Robert Rhodes borrows $10,000 at an annual effective interest rate of 4% and agr

Question

Robert Rhodes borrows $10,000 at an annual effective interest rate of 4% and agrees to repay it with installments at the end of each year for 30 years. The first 15 payments are each 2R, and the last 15 payments are each R. At the time of the 10th payment, Robert is given the option to repay the remainder of the loan with a final payment of X in addition to the regular payment then due. However, X is determined so that the lender will realize an annual effective interest rate of 4.5% over the entire 10-year investment period. Find X. (Answer should be: $6,879)

Explanation / Answer

Option-1: Repayment of loan in 30 years: Year end Cash Flows PVF (4%) PV = CF * PVF 1 to 15 2R    11.11839 2R*11.11839 16 to 30 R      6.17365 R*6.17365 Total Present value R*28.41042 Total Present value Should be equal to amount of loan : Hence , R*28.41042 = 10000 R = 10000/ 28.41042 $                                                                                                            351.9835 Option-1: Repayment of loan in 10 years: We know that R = 351.9835 hence 2R = 351.9835*2 = $703.9671 or say the payment for first 10 years shall be =$703.9671 Year end Cash Flows PVF (4.5%) PV = CF * PVF 1 to 10 703.9671      7.91272 $                   5,570.29 10 X      0.64393 X *0.64393 Total Present value X *0.64393 +5570.29 Total Present value Should be equal to amount of loan : Hence, X *0.64393 +5570.29 = 10000 X *0.64393 = 10000-5570.29 X = (10000-5570.29)/0.64393 $                                                                                                             6,879.18 Hence the additional Payment should be $6879.18

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