It seems like you want a complicated example.
The Federal Reserve only buys back about 10% of Treasury debt before maturity. The rest is owned by banks and individuals.
Bank reserves are always cash (physical Federal Reserve Notes or electronic credits). Bank reserves and assets are different. The Treasury debt does count as part of the banks’ assets. Treasury Debt is a “Level 1 asset”, because there’s a liquid market where they can be bought and sold. The Federal Reserve will always buy Treasury debt if the prices get too low (interest rates get too high).
Reserve ratio = (cash on hand) / (total debts + cash on hand). Reserve Ratio >= 0.1 at all times.
Leverage ratio = (total assets) / (total assets + total debt). Leverage Ratio >= 0.01 at all times. The legal leverage ratio varies by asset class; in this example, I’m using Treasury debt and assuming the legal leverage ratio is 0.01.
Assume banks are allowed to use reserve ratios of 10x. Assume banks are allowed to use leverage ratios of 100x when investing in Treasury debt. Banks will always have the maximum reserve ratio and leverage ratio that the law allows; otherwise, they’re not maximizing their profits!
Suppose Bank A has: $1B in Treasury Debt (yielding 2.2%) $1.10B in debt (at the Fed Funds Rate of 2%) $130M cash on hand
The bank has a leverage ratio of (approximately) 0.01. The bank has a reserve ratio of 0.1. The bank has a net worth of $30M. The bank is making a guaranteed riskless profit, because it’s borrowing at 2% and investing in debt yielding 2.2%.
Suppose Bank B has the same balance sheet as A: $1B in Treasury Debt (yielding 2.2%) $1.10B in debt (at the Fed Funds Rate of 2%) $130M cash on hand
The Federal government decides to auction another $1B on Treasury debt. Bank A buys. Now, Bank A has a balance sheet of: $2B in Treasuy Debt (yielding 2.2%) $1.10B in debt (at the Fed Funds Rate of 2%) -$870M cash on hand.
Bank A needs to come up with another $1B so in can pay the Federal government and not be violating its reserve requirement rules. What does Bank A do? It borrows from Bank B.
However, Bank B doesn’t have enough money to lend Bank A. Oh no! The Fed Funds rate is rising! It’s time for the Federal Reserve to “monetize the debt”. The Federal Reserve buys $100M in Treasury Debt from Bank B.
Now, Bank B’s balance sheet is: $900M in Treasury Debt $1.10B in debt $230M cash on hand.
Bank B lends $100M to bank A.
Now, Bank B’s balance sheet is: $900M in Treasury Debt (yielding 2.2%) $1.10 in debt (at the Fed Funds Rate of 2%) $100M loan to Bank A (at the Fed Funds Rate of 2%) (an asset) $130M cash on hand.
Bank A now has $100M, which it gives to the Federal government. The Federal government immediately deposits this cash in its account at Bank B.
Bank B now can loan another $90M to Bank A. Bank A pays this $90M to the Federal government. The government deposits this money in its account at Bank B.
Summing up the infinite series, Bank B creates another $900M in new money via fractional reserve banking. In practice, there’s a little “wiggle room” in the bank balance sheets, so the whole process occurs in only one step.
At the end, Bank B’s balance sheet is: $900M in Treasury Debt (yielding 2.2%) $1.10 in debt (at the Fed Funds Rate of 2%) $1B loan to Bank A (at the Fed Funds Rate of 2%) (asset) $1B in Federal Government’s account (yielding 0%) (debt) $230M cash on hand
Bank A’s balance sheet is: $2B in Treasury Debt (yielding 2%) $1.10B in debt (at the Fed Funds Rate of 2%) $1B in debt to Bank B (at the Fed Funds Rate of 2%) $130M cash on hand.
Does this help?