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A savings account (Mango) offers 6% APY but has a nonwaivable $3 monthly fee.

This is my best shot determining what the account's after-fee APR is:

If you have a $5,000 balance, this means that the APY drops to 5.44%. To find this: You’d earn $308.39 given that the 6% APY compounds monthly, which I found using this calculator https://www.bankofinternet.com/calculators/apy-interest-calculator. Subtract the total yearly fees of $36 from the $308.39 to get $272.39. To find out what percent $272.39 is of $5,000, I simply divided $272.39 by $5,000 and got 0.0544. To convert that to a percent, I moved the decimal two places to the right.

I'm not sure I did the calculation correctly. The site Doctor of Credit states that Mango's after-fee APY is 5.28%, not 5.44%.

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Taking the understanding from the OP's question that the 6% includes the fee and a lower APY before fees is required, and that the APY is nominal, compounded monthly.

apy = 0.06

The effective annual rate r = (1 + apy/12)^12 - 1 = 0.0616778

For a loan of $5000 with a monthly fee of $3 the effective rate before fees is

r2 = (loan*(1 + r) - 12*fee)/loan - 1

   = r - (12*fee)/loan = 0.0544778

This is the figure the OP calculated, but it is an effective annual rate.

Converting back to a nominal rate compounded monthly the actual APY apy2 is

apy2 = 12*((1 + r2)^(1/12) - 1) = 5.31631 %

Putting all the steps together for a general function of the loan amount

apy2 = f(loan) = 12*((1 + (1 + apy/12)^12 - 1 - (12 fee)/loan)^(1/12) - 1)

               = 12*(((1 + apy/12)^12 - (12 fee)/loan)^(1/12) - 1)

The APY before fees varies with the loan amount since the fee is constant.

Plotting f(loan) with

apy = 6 %
fee = $3

enter image description here

The Doctor of Credit is assuming the 6% APY is an effective annual rate.

apy = 0.06

loan = 5000
fee = 3

apy2 = (loan*(1 + apy) - 12*fee)/loan - 1

     = apy - (12 fee)/loan = 5.28 %

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