Chapter008

# Chapter008 - The Theory of Interest Solutions Manual...

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The Theory of Interest - Solutions Manual 87 Chapter 8 1. Let X be the total cost. The equation of value is 12 where is the monthly rate of interest 10 j X Xa j ⎛⎞ = ⎜⎟ ⎝⎠ ±± or 12 10. j a = The unknown rate j can be found on a financial calculator as 3.503%. The effective rate of interest i is then () 12 12 1 1 1.03503 1 .512, or 51.2% ij = + −= . 2. Per dollar of loan we have 1 .12 18 1.12/18 LK n R == = = and the equation of value 18 18 1.12 1 or 16.07143. 18 jj aa The unknown rate j can be found on a financial calculator as .01221. The APR is then ( )( ) APR 12 12 .01221 .1465, or 14.65%. j = 3. The equation of value is 16 16 7.66 100 or 13.05483. j = = The unknown rate j can be found on a financial calculator as .025. The APY is then 12 12 APY 1 1 1.025 1 .3449, or 34.49%. j =+ −= 4. ( a ) Amount of interest = Total payments Loan amount Option A: ( ) 13 1000 12,000 1,000.00. Option B: 12 .01 12,000 12 12,000 794.28. a ⋅− = Difference in the amount of interest = 1,000.00 794.28 = \$205.72. ( b ) The equation of value is 12 12 12,000 1000 1000 or 11. = Using a financial calculator .013647 j = and APR 12 .1638, j = = or 16.38%.

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The Theory of Interest - Solutions Manual Chapter 8 88 ( c ) ( ) APR 12 .01 12.00%, == since the amortization rate directly gives the APR in the absence of any other fees or charges. 5. Bank 1: ( )( ) 2. 0 6 5 .04708 . 24 LL X L + Bank 2: We have () 1 12 1.126 1 .00994 j =− = so that 24 .04704 . j L YL a Bank 3: We have .01 j = and 24 .04707 . j L Z L a Therefore . YZX << 6. ( a ) The United States Rule involves irregular compounding in this situation. We have ( )( ) [ ] ( ) [] 3 9 12 8000 2000 8000 .03 6240 6240 4000 6240 .06 2614.40 2614.40 1.03 \$2692.83. B B XB = = = ( b ) The Merchant’s Rule involves simple interest throughout. We have ( ) ( ) ( ) 8000 1.12 2000 1.09 4000 1.03 \$2660.00. X =−− = 7. ( a ) The interest due at time 1 t = is ( ) 10,000 .1 1000 = . Since only 500 is paid, the other 500 is capitalized. Thus, the amount needed to repay the loan at time 2 t = is ( ) 10,500 1.1 \$11,550. = ( b ) Under the United States Rule, the interest is still owed, but is not capitalized. Thus, at time 2 t = the borrower owes 500 carryover from year 1, 1000 in interest in year 2, and the loan repayment of 10,000 for a total of \$11,500. 8. ( a ) The equation of value is 2 2 200 1 1000 1 1000 0 15 1 5 0 . ii + −+ + = + + = Now solving the quadratic we obtain 2 5 5 415 5 5 1 21 2 1.382 and 2.618 i ±− ± += = = s o t h a t .382 and 2.618 i = , or 38.2% and 261.8%.
The Theory of Interest - Solutions Manual Chapter 8 89 ( b ) The method of equated time on the payments is ( )( ) ( 200 0 1000 2 5 . 1200 3 t + == This method then uses a loan of 1000 made at time 1 t = repaid with 1200 at time 5 . 3 t = The equation of value is ( ) 1000 1 1200 j += or .20 j = for 2/3 of a year. Thus, the APR 3/ 2 .30, j or 30%. 9. Consider a loan n L a = with level payments to be repaid by n payments of 1 at regular intervals. Instead the loan is repaid by A payments of 1 each at irregular intervals. Thus, n A a represents the finance charge, i.e. total payments less the amount of loan. If B is the exact single payment point then () 1 B A i + is the present value of total payments or the amount of the loan. Thus, 1 B A Ai −+ is again the finance charge. /1000 C is the finance charge per 1000 of payment and there are A payments. Thus, 1000 A C ⎛⎞ ⎜⎟ ⎝⎠ is the total finance charge.

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Chapter008 - The Theory of Interest Solutions Manual...

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