Following a Deposit Through Multiple Banks
When a bank lends part of a deposit and the loan proceeds are deposited at another bank, the funds can support another round of lending. Tracking that sequence makes clear how deposits and loans build up across banks—and why a new deposit at one bank is not the same thing as new reserves for the banking system.
This tutorial uses a simplified model to follow a specific chain of deposits. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. The model is useful for learning the mechanics, but it is not a claim that banks in practice lend every available dollar or that deposits always remain in the banking system.
The setup assumes that each bank applies the same reserve ratio, lends all funds above required reserves, and that every loan is redeposited in full at another bank. There are no cash withdrawals, reserve leakages, loan repayments, or changes in the reserve ratio during the sequence. The final bank shown in a finite calculation has received a deposit but has not yet made its own loan unless the problem says otherwise.
These assumptions describe a teaching model, not a complete account of bank lending. In particular, reserves are assets held by banks; they are not created each time a loan is made. A payment to a borrower at another bank transfers reserves between banks, while the loan creates a deposit for the borrower.
The Round-by-Round Relationship
At a bank with deposit \(D_k\), required reserves are \(rD_k\). The amount available for lending is therefore the deposit less those required reserves:
When the loan is deposited at the next bank, it becomes that bank’s deposit. Thus \(D_{k+1}=L_k\), giving the round-to-round rule:
Why this relationship holds. The bank starts with a deposit of \(D_k\). By assumption, it retains exactly the fraction \(r\), or \(rD_k\), and lends the balance. Subtracting the retained amount from the original deposit gives \((1-r)D_k\). If all proceeds are then deposited at the next bank, that balance is exactly \(D_{k+1}\). Repeating the same reasoning shows that, after \(k\) transitions, \(D_k=(1-r)^kD_0\). This expression describes an individual round; the examples below add the relevant finite rounds explicitly.
A round’s deposit is not the same as total deposits in the chain. If the initial deposit is \(D_0\) and \(N\) loans have been made, the deposits recorded include \(D_0\) and the \(N\) loan proceeds deposited at later banks. Total loans are the sum of those \(N\) loans. Counting the starting deposit and each new deposit once gives total deposits equal to the initial deposit plus the loans made.
Worked Example: Four Loans from an Initial Deposit
A hypothetical banking system receives an initial deposit of $80,000. Each bank has a required-reserve ratio of 25 percent, lends all funds above required reserves, and has its loan proceeds deposited in full at the next bank. Follow four loans. Find the deposit, required reserves, and loan in each originating bank, then find total deposits, total loans, and total system reserves.
Method. For each originating bank, calculate required reserves as \(0.25D_k\), then calculate the loan as \(0.75D_k\). The next bank’s deposit is the preceding bank’s loan.
| Originating bank | Deposit | Required reserves | Loan |
|---|---|---|---|
| Bank 1 | $80,000 | $20,000 | $60,000 |
| Bank 2 | $60,000 | $15,000 | $45,000 |
| Bank 3 | $45,000 | $11,250 | $33,750 |
| Bank 4 | $33,750 | $8,437.50 | $25,312.50 |
After the fourth loan is deposited, Bank 5 receives $25,312.50. The deposits counted are $80,000, $60,000, $45,000, $33,750, and $25,312.50. Their total is:
The four loans total:
The initial $80,000 of reserves is transferred as the loan proceeds move from bank to bank; it is not multiplied by the lending process. The system still has $80,000 in reserves. As a balance-sheet check, total deposits equal total loans plus total reserves: $164,062.50 + $80,000 = $244,062.50.
Bank 5 has not yet made a loan in this calculation. It holds the $25,312.50 it received. Of that balance, $6,328.125 is the required reserve at 25 percent, and the remaining $18,984.375 is available for a possible later loan. The remaining funds explain why, at this stopping point, Bank 5’s actual reserves exceed its required reserves.
Common mistake avoided: Counting only the initial deposit and the first three loans would omit the fourth loan and the deposit it creates. Four loans require four loan amounts in the total, and the final loan proceeds are also part of total deposits.
Reserves Are Transferred, Not Recreated
A loan paid to someone who banks elsewhere generally results in a reserve transfer between banks as the payment settles. The lending bank’s reserves fall by the payment amount; the receiving bank obtains those reserves along with the customer’s deposit. For the banking system as a whole, however, a transfer from one bank to another does not change the total reserves.
This distinction matters when comparing three quantities:
- Round deposits: the amount deposited at one particular bank in the sequence.
- Cumulative deposits and loans: totals across the banks included in the specified rounds.
- Total system reserves: the reserves held across all banks. In this model, the amount remains equal to the initial reserve deposit as funds move through the chain.
Required reserves are calculated against each bank’s deposits, but the reserves held at a bank need not equal its required amount at every point in a stopped chain. Banks that have already lent hold the required portion under the model. The last receiving bank holds the incoming funds until it makes a loan, so some of its reserves may be excess reserves. Do not add the required reserves at each bank to the original reserves as if they were newly created; those reserves are portions of the same transferred funds.
Worked Example: Tracking a Chain with a Different Reserve Ratio
A hypothetical system starts with a $50,000 deposit. The reserve requirement is 20 percent, and four banks each make one loan whose full proceeds are deposited at the next bank. Calculate the deposits, loans, total deposits, total loans, and total reserves. Also identify the required and excess reserves held by the last bank after it receives the fourth loan.
Method. Each originating bank lends 80 percent of its deposit. The successive deposits are therefore $50,000, then $40,000, then $32,000, then $25,600; the fourth loan creates a deposit of $20,480 at the next bank.
| Originating bank | Deposit | Required reserves at 20% | Loan at 80% |
|---|---|---|---|
| Bank 1 | $50,000 | $10,000 | $40,000 |
| Bank 2 | $40,000 | $8,000 | $32,000 |
| Bank 3 | $32,000 | $6,400 | $25,600 |
| Bank 4 | $25,600 | $5,120 | $20,480 |
The five deposits, including the deposit at the final receiving bank, sum to $50,000 + $40,000 + $32,000 + $25,600 + $20,480 = $168,080. The four loans sum to $40,000 + $32,000 + $25,600 + $20,480 = $118,080. Total system reserves remain $50,000, so the check is $118,080 + $50,000 = $168,080.
The final receiving bank holds $20,480 in reserves before it makes its own loan. Its required reserves are \(0.20 \times \$20,480=\$4,096\), leaving $20,480 − $4,096 = $16,384 in excess reserves. The reserves held by the four originating banks total $10,000 + $8,000 + $6,400 + $5,120 = $29,520. Adding the final bank’s $20,480 gives $50,000 in system reserves.
Common mistake avoided: The required reserves held by the originating banks are not the system’s total reserves. The final receiving bank also holds reserves, including the portion not yet lent onward.
Inferring the Pattern from Observed Deposits
Sometimes a problem gives deposits or loans in the first rounds and asks you to continue the chain. If a bank’s deposit and loan are known, their difference is the amount it retained. Dividing that retained amount by the deposit gives the reserve ratio. Once the ratio is established, the same round-by-round rule applies to later deposits.
Worked Example: Infer the Reserve Ratio and Continue the Chain
A hypothetical system receives an initial deposit of $90,000. Bank 1 lends $72,000, and the full loan proceeds are deposited at Bank 2. Assume the same reserve ratio at each bank and that each bank lends all funds above required reserves. Find the reserve ratio, follow three loans, and calculate total deposits, total loans, and total system reserves.
Step 1: Infer the ratio. Bank 1 retains $90,000 − $72,000 = $18,000. The reserve ratio is $18,000 ÷ $90,000 = 0.20, or 20 percent.
Step 2: Calculate the next rounds. Bank 2 receives $72,000 and lends 80 percent: \(0.80 \times \$72,000=\$57,600\). Bank 3 receives $57,600 and lends \(0.80 \times \$57,600=\$46,080\). Thus the three loans are $72,000, $57,600, and $46,080; the final loan is deposited at Bank 4.
Step 3: Total the deposits and loans. Deposits are $90,000 + $72,000 + $57,600 + $46,080 = $265,680. Loans are $72,000 + $57,600 + $46,080 = $175,680. System reserves remain $90,000, and $175,680 + $90,000 = $265,680.
Bank 4 has received $46,080 but has not yet lent it. Its required reserves at 20 percent are $9,216, and its excess reserves are $46,080 − $9,216 = $36,864. This final check confirms both the inferred reserve ratio and the distinction between required and total reserves at a finite stopping point.
Common mistake avoided: Dividing the loan by the original deposit gives the lending fraction, 80 percent—not the reserve ratio. The retained fraction is the reserve ratio: \((\$90,000-\$72,000)/\$90,000=20\%\).
How to Organize a Multi-Bank Calculation
A short schedule reduces errors, particularly when a problem specifies several loans or asks for both round amounts and cumulative totals. Use the same sequence each time:
Label it \(D_0\). Confirm the reserve ratio and the number of loans required.
For each deposit \(D_k\), compute \(rD_k\) and \(L_k=(1-r)D_k\).
If the full loan is redeposited, set the next bank’s deposit equal to that loan. Continue for exactly the stated number of loans.
If the last loan is deposited at another bank, include that deposit in total deposits even when that bank has not yet made a loan.
In this model, total deposits equal the initial reserve deposit plus total loans. Total system reserves remain equal to the initial deposit.
Common Exam Traps
- Stopping one deposit too early. If the problem says that the proceeds of the final loan are deposited at another bank, include that final deposit in total deposits.
- Confusing the lending fraction with the reserve ratio. When \(r=0.20\), the loan fraction is \(1-r=0.80\). The two fractions are complements, not the same quantity.
- Adding reserves as if each bank created them. A reserve transfer changes where reserves are held, not the banking system’s total reserve balance.
- Reporting only round amounts when asked for totals. A bank’s deposit and loan are individual-round figures; cumulative totals require adding the deposits or loans across all specified rounds.
- Assuming the last bank has already lent. Unless another loan is specified, a final bank that has just received loan proceeds still holds the full deposit in reserves.
Check Your Understanding
For each problem, follow the stated number of loans and include the final loan proceeds in deposits when they are deposited at another bank. Unless a problem says otherwise, assume all loan proceeds are redeposited in full, the reserve ratio is constant, and the final receiving bank has not yet made a loan.
- A hypothetical system receives $64,000 with a reserve ratio of 25 percent. Two loans are made, and each is deposited at the next bank. Find total deposits, total loans, and total system reserves. (Answer: Deposits $148,000; loans $84,000; reserves $64,000.)
- An initial deposit of $125,000 is subject to a 20 percent reserve ratio. Follow three loans. Find the four deposits in the chain, total deposits, total loans, and total reserves. (Answer: Deposits by bank: $125,000, $100,000, $80,000, $64,000; total deposits $369,000; loans $244,000; reserves $125,000.)
- A $90,000 initial deposit produces a first loan of $72,000. With a constant reserve ratio, follow three loans. Find the reserve ratio, total deposits, total loans, and total reserves. (Answer: Reserve ratio 20%; deposits $90,000, $72,000, $57,600, $46,080; total deposits $265,680; loans $175,680; reserves $90,000.)
- A system begins with $200,000 and a 10 percent reserve ratio. After two loans are deposited at the next bank, find the deposit received by the third bank, the second originating bank’s required reserves and loan, and the cumulative totals. (Answer: Third-bank deposit $162,000; second-bank required reserves $18,000; second loan $162,000; total deposits $542,000; total loans $342,000; reserves $200,000.)
- A $128,000 initial deposit is subject to a 25 percent reserve ratio. Four loans are made, with each loan deposited at the next bank. Find total deposits, total loans, and total reserves. (Answer: Deposits $390,500; loans $262,500; reserves $128,000.)