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Okay I know that the effective annual interest rate takes into account the effects of compounding, which reflects the real return. I have 2 word problems:

A firm is considering purchasing or leasing a luxury automobile for its CEO. The vehicle is expected to last 3 years. You can buy the car for $65,000 up front, or you can lease it for $1,800 per month for 36 months (payment at the end of the month). The firm can borrow at an interest rate of 8% p.a. with quarterly compounding. Should you purchase the vehicle outright or pay $1,800 per month?

Consider a $30,000 car loan with 60 equal monthly payments, computed using a 6.75% APR with monthly compounding. What is the one-month discount rate

Now I only want to understand how to use the appropriate rate for each problem.

For the first problem: The EAR of a APR of 8% with quarterly compounding is calculated to take in the effects of compounding. enter image description here

Then the monthly periodic rate of this EAR is calculated. enter image description here

But for problem 2, (the loan payment) the answer is

6.75% APR with monthly compounding corresponds to a one-month discount rate of 6.75% / 12 = 0.5625%

So my question is why don't I need to convert the APR to EAR for problem 2 to take into the account of compounding? When should I use APR or EAR?

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When should I use APR or EAR?

The EAR formula is used to convert a rate compounded at one frequency into am equivalent rate compounded at another frequency. So in the first example, you took a rate that's compounded quarterly and converted it the the equivalent rate is compounded annually. You then converted that rate to one to the equivalent rate for monthly compounding.

Note that you could have done it in one step with the formula

rm = (1+APR/4)^(1/3) - 1 = 0.66227%

which converts a quarterly-compounded rate (quoted as an APR) into a monthly-compounded rate.

In the second example, you're looking for the monthly discount rate, but already have the annual rate based on monthly compounding, so there's no need to convert to the EAR and convert back to monthly (you'd get the same result).

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