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Chapter 5 - Discounted Cash Flows

Chapter 5 - Discounted Cash Flows. Annuities. Elements: Multiple Cash flows Identical (or constant) Cash Flows Finite (for a fixed number of periods) Types of annuities Ordinary Annuities – Cash flows occur at the end of a period

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Chapter 5 - Discounted Cash Flows

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  1. Chapter 5 - Discounted Cash Flows

  2. Annuities • Elements: • Multiple Cash flows • Identical (or constant) Cash Flows • Finite (for a fixed number of periods) • Types of annuities • Ordinary Annuities – Cash flows occur at the end of a period • Annuities Due – Cash flows occur at the beginning of a period.

  3. Annuities • We deal with constant multiple finite cash flows frequently. • Examples: • Car loans • Mortgages • Trust fund pmts

  4. Annuities • Example: What is the future value of a 3 year ordinary annuity of $100 @ 10%? r = 10% 3 1 2 0 $100 $100 $100

  5. Annuities 100 100 100 r= 10% 3 0 2 1 FV = ? CF2(1 + r)1 CF1(1 + r)2

  6. Annuities Using a financial Calculator • Important: • Set to “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 0, then press PV • Enter 100, then press pmt • Enter 10, then press i • Enter 3, then press n • Press FV and viola FV = ____331.00 “Time Value of Money” Menu n i PV pmt FV

  7. Annuities • Example: What is the Present Value of a 3 year ordinary annuity of $100 @ 10%? r = 10% 3 1 2 0 $100 $100 $100

  8. Annuities 100/(1+i) 100 100 100 r= 10% 3 0 2 1 100/(1+i)1 100/(1+i)2 100/(1+i)3

  9. Annuities Using a financial Calculator • Important: • Set to “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 0, then press FV • Enter 100, then press pmt • Enter 10, then press i • Enter 3, then press n • Press PV and viola PV = ____-$248.68 “Time Value of Money” Menu n i PV pmt FV

  10. Annuities • What is the Future Value of a 3 year annuity due of $100 at 10% r = 10% 3 0 $100 $100 $100

  11. Annuities 100 100 100 r= 10% 3 0 2 1 FV = ? CF2(1 + r)1 CF1(1 + r)2 CF1(1 + r)3

  12. Annuities Using a financial Calculator • Important: • Set To “Begin” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 0, then press PV • Enter 100, then press pmt • Enter 10, then press i • Enter 3, then press n • Press FV and viola FV = ____-$364.10 “Time Value of Money” Menu n i PV pmt FV

  13. Annuities • What is the Present Value of a 3 year annuity due of $100 at 10% r = 10% 3 0 $100 $100 $100

  14. Annuities 100/(1+i) 100 100 100 r= 10% 3 0 2 1 100/(1+i)1 100/(1+i)2

  15. Annuities Using a financial Calculator • Important: • Set To “Begin” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 0, then press FV • Enter 100, then press pmt • Enter 10, then press i • Enter 3, then press n • Press PV and viola PV = ____-$273.55 “Time Value of Money” Menu n i PV pmt FV

  16. Annuities • Example: You need $1 million in exactly 40 years to retire. If you can earn 10% per year, how much do you have to invest each year, assuming you make your first payment in exactly one year? • What type of annuity is this?

  17. Annuities Using a financial Calculator • Important: • Set To “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 0, then press PV • Enter $1m, then press FV • Enter 10, then press i • Enter 40, then press n • Press pmt and viola pmt = ____-$2259.41 “Time Value of Money” Menu n i PV pmt FV

  18. Annuities • Example: You need $1,000 today to buy a PC. IF you can get a loan at 10% per year for 3 years, how much will your annual payments be, assuming you make your 1st payment in one year? • What type of annuity is this?

  19. Annuities Using a financial Calculator • Important: • Set To “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter 1,000, then press PV • Enter 0, then press FV • Enter 10, then press i • Enter 3, then press n • Press pmt and viola pmt = ____-$402.11 “Time Value of Money” Menu n i PV pmt FV

  20. Annuities • Example: You pay $864.80 for an investment that promises to pay you $250 per year for the next 4 years, with payments made at the end of the year. What interest rate will you earn on this investment? • What type of annuity is this?

  21. Annuities Using a financial Calculator • Important: • Set To “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter -$864.80, then press PV • Enter 0, then press FV • Enter $250, then press pmt • Enter 4, then press n • Press i and viola i = ____6.07 “Time Value of Money” Menu n i PV pmt FV

  22. Perpetuities • Elements of a Perpetuity • Muliple Cash Flows • Identical (or constant) Cash Flows • Infinite – Cash Flows continue forever

  23. Perpetuities • Example: What is the present value (or market price) of each share of a preferred stock that pays out $4 per year if the opportunity cost rate is 8%?

  24. Perpetuities PV = payment / Interest Rate = pmt / i

  25. Uneven cash flows • Payment (pmt) – designates constant cash flows • Cash flow (CF) – designates cash flows in general, including uneven cash flows

  26. General Pricing equations: • PV = CF1[1/(1+r)1] + CF2[1/(1+r)2]+ …+ CFn[1/(1+r)n] • FV = CF1(1+r)n-1 + CF2(1+r)n-2 + …+ CFn(1+r)n

  27. Uneven Cash Flow • Example: What is the PV of this uneven cash flow stream? Assuming the opportunity cost rate is 10%? timeline: r = 10% 5 0 $100 $300 $300 $-50 PV = $?

  28. Uneven Cash Flow • Write an Equation for PV that you could solve using any calculator. • Financial Calculator Solution: • Input in Cash Flow register • CF0 = 0 • CF1 = 100 • CF2 = 300 • CF3 = 300 • CF4 = -50 • Then enter 10% for Interest rate, and press NPV

  29. Uneven Cash Flow • In the last example, would PV be larger or smaller if the opportunity cost rate had been bigger? • Why? • Check your answer using a bigger opportunity cost, say 20% PV @ 20% = ___________

  30. Uneven Cash Flow • In the last example, would PV be larger or smaller if the opportunity cost rate had been smaller? • Why? • Check your answer using a bigger opportunity cost, say 5% PV @ 5% = ___________

  31. Complications: Semi-annual and other compounding periods • Example: Calculate the FV of $100 invested for 2 years at 8% if interest is compounded annually, semiannually and monthly. • Annually: r = 8% 1 2 0 $100 FV = ?

  32. Complications: Semi-annual and other compounding periods On the financial calculator: n = i = PV = Pmt = FV =

  33. Complications: Semi-annual and other compounding periods • Semiannually: r = 8% 1 1/2 1 1/2 2 0 $100 FV = ?

  34. Complications: Semi-annual and other compounding periods On the financial calculator: n = i = PV = Pmt = FV =

  35. Complications: Semi-annual and other compounding periods • Monthly: n = i = PV = Pmt = FV =

  36. Complications: Semi-annual and other compounding periods • Note: • Simple (or quoted) interest rate is often what people quote as your interest rate for loans, bank accounts, credit cards and bonds. In the last example, the simple or quoted interest rate was 8% • Effective annual rate (or ERA) interest rate is the annual rate actually being earned or paid, when compounding is considered.

  37. Complications: Semi-annual and other compounding periods Effective Annual Rate (EAR) = (1+(isimple/n)n - 1

  38. Complications: Semi-annual and other compounding periods (n is the number of times interest ins compounded each year) • What was the EQAR for each compounding frequency in the last Example • Annual: EAR = • Semiannual: EAR = • Monthly: EAR =

  39. Fractional Time Periods • Example: Calculate the PV of $100 invested for half a year in a bank account that pays an EAR of 8%

  40. Fractional Time Periods • Example: Calculate the FV of $100 invested for half a year in a bank account that pays an EAR of 8%

  41. Fractional Time Periods • Example: Calculate the PV of $100 invested for 9 months in a bank account that pays an EAR of 8%

  42. Fractional Time Periods • Example: Calculate the FV of $100 invested for 9 months in a bank account that pays an EAR of 8%

  43. Amortized Loans • These are loans that are repaid in a series of equal payments. • Example: You need $1,000 today to buy a PC. IF you can get a loan at 10% per year for 3 years, how much will you annual payments be, assuming the first payment is in exactly one year. • What type of annuity is this?

  44. Amortized Loans • Draw a timeline Loan Amortization Schedule Year BegBal Pmt Int Prin EndBal 1 2 3 Total

  45. Comparison of Different types of Interest Rates • Simple (or quoted) interest rate is often what people quote as your interest rate for loans, bank accounts, credit cards and bonds. • Effective Annual rate (or EAR) interest rate is the annual rate actually being earned or paid, when compounding is considered. • Periodic rate is the rate collected or paid each period. • Annual percentage rate (APR) is the rate reported (as required by law) to borrowers - it is the periodic rate times the number of periods in the year. So the actual effects of compounding are not considered. (Same as the simple rate)

  46. Comparison of Different types of Interest Rates • Periodic Rate = isimple / n • isimple = Periodic rate x n • Effective Annual Rate (EAR) = (1+(isimple/n)n - 1

  47. Comparison of Different types of Interest Rates • Example: Suppose your Credit Card has an APR of 18%. What is your effective annual interest rate? (Interest is compounded once per month)

  48. Comparison of Different types of Interest Rates • Example: Suppose you have a house mortgage at 7.5%. What is your effective annual interest rate? (you make payments monthly)

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