Question

You have just purchased a new warehouse. To finance the purchase, you've arranged for a 25-year mortgage for 80 percent of the $1,800,000 purchase price. The monthly payment on this loan will be $10,800. What is the APR

Answer 1

Answer:

• 7.67%

Explanation:

Monthly payments from mortgages are calculated with the compounding montly interest rate.

Thus, you can "calculate" the monthly rate and the multiply by 12 to obtain the APR (annual percentage rate).

The equation for the monthly payment is:

$Monthly\text{ }Payment=Loan\times \bigg[\dfrac{r(1+r)^t}{(1+r)^t-1}\bigg]$

• Loan = 80% × $1,800,00 = $1,440,000
• Monthly payment = $10,800
• t = number of months = 25 × 12 = 300

Substitute:

      $\ 10,800=\ 1,440,000\times \bigg[\dfrac{r(1+r)^{300}}{(1+r)^{300}-1}\bigg]$

You must find r but it is very difficult to make it the subject of the equation; thus, the best is to do succesive calculations:

Tests:

          r                     monthyly payment

• 0.01                       $15,166.43     > $10,800 ⇒ lower
• 0.005                    $ 9,277.94    < $10,800 ⇒ increase
• 0.006                    $10,362.08    pretty close; increase a little bit
• 0.00639059         $10,800          ↔ this is the number

Multiply the rate by 12 (to obtain the APR): 0.00639059 × 12 = 0.07668708 = 7.67%.

• APR = 7.67% ← answer