Two banks offer 30-year $150,000 mortgages at 8.5% and charge $1,000
loan application fee. Bank X refunds the fee if the application is
denied, bank Y does not. The current disclosure law requires that any
fees that will be refunded if the applicant is rejected be included in
calculating APR, but this is not required with nonrefundable fees
(presumably because refundable fees are part of the loan rather than a
fee). Now I know that for the refundable rate EAR=8.92% APR=8.57% and
for the nonrefundable rate EAR=8.84% and APR=8.50%. But how do I
actually calculate these here EARs and APRs on these two loans. So
please could you provide me with a workout for my problem.
Thanks for the attention.

If a nonrefundable fee is not part of the loan (this is your hypothesis - I don't know that piece of this), then it can be ignored. The 8.5% APR compounded monthly is equivalent to an effective rate of 8.84% (rounded):

(1+0.085/12)^12-1=0.0884

Let's calculate the monthly payment, which we'll need for the next part, using PV = PMT*[(1-(1/(1+i)^n))/i]

150000 = PMT*[(1-(1/(1+.085/12)^360))/(.085/12)]
150000 = PMT * 130.05
PMT = 1153.40

If a refundable fee is part of the loan, then the loan is for 149000.

PV = PMT*[(1-(1/(1+i)^n))/i]
149000 = 1153.4*[(1-(1/(1+i/12)^360))/(I/12)]
0 = 149000 - 1153.4*[(1-(1/(1+i/12)^360))/(I/12)]
We need to solve for I
By trial and error with a spreadsheet, I = .0857 gives the closest result.
This is the APR.

(1+0.0857/12)^12-1 = .0891 is the APR.

The solution workout is exactly what I needed.
Thanks a lot. I appreciate It.

Yours truly

Kevin Ramakers
The Netherlands