fractional musings

Here is a question I pose to the more accountant-minded individuals out there-- does the policy of fractional reserve subsidize bank promotions? For example, if a bank offers you $100 to open a checking account with them, is this promotion somehow subsidized by the fact that your $100 promotional gift redeposited into the new account is worth another $900 to the bank in loanable funds?

My first question is then, why does the bank need to give the money to the customer in the first place, instead of "depositing" it themselves, and reaping the full benefit. Then again, I'm not an accountant, nor a banker, so I don't know if these funds are coming from bank profits or from another depositors account, and if from the latter, I would think would just be a wash as far as the total loanable funds and gain the bank perhaps nothing besides another customer.

Can someone help me shed some light on this question?