Question

You have just purchased a new warehouse. To finance the purchase, you’ve arranged for a 35-year mortgage loan for 75 percent of the $3,250,000 purchase price. The monthly payment on this loan will be $15,800.

Requirement 1:
What is the APR on this loan? (Round your answer as directed, but do not use rounded numbers in intermediate calculations. Enter your answer as a percent rounded to 2 decimal places (e.g., 32.16).)


Requirement 2:
What is the EAR on this loan? (Round your answer as directed, but do not use rounded numbers in intermediate calculations. Enter your answer as a percent rounded to 2 decimal places (e.g., 32.16).)

Answer 1

PV of loan = .75(3250000) = $2,437,500
paym = 15800
n = 12(35) =420
let the monthly rate be i
2437500 = 15800( 1 - (1+i)^-420)/i
154.2721519 i = 1 - (1+i)^-420

This is a very nasty equation to solve.
Back in the "good ol' days", we used a method called interpolation, and it took forever to do something like this.
Often there there tables and charts, but that n = 420 would have been beyond most of the n values in the chart, so that caused further problems.

I assume you have access to some software or calculator technology that will solve this for you.
Using Wolfram, I got i = .00594417
http://www.wolframalpha.com/input/?i=154.2721519+x+%3D+1+-+(1%2Bx)%5E-420
- notice I changed the i to x, it assumed i was a complex number.

check:
15800( 1-1.00594417)^-420)/.00594417
= 2,437,501.39
not bad, off by about $1 in 2 1/2 million!!

so if i = .00594417, then 12i = .07133
or 7.13% per annum compounded monthly

for b) I would simply find the annual rate j, so that

1+j = 1.00594417^12
j = .0737088...
So the effective annual rate is 7.37%