The Payans have just learned that the bank will approve them for a mortgage at an APR of [tex]$4.3 \%$[/tex] for 30 years if they meet the back-end ratio requirement. To determine whether they'll meet the requirement, the back-end ratio needs to be calculated with the actually monthly payment rather than the estimate used in part A. Use this monthly payment formula to calculate the Payans' monthly mortgage payment. Monthly payment formula: [tex]$M=\frac{p\left(\frac{r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}}{\left(1+\frac{r}{12}\right)^{12 t}-1}$[/tex], where [tex]$M=$[/tex] monthly payment [tex]$p=$[/tex] principal [tex]$r=$[/tex] interest rate [tex]$t=$[/tex] number of years A. $895.83 B. $935.12 C. $1,135.17 D. $1,237.18
Answer (2)
The Payans' monthly mortgage payment is approximately $1237.18, calculated using the given formula and assuming a principal of $250,000 at an APR of 4.3% over 30 years. Steps included substituting values into the formula, calculating the required powers and fractions, and simplifying to find the monthly payment. Therefore, the chosen answer is D: $1,237.18.
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Substitute the principal amount, interest rate, and loan term into the monthly payment formula.
Calculate the terms inside the parentheses, including the interest rate divided by 12 and 1 plus the result.
Raise the result to the power of 12 times the number of years.
Multiply the principal by the interest rate divided by 12 and the result from the previous step.
Divide the result by the result from step 3 minus 1, obtaining the monthly payment: $1237.18 .
Explanation
Understanding the Problem and Given Data We are given the monthly payment formula: M = ( 1 + 12 r ) 12 t − 1 p ( 12 r ) ( 1 + 12 r ) 12 t where: M = monthly payment p = principal r = interest rate t = number of years
We are given: r = 4.3% = 0.043 t = 30 years
We need to find the monthly payment M for a principal p = 250000 .
Substituting Values into the Formula Substitute the given values into the formula: M = ( 1 + 12 0.043 ) 12 × 30 − 1 250000 ( 12 0.043 ) ( 1 + 12 0.043 ) 12 × 30 Now, we simplify the expression step by step. First, calculate 12 0.043 :
12 0.043 ≈ 0.0035833 Next, calculate 1 + 12 0.043 :
1 + 0.0035833 = 1.0035833 Then, calculate 12 × 30 :
12 × 30 = 360 Now, we have: M = ( 1.0035833 ) 360 − 1 250000 ( 0.0035833 ) ( 1.0035833 ) 360
Calculating the Monthly Payment Calculate ( 1.0035833 ) 360 :
( 1.0035833 ) 360 ≈ 3.61177 Now, we have: M = 3.61177 − 1 250000 ( 0.0035833 ) ( 3.61177 ) Multiply 250000 × 0.0035833 × 3.61177 :
250000 × 0.0035833 × 3.61177 ≈ 3233.02 Subtract 3.61177 − 1 :
3.61177 − 1 = 2.61177 Now, we have: M = 2.61177 3233.02 Divide 3233.02 by 2.61177 :
M ≈ 1237.18
Final Answer The monthly mortgage payment is approximately $1237.18 .
Examples
Understanding mortgage calculations is essential for making informed financial decisions. For instance, if you're planning to buy a house, knowing how to calculate your monthly mortgage payment helps you determine your budget and affordability. This calculation involves the principal amount, interest rate, and loan term. By using the monthly payment formula, you can accurately estimate your monthly expenses and plan your finances accordingly, ensuring you can comfortably manage your mortgage payments over the loan's duration. This knowledge empowers you to make sound financial decisions and avoid potential financial strain.
