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 an auto loan has an APR of 6% with monthly payment how do I find the EAR

(1) First, if the fees are non-refundable, then they shouldn't affect the loan as they are not part of the loan.
(2) However, if the fees are refundable, then they are regarded as part of the loan.
Let's look at the first case. APR is 8.5%. Thus, the effective monthly rate is .7083% (8.5/12). EAR = (1+ APR/12)^12-1 = 8.839%. There's no problem so far because the calculation follows the normal process. Here, we need to calculate the monthy payments. PV=C*(1-1/(1+r)^t)/r where PV=proceeds, C= mothly amount, r=monthly rate, and t=number of months. Hence, 150,000=C*(1-1/0.7083^360)/0.7083. C=1,153.37
Now, you have refundable application fee of $1,000. You should include this in your calculation to get the right APR and EAR. One thing to notice is that even if you put the money upfront, you will have to pay the same amount of money on a monthly basis. In this case, proceeds = 149,000(150,000-1,000), monthly payment = 1,153.37, number of months = 360, and monthly rate=you don't know yet.
Again, PV=C*(1-1/(1+r)^t)/r. Plug in everything you've got. 149,000 = 1153.37 * (1-1/r^360)/r. This formula will give you 0.714394, which is the effictive monthly rate. Multiply it with 12, the you will get APR 8.5727%. And then use the formula EAR = (1+APR/12)^12 - 1, then it will give you EAR 8.9177%. Hope this helps!!