There is a academic paper here explaining how it works.

https://www.sciencedirect.com/science/article/pii/S105752191...

Essentially the bank can make the loan with a zero balance sheet.

If you work at the back end of a retail bank operation then you'll know that the loan department and the treasury department do not talk to each other about quantities. They talk to each other about price. Other than that the two operations operate in parallel. 'Funding' is just swapping the liabilities around in the market to try and reduce the amount the bank pays out in interest.

“I find it difficult to understand how this flow of funds analysis validates an assertion of general principle. What is described is a special case where a highly leveraged bank could find itself temporarily insolvent, with a very limited window of exposure before the bank would be found guilty of criminal fraud.”

https://www.cooperative-individualism.org/dodson-edward_revi...

I think the problem with what you're saying is the ambiguous use of the word 'cash'. Deposits, reserves, and physical currency are all 'cash' in some sense, but here it helps to be precise.

Banks don't 'lend out cash', they create deposits. Deposits are liabilities for banks. When a bank takes on a deposit, it also takes on a corresponding asset: either a loan in its books, or an increase in its reserve account.

In practice the reserve ratio is unimportant for a bank: in Canada and the UK the ratio is 0%. The real constraint on a banks ability to create money is it's leverage ratio, i.e. (assets - liabilities)/assets. In reality the formula is more complicated and uses risk weighting by asset class, but that's the crux of it.