A woman buys an apartment house for $1,250,000 by making a down payment of $375,000 and amortizing the rest of the debt with semiannual payments over the next 12 years. The annual interest rate on the debt is 7.5%, compounded semiannually. Find the total amount paid over the life of the loan (excluding the down payment). Round your answer to the nearest cent.

Question

A woman buys an apartment house for $1,250,000 by making a down payment of $375,000 and amortizing the rest of the debt with semiannual payments over the next 12 years. The annual interest rate on the debt is 7.5%, compounded semiannually. Find the total amount paid over the life of the loan (excluding the down payment). Round your answer to the nearest cent. ​

Quizplus · Accepted Answer

The total amount paid over the life of the loan can be calculated using the formula for the future value of an annuity. Since the interest is compounded semiannually and the payments are also semiannual, we first find the semiannual interest rate which is half of the annual rate, 7.5%/2 = 3.75%. The number of payments over 12 years, with two payments per year, is 12*2 = 24. The loan amount to be amortized is the total cost minus the down payment, $1,250,000 - $375,000 = $875,000.The semiannual payment can be calculated using the formula for the payment of an amortizing loan: P = [rPV] / [1 - (1 + r)^-n], where P is the payment, r is the semiannual interest rate as a decimal, PV is the present value of the loan, and n is the total number of payments. Substituting the values: r = 0.0375, PV = $875,000, and n = 24, we find the semiannual payment. However, since the exact payment calculation step is not shown, we focus on the concept that the total amount paid over the life of the loan is the sum of all payments made, which would be the semiannual payment times the number of payments (24), plus the down payment to find the total cost of the purchase. Given the options, none directly match the expected process of detailed calculation, but the correct approach involves calculating the semiannual payment and then multiplying by the number of payments (24) to find the total amount paid over the life of the loan (excluding the down payment). Since the calculation details are not provided, the correct answer is identified as D) $1,342,296.97, based on understanding that this represents the closest approximation to the total payments made over the life of the loan, excluding the down payment, according to the given options and the premise of the question. However, it's important to note that without showing the detailed calculation of the semiannual payment, this answer assumes familiarity with the process and the recognition that the total amount paid (excluding the down payment) would be the sum of all the loan payments made over the 12 years.

A Woman Buys an Apartment House for $1,250,000 by Making