Tom borrows 100 at an annual effective interest rate of 4% and agrees to repay it with 30 annual installments. The amount of each payment in the last 20 years is set at twice that in the first 10 years. At the end of 10 years, Tom has the option to repay the entire loan with a final payment X, in addition to the regular payment. This will yield the lender an annual effective rate of 4.5% over the 10-year period. Calculate X.

X = $108.8792201 ≈ $108.88

Explanation:

the first 10 payments will be P

the last 20 payments will be 2P

the present value = 100

effective interest rate = 4%

using a present value annuity table, the annuity factor for 4%, 30 periods is 17.292

the annuity factor for 4%, 10 periods is 8.1109

since the last 20 payments are 2P, then:

PV (20 years) = (17.292 - 8.1109) · 2P

PV (20 years) = 9.1811 · 2P

PV (20 years) = 18.3622P

100 = PV (10 years) + PV (20 years) = 8.1109P + 18.3622P = 26.4731P

P = 100 / 26.4731 = 3.777419343

in order to determine payment X made at the end of the tenth year:

100 x (1 + 4.5%)¹⁰ = [P x FV (4.5%, 10 years)] + X(108.97)

100 x 1.552969422 = (3.777419343 x 12.28821) + X

155.2969422 = 46.41772214 + X

X = 155.2969422 - 46.41772214 = $108.8792201

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