Why Ledgers and Multipliers Belong Together
A money-creation problem can ask for both a bank’s balance sheet after a transaction and the resulting change in deposits across the banking system. These are connected questions, but they are not interchangeable. A loan recorded at one bank changes that bank’s assets and liabilities; the simple multiplier describes a possible system-wide expansion under specific assumptions. Keeping the ledger and the system model distinct prevents a common error: treating every new loan as an automatic, fixed multiple of reserves.
The balance-sheet identity is the starting point. For a commercial bank, assets include reserves, loans, and securities; liabilities include deposits and other borrowing; equity is the residual claim of owners. Every transaction must preserve the identity. The reserve requirement determines how much of a bank’s deposits must be supported by required reserves; reserves above that amount are excess reserves.
A bank may create a deposit when it grants a loan: the loan asset and the deposit liability increase together. If the borrower then pays someone who banks elsewhere, the payment is settled through reserves. The originating bank loses reserves and the deposit leaves its books. The loan remains until it is repaid. By contrast, when a borrower repays a loan using a deposit at the same bank, the bank reduces both the loan asset and deposit liability; reserves do not necessarily move.
Worked Example: A Loan, a Payment, and a Balance-Sheet Check
Worked Example: A Loan, a Payment, and a Balance-Sheet Check
A hypothetical bank has reserves of $120,000, loans of $800,000, and securities of $50,000. It has deposits of $900,000 and equity of $70,000. The reserve requirement is 10%. The bank grants a $20,000 loan by crediting the borrower’s account; the borrower immediately pays a supplier who banks elsewhere. Track the loan and clearing stages, then check the ending balance sheet and reserve position.
Method. Record loan origination first, when the bank credits its own customer’s deposit. Then record the outgoing payment when it is settled with the other bank. The two stages matter because the deposit exists on the originating bank’s books between origination and clearing.
Initial check. Assets are \(120{,}000+800{,}000+50{,}000=970{,}000\). Liabilities plus equity are \(900{,}000+70{,}000=970{,}000\), so the initial balance sheet balances. Required reserves are \(0.10(900{,}000)=90{,}000\), and initial excess reserves are \(120{,}000-90{,}000=30{,}000\).
After loan origination. Loans rise by $20,000, from $800,000 to $820,000. Deposits rise by $20,000, from $900,000 to $920,000. Reserves and securities are unchanged. Assets are \(120{,}000+820{,}000+50{,}000=990{,}000\); deposits plus equity are \(920{,}000+70{,}000=990{,}000\). Required reserves at this intermediate stage are \(0.10(920{,}000)=92{,}000\), so the bank still holds $28,000 of excess reserves.
After payment clears. The bank’s reserves fall by $20,000, to $100,000, and its deposits fall by $20,000, back to $900,000. Loans remain $820,000, because making a payment is not the same as repaying the loan. Ending assets are \(100{,}000+820{,}000+50{,}000=970{,}000\); liabilities plus equity are \(900{,}000+70{,}000=970{,}000\). Required reserves are again $90,000, leaving \(100{,}000-90{,}000=10{,}000\) of excess reserves.
Check and trap. Both stages preserve the balance-sheet identity. A common mistake is to record the $20,000 payment as a reduction in loans. The payment transfers funds to another bank; it reduces the originating bank’s reserves and deposits, while the borrower’s loan remains outstanding.
Repayment and Reserve Changes Are Different Transactions
A repayment funded from a deposit held at the same bank reduces that bank’s deposits and loans by the same amount. It does not directly change the bank’s reserve assets. Consequently, the reserve ratio can rise even though the bank has received no additional reserves: the denominator, deposits, has fallen. This is an accounting result, not evidence that the bank has received new central-bank money.
A reserve injection is different. If a bank receives reserves from the central bank while its deposits are unchanged, its excess reserves rise by the amount of the injection. In a system-wide multiplier model, the injection can support further deposit expansion if banks lend, borrowers accept deposits rather than currency, and the other assumptions of the model hold. The model does not determine how much a particular bank will lend.
Worked Example: Repayment from an Existing Deposit
A hypothetical bank initially has reserves of $100,000, loans of $650,000, securities of $100,000, deposits of $800,000, and equity of $50,000. Its reserve requirement is 10%. A customer repays $30,000 of a loan using a deposit at this same bank. Find the ending balance sheet, required and excess reserves, and the change in deposits.
Method. Because the payment is internal to the bank, extinguish the customer’s deposit liability and the corresponding loan asset. Do not change reserves.
Initial check. Assets equal \(100{,}000+650{,}000+100{,}000=850{,}000\). Deposits plus equity equal \(800{,}000+50{,}000=850{,}000\).
Record repayment. Loans become \(650{,}000-30{,}000=620{,}000\). Deposits become \(800{,}000-30{,}000=770{,}000\). Reserves remain $100,000, securities remain $100,000, and equity remains $50,000. Ending assets are \(100{,}000+620{,}000+100{,}000=820{,}000\); deposits plus equity are \(770{,}000+50{,}000=820{,}000\).
Reserve position and deposit change. Required reserves are \(0.10(770{,}000)=77{,}000\). Excess reserves are \(100{,}000-77{,}000=23{,}000\). Deposits have fallen by $30,000. The reserve ratio has risen from \(100{,}000/800{,}000=12.5\%\) to \(100{,}000/770{,}000\approx12.99\%\), because reserves stayed constant while deposits decreased.
Check and trap. The balance sheet contracts by $30,000 on each side. Subtracting $30,000 from reserves as well as from loans and deposits would make the ledger inconsistent with the stated internal repayment: no reserve payment to another bank was specified.
From a Simple Multiplier to an Adjusted Multiplier
The simple deposit multiplier is a system-wide model. Let \(R\) be reserves, \(D\) deposits, and \(r\) the required reserve ratio. If banks hold no excess reserves and each dollar of lending is redeposited in the banking system, the required-reserve relationship is \(R=rD\). Therefore \(D=R/r\), and a change in reserves supports a modeled change in deposits of \(\Delta D=\Delta R/r\). This result assumes a fixed reserve ratio, no currency leakage, no excess reserves, and banks willing and able to lend the available funds. It is not a guarantee about actual lending or a description of one bank’s individual decision.
When households hold some money as currency and banks hold excess reserves, an adjusted multiplier gives a more complete system relationship. Define \(C\) as currency held outside banks, \(D\) as deposits, \(RR\) as required reserves, \(ER\) as excess reserves, and \(R=RR+ER\) as total reserves. The money supply in this model is \(M=C+D\), and the monetary base is \(B=C+R\). If the ratios \(c=C/D\), \(r=RR/D\), and \(e=ER/D\) are fixed, then \(R=(r+e)D\).
Why the Adjusted Multiplier Has This Form
The multiplier follows directly by expressing money and the monetary base in terms of deposits. Since \(C=cD\), total money is \(M=C+D=(1+c)D\). Since \(R=RR+ER=(r+e)D\), the base is \(B=C+R=(c+r+e)D\). Dividing the first identity by the second gives the multiplier. The same fixed ratios allow the relationship to be applied to changes.
Setting \(c=0\) and \(e=0\) gives \(m=1/r\), the simple deposit multiplier. Currency held by the public and excess reserves both reduce the portion of the monetary base that supports deposits, relative to the no-leakage, no-excess-reserve case. Always identify whether a question asks for deposits, total money, or the monetary base: they are different quantities.
Worked Example: An Adjusted Multiplier with a Base Increase
A hypothetical banking system has a required-reserve ratio of 8%, a currency-to-deposit ratio of 15%, and an excess-reserve-to-deposit ratio of 2%. Suppose the monetary base increases by $50 million and these ratios remain fixed. Calculate the adjusted multiplier, the changes in deposits, currency, reserves, and total money, and the implied change in bank loans if banks’ other assets and equity are unchanged.
Method. Use the adjusted multiplier for total money, then use \(B=(c+r+e)D\) to find deposits. Separate currency from deposits and reserves, and use the bank balance-sheet identity to infer the loan change.
Multiplier and deposits. Here \(c=0.15\), \(r=0.08\), and \(e=0.02\). Thus \(m=(1+0.15)/(0.15+0.08+0.02)=1.15/0.25=4.6\). The deposit change is \(\Delta D=\Delta B/(c+r+e)=50/0.25=\$200\) million.
Currency, reserves, and money. The currency change is \(\Delta C=c\Delta D=0.15(200)=\$30\) million. The total-reserve change is \(\Delta R=(r+e)\Delta D=(0.08+0.02)(200)=\$20\) million. Therefore \(\Delta M=\Delta C+\Delta D=30+200=\$230\) million. Independently, \(m\Delta B=4.6(50)=\$230\) million, which checks the result.
Implied loan change and base check. If other bank assets and equity do not change, the change in bank loans equals the change in deposits minus the change in reserves: \(\Delta L=200-20=\$180\) million. The monetary-base check is \(\Delta C+\Delta R=30+20=\$50\) million, as specified.
Check and trap. Deposits increase by $200 million, but total money increases by $230 million because currency held by the public also rises. Reporting $200 million as the change in total money would confuse deposit expansion with money-supply expansion.
A Reliable Sequence for Integrated Problems
For a connected problem, solve the accounting before applying a multiplier. A multiplier cannot repair a balance sheet that has been recorded incorrectly. The following sequence makes each quantity and assumption explicit.
List assets, liabilities, and equity, and verify that assets equal liabilities plus equity.
Separate loan origination, payment clearing, deposit withdrawal, and repayment. State which asset and liability change at each stage.
Find required reserves from the current deposit balance, then calculate excess reserves as actual reserves minus required reserves.
Use the simple multiplier only when its no-currency, no-excess-reserve assumptions apply. Use the adjusted relationship when the specified currency and excess-reserve ratios are included.
Check both assets against liabilities plus equity and, for a multiplier calculation, verify the base, deposits, currency, and total money separately.
Common Traps and Why the Distinction Matters
The most important distinction is between a transaction recorded at one bank and a modeled change across the banking system. A bank’s loan origination creates a loan asset and deposit liability; an outgoing payment can then reduce its reserves and deposits. A repayment from a deposit at that bank reduces loans and deposits. Neither transaction, by itself, licenses multiplying its dollar amount by a system-wide multiplier.
- Changing only one side of a balance sheet. A new loan credited to a customer increases both loans and deposits. A repayment from an existing deposit reduces both.
- Treating every payment as repayment. A borrower’s payment to a supplier at another bank transfers reserves; it does not extinguish the borrower’s loan.
- Using the wrong reserve base. Required reserves are based on the current deposits specified by the problem, not the initial deposits after a transaction has changed them.
- Confusing deposits with total money. If the public holds currency, \(M=C+D\); the deposit change alone is not the total-money change.
- Applying a multiplier without its assumptions. Fixed reserve and currency ratios, lending behavior, and redepositing are model assumptions. A bank may retain reserves, face capital or liquidity constraints, or choose not to lend.
For actuarial and financial analysis, the practical value of this discipline is that it separates what is certain from what is modeled. Balance-sheet entries follow accounting identities. Multiplier results follow from behavioral and ratio assumptions. A sound answer states which kind of result it has calculated, checks the relevant identities, and does not present a conditional system estimate as a guaranteed outcome. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, keep the bank ledger distinct from any system-wide multiplier calculation.
- A hypothetical bank has deposits of $600,000 and reserves of $72,000. The required-reserve ratio is 10%. Find required reserves and excess reserves. (Answer: Required reserves $60,000; excess reserves $12,000.)
- A bank grants an $18,000 loan by crediting a customer’s account, and the customer immediately pays a recipient at another bank. Starting from reserves of $90,000 and deposits of $700,000, find the originating bank’s reserves and deposits after clearing, assuming no other transactions. (Answer: Reserves $72,000; deposits $700,000.)
- A bank initially has deposits of $800,000 and reserves of $90,000. A customer withdraws $25,000 in currency, and no reserve injection or other transaction occurs. With a 10% reserve requirement, find ending deposits, reserves, required reserves, and any reserve shortfall. (Answer: Deposits $775,000; reserves $65,000; required reserves $77,500; shortfall $12,500.)
- A bank has reserves of $100,000, loans of $650,000, securities of $100,000, deposits of $800,000, and equity of $50,000. A borrower repays $30,000 from a deposit at that bank. With a 10% reserve requirement, give the ending loans, deposits, excess reserves, and change in deposits. (Answer: Loans $620,000; deposits $770,000; excess reserves $23,000; deposits decrease by $30,000.)
- A banking system has \(r=0.08\), \(c=0.15\), and \(e=0.02\). If the monetary base increases by $50 million and the ratios remain fixed, find the adjusted multiplier and the changes in total money, deposits, currency, and reserves. (Answer: Multiplier 4.6; total money increases $230 million; deposits $200 million; currency $30 million; reserves $20 million.)