Why a Single Bank’s Lending Capacity Depends on What Happens Next
A bank can originate a loan by recording a loan asset and crediting the borrower’s deposit. That creates a deposit liability, but it does not automatically add reserves to the lending bank. If the borrower pays someone who banks elsewhere, settlement transfers reserves out of the lending bank. If the funds stay with the bank, its reserves remain in place while its deposit liabilities rise. These different outcomes change the bank’s excess reserves by different amounts.
This distinction matters when assessing immediate lending capacity. A reserve-based calculation for one bank is conditional on what borrowers do with loan proceeds and on the assumptions used about settlement. It is not, by itself, a calculation of how much deposits can expand across the whole banking system.
The reserve requirement is applied to deposits, not directly to loans. A loan that remains as a deposit at the lending bank increases the deposit base subject to the requirement. A payment to another bank reduces the lending bank’s reserves through settlement; when the recipient’s deposit is held elsewhere, the payment also removes the borrower’s deposit from the lending bank. The loan asset remains in either case.
Two Settlement Outcomes
Begin with a bank whose required reserves are \(rD\), whose reserves are \(R\), and whose excess reserves are \(E=R-rD\). Consider a new loan of amount \(L\). First, the bank records a loan asset of \(L\) and credits a deposit of \(L\). What happens to that deposit determines the reserve effect.
| Use of loan proceeds | Change in reserves at lending bank | Change in deposits at lending bank | Change in excess reserves |
|---|---|---|---|
| Proceeds remain deposited at the lending bank | 0 | Increase by \(L\) | Decrease by \(rL\) |
| Proceeds are paid to a customer of another bank | Decrease by \(L\) | No net change after settlement | Decrease by \(L\) |
In the first row, the loan creates a deposit that remains on the bank’s balance sheet. Since reserves do not change, the required reserve amount rises by \(rL\), reducing excess reserves by that amount. In the second row, the payment first creates a deposit and then transfers it to the recipient’s bank. The lending bank ends with its original deposit total, but \(L\) fewer reserves. Its required reserves therefore return to their original level while its excess reserves fall by \(L\).
Worked Example: Proceeds Stay at the Lending Bank
Worked Example: Proceeds Stay at the Lending Bank
A bank has customer deposits of $500,000 and reserves of $130,000. Its required reserve ratio is 20%. A borrower receives a loan that remains deposited at the same bank. Find the bank’s initial excess reserves, the largest additional loan supported by those reserves under this retained-deposit assumption, and the ending reserve position if that maximum loan is made.
Method. Calculate the starting reserve requirement and excess reserves, then recognize that each dollar of retained loan proceeds adds a dollar of deposits. The reserve requirement therefore rises by 20 cents per dollar lent.
Initial required reserves. The bank must hold \(0.20\times \$500{,}000=\$100{,}000\).
Initial excess reserves. Subtract the requirement from reserves: \(\$130{,}000-\$100{,}000=\$30{,}000\).
Maximum loan with proceeds retained. A loan of \(L\) raises required reserves by \(0.20L\) and does not change reserves. All initial excess reserves are used when \(0.20L=\$30{,}000\), so \(L=\$30{,}000/0.20=\$150{,}000\).
Check the ending position. Deposits become \(\$500{,}000+\$150{,}000=\$650{,}000\). Required reserves become \(0.20\times \$650{,}000=\$130{,}000\). Reserves remain $130,000, so ending excess reserves are \(\$130{,}000-\$130{,}000=\$0\). The calculation is consistent: the bank uses all its initial reserve headroom, and the new deposit requirement exactly matches its reserves.
Common mistake avoided: Subtracting the entire $150,000 loan from reserves would be wrong in this scenario. The loan creates a deposit, but the proceeds stay at this bank, so there is no settlement outflow of reserves; the reserve effect is the increased requirement of $30,000.
Worked Example: Proceeds Transfer to Another Bank
Worked Example: Proceeds Transfer to Another Bank
A bank has deposits of $800,000 and reserves of $112,000. Its required reserve ratio is 10%. It makes a $25,000 loan, and the borrower immediately pays a supplier whose account is at another bank. Assume the payment settles in reserves and the supplier’s deposit is not at the lending bank. Find the lending bank’s excess reserves before and after the loan and settlement.
Initial requirement and headroom. Required reserves are \(0.10\times \$800{,}000=\$80{,}000\). Initial excess reserves are \(\$112{,}000-\$80{,}000=\$32{,}000\).
Record the loan. Loan origination adds a $25,000 loan asset and initially adds a $25,000 deposit liability. When the payment settles to the other bank, the lending bank’s reserves fall by $25,000 and the deposit leaves the lending bank.
Ending balances relevant to reserves. Reserves are \(\$112{,}000-\$25{,}000=\$87{,}000\). Deposits return to $800,000: the loan added $25,000, and the payment removed the same amount. Required reserves are therefore still \(0.10\times \$800{,}000=\$80{,}000\).
Ending excess reserves. Excess reserves are \(\$87{,}000-\$80{,}000=\$7{,}000\). The bank began with $32,000 of excess reserves and used $25,000 in settlement, leaving $7,000.
Check. The loan asset has increased by $25,000, reserves have decreased by $25,000, and deposits have no net change. The asset increase is matched by the reserve decrease, so the bank’s balance sheet remains balanced. Under the same full-outflow assumption, any further loan paid to another bank would use reserves dollar for dollar; the remaining $7,000 is the simplified reserve-based headroom.
Common mistake avoided: Applying the reserve ratio to deposits of $825,000 at the end would be wrong. The $25,000 deposit was transferred away, so the lending bank’s ending deposits are $800,000, not $825,000.
Worked Example: Some Proceeds Stay and Some Leave
Worked Example: Some Proceeds Stay and Some Leave
A bank has deposits of $500,000, reserves of $130,000, and a required reserve ratio of 20%. It has $30,000 in initial excess reserves. For a new loan, half of the proceeds remain deposited at the lending bank and half are paid to customers of other banks. Find the largest loan supported by the initial excess reserves under this pattern, and verify the ending balances.
Method. For a loan \(L\), half the amount, \(0.5L\), leaves as a reserve transfer. The remaining \(0.5L\) stays as an added deposit, raising required reserves by \(0.20(0.5L)\). Thus excess reserves decline by \(0.5L+0.20(0.5L)=0.60L\).
Find the maximum. Set the reduction in excess reserves equal to the initial $30,000: \(0.60L=\$30{,}000\). Therefore \(L=\$30{,}000/0.60=\$50{,}000\).
Verify reserves and deposits. Half of the $50,000, or $25,000, leaves in settlement. Ending reserves are \(\$130{,}000-\$25{,}000=\$105{,}000\). The other $25,000 remains deposited, so ending deposits are \(\$500{,}000+\$25{,}000=\$525{,}000\).
Verify the requirement and excess. Required reserves are \(0.20\times \$525{,}000=\$105{,}000\). Ending excess reserves are \(\$105{,}000-\$105{,}000=\$0\). The bank has used its entire initial reserve headroom, as required by the maximum-loan calculation.
Common mistake avoided: Treating the full $50,000 as a reserve outflow ignores the portion retained at the bank. Treating the full amount as a new deposit ignores the portion transferred away. Both effects must be included.
Why the Capacity Formula Holds
The mixed-settlement calculation can be generalized. Let \(q\) be the fraction of the loan proceeds that remains deposited at the lending bank, where \(0\leq q\leq 1\). The fraction \(1-q\) leaves through settlement. Assume no other deposits, reserve flows, or balance-sheet changes occur.
The bank’s ending excess reserves are its ending reserves less its ending required reserves. Substitute the expressions above and use \(E=R-rD\):
The coefficient \(1-q(1-r)\) is the reduction in excess reserves per dollar lent under this specific settlement pattern. If \(q=1\), all proceeds stay at the bank and the coefficient becomes \(r\), giving \(E'=E-rL\) and a reserve-based maximum of \(E/r\). If \(q=0\), all proceeds leave and the coefficient becomes 1, giving \(E'=E-L\) and a maximum of \(E\). Intermediate values lie between these cases. The result is a balance-sheet identity under the stated assumptions; it does not establish that the bank will choose to lend that much.
Common Traps and Practical Limits
- Confusing loan creation with reserve creation. Loan origination adds a loan asset and a deposit liability; it does not itself add reserves to the bank.
- Assuming every loan deposit stays. A borrower’s payment can transfer both the deposit claim and settlement reserves to another bank. Track the destination of the funds.
- Using the wrong capacity calculation. Under the retained-deposit assumption, the reserve requirement rises by \(rL\), so the reserve-only headroom is \(E/r\). Under the full-outflow assumption, reserves fall by \(L\), so the corresponding headroom is \(E\).
- Calling reserve headroom an unconditional lending limit. This calculation isolates a reserve requirement and settlement pattern. It does not include other factors that may affect a real bank’s ability or willingness to lend.
- Inferring system-wide deposit expansion from one bank. A reserve transfer leaving one bank arrives at another institution. Understanding the resulting system-wide sequence requires tracing multiple banks, rather than treating the first bank’s calculation as the whole outcome.
Check Your Understanding
For each problem, state the relevant settlement assumption and track deposits and reserves rather than applying one capacity formula automatically.
- A bank has deposits of $400,000, reserves of $58,000, and a 12% required reserve ratio. If all new loan proceeds remain at the bank, what is its initial excess reserve amount and its reserve-based maximum additional loan? (Answer: Initial excess reserves are $10,000; maximum loan is $83,333.33.)
- A bank has $18,000 in excess reserves. A new loan is paid entirely to a customer of another bank and settles in reserves. How much reserve-based headroom remains after a $14,000 loan? (Answer: $4,000.)
- A bank has $20,000 in excess reserves and a 25% reserve ratio. For each loan, 40% of proceeds remain at the lending bank and 60% leave through settlement. What is the reserve-based maximum loan under this pattern? (Answer: $28,571.43.)
- A bank has $9,000 in excess reserves and a 10% reserve ratio. It lends $30,000, and all proceeds stay deposited at the bank. By how much do required reserves rise, and what excess reserves remain? (Answer: Required reserves rise by $3,000; $6,000 in excess reserves remain.)
- A bank has $15,000 in excess reserves, a 20% reserve ratio, and a lending pattern in which 25% of proceeds stay as deposits while 75% leave through settlement. If it makes a $15,000 loan, how much excess reserve remains? (Answer: $3,000.)