First banks acquire deposits from customers, say savings accounts, and pay their customers interest (say it's an online bank like Ally that pays a high interest rate of 1.6%, the national average savings account now pays 0.09%). Through the magic of fractional banking the bank may lend up to 10 times the amount of deposits it holds. So the leverage is 10:1. The bank's cost of the money it's loaning is 1/10 * 1.6% = 0.16%.
No. A bank may not lend 10x the amount of deposits it holds.
Fractional reserve banking is widely misunderstood and incorrectly described on the internet.
It is NOT some magic thing that lets banks multiply up their returns.
Banks take in X in deposits, and are then allowed to lend something like 0.9 X. The remaining part of the deposits is held in reserve. That is fractional reserve banking. A fraction of what they take in as deposits must be held in reserve.
The talk of multiplier effect comes from academic economists who think about the effect of these loans on the money supply. That's an entirely different discussion, and those equations don't apply to the bank's returns.
When talking about the bank's returns, note that the bank has to pay out interest on those deposits ... ALL of those deposits, the entire X. Right now interest rates are low, so the bank pays out maybe 1% interest on X deposits, but it is still on the entire X. No magic multiplier.
The bank receives interest income on the amount they loaned out, which is 0.9 X . This is LESS than X. Not some magic multiplier MORE.
In this sense banks are EXACTLY like you and me. I don't lend out ALL the cash I have. I reserve some for a rainy day.
In the above discussion I presumed the reserve requirement was 10%, but in actual practice it is not a single number, but thousands of pages of rules. The reserve requirements are different for different kinds of accounts (ie saving vs checking) and for different kinds of loans (ie mortgages vs line-of-credit) and for different kinds of investments (ie treasuries vs other kinds of bonds vs money the bank has deposited at the federal reserve).
You are correct to say that banks are paying very low rates on deposits these days. The rates vary wildly between 0.05% at some stock brokers to 1% or 1.5% at other institutions, with the occasional bank offering teaser rates on CDs over 2%.
So you can see that what banks earn is actually LESS than the difference between the incoming and outgoing interest rates multiplied by the amount of deposits. It is certainly not some multiple of this difference.
So where does all the talk of a "multiplier" effect come from?
When economists think about the money supply, ie the available amount of money for all of us to use, they note that making loans CREATES money. It works like this. (To keep this simple, lets assume the reserve requirement is 0%, and I deposit $1 in my account.) Money in my account allows the bank to lend money (perhaps you borrow $1 for the new home you're building). You don't spend that all right away. The day you take out the loan, you deposit the money in your bank account. Now there are two bank accounts with $1 in them. But now the bank has $2 on deposit and has loaned out only $1, so they can make another $1 loan to Joe. He begins by putting that in his bank account, and the process repeats. The amount of money in all our bank accounts is larger than the amount of money we started with. Even when you eventually take money out of your account to pay your architect, this just gets deposited in the architect's account, so is still counted in the total of all bank deposits. (Might be at a different bank, but economists don't care, as they study the economy as a whole.) In this simple example, the reserve requirement is 0% and the multiplier is infinite, as this can go on forever. In actual practice the reserve requirement is greater than 0, say maybe 10%, so the multiplier is something like maybe 10. In general, the multiplier is the reciprocal of the reserve requirement.
It is important to understand that this notion of a "multiplier" is not science. It is just a heuristic that economists have devised to explain the effect that lending has on the economy as a whole. It teaches, for example, that one can have some control on the money supply by controlling bank reserve requirements. This is of interest to economists, but mostly not to you and me.
More info (without conspiracy theory hype) at...
https://en.wikipedia.org/wiki/Fractional-reserve_banking