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I'm trying to figure out the fee-adjusted interest rate paid when incurring a balance transfer fee on a loan, not accounting for inflation.

If the APR is 4%, the loan 5000, the transfer 2% and you pay it off in 200 days (54.8% of the year when rounded to 2 decimals), I see it this way:
(APR: 4% Loan: 5,000 Transfer fee: 2% Days: 200)

Present Value: 4,900 (Loan - transfer fee %)
Future Value: 5,109.589041095890411 (loan * ( APR * (200/365) ) )
Real Rate Over 200 Days: 4.2773273693039% ( (FV - PV) / PV )
Real Rate Over 365 days: 7.806122448979618% ( ( (FV - PV) / PV ) * (365/200) )

Is there something missing?

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I've never done a balance transfer but I don't think you'll end up owing less immediately after the transfer. If you're transferring $5,000, and the transfer fee is 2% ($100), you'd either send them a check for $100 to transfer the $5,000, or you'd end up with a balance of $5,100 immediately after the transfer. It wouldn't be $4,900.

Your effective rate will also depend on how you pay off the loan. If you pay off just the minimum amount each time it's due, and then pay off the rest at the end of the 200 days (or 365 days) then you'll pay more than if you throw a big chunk at the balance at the beginning.

Let's say that you amortize this over 12 months, and make 12 equal payments. Using this calculator, with a balance of $5,100 and an interest rate of 4%, you'd make twelve payments of $434.26 (plus or minus) and pay $111.12 in interest. Add this to the $100 fee, and your effective rate is about 4.1%. Now, if you make twelve payments of $102 (2% of the initial balance), and pay off the balance (a little over $4000) in the 12th month, you'll pay almost twice as much interest ($198.63) and your effective APR will be almost 6%.

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