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# Time volue of money

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### Time volue of money

1. 1. 1 Chapter 3 Time Value of Money
2. 2. 2 The Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan
3. 3. 3 The Interest Rate Which would you prefer -- \$10,000 today or \$10,000 in 5 years? years Interest is the money paid for the use of money. Obviously, \$10,000 today. today You already recognize that there is TIME VALUE TO MONEY!! MONEY
4. 4. 4 Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. INTEREST
5. 5. 5 Types of Interest Simple Interest Interest paid (earned) on only the original amount, or principal borrowed (lend). Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).
6. 6. 6 Simple Interest Formula Formula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods
7. 7. 7 Simple Interest Example Assume that you deposit \$1,00 in an account earning 8% simple interest for 10 years. What is the accumulated interest at the end of the 10nd year? SI = P0(i)(n) = \$1,00(.08)(10) = \$80
8. 8. 8 Simple Interest (FV) What is the Future Value (FV) of the FV deposit? FV = P0 + SI = \$1,00 + \$80 = \$180 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
9. 9. 9 Simple Interest (PV) What is the Present Value (PV) of the PV previous problem? The Present Value is simply the \$1,00 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.
10. 10. 10 Compound interest Interest that is earned on a given deposit and has become part of principal at the end of a specified period Future value of a present amount at a future date, found by applying compound interest over a specified period of time.
11. 11. 11 The equation for future value FV=PV*(1+i)n If Fred places \$100 in a savings account paying 8% interest compounded annually, at the end of 1 year he will have \$108 in the account.<100*(1.08)=\$108> If Fred were to leave this money in the account for another year, he would be paid interest at the rate of 8% on the new principal of \$108.At the end of this second year there would be \$116.64 in the account.<108*(1.08)=116.64> or <100*(1.08)2=116.64>
12. 12. 12 General Future Value Formula FV1 = P0(1+i)1 FV2 = P0(1+i)2 etc. General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) -- See Table I
13. 13. 13 Valuation Using Table I FVIFi,n is found on Table I at the end of the book or on the card insert. Period 1 2 3 4 5 6% 1.060 1.124 1.191 1.262 1.338 7% 1.070 1.145 1.225 1.311 1.403 8% 1.080 1.166 1.260 1.360 1.469
14. 14. 14 Using Future Value Tables FV2 = \$1,000 (FVIF7%,2) = \$1,000 (1.145) = \$1,145 [Due to Rounding] Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469
15. 15. 15 Story Problem Example Julie Miller wants to know how large her deposit of \$10,000 today will become at a compound annual interest rate of 10% for 5 years. years 0 10% 1 2 3 4 5 \$10,000 FV5
16. 16. 16 Story Problem Solution Calculation based on general formula: FVn = P0 (1+i)n FV5 = \$10,000 (1+ 0.10)5 = \$16,105.10 Calculation based on Table I: FV5 = \$10,000 (FVIF10%, 5) = \$10,000 (1.611) = \$16,110 [Due to Rounding]
17. 17. 17 Present value of a single amount The current dollar value of a future amount-the amount of money that would have to be invested today at a given interest rate over a specified period to equal the future amount.
18. 18. 18 Concept of present value The process of finding present value is often referred to as discounting cash flows. It is concerned with answering the following question:" if I can earn i percent on my money, what is the most I would be willing to pay now for an opportunity to receive FV n dollars n periods from today?”
19. 19. 19 Equation for calculating PV PV*(1+i)n=FV PV=FV/(1+i)n
20. 20. 20 Present Value Single Deposit (Graphic) Assume that you need \$1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 7% 1 2 \$1,000 PV0 PV1
21. 21. 21 Present Value Single Deposit (Formula) PV0 = FV2 / (1+i)2 = \$1,000 / (1.07)2 = FV2 / (1+i)2 = \$873.44 0 7% 1 2 \$1,000 PV0
22. 22. 22 General Present Value Formula PV0 = FV1 / (1+i)1 PV0 = FV2 / (1+i)2 etc. General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVIFi,n) -- See Table II
23. 23. 23 Valuation Using Table II PVIFi,n is found on Table II at the end of the book or on the card insert. Period 1 2 3 4 5 6% .943 .890 .840 .792 .747 7% .935 .873 .816 .763 .713 8% .926 .857 .794 .735 .681
24. 24. 24 Using Present Value Tables PV2 = \$1,000 (PVIF7%,2) = \$1,000 (.873) = \$873 [Due to Rounding] Period 6% 7% 1 .943 .935 2 .890 .873 3 .840 .816 4 .792 .763 5 .747 .713 8% .926 .857 .794 .735 .681
25. 25. 25 Story Problem Example Julie Miller wants to know how large of a deposit to make so that the money will grow to \$10,000 in 5 years at a discount rate of 10%. 0 10% 1 2 3 4 5 \$10,000 PV0
26. 26. 26 Story Problem Solution Calculation based on general formula: PV0 = FVn / (1+i)n PV0 = \$10,000 / (1+ 0.10)5 = \$6,209.21 Calculation based on Table I: PV0 = \$10,000 (PVIF10%, 5) = \$10,000 (.621) = \$6,210.00 [Due to Rounding]
27. 27. 27 Graphical view of FV GRAPHICAL VIEW OF FV
28. 28. 28 Future value relationship Higher the interest rates, higher the future value Longer the period of time, higher the future value For an interest rate of 0% the FV is always equal to its PV(1.00). But for any interest rate greater than zero, future value is greater than the present value
29. 29. 29 A graphical view of present value
30. 30. 30 Present value relationship The higher the discount rate, the lower the present value The longer the period of time, the lower the present value At the discount rate 0%,the present value is always equal to its future value
31. 31. 31 Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Ordinary Annuity: Payments or receipts Annuity occur at the end of each period. Annuity Due: Payments or receipts Due occur at the beginning of each period.
32. 32. 32 Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings
33. 33. 33 Parts of an Annuity (Ordinary Annuity) End of Period 1 0 End of Period 2 End of Period 3 1 2 3 \$100 \$100 \$100 Today Equal Cash Flows Each 1 Period Apart
34. 34. 34 Parts of an Annuity (Annuity Due) Beginning of Period 1 Beginning of Period 2 0 1 2 \$100 \$100 Beginning of Period 3 \$100 Today 3 Equal Cash Flows Each 1 Period Apart
35. 35. 35 Overview of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 n . . . i% R R R R = Periodic Cash Flow FVAn = R(1+i) + R(1+i) + ... + R(1+i)1 + R(1+i)0 n-1 n-2 FVAn n+1
36. 36. 36 Example of an Ordinary Annuity -- FVA Cash flows occur at the end of the period 0 1 2 3 \$1,000 \$1,000 4 \$1,000 7% \$1,070 \$1,145 FVA3 = \$1,000(1.07)2 + \$1,000(1.07)1 + \$1,000(1.07)0 = \$1,145 + \$1,070 + \$1,000 = \$3,215 \$3,215 = FVA3
37. 37. 37 Future value interest factor for an ordinary annuity FVIFi,n=1/i*<(1+i)n-1> FVA=PMT*(FVIFAi,n)
38. 38. 38 Hint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.
39. 39. 39 Valuation Using Table III FVAn FVA3 = R (FVIFAi%,n) = \$1,000 (FVIFA7%,3) = \$1,000 (3.215) = \$3,215 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
40. 40. 40 Overview View of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 R R R i% R . . . FVADn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1 = FVAn (1+i) n-1 n R FVADn
41. 41. 41 numerical Martin has \$10000 that she can deposit in any of three saving counts for a 3 year period. Bank A compounds interest on an annual basis, bank B compounds interest twice each year, Bank C compounds interest each quarter. All three banks have a stated annual interest rate of 4% What amount would Ms.Martin have at the end of third year? On the basis of your findings in banks, which bank should she prefer.
42. 42. 42 NUMERICALS Ramish wishes to choose the better of two equally costly cash flow streams: annuity X and annuity Y.X is an annuity due with a cash inflow of \$9000 for each of 6 years is an ordinary annuity with cash inflow of 410000 or each of 6 years. Assume that he can earn 15% on his investment. On a subject basis, which annuity do you think is more attractive and why? Find the future value at the end of year 6,for both annuity x and Y.
43. 43. 43 NUMERICALS what is the present value of \$ 6000 to be received at the end of 6 years if the discount rate is 12%? \$100 at the end of three years is worth how much today, assuming a discount rate of 100% 10% 0%
44. 44. 44 Example of an Annuity Due -- FVAD Cash flows occur at the beginning of the period 0 1 2 3 \$1,000 \$1,000 4 \$1,070 7% \$1,000 \$1,145 \$1,225 FVAD3 = \$1,000(1.07)3 + \$1,000(1.07)2 + \$1,000(1.07)1 = \$1,225 + \$1,145 + \$1,070 = \$3,440 \$3,440 = FVAD3
45. 45. 45 Valuation Using Table III FVADn FVAD3 = R (FVIFAi%,n)(1+i) = \$1,000 (FVIFA7%,3)(1.07) = \$1,000 (3.215)(1.07) = \$3,440 Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
46. 46. 46 Present value of ordinary annuity PVA=PMT(PVIFAi,n) PVIF= 1 - 1 (1+i)n i
47. 47. 47 PRESENT VALUE OF ANNUITY DUE PVIF=PVIFA*(1+i)
48. 48. 48 PV of an ordinary annuity Braden company a small producer of toys wants to determine the most it should pay to purchase a particular ordinary annuity. the annuity consist of cash flows of 700v at the end of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%.
49. 49. 49 Long method for finding the present value of an ordinary annuity Year CF PVIF8%,n (1) (2) 1 700 0.926 2 700 0.857 3 700 0.794 4 700 0.735 5 700 0.681 present value of annuity=2795.10 PVIF=1/i*(1-1/(1+i)n) PV (1*2) 648.20 599.90 555.80 514.50 476.70
50. 50. 50 Finding present value of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 n n+1 . . . i% R R R R = Periodic Cash Flow PVAn PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n
51. 51. 51 Example of an Ordinary Annuity -- PVA Cash flows occur at the end of the period 0 1 2 3 \$1,000 \$1,000 \$1,000 7% \$ 934.58 \$ 873.44 \$ 816.30 \$2,624.32 = PVA3 PVA3 = \$1,000/(1.07)1 + \$1,000/(1.07)2 + \$1,000/(1.07)3 = \$934.58 + \$873.44 + \$816.30 = \$2,624.32 4
52. 52. 52 Hint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the end of the first cash flow period.
53. 53. 53 Valuation Using Table IV PVAn PVA3 = R (PVIFAi%,n) = \$1,000 (PVIFA7%,3) = \$1,000 (2.624) = \$2,624 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
54. 54. 54 Overview of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 PVADn n . . . i% R n-1 R R R R: Periodic Cash Flow PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAn (1+i)
55. 55. 55 Example of an Annuity Due -- PVAD Cash flows occur at the beginning of the period 0 1 2 \$1,000 3 \$1,000 7% \$1,000.00 \$ 934.58 \$ 873.44 \$2,808.02 = PVADn PVADn = \$1,000/(1.07)0 + \$1,000/(1.07)1 + \$1,000/(1.07)2 = \$2,808.02 4
56. 56. 56 Valuation Using Table IV PVADn = R (PVIFAi%,n)(1+i) PVAD3 = \$1,000 (PVIFA7%,3)(1.07) = \$1,000 (2.624)(1.07) = \$2,808 Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
57. 57. 57 Steps to Solve Time Value of Money Problems 1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional)
58. 58. 58 Mixed Flows Example Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%? 10% 0 1 10% \$600 PV0 2 3 4 5 \$600 \$400 \$400 \$100
59. 59. 59 How to Solve? 1. Solve a “piece-at-a-time” by piece-at-a-time discounting each piece back to t=0. 2. Solve a “group-at-a-time” by first group-at-a-time breaking problem into groups of annuity streams and any single cash flow group. Then discount each group back to t=0.
60. 60. 60 “Piece-At-A-Time” 0 1 10% \$600 2 3 4 5 \$600 \$400 \$400 \$100 \$545.45 \$495.87 \$300.53 \$273.21 \$ 62.09 \$1677.15 = PV0 of the Mixed Flow
61. 61. 61 “Group-At-A-Time” (#1) 0 10% 1 \$600 2 3 4 5 \$600 \$400 \$400 \$100 \$1,041.60 \$ 573.57 \$ 62.10 \$1,677.27 = PV0 of Mixed Flow [Using Tables] \$600(PVIFA10%,2) = \$600(1.736) = \$1,041.60 \$400(PVIFA10%,2)(PVIF10%,2) = \$400(1.736)(0.826) = \$573.57 \$100 (PVIF10%,5) = \$100 (0.621) = \$62.10
62. 62. 62 “Group-At-A-Time” (#2) 0 Plus \$347.20 Plus \$62.10 0 3 \$400 \$400 1 2 \$200 0 2 \$400 \$1,268.00 1 \$200 1 2 4 \$400 PV0 equals \$1677.30. 3 4 5 \$100
63. 63. 63 Frequency of Compounding General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today
64. 64. 64 Impact of Frequency Julie Miller has \$1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) 1,000 = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) 1,000 = 1,262.48
65. 65. 65 Impact of Frequency Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) 1,000 = 1,266.77 Monthly FV2 = 1,000(1+ [.12/12])(12)(2) 1,000 = 1,269.73 Daily FV2 = 1,000(1+[.12/365])(365)(2) 1,000 = 1,271.20
66. 66. 66 Present value of perpetuity An annuity with an infinite life, providing continual annual cash flow PVIF=1/i
67. 67. 67 Effective Annual Interest Rate The annual rate of interest actually paid or earned The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] )m - 1
68. 68. 68 Nominal annual rate Contractual annual rate of interest charged by a lender or promised by a borrower.
69. 69. 69 BW’s Effective Annual Interest Rate Basket Wonders (BW) has a \$1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR EAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%!
70. 70. 70 Loan amortization The determination of the equal periodic loan payments necessary to provide lender with a specified interest return and to repay the loan principal over a specified period.
71. 71. 71 Steps to Amortizing a Loan 1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat.
72. 72. 72 Amortizing a Loan Example Julie Miller is borrowing \$22,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PV0 = R (PVIFA i%,n) \$22,000 = R (PVIFA 12%,5) \$22,000 = R (3.605) R = \$22,000 / 3.605 = \$5351
73. 73. 73 Amortizing a Loan Example End of Payment Year 0 --- Interest Principal Ending (Pmt-int) Balance ----\$22,000 1 \$5351 2640 2711 19289 2 5351 2315 3036 16253 3 5351 1951 3400 12853 4 5351 1542 3809 9044 5 5351 1085 4266 4778 6 5351 573 4778 0
74. 74. 74 Usefulness of Amortization 1. 2. Determine Interest Expense -Interest expenses may reduce taxable income of the firm. Calculate Debt Outstanding -- The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.