20,000 PV; 8 I/Y; • CPT PMT  -488.26

  31. Credit Cards • You realize that you have a $5000 balance on your credit card, which is being assessed 18% yearly interest. If you cut the credit card up and make $100 payments every month on it, how long until you’ve paid it off?

  32. Finding the Rate • Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the interest rate?

  33. Finding the Rate • 2ND P/Y 1 ENTER, 2ND QUIT, 60 N, 10000 PV, -207.58 PMT • CPT I/Y  0.75% per month • 0.75% * 12 = 9.0% annually • 2ND P/Y 1 ENTER, 2ND QUIT, 60 N, 10000 PV, -207.58 PMT • CPT I/Y  9.0% per year

  34. Future Values with Monthly Compounding • Suppose you deposit $50 a month into an account that makes 9%. How much will you have in the account in 35 years?

  35. Future Values with Monthly Compounding • P/Y = 1  420 N, 0.75 I/Y, 50 PMT, CPT FV --- or --- • P/Y = 12  420 N, 9 I/Y, 50 PMT, CPT FV

  36. Quick Quiz – Part 2 • You want to have $1 million to use for retirement in 35 years. If you can earn 12% annually, how much do you need to deposit on a monthly basis if the first payment is made in one month? • What if the first payment is made today? • You are considering preferred stock that pays a yearly dividend of $6.00 and costs $75. What return does this imply?

  37. Annual Percentage Rate • This is the annual rate that is quoted by law • By definition APR = period rate (ie, 1% per month) times the number of periods per year (1% * 12 = 12%) • Also called “simple rate” or “nominal rate”

  38. Computing APRs • What is the APR if the monthly rate is .5%? • .5(12) = 6% • What is the APR if the semiannual rate is .5%? • .5(2) = 1% • What is the monthly rate if the APR is 12% with monthly compounding? • 12 / 12 = 1%

  39. Compounding Periods • Which would you rather have: 1. $100 compounded yearly at 10%? 2. $100 compounded semiannually at 10%? • Both have 10% APR • Assume 20-year investment, and find FV

  40. Compounding Periods #1: P/Y = 1 20 N 10 I/Y -100 PV CPT FV $672.75

  41. Compounding Periods #2: 2ND P/Y 2 ENTER, 2ND QUIT 40 N  N = #periods  40 6-month periods 10 I/YR (Still use yearly rate) -100 PV CPT FV $704.00 Versus $672.75 for yearly compounding

  42. Interest Rates • APR = Simple (Quoted) Interest Rate • rate used to compute the interest payment paid per period • Effective Annual Rate (EAR) • annual rate of interest actually being earned, considering the compounding of interest

  43. Interest Rates • With annual compounding: • APR = Effective Rate • With semiannual/monthly compounding: • APR < Effective Rate

  44. Interest Rates • Effective annual rate on TIBA/LW 2ND ICONV 10 ENTER  NOM% = 10 ↑ 2 ENTER  2 C/Y ↑ CPT  EFF% = 10.25 • If compounded monthly, what’s the effective annual rate on 10%?

  45. Decisions, Decisions II • You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?

  46. Decisions, Decisions II Continued • To verify… Suppose you invest $100 in each account. How much will you have in each account in one year? • First Account: • 2ND P/Y 365 ENTER 2ND QUIT, 365 N; 5.25 I/Y; 100 PV • CPT FV  105.39 • Second Account: • 2ND P/Y 2 ENTER 2ND QUIT, 2 N; 5.3 I/YR; 100 PV • CPT FV  105.37

  47. APR - Example • Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?

  48. Amortized Loan --Home Mortgage 30-year, $100K loan @ 7%  Compute PMT 2ND P/Y 12 ENTER 2ND QUIT 30 yrs * 12 months = 360 payments  360 N 7 I/Y $100,000 PV (0 FV) Hit CPT PMT to compute PMT = $665.30

  49. Home Mortgage Each payment = $665.30 = principal + interest 1st payment: 2ND AMORT P1=1 ENTER ↓ P2=1 ENTER ↓ ↓ ↓ $665.30 = $81.97 principal + $583.33 interest New loan balance = $100K - $81.97 = $99,918

  50. Home Mortgage Each payment = $665.30 = principal + interest 2nd payment: 2ND AMORT P1=2 ENTER ↓ P2=2 ENTER ↓ ↓ ↓ $665.30 = $82.45 principal + $582.85 interest New loan balance = $99,918 - $82.45 = $99,836

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