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Time Value of Money

Time Value of Money. One Period. Interest Rate (per period). 5%. Time:. 0. 1. 2. 3. 4. ?. CF:. -100. + indicates Cash Inflow - indicates Cash Outflow. Cash Flow Time Lines. CF Time Lines are a graphical representation of cash flows associated with a particular financial option.

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Time Value of Money

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  1. Time Value of Money Besley Ch. 6

  2. One Period Interest Rate (per period) 5% Time: 0 1 2 3 4 ? CF: -100 + indicates Cash Inflow - indicates Cash Outflow Cash Flow Time Lines CF Time Lines are a graphical representation of cash flows associated with a particular financial option. NOTE: Each tick mark denotes the end of one period. Besley Ch. 6

  3. Cash Flow Time Lines Outflow: A payment or disbursement of cash, such as for investment, or expenses. Inflow: A receipt of cash, can be in the form of dividends, principal, annuity payments, etc. Besley Ch. 6

  4. 5% 0 1 2 3 ? -100 Future Value (FV) Future Value (FV): The ending value of a cash flow (or series of cash flows) over a given period of time, when compounded for a specified interest rate. Compounding: The process of calculating the amount of interest earned on interest. Present Value (PV) FV Besley Ch. 6

  5. 5% 0 1 2 3 ? -100 FV Calculations Given: PV: $100 i: 5% n: 1 INT: (PV x i) Solution: FVn = PV+INT = PV + (PV x i) = PV(1+i) Solution: FVn = 100+INT = 100 + (100 x 5%) = 100(1+ 0.05) = 105 Besley Ch. 6

  6. FV Calculations 5% 0 1 2 3 FV1 = PV(1+i) FV2 = FV1(1+i) = [PV(1+i)](1+i) FV3 = FV2(1+i) = {[PV(1+i)](1+i)}(1+i) FVn = PV(1+i)n ? -100 INT2 INT1 INT3 5.00 5.25 5.51 S=15.76 Value at end of Period: 110.25 115.76 105.00 Besley Ch. 6

  7. FV Calculations Three ways to calculate Time Value of Money (TVM) solutions: • Numerical Solution:Calculate solution with formula • Tabular Solutions:Use Interest Factor tables to calculate • Financial Calculator Solutions:Use calculator Besley Ch. 6

  8. Numerical Solution Future Value Interest Factor for i and n (FVIFi,n) is the factor by which the principal grows over a specified time period (n) and rate (i). FVIFi,n = (1+i)n Given: Solution: PV: $1 FVn = PV(1+i)n = PV(FVIFi.n) i: 5% FV5 = 1(1+.05)5 n: 5 FV5 = 1.2763 Besley Ch. 6

  9. Tabular Solution • Given: • i: 5% • n: 5 FVn = PV(1+i)n = PV(FVIFi.n) FVIFi,n = (1 + i)n Besley Ch. 6

  10. Input: 5 5 -1 0 N I/Y PV PMT FV Output: 1.2763 Financial Calculator Points to remember when using your Financial Calculator: • Check your settings: • END / BGN • P/Y • Clear TVM memory • Five Variables (N, I/Y, PV, PMT, FV) - with any 4 the 5th can be calculated Given: N: 5 I/Y: 5% PV: $1 PMT:0 FV: ? Besley Ch. 6

  11. Present Value (PV) 5% 0 1 2 3 Present Value (PV): The current value of a future cash flow (or series of cash flows), when discounted for a specified period of time an rate. Discounting: The process of calculating the present value of a future cash flow or series of cash flows. ? 105 PV FV Besley Ch. 6

  12. PV Calculations 5% 0 1 2 3 Given: FV: 105 i: 5% n: 1 Solution: FVn = PV(1+i)n Solve for PV PVn = FVn / (1+i)n = FVn[1/(1+i)n] = FVn(PVIFi,n) ? 105 Solution: PVn = 105/(1+0.05)1 = 100 Besley Ch. 6

  13. 1 (1+i)n PV Calculations Given FV 5% 0 1 2 3 FV2 = FV3/(1+i) FV1 = FV2/(1+i) = [FV3/(1+i)]/(1+i) PV = FV1/(1+i) = {[FV1/(1+i)]/(1+i)}/(1+i) PVn = FVn 105.00 110.25 115.7625 Value at end of Period: ? • 1.05 • 1.05 • 1.05 Besley Ch. 6

  14. Numerical Solution Present Value Interest Factor for i and n (PVIFi,n) is the discount factor applied to the FV in order to calculate the present value for a specific time period (n) and rate (i). PVIFi,n = 1/(1+i)n Given: Solution: FV: $1 PVn = FV[1/(1+i)n] = FV(PVIFi.n) i: 5% PV5 = 1 [1/(1+.05)5] n: 5 PV5 = 0.7835 Besley Ch. 6

  15. Tabular Solution • Given: • i: 5% • n: 5 PVn = FV[1/(1+i)n] = FV(PVIFi.n) PVIFi,n = (1 + i)n Besley Ch. 6

  16. Financial Calculator Given: N: 5 I/Y: 5% PV: ? PMT:0 FV: -1 Input: -1 5 5 0 N I/Y PV PMT FV Output: .7835 Besley Ch. 6

  17. Annuities Annuity: a series of equal payments made at specific intervals for a specified period. Types of Annuities: • Ordinary (Deferred) Annuity - is an annuity in which the payments occur at the end of each period. • Annuity Due - is an annuity in which the payments occur at the beginning of each period. Besley Ch. 6

  18. 8% 0 1 2 3 4 1,000 1,000 1,000.00 1,000 1,080.00 1,166.40 1,259.71 S 4,506.11 FV Ordinary Annuities Example:You decide that starting a year from now you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years? FVn=PV(1+i)n FVAn = PMT(1+i)0 + PMT(1+i)1 + PMT(1+i)2 + . . . + PMT(1+i)n-1 Besley Ch. 6

  19. n n n S S S t=1 t=1 t=1 (1+i)n - 1 i FV Ordinary Annuities FVAnrepresents the future value of an ordinary annuity over n periods. FVAn = PMT(1+i)0 + PMT(1+i)1 + PMT(1+i)2 + . . . + PMT(1+i)n-1 = PMT (1+i)n-t = PMT (1+i)t = PMT (1+i)n-1 = PMT Besley Ch. 6

  20. FVIFAi,n = (1+i)n-t = n S t=1 (1+i)n - 1 i FV Ordinary Annuities Future Value Interest Factor for an Annuity (FVIFAi,n) is the future value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent. Besley Ch. 6

  21. 8% 0 1 2 3 4 1,000 1,000 1,000.00 1,000 FVAn = PMT (1+i)n - 1 i Numerical Solution Given: PMT: $1,000 I: 8% N: 4 Solution: FVAn = PMT {[(1+i)n – 1]/i} = 1,000 {[(1+0.08)4 – 1]/0.08] = 1,000 {4.5061} = $4,506.11 Besley Ch. 6

  22. Tabular Solution Given: PMT: $1,000 I: 8% N: 4 FVAn = PMT(FVIFAi,n) PVIFAi,n Besley Ch. 6

  23. Financial Calculator Given: N: 4 I/Y: 8% PV: 0 PMT:1,000 FV: ? Input: 0 4 8 1,000 N I/Y PV PMT FV Output: 4,506.11 Besley Ch. 6

  24. 0 1 2 3 4 n n S S t=1 t=1 FVA(Due)n =PMT (1+i)t = PMT (1+i)n-t x (1+i) (1+i)n - 1 FVA(Due)n = PMT x (1+i) i FV Annuity Due Example:You decide that starting today you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years? 8% 1,000 1,000 1,000 1,080.00 1,000 FVn=PV(1+i)n 1,166.40 1,259.71 1,360.49 4,866.60 S Besley Ch. 6

  25. = PMT (1+i)n-t x (1+i) n n S S t=1 t=1 (1+i)n - 1 = PMT x (1+i) i FV Annuity Due FVA(Due)nrepresents the future value of an annuity due over n periods. FVA(Due)n = PMT (1+i)t Besley Ch. 6

  26. (1+i)n - 1 FVIFA(Due)i,n = x (1+i) i FV Annuity Due Future Value Interest Factor for an Annuity Due (FVIFA(Due)i,n) is the future value interest factor for an annuity due of n periods compounded at i percent. Besley Ch. 6

  27. 0 1 2 3 4 FVA(Due)i,n (1+i)n - 1 = PMT x (1+i) i Numerical Solution 8% Given: PMT: $1,000 - BGN I: 8% N: 4 1,000 1,000 1,000 1,000 Solution: FVA(Due)n = PMT [{((1+i)n – 1)/i}x (1+i)} = 1,000 [{((1+0.08)4 – 1)/0.08}x (1+0.08)} = 1,000 {4.8666} = $4,866.60 Besley Ch. 6

  28. Tabular Solution Given: PMT: $1,000 - BGN I: 8% N: 4 FVA(Due)n = PMT[(FVIFAi,n)(1+i)] PVIFAi,n Besley Ch. 6

  29. Financial Calculator Given: N: 4 I/Y: 8% PV: 0 PMT:1,000 - BGN FV: ? BGN Input: 0 4 8 1,000 N I/Y PV PMT FV Output: -4,866.60 Besley Ch. 6

  30. 8% 0 1 2 3 4 1,000 1,000 1,000 1,000 (925.93) (857.34) (793.83) (735.03) (3,312.13) PV Ordinary Annuities Example:You decide that starting a year from now you will withdraw $1,000 each year for the next 4 years from a savings account which earns 8% interest per year. How much do you need to deposit today? The present value of an annuity is calculated by adding the PV of the individually discounted/compounded cash flows. PVAn = PMT[1/(1+i)1] + PMT[1/(1+i)2] + . . . + PMT[1/(1+i)n] Besley Ch. 6

  31. 1 n S t=1 1 1 - (1+i)n i PV Ordinary Annuities PVAnrepresents the present value of an ordinary annuity over n periods. PVAn = PMT[1/(1+i)1]+ PMT[1/(1+i)2]+ . . . + PMT[1/(1+i)n] = PMT (1+i)t = PMT Besley Ch. 6

  32. 1 1 - (1+i)n PVIFAi,n = i PV Ordinary Annuities Present Value Interest Factor for an Annuity (PVIFAi,n) is the present value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent. Besley Ch. 6

  33. 8% 0 1 2 1 3 4 1,000 1,000 1,000 1,000 1 - (1+i)n PVAn = PMT i Numerical Solution Given: PMT: $1,000 I: 8% N: 4 Solution: PVAn = PMT {1-[1/(1+i)n]/i} = 1,000 {1-[1/(1+0.08)4]/0,08} = 1,000 {3.3121} = $3,312.13 Besley Ch. 6

  34. Given: PMT: $1,000 I: 8% N: 4 PVAn = PMT(PVIFAi,n) Tabular Solution PVIFAi,n Besley Ch. 6

  35. Financial Calculator Given: N: 4 I/Y: 8% PV: ? PMT:1,000 FV: 0 Input: 0 4 8 1,000 N I/Y PV PMT FV Output: -3,3121.13 Besley Ch. 6

  36. 8% 0 1 2 3 4 (1,000.00) 1,000 1,000 1,000 (925.93) (857.34) (793.83) (3,577.10) PV Annuity Due Example:You decide that starting today you will withdraw $1,000 each year for the next four years from a savings account earning 8% interest per year. How much do you need today? Besley Ch. 6

  37. 1 n-1 S PVA(Due)n = PMT (1+i)t t=0 n 1 S = PMT x (1+i) (1+i)t t=1 1 1 - (1+i)n = PMT x (1+i) i PV Annuity Due PVA(Due)nrepresents the future value of an annuity due over n periods. Besley Ch. 6

  38. 1 1 - (1+i)n PVIFA(Due)i,n= PMTx (1+i) i PV Annuity Due Present Value Interest Factor for an Annuity Due (PVIFA(Due)i,n) is the present value interest factor for an annuity due of n periods compounded at i percent. Besley Ch. 6

  39. 0 1 2 3 4 1 1 - (1+i)n PVIFA(Due)i,n= PMT x (1+i) i Numerical Solution 8% Given: PMT: $1,000 - BGN I: 8% N: 4 1,000 1,000 1,000 1,000 Solution: PVA(Due)n = PMT [{(1-1/(1+i)n)/i}x (1+i)] = 1,000 [{(1-1/(1+0.08)4)/0.08}x (1+0.08)] = 1,000 {3.5771} = $3,577.10 Besley Ch. 6

  40. Tabular Solution Given: PMT: $1,000 - BGN I: 8% N: 4 PVA(Due)n = PMT[(PVIFAi,n)(1+i)] PVIFAi,n Besley Ch. 6

  41. Financial Calculator Given: N: 4 I/Y: 8% PV: ? PMT:1,000 - BGN FV: 0 BGN Input: 0 4 8 1,000 N I/Y PV PMT FV Output: -3,577.10 Besley Ch. 6

  42. ?% 1,000 1,000 1,000 -3,239.72 1,000 0 1 2 3 4 1 1 - (1+i)n PVIFAi,n = i Solving for Interest Rates with Annuities PVAn=PMT(PVIFAi,n) -3,239.72 = 1,000(PVIFAi,n) -3.2397 = PVIFAi,n Numerical Solution: Trial & Error  Solve for PVIFA Besley Ch. 6

  43. ?% 1,000 1,000 1,000 -3,239.72 1,000 0 1 2 3 4 Solving for Interest Rates with Annuities Tabular Solution: -3.2397 = PVIFAi,n PVIFAi,n Besley Ch. 6

  44. ?% 1,000 1,000 1,000 -3,239.72 1,000 0 1 2 3 4 Solving for Interest Rates with Annuities Financial Calculator: N: 4 I/Y: ? PV: -3,239.72 PMT:1,000 FV: 0 Input: 0 4 -3, 239.72 1,000 N I/Y PV PMT FV Output: 9 Besley Ch. 6

  45. Perpetuities Perpetuity: A perpetual annuity, an annuity which continues forever. ConsolA perpetual bond issued by the British government where the proceeds were used to consolidate past debts. PVP = PMT / i Besley Ch. 6

  46. Perpetuities PVA5%,100 = $19,848 Besley Ch. 6

  47. 0 1 2 3 4 Uneven Cash Flow Streams 8% 1,000 750 750 250 (231.48) (643.00) (595.37) (735.03) PVn = FV[1/(1+i)n] = FV(PVIFi.n) (2,204.88) Besley Ch. 6

  48. Semiannual and Other Compounding Periods Simple Interest Rate: The interest rate used to compute the interest rate per period; the quoted interest rate is always in annual terms. Effective Annual Rate (EAR): The actual interest rate being earned during a year when compounded interest is considered. Besley Ch. 6

  49. Semiannual and Other Compounding Periods Types of Compounding: • Annual Compounding • Semiannual Compounding (Bonds) • Quarterly (Stock Dividends) • Daily (Bank Accounts/Credit Cards) EAR Formula isimple m EAR = 1+ -1 m Besley Ch. 6

  50. Semiannual and Other Compounding Periods Annual Percentage Rate (APR): the periodic rate multiplied by the number of period per year. Besley Ch. 6

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