Chapter 3 FIN3414

Chapter 3 FIN3414 - 1 2 Chapter Three 3 SIMPLE RATE OF...

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1 Chapter Three 1 2 SIMPLE RATE OF INTEREST 3 4 FUTURE VALUE OF AN AMOUNT 5 6 Let a principal amount $ P be deposited in an account that has a stated or nominal 7 interest rate of k or 100 k percent per annum. What will be the future value of these P 8 dollars if left in the account for n years? The mathematical basis can be developed with 9 the help of the following diagram: 10 11 Deposit $ P 12 13 Year 0 1 2 3 4 n-1 n 14 Today 15 16 Remembering that with simple interest only the principal ( P , in this case) earns interest, 17 we have the following formulas for interest earned at the end of 18 19 Year 1 = kP 20 Year 2 = 2 kP 21 Year 3 = 3 kP 22 . 23 . 24 25 Year n = nkP 26 27 Therefore, the principal P will grow to the future value ( FV ) as follows: 28 29 FV = P + nkP = P (1+ nk ) (3.1) 30 31 32 38 39 What if the interest rate does not stay the same over time? For example, the simple 40 interest rate is k 1 , for the first n 1 years, k 2 for the next n 2 years, k 3 for the next n 3 years, 41 etc. Then, the principal P will accrue interest Pk 1 n 1 for the first n 1 years, Pk 2 n 2 for the 42 The future value (FV) of an amount P at an annual simple interest rate of k for n years is given by FV = P(1+ nk).
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2 next n 2 years, and Pk 3 n 3 for the next n 3 years, etc. This implies that the future value 43 under changing simple interest rates is 44 45 FV = P (1 + n 1 k 1 + n 2 k 2 + n 3 k 3 + . ...) 46 47 Example 3.1 : Suppose an account earns 15 percent simple interest per year. What would 48 the future value be of a deposit of: 49 50 (a) $15 left for 10 years? 51 (b) $0.15 left for 5 years? 52 (c) $1 left for 2,000 years? 53 54 Answers 55 56 2625 . 0 $ ) 5 15 . 1 ( 15 . 5 , 15 . , 15 . ) ( FV n k P b 57 58 59 301 $ ) 000 , 2 15 . 1 ( 1 000 , 2 , 15 . , 1 ) ( n k P c 60 61 62 Example 3.2 : Suppose you deposit $1,000 in an account that pays simple rate of interest. 63 What will be the future value of the account if: 64 65 (a) the annual simple interest rate is 7% for the first 5 years, 10% for the next 10 years, 66 and 12% for the last 5 years? 67 (b) 5% for the first 10 years, 10% for the next 10 years, 15% for the last 10 years? 68 69 Answers: 70 71 (a) P = $1,000, k 1 = 7%, n 1 = 5 72 k 2 = 10%, n 2 = 10 73 k 3 = 12%, n 3 = 5 74 FV = $1,000(1 + .07 × 5 + .10 × 10 + .12 × 5) 75 = $1,000(2.95) = $2,950. 76 77 (b) P = $1,000, k 1 = 5%, n 1 = 10 78 5 . 37 $ ) 10 15 . 1 ( 15 10 , 15 . , 15 ) ( n k P a
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3 k 2 = 10%, n 2 = 10 79 k 3 = 15%, n 3 = 10 80 FV = $1,000(1 + .05 × 10 + .10 × 10 + .15 × 10) 81 = $1,000(4) = $4,000. 82 83 84 PRESENT VALUE OF AN AMOUNT 85 86 The formula for the present value of some assets to be received in the future can be 87 directly obtained from expression (3.1). For an amount FV to be received after n years, 88 the present value is P -the amount initially deposited in the account. Thus, solving for P 89 and calling it instead the present value, PV , we have 90 91 nk F PV 1 (3.2) 92 93 In the above expression, F is equal to FV , the amount to be received after n years. 94 95 103 The formula in the box assumes that rate of interest k will remain constant in each of 104 the periods. However, if there are more than one simple interest rate over time, that is, k 1 , 105 n 1 , k 2 , n 2 , k 3 , n 3 , . ..etc. Then 106 107 ... 1 3 3 2 2 1 1 k n k n k n F 108 109 Example 3.3 : Suppose an account carries a 15 percent annual simple interest rate. What
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Chapter 3 FIN3414 - 1 2 Chapter Three 3 SIMPLE RATE OF...

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