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How Much House?

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Note: You can use any financial calculator to do this problem, but if you want the BEST, you can get our 10bii Financial Calculator for iOS, Android, Mac, and Windows!

When people go to buy a house, they often focus on their maximum price tag. That makes sense; most of the time house shoppers have a pre-approval from a bank which specifies the most money the bank is willing to lend. The amount banks will lend a person depends on a lot of factors, but one of those factors is the size of a person's income. The more income a person makes, the more money they'll be able to devote to a loan payment, and the higher the loan balance can be. As a completely notional example, let's say that Sarah makes $78,000 per year, and pays 40% of that in taxes. A bank will loan her money at 4.75%, amortizing over 30 years, so long as the payment on that loan (principal and interest only, excluding taxes and insurance, which are often bundled into a house payment) does not exceed 35% of Sarah's after-tax income. The Question: What's the most money the bank will be willing to lend Sarah based on her income and the bank's lending guidelines? For the purposes of this question, assume that no down payment is required for the bank to lend Sarah the money. THE SOLUTION First of all, we need to know how much after-tax income Sarah has each month, so we can figure out what her maximum loan payment will be. $78,000 annually = $78,000 ÷ 12 = $6,500 monthly, before taxes. $6,500 x (100% - 40% taxes) = $3,900 monthly, after taxes. $3,900 x 35% = $1,365 maximum monthly loan payment. Now on to the calculation... First things first, make sure the calculator is using 12 Payments per Year. N: 360 (The loan will last for 30 years) I/YR: 4.75 (The lender will lend at 4.75%) PV: (This is what I'm trying to find) PMT: -1,365 (Sarah's maximum loan payment is $1,365 per month) FV: 0 (The loan amortizes fully, so nothing is left at the end)

The most the bank will lend Sarah is $361,671.04. If she wants to buy a house for more than the $361K, she can do that, but she'll need to make up the difference between the purchase price and the loan amount with a down payment.

What do you think? Does the bank's method of determining the amount it's willing to lend to Sarah make sense? Do you think Sarah should borrow the full amount, or borrow less and save money each month? Why? Let us know in the comments!