You are purchasing a home for 120,000 and are shopping for...

You are purchasing a home for $\$ 120,000$ and are shopping for a loan. You have a total of $\$ 31,000$ to put down, including the closing costs of $\$ 1000$ and any loan fee that might be charged. Bank $A$ offers a $10 \%$ APR amortized over 30 years with 360 equal monthly payments. There is no loan fee. Bank $B$ offers a $9.5 \%$ APR amortized over 30 years with 360 equal monthly payments. There is a $3 \%$ loan fee (i.e., a one-time up-front charge of $3 \%$ of the loan). Which loan is better?

Key Concepts

Total Cost of Borrowing

The total cost of borrowing considers all expenses over the life of the loan, including interest, fees, and other charges. Comparing the total cost helps borrowers understand which loan is more economical in the long run, beyond just the nominal interest rate or monthly payment amounts.

Loan Fees

Loan fees are one-time charges associated with originating a loan, such as a loan processing fee or origination fee. These fees affect the effective cost of a loan and can influence the overall affordability when comparing loan options, even if they may seem small in percentage terms.

Down Payment and Closing Costs

A down payment is the part of the home's purchase price that the buyer pays upfront, and closing costs are the fees and expenses paid at the completion of a real estate transaction. Both factors reduce the amount of money that needs to be financed and are critical in determining the principal of the loan and the financial feasibility of different mortgage options.

Loan Amortization

Loan amortization refers to the process of spreading out a loan into a series of fixed payments over time. Each payment covers both interest and a portion of the principal, and understanding this process is important in evaluating how quickly one repays the borrowed amount under different loan terms.

Annual Percentage Rate (APR)

APR is the annual rate charged for borrowing expressed as a single percentage number that represents the actual yearly cost of funds over the term of a loan. It helps borrowers compare different loan offers by taking into account not only the interest rate but also some additional fees and compounding effects, making it a crucial metric in mortgage comparisons.

Transcript

0:00 All right.

00:01 So we're given a function m of r that tells us the monthly payment associated with a given interest rate.

00:12 And then we know that there's rates offered in the range from 0 .04 to 0 .05.

00:21 In fact, we can make those inclusive with square brackets.

00:26 And we're interested in finding a monthly payment of $1 ,300.

00:32 So we want to know if there's a rate in that, in the given range where the monthly payment will be $1 ,300.

00:40 So the first thing we're going to do is try to show that there is, in fact, a solution.

00:49 And in order to do that, the goal is to, of course, use the intermediate value theorem.

00:53 So in order to satisfy the intermediate value theorem, we need to know that that m of r is continuous on this range from 0 .04 to 0 .05.

01:07 So to do that, since it's more or less a giant fraction, we just need to check that the denominator is never zero in that range.

01:21 So we'll write out the denominator here, 1 minus 1 plus r over 12 to the negative 360.

01:32 We're going to set that equal to zero.

01:34 And so we want to show that this equation doesn't have any solutions.

01:39 So we'll just move the large term to the other side, adding it to both sides.

01:45 One plus r over 12 to the negative 360.

01:51 And then what we can do is raise both sides of the equation to the negative 1 over 360 of power to get rid of this exponent.

02:04 Since the left -hand side is just one, this won't be too much trouble.

02:12 So on the right -hand side, we can cross those out.

02:14 But remember that we're taking an even root here, 360.

02:20 So that's going to give us plus or minus one as opposed to just one as the root.

02:29 And then the negative exponent just means that we're basically taking one over the result, which in this case is still just one or negative one.

02:39 So to write that out, we're going to have either plus or minus one equal to what's left on the right hand side.

02:52 So we can split this up.

02:59 So either positive 1 equals 1 plus r over 12, or negative 1 equals 1 plus r over 12.

03:12 If either of these has a solution, then there's potentially a continuity problem in our function.

03:18 So starting with the first one, we subtract one from both sides.

03:22 0 equals r over 12.

03:26 Since r is between 0 .04 and 05, this obviously has no solutions.

03:31 R is never zero in that range.

03:33 And then for the second equation, we can add one to both sides.

03:37 We get 0 equals 2 plus r over 12...