Chapter 4 FIN3414

Chapter 4 FIN3414 - Intermediate Financial Management 1 2...

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Intermediate Financial Management 1 Chapter Four 1 2 THE TIME VALUE OF MONEY WITH ANNUAL COMPOUNDING 3 4 By convention, the interest rate is always quoted in annual terms, but the 5 compounding interval can be of any length of time. Typically, it is less than or equal to a 6 year. The mathematical formulas that follow determine future and present values with 7 compounded rates that assume ANNUAL COMPOUNDING. 8 9 Future Value of an Amount 10 11 Let an amount P be deposited in an account that earns an annual rate of interest k .What 12 will its future value be at the end of n years? The mathematical formula can be developed 13 as follows: 14 15 P 16 Deposit 17 18 Year 0 1 2 3 4 …… n 19 20 Since the interest earned on principal P together with the principal starts earning interest 21 k at the end of each year, we have the following situation: 22 23 Principal at the end of the year: 24 25 0 (today) P 26 1 P + kP = P (1 + k ) 27 2 P (1+ k ) + kP (1+ k ) = P (1 + k ) 2 28 3 P (1+ k ) 2 + kP (1+ k ) 2 = P (1 + k ) 3 29 . 30 . 31 . 32 n P (1+ k ) n -1 + kP (1+ k ) n -1 = P (1 + k ) n 33 34 35 But the amount at the end of the n th year is the future value of $ P deposited today. 36 Therefore, the future value of P is 37 38 FV = P (1 + k ) n . (4.1) 39 40
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Intermediate Financial Management 2 41 42 43 The future value ( FV ) at the end of n years of an amount P deposited today at an 44 annual rate of interest k compounded annually is 45 46 FV = P(1 + k) n . 47 48 49 Example 4.1: Suppose at the time of your birth, 25 years ago, your father deposited 50 $1,200 in an account at an annual interest rate of 15 percent. How much money would 51 exist in that account today? 52 53 Answer: P = 1,200, k = .15, n = 25 54 55 FV = 1,200(l + .15) 25 56 = $39,502.74. 57 58 This is a tidy sum to say the least. 59 60 Example 4.2 : Suppose at the time of your birth, 35 years ago, your father deposited 61 $1,200 in an account that earns an annual interest rate of 15 percent with the following 62 stipulations: 63 64 (a) You must withdraw half the accumulated amount on your eighteenth birthday; 65 and 66 (b) On or after your thirty-fifth birthday, you can close the account. 67 68 Today is your thirty-fifth birthday. How much money is left in the account to withdraw? 69 Answer: 70 71 $1,200 72 Amount deposited 73 Birth year 0 1 2 18 th birthday 19 ... . ....... 35 74 75 The amount of money in the account on the eighteenth birthday is: 76 77 FV = 1,200(1 + .15) 18 = $14,850.54. 78 79 Since half of that must be withdrawn: 80 81 Amount withdrawn = 14,850.54 2 = $7,425.27. 82 83
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Intermediate Financial Management 3 The future value of the remaining balance of $7,425.27, which by the thirty-fifth birthday 84 will have accrued interest for 17 years, is 85 86 FV = 7,425.27(l + .15) 17 = $79,905.29. 87 88 Notice how the amount grows at an exponential rate. The last 17 years accumulate nearly 89 5 times as much as the first 18 years. 90 91 Example 4.3 : Suppose Mr. Smith contracted with Roles Rice, Inc., to take delivery on a
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Chapter 4 FIN3414 - Intermediate Financial Management 1 2...

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