Sandy finds a bank willing to give her a 30-year, fixed-rate mortgage. They require her to give a 20% down payment on the purchase of her home. The interest rate (APR) is 8.4%, and she must pay 1.5 points, which will be rolled into the mortgage. After property taxes, PMI, etc., Sandy can afford $1,500 per month on the mortgage payment. What is the maximum price of the home that she can purchase? ($212,000, $222,000, $232,000, or $242,000)
The key to solving this problem is taking in piece-by-piece.
Step 1: How can Sandy repay with $1,500 per month, at 8.4%, for 30 years?
Using the DebtSmart Loan Calculator you can easily find that a mortgage principal amount of $196,892.35 satisfies the above conditions. Therefore, $196,892.35 is the total amount Sandy can afford to mortgage give $1,500 per month.
Alternately, you can use the loan formula. I talk about that in detail in my articles, Amortized Loans and How Do You Get that Loan Formula?
Step 2: Find the relationship between the mortgage principal, the amount to be financed, and cost of points.
Points = 1.5% of the amount to be financed.
The relationship is this: The total mortgage amount equals amount to be financed plus points.
Let’s say “p” is the amount to be financed of the principal before the points are added.
The equation based on the above statement is:
$196,892.35 = p + 0.015p
$196,892.35 = 1.015p
p = $196,892.35/1.015
p = $193,982.61 (Amount to be financed before points are included)
Step 3: Find the relationship between amount to be financed and the price of the house.
The relationship is this: The amount to be financed plus the down payment is equal to the maximum price of the house.
So let’s say “x” is the maximum value of the house Sandy can afford. The equation based on the above statement is:
$193,982.61 + 0.2x = x
Now it’s easy to find the original price of the house by solving for x:
$193,982.61 = x – 0.2x
$193,982.61 = 0.8x
x = $193,982.61/0.8
x = $242,478.26 (Maximum Price)
Step 4: Check the solution
If the price of a house is $242,478.26 then:
The 20% down payment is $48,495.65
Subtract the down payment from the price of the house:
Amount to be financed = $242,478.26 – $48,495.65 = $193,982.61
Points = 1.5% of $193,982.61 = $2,909.74
Roll the points into amount to be financed:
$193,982.61 + $2,909.74 = $196,892.35
Using the DebtSmart Loan Calculator you find that a principal of $196,892.35 at 8.4% for 30 year requires a monthly payment of $1,500.