Question

A loan of $30,000 is paid off in 36 payments at the end of each month in the following way: Payments of $750 are made at the end of the month for the first 12 months. Payments of $750 + x are made at the end of the month for the second 12 months. Payments of $750 + 2x are made at the end of the month for the last 12 months. What should x be if the nominal monthly rate is 8%?

Answer 1

The nominal monthly rate is 8%. Hence, the monthly rate, r = 8%/12 = 0.6667%

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PV today of first 12 payments = PV of annuity = A/ r x [1 - (1 + r)^-n] = 750 / 0.6667% x [1 - (1 + 0.6667%)^-12] = 8,621.84

PV at the end of 12 months of the second 12 months payments = PV of annuity = A/ r x [1 - (1 + r)^-n] = (750 + x) / 0.6667% x [1 - (1 + 0.6667%)^-12] = 11.4958 x (750 + x) = 8,621.84 + 11.4958x

PV today = [8,621.84 + 11.4958x] x (1 + r)^-n = [8,621.84 + 11.4958x] x (1 + 0.6667%)^-12 = 7,961.07 + 10.6148x

PV at the end of 24 months of the third 12 months payments = PV of annuity = A/ r x [1 - (1 + r)^-n] = (750 + 2x) / 0.6667% x [1 - (1 + 0.6667%)^-12] = 11.4958 x (750 + 2x) = 8,621.84 + 22.9916x

PV today = [8,621.84 + 22.9916x] x (1 + r)^-n = [8,621.84 + 22.9916x] x (1 + 0.6667%)^-24 = 7,350.95 + 19.6025x

Hence, total PV = 8,621.84 + 7,961.07 + 10.6148x + 7,350.95 + 19.6025x = 23,933.85 +30.2173x = value of the loan today = 30,000

Hence, x = (30,000 - 23,933.85) / 30.217 = 200.75 (Please do round it off as per your requirement)