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Math 618 Solution to Problem 2.2.15 r(12) = 12% = .12 implies monthly rate is 12%12 = 1% = .01. The quarterly rate is j = (1.01)3−1 = 0.030301, and the annual rate is i = (1.01)12−1 = 0.1268250301. For each year, the three payments on 4/1, 7/1, and 10/1 of $500 each are equivalent to an annual payment at the end of the year with the amount 500(1.030301)3 + 500(1.030301)2 + 500(1.030301) = 1592.753212. (This annual amount can also be obtained by the accumulated amount of four quarterly payments of $500 each subtract the last: 500s4|.030301 −500 = 1592.7532.) Using a calculator with equivalent annual payment of $1592.75 and annual rate of i = 0.1268250301, we can find the number of years needed to payoff the loan ($10,000): PMT =−1592.75, FV = 0, PV = 10000, T/Y = 12.6825% =⇒ N = 13.3239. So it takes a little over 13 years to payoff the loan. Using a calculator, we can find the present value of 13 annual payments: PMT =−1592.75, FV = 0, N = 13, T/Y = 12.6825% =⇒ PV = 9899.0912. So the 13 annual payments paid $9899.09 of the total $10,000. There is 10000−9899.09 = 100.91 to be paid by the last fractional payment. The last payment date is 13 years and 1 quarter from January 1, 2005 (or 53 quarters from January 1, 2005); it is April 1, 2018. The last payment is 100.91(1.030301)53 = 490.94. Answer: The last payment is $490.94 on 4/1/2018 Another Solution to Problem 2.2.15 r(12) = 12% = .12 implies monthly rate is 12%12 = 1% = .01. The quarterly rate is j = (1.01)3−1 = 0.030301, and the annual rate is i = (1.01)12−1 = 0.1268250301. For each year, the three payments on 4/1, 7/1, and 10/1 of $500 each are equivalent to an annual payment at the beginning of the year with the amount 500a3|.030301 = 1413.4876. Using a calculator with equivalent annual payment of $1413.49 at the beginning of the year and annual rate of i = 0.1268250301, we can find the number of years needed to payoff the loan ($10,000): (This is an annuity-due; set to BGN!) PMT =−1413.49, FV = 0, PV = 10000, T/Y = 12.6825% =⇒ N = 13.3238. So it takes a little over 13 years to payoff the loan. Using a calculator, we can find the present value of 13 annual payments: PMT =−1413.49, FV = 0, N = 13, T/Y = 12.6825% =⇒ PV = 9899.1277. So the 13 annual payments paid $9899.13 of the total $10,000. There is 10000−9899.13 = 100.87 to be paid by the last fractional payment. The last payment date is 13 years and 1 quarter from January 1, 2005 (or 53 quarters from January 1, 2005); it is April 1, 2018. The last payment is 100.87(1.030301)53 = 490.75. Answer: The last payment is $490.75 on 4/1/2018

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