Chapter004

# Chapter004 - The Theory of Interest Solutions Manual...

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The Theory of Interest - Solutions Manual 34 Chapter 4 1. The nominal rate of interest convertible once every two years is j , so that () 4 4 .07 1 1 and 1.035 1 .14752. 2 jj ⎛⎞ += + = −= ⎜⎟ ⎝⎠ The accumulated value is taken 4 years after the last payment is made, so that ( ) ( ) 2 8 2000 1 2000 13.60268 1.31680 \$35,824 to the nearest dollar. j sj += = 2. The quarterly rate of interest j is obtained from 4 1 1.12 so that .02874. = The present value is given by 40 20 600 200 600 24.27195 200 15.48522 \$11,466 to the nearest dollar. aa =− = ±± 3. The equation of value at time 8 t = is ( ) ( ) ( ) ( ) [ ] 100 1 8 1 6 1 4 1 2 520 iiii ++ + = so that 4 20 5.2, or 20 1.2, and .06, or 6%. ii i = = 4. Let the quarterly rate of interest be j . We have 40 40 400 10,000 or 25. = = Using the financial calculator to find an unknown j , set N4 0P V2 5P M T 1 = == and CPT I to obtain .02524, j = or 2.524%. Then 12 12 4 12 1 1.02524 and .100, or 10.0%. 12 i i = 5. Adapting formula (4.2) we have 8 32 .035 4.035 2000 1.035 57.33450 2000 1.31681 \$35,824 to the nearest dollar. 4.21494 s s

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The Theory of Interest - Solutions Manual Chapter 4 35 6. ( a ) We use the technique developed in Section 3.4 that puts in imaginary payments and then subtracts them out, together with adapting formula (4.1), to obtain () 176 32 4 200 . aa s Note that the number of payments is 176 32 36, 4 = which checks. ( b ) Similar to part ( a ), but adapting formula (4.3) rather than (4.1), we obtain 180 36 4 200 . a Again we have the check that 180 36 36. 4 = 7. The monthly rate of discount is 12 .09 .0075 12 12 j d d == = and the monthly discount factor is 1 .9925. jj vd =− = From first principles, the present value is 120 61 2 1 1 4 6 1 .9925 300 1 .9925 .9925 .9925 300 1 .9925 ⎡⎤ ++ + + = ⎣⎦ " upon summing the geometric progression. 8. Using first principles and summing an infinite geometric progression, we have 3 369 3 3 11 2 5 19 1 v vvv v i +++= = = +− and 33 91 216 o r 1 125 125 ii = += 1 3 216 6 and 1 1.2 which gives .20, or 20%. 125 5 ⎛⎞ = = = ⎜⎟ ⎝⎠ 9. Using first principles with formula (1.31), we have the present value [ ] .02 .04 .38 100 1 ee e −− +++ + " and summing the geometric progression .4 .02 1 100 . 1 e e
The Theory of Interest - Solutions Manual Chapter 4 36 10. This is an unusual situation in which each payment does not contain an integral number of interest conversion periods. However, we again use first principles measuring time in 3-month periods to obtain 81 4 0 4 33 3 1 vv v +++ + " and summing the geometric progression, we have 4 3 48 1 . 1 v v 11. Adapting formula (4.9) we have ( ) ( ) 44 10 .12 5 .12 2400 800 . aa ±± Note that the proper coefficient is the “annual rent” of the annuity, not the amount of each installment. The nominal rate of discount ( ) 4 d is obtained from () 1 4 4 4 4 1 1 1.12 and 4 1 1.12 .11174. 4 d id ⎛⎞ ⎡⎤ −= + = = = ⎣⎦ ⎜⎟ ⎝⎠ The answer is 10 5 1 1.12 1 1.12 2400 800 \$11,466 to the nearest dollar. .11174 .11174 −− ⋅− = 12. ( a ) 1 1 11 1 1 1 . t mm nn n n tt vvv va a v a d ii == −−− = = = ∑∑ ( b ) The first term in the summation is the present value of the payments at times 1 ,1 , , 1 . n m +− + The second term is the present value of the payments at times 22 2 ,1 , , 1 . n m + This continues until the last term is the present value of the payments at times 1, 2, , .

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Chapter004 - The Theory of Interest Solutions Manual...

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