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Chapter001

# Chapter001 - The Theory of Interest Solutions Manual...

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The Theory of Interest - Solutions Manual Chapter 1 1. ( a ) Applying formula (1.1) ( 29 ( 29 2 2 3 and 0 3 A t t t A = + + = so that ( 29 ( 29 ( 29 ( 29 ( 29 2 1 2 3 . 0 3 A t A t a t t t k A = = = + + ( b ) The three properties are listed on p. 2. (1) ( 29 ( 29 1 0 3 1. 3 a = = (2) ( 29 ( 29 1 2 2 0 for 0, 3 a t t t = + so that ( 29 a t is an increasing function. (3) ( 29 a t is a polynomial and thus is continuous. ( c ) Applying formula (1.2) ( 29 ( 29 [ ] ( 29 ( 29 2 2 2 2 1 2 3 1 2 1 3 2 3 2 1 2 2 3 2 1. n I A n A n n n n n n n n n n n = - - = + + - - + - + = + + - + - - + - = + 2. ( a ) Appling formula (1.2) ( 29 ( 29 [ ] ( 29 ( 29 [ ] ( 29 ( 29 [ ] ( 29 ( 29 1 2 1 0 2 1 1 0 . n I I I A A A A A n A n A n A + + + = - + - + + - - = - K L ( b ) The LHS is the increment in the fund over the n periods, which is entirely attributable to the interest earned. The RHS is the sum of the interest earned during each of the n periods. 3. Using ratio and proportion ( 29 5000 12,153.96 11,575.20 \$260 11,130 . - = 4. We have ( 29 2 , a t at b = + so that ( 29 ( 29 0 1 3 9 1.72. a b a a b = = = + = 1

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The Theory of Interest - Solutions Manual Chapter 1 Solving two equations in two unknowns .08 and 1. a b = = Thus, ( 29 ( 29 2 5 5 .08 1 3 a = + = ( 29 ( 29 2 10 10 .08 1 9 a = + = . and the answer is ( 29 ( 29 10 9 100 100 300. 5 3 a a = = 5. ( a ) From formula (1.4 b ) and ( 29 100 5 A t t = + ( 29 ( 29 ( 29 5 5 4 125 120 5 1 . 4 120 120 24 A A i A - - = = = = ( b ) ( 29 ( 29 ( 29 10 10 9 150 145 5 1 . 9 145 145 29 A A i A - - = = = = 6. ( a ) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 5 4 5 4 100 1.1 and 5 4 100 1.1 1.1 1.1 1 .1. 4 100 1.1 t A t A A i A = - - = = = - = ( b ) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 10 9 10 9 10 9 100 1.1 1.1 1.1 1 .1. 9 100 1.1 A A i A - - = = = - = 7. From formula (1.4 b ) ( 29 ( 29 ( 29 1 1 n A n A n i A n - - = - so that ( 29 ( 29 ( 29 1 1 n A n A n i A n - - = - and ( 29 ( 29 ( 29 1 1 . n A n i A n = + - 8. We have 5 6 7 .05, .06, .07, i i i = = = and using the result derived in Exercise 7 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 5 6 7 7 4 1 1 1 1000 1.05 1.06 1.07 \$1190.91. A A i i i = + + + = = 9. ( a ) Applying formula (1.5) ( 29 615 500 1 2.5 500 1250 i i = + = + so that 2
The Theory of Interest - Solutions Manual Chapter 1 1250 115 and 115/1250 .092, or 9.2%. i i = = = ( b ) Similarly, ( 29 630 500 1 .078 500 39 t t = + = + so that 1 3 39 130 and 130/39 10/3 3 years. t t = = = = 10. We have ( 29 1110 1000 1 1000 1000 it it = + = + 1000 110 and .11 it it = = so that ( 29 ( 29 [ ] ( 29 ( 29 [ ] 3 500 1 2 500 1 1.5 4 500 1 1.5 .11 \$582.50. i t it + = + ÷ = + = 11. Applying formula (1.6) ( 29 ( 29 .04 and .025 1 1 1 .04 1 n i i i n n = = + - + - so that ( 29 .025 .001 1 .04, .001 .016, and 16 . n n n + - = = = 12. We have 1 2 3 4 5 .01 .02 .03 .04 .05 i i i i i = = = = = and adapting formula (1.5) ( 29 ( 29 1 2 3 4 5 1000 1 1000 1.15 \$1150. i i i i i + + + + + = = 13. Applying formula (1.8) ( 29 2 600 1 600 264 864 i + = + = which gives ( 29 2 1 864/ 600 1.44, 1 1.2, and .2 i i i + = = + = = so that ( 29 ( 29 3 3 2000 1 2000 1.2 \$3456. i + = = 14. We have ( 29 ( 29 ( 29 1 1 1 and 1 1 1 n n n i i r r j j + + = + + = + + 3

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The Theory of Interest - Solutions Manual Chapter 1 so that ( 29 ( 29 1 1 1 1 . 1 1 1 i j i i j r j j j + - + + - = - = = + + + This type of analysis will be important in Sections 4.7 and 9.4. 15. From the information given: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 2 1 3 1 3/ 2 3 1 15 1 5 6 1 10 1 5/3. a a b b c c n n i i i i i i i i + = + = + = + = + = + = + = + = By inspection 5 2 1 5 . 3 3 2 = × × Since exponents are addictive with multiplication, we have . n c a b = - - 16. For one unit invested the amount of interest earned in each quarter is: Quarter: Simple: Compound: ( 29 ( 29 ( 29 ( 29 ( 29 2 3 2 4 3 1 2 3 4 .03 .03 .03 .03 1.03 1 1.03 1.03 1.03 1.03 1.03 1.03 - - - - Thus, we have ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 4 3 3 2 4 1.03 1.03 .03 1.523.
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