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# Time Value of Money

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Timeline, time value of money, simple interest rate, compound interest rate, annuities, perpetuities, loan amortization, uneven cash flow stream

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### Time Value of Money

1. 1. Chapter 2: The Time Value of Money Instructor: Fahim Muntaha
2. 2. Outline • Time value of Money • Simple interest rate • Compound interest rate • Annuities • Perpetuities • Loan amortization • Uneven cash flow stream
3. 3. Learning Outcomes • Will a clear concept of Time value of Money • Will learn how to calculate Simple interest rate and Compound interest rate • Learn about Annuities and Perpetuities • Learn to calculate and take decision from Loan amortization schedule
4. 4. Learning objective How to draw time lines
5. 5. Time lines • Show the timing of cash flows. • Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. CF0 CF1 CF3CF2 0 1 2 3 i%
6. 6. Drawing time lines: i. \$100 lump sum due in 2 years; ii. 3-year \$100 ordinary annuity 100 100100 0 1 2 3 i% ii. 3 year \$100 ordinary annuity 100 0 1 2 i% i. \$100 lump sum due in 2 years
7. 7. Drawing time lines: Uneven cash flow stream; CF0 = -\$50, CF1 = \$100, CF2 = \$75, and CF3 = \$50 100 5075 0 1 2 3 i% -50 Uneven cash flow stream
8. 8. Drawing time lines: Uneven cash flow stream; CF0 = -\$50, CF1 = \$100, CF2 = - \$75, and CF3 = \$50 100 50-75 0 1 2 3 i% -50 Uneven positive and negative cash flow stream
9. 9. Draw a timeline by yourself Mr. X invested in his business AED 20,000 at the beginning of the year and expecting that the business will need to reinvest AED 3000 in the next year. After that the business will generate AED 5000 cash inflows at the end of each year. But in the last year the cash inflow will be 10,000. Business will run for 6 years. Draw a Time line by yourself.
10. 10. Drawing time lines: Uneven cash flow stream; CF0 = -AED 20,000, CF1 = - 3000, CF2 = 5000, CF3 = 5000, CF4 = 5000, CF5= 5000, and CF6= 5000, -3000 10,0005000 0 1 2 6 i% -20,000 Uneven positive and negative cash flow stream 5000 5000 5000 43 5
11. 11. Learning objective Understanding the concept The Time value of Money
12. 12. Definition The Time Value of Money mathematics quantifies the value of a dollar through time. • A dollar on hand today is worth more than a dollar to be received in the future because the dollar on hand today can be invested to earn interest to yield more than a dollar in the future. • This depends upon the rate of return or interest rate which can be earned on the investment.
13. 13. Uses of the Time Value of Money The Time Value of Money has applications in many areas of Corporate Finance including: • Capital Budgeting, • Bond Valuation, and • Stock Valuation. • Loan amortization. • Forecast investment decision.
14. 14. Areas of Time Value of Money The Time Value of Money concepts will be grouped into two areas: Future Value describes the process of finding what an investment today will grow to in the future. Present Value describes the process of determining what a cash flow to be received in the future is worth in today's dollars. FV = ? 0 1 2 310% 100 133 0 1 2 310% PV = ?
15. 15. Financial Equation to solve FV FVn = PV ( 1 + i )n Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years
16. 16. What is the future value (FV) of an initial \$100 after 3 years, if I/YR = 10%? Compounding: Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding. Compound Interest rate: Interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Compound interest can be thought of as “interest on interest,” and will make a deposit or loan grow at a faster rate than simple interest, which is interest calculated only on the principal amount. FV = ? 0 1 2 3 10% 100
17. 17. Solving for FV: The arithmetic method • After 1 year: – FV1 = PV ( 1 + i ) = \$100 (1.10) = \$110.00 • After 2 years: – FV2 = PV ( 1 + i )2 = \$100 (1.10)2 =\$121.00 • After 3 years: – FV3 = PV ( 1 + i )3 = \$100 (1.10)3 =\$133.10 • After n years (general case): – FVn = PV ( 1 + i )n
18. 18. Others methods to solve FV FV can be solved by using the • arithmetic, • financial calculator, • Compounding and discounting table and • spreadsheet methods.
19. 19. Solving FV using Compounding Tables Financial Equation is FVn = PV ( 1 + i )n Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years Using compounding Table FVn = PV X FVIFi,n Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years FVIFi,n = Future value interest factor for i% n years
20. 20. Use any of these equations Using Financial Equation FVn = PV ( 1 + i )n FV3 = PV ( 1 + i )3 = \$100 (1.10)3 =\$133.10 Using Compounding table FVn = PV X FVIFi,n FV3 = \$100 (1.3310) =\$133.10 What is the future value (FV) of an initial \$100 after 3 years, if I/YR = 10%?
21. 21. Solve the Problems by yourself 1. You currently have \$ 2,000 on hand that you plan to use to purchase a three years bank certificate of deposit (CD). The CD pays 4% interest rate annually. a. How much money will you have when the CD matures? b. What would be the FV if the CD will mature after 5 years? c. What if the interest rate increased from 4% to 6% , but the CD maturity was still three years? Answers: \$2,249.73, \$2,433.31, \$2,382.03.
22. 22. What is Present Value Present Value describes the process of determining what a cash flow to be received in the future is worth in today's dollars. The PV shows the value of cash flows in terms of today’s purchasing power. Discounting: Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding). 133 0 1 2 310% PV = ?
23. 23. Financial Equation to solve PV FVn = PV ( 1+ i)n Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years PV = FVn/( 1 + i )n Or PV = FVn X PVIFi, n PVIFi, n= Present Value interest factor for i% n years
24. 24. What is the present value (PV) of \$100 due in 3 years, if I/YR = 10%? Using Financial Equation PV = FVn / ( 1 + i )n = \$100 / ( 1 + 0.10 )3 = \$75.13 Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years Using Discounting table PV = FVn X PVIFi, n = \$100 X 0.7513 = \$75.13 PVIFi, n= Present Value interest factor for i% n years = 0.7513 (derived from PV table)
25. 25. Solve the Problems by yourself 1. What amount must pay today in a three year CD paying 4% interest annually to provide you with \$2,249.73 at the end of CD maturity? a. What would be the PV if the CD will mature after 5 years? b. What if the interest rate increased from 4% to 6% , but the CD maturity was still three years? Answers: \$2,000, \$1,849.11, \$1,888.92.
26. 26. Learning objective Understanding semiannual and other compounding period
27. 27. Use this formula FVn = PV ( 1 + i/m )n X m Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years M = number of the month interest compounded in a year
28. 28. What is the FV of \$100 after 3 years under 10% semiannual compounding? Quarterly compounding?
29. 29. Solution \$134.49(1.025)\$100FV \$134.01(1.05)\$100FV ) 2 0.10 1(\$100FV ) m i 1(PVFV 12 3Q 6 3S 32 3S nm n       Here, FV = Future value PV = Present Value i= Interest rate per year n = Number of years M = number of the month interest compounded in a year
30. 30. What’s the FV of a 3-year \$100 annuity, if the quoted interest rate is 10%, compounded semiannually? • Payments occur annually, but compounding occurs every 6 months. • Cannot use normal annuity valuation techniques. 0 1 100 2 3 5% 4 5 100 100 6 1 2 3
31. 31. Monthly compound interest 1. Calculate the present value on Jan 1, 2011 of \$1,500 to be received on Dec 31, 2011. The market interest rate is 9%. Compounding is done on monthly basis. 2. A friend of you has won a prize of \$10,000 to be paid exactly after 2 years. On the same day, he was offered \$8,000 as a consideration for his agreement to sell the right to receive the prize. The market interest rate is 12% and the interest is compounded on monthly basis. Help him by determining whether the offer should be accepted or not.
32. 32. Solution 1. We have, Future Value FV = \$1,500 Compounding Periods n = 12 Interest Rate i = 9%/12 = 0.75% Present Value PV = \$1,500 / ( 1 + 0.75% )^12 = \$1,500 / 1.0075^12 = \$1,500 / 1.093807 = \$1,371.36 2. We have, Future Value FV = \$10,000 Compounding Periods n = 2 × 12 = 24 Interest Rate i = 12%/12 = 1% Present Value PV = \$10,000 / ( 1 + 1% )^2X12) = \$10,000 / 1.01^24 = \$10,000 / 1.269735 = \$7,875.66 Since the present value of the prize is less than the amount offered, it is good to accept the offer.
33. 33. Learning objective Understanding Annuities and Perpetuities (EVEN CASH FLOW STREAM)
34. 34. Annuity An annuity is a series of equal dollar payments that are made at the end or beginning of equidistant points in time. Characteristics • An annuity is a series of payments • It has fixed intervals of time. • It has a limited life time. • The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time. • The valuation of an annuity entails concepts such as time value of money, interest rate, and future value.
35. 35. Example of Annuity Examples of annuities are • regular deposits to a savings account, • monthly home mortgage payments, • monthly insurance payments and • pension payments etc. Annuity PMT PMTPMT 0 1 2 3 i%
36. 36. Types of Annuity • There are two types of annuity formulas. 1. Ordinary Annuity 2. Annuity Due: We will derive the ordinary annuity formula first.
37. 37. Understanding Ordinary Annuity
38. 38. Ordinary Annuity If a series of payment is made at the end od each payment period then it is an Ordinary Annuity. One formula is based on the payments being made at the end of the payment period. This called ordinary annuity. Ordinary Annuity PMT PMTPMT 0 1 2 3 i%
39. 39. FV of Ordinary Annuity
40. 40. 10-40 \$1000 \$1000 (1.04)1n = 1 Sum = FV of annuity 0 1 2 3 4 Interval number \$1000 \$1000 \$1000 \$1000 (1.04)2 n = 2 \$1000 (1.04)3 n = 3 …the sum of the future values of all the payments Assume that there are four(4) annual \$1000 payments with interest at 4% Future Value of an Ordinary Simple Annuity
41. 41. • FVn = FV of annuity at the end of nth period. • PMT = annuity payment deposited or received at the end of each period • i = interest rate per period • n = number of periods for which annuity will last FV of an Ordinary Annuity
42. 42. • Example 6.1 How much money will you accumulate by the end of year 10 if you deposit \$3,000 each for the next ten years in a savings account that earns 5% per year? FV of an Ordinary Annuity
43. 43. • FV = \$3000 {[ (1+.05)10 - 1] ÷ (.05)} = \$3,000 { [0.63] ÷ (.05) } = \$3,000 {12.58} = \$37,740 FV of an Ordinary Annuity
44. 44. Solving for PMT in an Ordinary Annuity • Instead of figuring out how much money you will accumulate (i.e. FV), you may like to know how much you need to save each period (i.e. PMT) in order to accumulate a certain amount at the end of n years. • In this case, we know the values of n, i, and FVn in equation 6-1c and we need to determine the value of PMT.
45. 45. Solving for PMT in an Ordinary Annuity (cont.) • Example 6.2: Suppose you would like to have \$25,000 saved 6 years from now to pay towards your down payment on a new house. If you are going to make equal annual end-of- year payments to an investment account that pays 7 percent, how big do these annual payments need to be?
46. 46. Solving for PMT in an Ordinary Annuity (cont.) Here we know, • FVn = \$25,000; • n = 6; and • i=7% and • we need to determine PMT.
47. 47. Solving for PMT in an Ordinary Annuity (cont.) • \$25,000 = PMT {[ (1+.07)6 - 1] ÷ (.07)} = PMT{ [.50] ÷ (.07) } = PMT {7.153} \$25,000 ÷ 7.153 = PMT = \$3,495.03
48. 48. You save \$500 every month for the next three years. Assume your savings can earn 3% converted monthly. Determine the total in your account three years from now. 18810.28 Careful about effective Interest Rate Solve the FV of Ordinary annuity by yourself
49. 49. You vow to save \$500/month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now 2007.51 Solve the FV of Ordinary annuity by yourself
50. 50. PV of Ordinary Annuity
51. 51. The Present Value of an Ordinary Annuity • The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future.
52. 52. The Present Value of an Ordinary Annuity (cont.) • Figure 6-2 shows the lump sum equivalent (\$2,106.18) of receiving \$500 per year for the next five years at an interest rate of 6%. • If you don’t know annuity formula you can use simple PV formula and Time line. PV = FVn/( 1 + i )n = FVn ( 1 + i )-n
53. 53. If you don’t know annuity formula you can use simple PV formula.
54. 54. The Formula of Present Value of an Ordinary Annuity (cont.) Here • PMT = annuity payment deposited or received at the end of each period. • i = discount rate (or interest rate) on a per period basis. • n = number of periods for which the annuity will last.
55. 55. Practice yourself using Ordinary annuity formula • Assume that there are four(4) annual \$1000 payments with interest at 4% annually compounded. According to formula • PV = PMT [1-{1/(1+i)n}]/ i =3629.90
56. 56. Understanding Loan Amortization schedule using Ordinary annuity concept
57. 57. Amortized Loans • An amortized loan is a loan paid off in equal payments – consequently, the loan payments are an annuity. • Examples: Home mortgage loans, Auto loans
58. 58. Amortized Loans (cont.) In an amortized loan, • the present value can be thought of as the amount borrowed, • n is the number of periods the loan lasts for, • i is the interest rate per period, • future value takes on zero because the loan will be paid of after n periods, and • payment is the loan payment that is made.
59. 59. Amortized Loans (cont.) • Example 6.5 Suppose you plan to get a \$9,000 loan from a furniture dealer at 18% annual interest with annual payments that you will pay off in over five years. What will your annual payments be on this loan?
60. 60. Amortized Loans (cont.) • Using FVOA or PVOA Formula solve the PMT first – N = 5 – i = 18.0 – PV = 9000 – FV = 0 – PMT = \$2,878.00
61. 61. The Loan Amortization Schedule Year Amount Owed on Principal at the Beginning of the Year (1) Annuity Payment (2) Interest Portion of the Annuity (3) = (1) × 18% Repaymen t of the Principal Portion of the Annuity (4) = (2) –(3) Outstanding Loan Balance at Year end, After the Annuity Payment (5) =(1) – (4) 1 \$2,878 2 \$2,878 3 \$2,878 4 \$2,878 5 \$2,878 \$0.00
62. 62. The Loan Amortization Schedule Year Amount Owed on Principal at the Beginning of the Year (1) Annuity Payment (2) Interest Portion of the Annuity (3) = (1) × 18% Repaymen t of the Principal Portion of the Annuity (4) = (2) –(3) Outstanding Loan Balance at Year end, After the Annuity Payment (5) =(1) – (4) 1 \$9,000 \$2,878 \$1,620.00 2 \$2,878 3 \$2,878 4 \$2,878 5 \$2,878 \$0.00
63. 63. The Loan Amortization Schedule Year Amount Owed on Principal at the Beginning of the Year (1) Annuity Payment (2) Interest Portion of the Annuity (3) = (1) × 18% Repaymen t of the Principal Portion of the Annuity (4) = (2) –(3) Outstanding Loan Balance at Year end, After the Annuity Payment (5) =(1) – (4) 1 \$9,000 \$2,878 \$1,620.00 \$1,258.00 2 \$2,878 3 \$2,878 4 \$2,878 5 \$2,878 \$0.00
64. 64. The Loan Amortization Schedule Year Amount Owed on Principal at the Beginning of the Year (1) Annuity Payment (2) Interest Portion of the Annuity (3) = (1) × 18% Repaymen t of the Principal Portion of the Annuity (4) = (2) –(3) Outstanding Loan Balance at Year end, After the Annuity Payment (5) =(1) – (4) 1 \$9,000 \$2,878 \$1,620.00 \$1,258.00 \$7,742.00 2 \$7,742.00 \$2,878 3 \$2,878 4 \$2,878 5 \$2,878 \$0.00
65. 65. The Loan Amortization Schedule Year Amount Owed on Principal at the Beginning of the Year (1) Annuity Payment (2) Interest Portion of the Annuity (3) = (1) × 18% Repaymen t of the Principal Portion of the Annuity (4) = (2) –(3) Outstanding Loan Balance at Year end, After the Annuity Payment (5) =(1) – (4) 1 \$9,000 \$2,878 \$1,620.00 \$1,258.00 \$7,742.00 2 \$7,742 \$2,878 \$1,393.56 \$1,484.44 \$6,257.56 3 \$6257.56 \$2,878 \$1,126.36 \$1,751.64 \$4,505.92 4 \$4,505.92 \$2,878 \$811.07 \$2,066.93 \$2,438.98 5 \$2,438.98 \$2,878 \$439.02 \$2,438.98 \$0.00
66. 66. The Loan Amortization Schedule (cont.) • We can observe the following from the table: – Size of each payment remains the same. – However, Interest payment declines each year as the amount owed declines and more of the principal is repaid.
67. 67. The Present Value of an Ordinary Annuity (cont.) • Note , it is important that “n” and “i” match. If periods are expressed in terms of number of monthly payments, the interest rate must be expressed in terms of the interest rate per month.
68. 68. You overhear your friend saying the he is repaying a loan at \$450 every month for the next nine months. The interest rate he has been charged is 12% compounded monthly. i. Calculate the amount of the loan, ii. Show Monthly Loan Amortization Schedule iii. and the amount of interest involved. PV= , Repaid 4,050.00, i = 131.76 Class work/ HW
69. 69. Amortized Loans with Monthly Payments • Many loans such as auto and home loans require monthly payments. This requires converting n to number of months and computing the monthly interest rate.
70. 70. Amortized Loans with Monthly Payments (cont.) Example 6.6 You have just found the perfect home. However, in order to buy it, you will need to take out a \$300,000, 30-year mortgage at an annual rate of 6 percent. What will your monthly mortgage payments be?
71. 71. Amortized Loans with Monthly Payments (cont.) • Mathematical Formula • Here annual interest rate = .06, number of years = 30, m=12, PV = \$300,000
72. 72. Amortized Loans with Monthly Payments (cont.) \$300,000= PMT \$300,000 = PMT [166.79] \$300,000 ÷ 166.79 = PMT \$1798.67 = PMT 1- 1/(1+.06/12)360 .06/12
73. 73. Understanding Annuity Due
74. 74. Annuity Due • Annuity due is an annuity in which all the cash flows occur at the beginning of the period. For example, rent payments on apartments are typically annuity due as rent is paid at the beginning of the month. PMT PMT 0 1 2 3 i% PMT Annuity Due
75. 75. Example of Annuity due If you lease equipment, a vehicle, or rent property, the typical lease contract requires payments at the beginning of each period of coverage
76. 76. 13 - 76 “Payments…in advance” “First payment … made today” “Payments at the beginning of each…..” “Payments ….. starting now” Clues to help identify annuities due
77. 77. What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity PMT PMTPMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due
78. 78. 13 - 78 FVdue = PMT (1+ i)n - 1[ i ]Formula * (1+ i) PVdue = PMT 1-(1+ i)-n [ i ]Formula * (1+ i)
79. 79. Annuities Due: Future Value • Computation of future value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity.
80. 80. Annuities Due: Future Value (cont.) • Recall Example 6.1 where we calculated the future value of 10-year ordinary annuity of \$3,000 earning 5 per cent to be \$37,734. • What will be the future value if the deposits of \$3,000 were made at the beginning of the year i.e. the cash flows were annuity due?
81. 81. Annuities Due: Future Value (cont.) • FV = \$3000 {[ (1+.05)10 - 1] ÷ (.05)} (1.05) = \$3,000 { [0.63] ÷ (.05) } (1.05) = \$3,000 {12.58}(1.05) = \$39,620
82. 82. Annuities Due: Present Value • Since with annuity due, each cash flow is received one year earlier, its present value will be discounted back for one less period.
83. 83. Annuities Due: Present Value (cont.) • Recall checkpoint 6.2 Check yourself problem where we computed the PV of 10-year ordinary annuity of \$10,000 at a 10 percent discount rate to be equal to \$61,446. • What will be the present value if \$10,000 is received at the beginning of each year i.e. the cash flows were annuity due?
84. 84. Annuities Due: Present Value (cont.) • PV = \$10,000 {[1-(1/(1.10)10] ÷ (.10)} (1.1) = \$10,000 {[ 0.6144] ÷ (.10)}(1.1) = \$10,000 {6.144) (1.1) = \$ 67,590
85. 85. future value and present value of an annuity due are larger than that of an ordinary annuity • The examples illustrate that both the future value and present value of an annuity due are larger than that of an ordinary annuity because, in each case, all payments are received or paid earlier.
86. 86. Calculate PV of ANNUITY DUE by yourself • 3 year annuity due of \$100 at 10% • 364.10
87. 87. Learning objective Understanding Perpetuity
88. 88. Perpetuities • A perpetuity is an annuity that continues forever or has no maturity. For example, a dividend stream on a share of preferred stock. There are two basic types of perpetuities: – Growing perpetuity in which cash flows grow at a constant rate, g, from period to period. – Level perpetuity in which the payments are constant rate from period to period.
89. 89. Other Perpetuity Examples • British Consol Bonds • Canadian Pacific 4% Perpetual Bonds • Endowments – How much can I withdraw annually without invading principal?
90. 90. Present Value of a Level Perpetuity • PV = the present value of a level perpetuity • PMT = the constant dollar amount provided by the perpetuity • i = the interest (or discount) rate per period
91. 91. Present Value of a Growing Perpetuity • In growing perpetuities, the periodic cash flows grow at a constant rate each period. • The present value of a growing perpetuity can be calculated using a simple mathematical equation.
92. 92. Present Value of a Growing Perpetuity (cont.) • PV = Present value of a growing perpetuity • PMTperiod 1 = Payment made at the end of first period • i = rate of interest used to discount the growing perpetuity’s cash flows • g = the rate of growth in the payment of cash flows from period to period
93. 93. Growing Perpetuity Example • Suppose the initial payment is \$100 and that this grows at 3% per year while the discount rate is 5% • The value of this growing perpetuity is: 000,5\$ 03.05. 100      gi PMT PV
94. 94. Other Growing Perpetuity Examples • Stock price = present value of growing dividend stream
95. 95. Lets review Today
96. 96. List all the equations
97. 97. FV & PV of lump sum Payment • FVn = PV ( 1 + i )n • FVn = PV ( 1 + i/m )n X m • PV = FVn/( 1 + i )n • PV =FVn/( 1 + i/m)nxm
98. 98. Annuities Ordinary Annuity Annuity Due
99. 99. Perpetuity Level Perpetuity Growing Perpetuity
100. 100. Learning objective Understanding uneven cash flow stream
101. 101. Complex Cash Flow Streams • The cash flows streams in the business world may not always involve one type of cash flows. The cash flows may have a mixed pattern. For example, different cash flow amounts mixed in with annuities. • For example, figure 6-4 summarizes the cash flows for Marriott.
102. 102. Year Cash Flow 1 500 2 200 3 -400 4 500 5 500 6 500 7 500 8 500 9 500 10 500 Uneven cash flow problem • i= 6% • Find out the present value of this project.
103. 103. Draw a time line by yourself
104. 104. Year Cash Flow 1 500 2 200 3 -400 4 500 5 500 6 500 7 500 8 500 9 500 10 500 Split the problem into different parts Uneven cash flows Negative Cash flow Annuity OA? Or AD?
105. 105. Year Cash Flow 1 500 2 200 Try to solve step by step Uneven cash flows Find the PV of each cash flow individually
106. 106. Year Cash Flow 1 500 2 200 Try to solve step by step Uneven cash flows Find the PV of each cash flow individually PV = FVn/( 1 + i )n = 500/(1+ 0.06) 1 =? = 300/(1+ 0.06) 2 =?
107. 107. Year Cash Flow 3 -400 Try to solve step by step Negative Cash flow Find the PV of this negative cash flow
108. 108. Year Cash Flow 3 -400 Try to solve step by step Negative Cash flow Find the PV of this negative cash flow PV = FVn/( 1 + i )n = -400/(1+ 0.06) 3 =?
109. 109. Year Cash Flow 4 500 5 500 6 500 7 500 8 500 9 500 10 500 Try to solve step by step Annuity OA? Or AD? Find the PV of this Annuity
110. 110. Year Cash Flow 4 500 5 500 6 500 7 500 8 500 9 500 10 500 Try to solve step by step Annuity Ordinary Annuity Find the PV of this Annuity = 500{ [1- 1/ (1+0.06)7 ] / 0.06 } = ?
111. 111. Complex Cash Flow Streams (cont.)
112. 112. Discount the present value of ordinary annuity back three years to the present. PV = FVn/( 1 + i )n = 2791 / (1+0.06) 3 =?
113. 113. Add all the PV together
114. 114. Complex Cash Flow Streams (cont.) • In this case, we can find the present value of the project by summing up all the individual cash flows by proceeding in four steps: 1. Find the present value of individual cash flows in years 1, 2, and 3. 2. Find the present value of ordinary annuity cash flow stream from years 4 through 10. 3. Discount the present value of ordinary annuity (step 2) back three years to the present. 4. Add present values from step 1 and step 3.
115. 115. Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%. Mixed Flows Example using tables 0 1 2 3 4 5 \$600 \$600 \$400 \$400 \$100 PV0 10%
116. 116. “Group-At-A-Time” 0 1 2 3 4 5 \$600 \$600 \$400 \$400 \$100 10% \$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
117. 117. Do it by yourself Year Cash Flows A Cash Flows B 1 100 300 2 400 400 3 400 400 4 400 400 5 300 100 Suppose you have two investment options A and B. Interest rate is 8% compounded annually. Find the Present value of these cash flows. Which one would you prefer and why?
118. 118. Summary • Time value of Money • Simple interest rate • Compound interest rate • Annuities • Perpetuities • Loan amortization QUESTION?
119. 119. Remarks • I, r, k can be used to represent interest rate • T, n can be used to represent time • PVOA = Present value of an ordinary annuity • FVOA =Future value of an ordinary annuity • PVAD = Present value of annuity due • FVAD = Future value of annuity due
120. 120. Suggestions i. What do you understand about FV and compounding? ii. What do you understand about PV and Discounting? iii. What is annuity? Describe the features of annuity. iv. What are different types of annuity? v. Differentiate between ordinary annuity and annuity due. vi. What is perpetuity? vii. What is loan amortization schedule? How this schedule can help us? (answer by yourself)
121. 121. Again • Assign home work • Fix a date for catch-up class. • Fix a date for Class test on this chapter
122. 122. Thank you! @EICabudhabiEuropean International College-AbuDhabiEIC Abu Dhabi