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Chapter 5: Problem 59

A real estate broker tells you that doubling the period of a mortgage halves the monthly payments. Is he correct? Support your answer by means of an example.

Short Answer

Expert verified
The real estate broker's claim is incorrect. In our example, a 20-year mortgage with a $100,000 loan and a 5% annual interest rate resulted in a monthly payment of approximately $659.96. When the mortgage period was doubled to 40 years, the monthly payment decreased to approximately $429.67, which is not half of the original monthly payment. Therefore, doubling the period of a mortgage does not halve the monthly payments.

Step by step solution

01

Mortgage Payment Formula

The mortgage payment formula is given by: \(P = L \times \frac{r(1+r)^n}{(1+r)^n -1}\) Where: P = monthly payment L = loan amount r = monthly interest rate = annual interest rate / 12 n = total number of monthly payments = years * 12 We will use this formula to calculate the monthly payment for a given mortgage amount and then check if doubling the mortgage period halves the monthly payment.
02

Choose a Mortgage Example

Let's choose a simple example for a mortgage, with the following values: Loan amount (L) = $100,000 Annual interest rate (R) = 5% Mortgage period (years) = 20 years First, we will calculate the monthly payment with these values, and then we will double the mortgage period to 40 years to see the effect on the monthly payment.
03

Calculate Monthly Payments for Original Mortgage

Using the mortgage payment formula, let's calculate the monthly payment for the original 20-year mortgage example: First, we need to find the values for r and n: r = 0.05 / 12 = 0.00416667 (monthly interest rate) n = 20 * 12 = 240 (total monthly payments) Now, we plug these values into the mortgage payment formula: P = $100,000 脳 \(\frac{0.00416667(1+0.00416667)^{240}}{(1+0.00416667)^{240} -1}\) After solving the formula, we get: P 鈮� $659.96 So, with a \(100,000 loan, a 5% annual interest rate, and a 20-year mortgage period, the monthly payment is approximately \)659.96.
04

Calculate Monthly Payments for Doubled Mortgage Period

Now, let's double the mortgage period to 40 years, recalculate r and n, and plug these new values into the mortgage payment formula: n = 40 * 12 = 480 (total monthly payments) P = $100,000 脳 \(\frac{0.00416667(1+0.00416667)^{480}}{(1+0.00416667)^{480} -1}\) After solving the formula, we get: P 鈮� $429.67 So, with a \(100,000 loan, a 5% annual interest rate, and a 40-year mortgage period, the monthly payment is approximately \)429.67.
05

Compare Results

With the 20-year mortgage period, the monthly payment was \(659.96. With the 40-year mortgage period, the monthly payment became \)429.67. If doubling the mortgage period halved the monthly payment, we would expect the monthly payment to be roughly half of \(659.96, which would be around \)329.98. However, the monthly payment we calculated is significantly greater than that, at $429.67. Therefore, we can conclude that the real estate broker's claim is incorrect 鈥� doubling the period of a mortgage does not halve the monthly payments, as illustrated by our example.

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Most popular questions from this chapter

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