A loan of $88,700 can be paid off by monthly payments of $1,000 for 11 years. What is the semi-annually compounded nominal interest rate?

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The semi-annually compounded nominal interest rate can be found using the formula for the present value of an annuity. Given the loan amount (PV) of $88,700, monthly payments (PMT) of $1,000, and the total number of payments (n) being 11 years * 12 months = 132, we need to find the interest rate (i) that satisfies the equation:$$PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$$However, since the interest is compounded semi-annually, we adjust the formula to account for semi-annual compounding. The effective monthly interest rate (i) is related to the semi-annual nominal rate (r) by the equation:$$i = \left(1 + \frac{r}{2}\right)^{\frac{2}{12}} - 1$$Solving this problem typically requires iterative methods or financial calculators, as there's no straightforward algebraic solution. By inputting the given values into a financial calculator or using an iterative approach (such as the Newton-Raphson method or trial and error in a spreadsheet), we find that the semi-annually compounded nominal interest rate that satisfies these conditions is closest to 7.86%, option D.

A Loan of $88,700 Can Be Paid Off by Monthly