Why a Reserve Number Does Not Dictate Lending
A simple deposit multiplier can be useful for showing how a change in reserves might support deposit expansion under a tightly specified model. It is not a mechanical rule that tells a bank how much it will lend. A bank needs creditworthy borrowers willing to borrow, and it must decide whether a proposed loan is attractive given its risk, costs, and funding options. Its balance sheet and payment flows also affect the decision.
The key distinction is between a model’s potential deposit expansion and a bank’s actual lending choices. Reserves are important for meeting reserve requirements and settling payments with other banks. But a bank does not ordinarily hand reserves to a borrower when it originates a loan: the loan creates a deposit, and later payments may cause reserves to move between banks.
What the Simple Multiplier Assumes
In the basic model, banks hold no excess reserves, the public holds no additional currency, the reserve requirement is fixed, and banks repeatedly lend all funds not required as reserves. The model also assumes that each loan is redeposited in the banking system and that banks can find willing, qualified borrowers at each step. Relaxing these assumptions changes the result or prevents the model’s lending sequence from continuing.
The model’s arithmetic can be seen as a sequence. Suppose the banking system receives an initial reserve increase \(\Delta R\). With a required reserve ratio \(r\), the first deposit increase is \(\Delta R\). The bank retains \(r\Delta R\) as required reserves and can lend the remainder, \((1-r)\Delta R\). If that amount is deposited at another bank, the next bank can lend the fraction \(1-r\) of it. Continuing under the model’s assumptions gives:
The final equality uses the sum of an infinite geometric series with common ratio \(1-r\), where \(0<r\leq 1\). This derivation explains why the multiplier works inside the model: each round is assumed to generate another deposit and another round of lending. It does not show that actual banks will make every loan in that sequence. If qualified loan demand runs out, a bank holds more excess reserves, or funds leave the banking system, the chain differs from the model.
Loan Demand and Credit Decisions
A bank cannot lend simply because a calculation suggests that reserves could support more deposits. A household or firm must want a loan on the available terms. The bank then assesses whether that applicant is likely to repay, what collateral or other protections are available, and whether the expected return compensates for credit and operating costs. If few borrowers apply, or few meet the bank’s standards, lending may remain below the amount implied by a multiplier.
A loan’s stated interest rate alone does not establish that it is profitable. The bank must account for the possibility of missed payments or default, recoveries after default, funding costs, and expenses associated with making and servicing the loan. A high stated rate may reflect greater risk; it does not guarantee a higher expected return.
Worked Example: Reserve Headroom and Qualified Loan Demand
A bank has deposits of $1,000,000 and reserves of $130,000. The required reserve ratio is 10%. A borrower group applies for $240,000 in loans, but only $180,000 of applications meet the bank’s stated credit standards. Assume the new loans are initially deposited at this bank, there is no currency withdrawal or outward payment, and the bank makes as many qualified loans as its reserve position permits. How much does it lend, and what is its reserve position afterward?
Method. Compare reserve-based deposit headroom with qualified loan demand. This calculation avoids treating the theoretical multiplier as a guarantee that borrowers will be available.
First calculate required reserves on the existing deposits:
The bank’s excess reserves are therefore:
If each dollar of new deposits requires 10 cents of reserves, this excess could support up to:
of additional deposits under these assumptions. But only $180,000 of loan applications qualify, so the bank lends $180,000, not $300,000. Loan origination creates a $180,000 deposit at the bank. New deposits are $1,180,000, and required reserves become:
Reserves remain $130,000 at origination, so the bank has $130,000 − $118,000 = $12,000 in excess reserves. The bank’s lending was limited by qualified demand in this example, not by the full reserve-based headroom. The check is that $12,000 is nonnegative and equals the original $30,000 excess less 10% of the $180,000 deposit increase.
Credit Risk Can Change Which Loans a Bank Makes
A bank may have enough reserves and applicants yet still decline some applications or choose one loan over another. Expected credit losses matter because default can prevent repayment of principal, and recovery after default may be less than the amount owed. A simplified expected-profit calculation can make the trade-off visible. It is only a screening illustration: real loan pricing and risk assessment may involve additional costs, timing, uncertainty, and contractual details.
Worked Example: Comparing Expected Loan Returns
A hypothetical bank is considering a one-year, $200,000 loan. The stated annual interest rate is 7%. Assume that if the borrower defaults, no interest is paid and the bank recovers 35% of the principal at the end of the year. The bank estimates a 3% probability of default. Its one-year funding cost is 4% of principal, and operating costs are 1.5% of principal. Assume no fees, taxes, or other costs. Estimate expected profit for the year.
Method. Calculate expected interest receipts, expected credit loss on principal, funding cost, and operating cost separately. This avoids counting the full stated interest as certain income.
If there is no default, the interest is:
The probability of no default is \(1-0.03=0.97\). Under the assumptions, expected interest is therefore:
The bank loses 65% of principal if default occurs, because it recovers 35%. The loss conditional on default is:
Multiplying by the 3% default probability gives an expected credit loss of \(0.03\times \$130{,}000=\$3{,}900\). The funding cost is \(0.04\times \$200{,}000=\$8{,}000\), and operating costs are \(0.015\times \$200{,}000=\$3{,}000\). Thus expected profit, before any costs excluded by the problem, is:
The simplified expected profit is −$1,320. The negative result does not prove that every bank would reject this loan; the assumptions omit potentially relevant factors. It does show why a bank does not necessarily lend simply because a borrower is willing and reserves are available. The check is that expected interest is below the sum of expected credit loss and the two stated costs: $13,580 < $3,900 + $8,000 + $3,000.
Funding and Payments Affect the Balance Sheet
A loan initially creates a deposit, but a borrower may use that deposit to pay someone who banks elsewhere. The payment then reduces the lending bank’s deposits and transfers reserves to the recipient’s bank through the settlement process. The system-wide deposit may remain, but the individual bank has fewer deposits and fewer reserves. That bank may seek deposits or other funding, manage its liquid assets, or adjust its lending plans.
A bank can obtain funding in more than one way. It may attract deposits, borrow in money markets, or use other available funding arrangements. Each option has costs and conditions. A bank weighing a loan therefore considers not just its initial reserves, but also the likely movement of funds and the price and availability of replacement funding. These choices do not make reserves irrelevant; they explain why reserves alone do not prescribe a fixed lending amount.
Worked Example: An Outward Payment and Reserve Funding
A hypothetical bank has deposits of $2,000,000 and reserves of $250,000. The required reserve ratio is 10%. It makes a $300,000 loan, and the borrower initially receives a deposit at this bank. The borrower then pays $120,000 to a supplier who banks elsewhere. Assume the payment is settled immediately, the reserve requirement applies to the bank’s deposits, and the bank borrows enough reserves to eliminate any resulting reserve shortfall. Track the balances and calculate the borrowing required.
Method. Record loan origination first, then separately record the outward payment. This avoids the mistake of assuming that reserves leave the bank as soon as it makes a loan.
Before lending, required reserves are \(0.10\times \$2{,}000{,}000=\$200{,}000\), so excess reserves are \(\$250{,}000-\$200{,}000=\$50{,}000\). At loan origination, deposits rise by $300,000 while reserves initially remain $250,000. Deposits become $2,300,000, and required reserves become:
Before the payment is made, the bank has \(\$250{,}000-\$230{,}000=\$20{,}000\) of excess reserves. The $120,000 payment to the supplier’s other bank reduces this bank’s deposits by $120,000 and transfers $120,000 of reserves out. After settlement, deposits are \(\$2{,}300{,}000-\$120{,}000=\$2{,}180{,}000\), and reserves are \(\$250{,}000-\$120{,}000=\$130{,}000\).
Required reserves on the new deposit balance are \(0.10\times \$2{,}180{,}000=\$218{,}000\). The reserve shortfall is \(\$218{,}000-\$130{,}000=\$88{,}000\), so the bank must borrow $88,000 in reserves under the problem’s assumption. After borrowing, reserves are $218,000, exactly equal to required reserves. This check confirms that the bank’s post-payment reserve position meets the stated requirement, but it also shows why a bank’s funding needs can change after loan origination.
What the Multiplier Can—and Cannot—Tell You
The multiplier remains a useful conditional calculation. It can answer a question such as, “Under the basic model, what deposit expansion is associated with this reserve change?” It cannot, by itself, answer, “How much will banks actually lend?” The latter requires information about borrower demand, credit quality, bank funding, payment flows, and willingness to hold excess reserves. Balance-sheet conditions, including the bank’s capacity to absorb losses, also influence decisions; the detailed role of capital requirements is a separate question.
For example, a low reserve ratio produces a large value of \(1/r\). That mathematical result does not establish that a bank can find enough sound borrowers or that it will want to use all available funding to make loans. Conversely, a bank may have ways to fund lending beyond its initial excess reserves, although those arrangements have costs and constraints. A single reserve-based number cannot capture all those choices.
This distinction is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus: candidates should be able to use the simple multiplier while recognizing the assumptions that make it valid. In a real-world explanation, treat it as a model benchmark, not as a mechanical lending law.
Common Mistakes
- Calling reserves the amount banks lend. A loan creates a deposit at origination; reserves are used to meet reserve requirements and settle payments.
- Applying \(1/r\) without checking the model’s assumptions. A multiplier calculation is conditional on continued lending and redepositing, among other assumptions.
- Assuming every applicant receives a loan. Banks screen applicants, and willingness to borrow is not the same as credit approval.
- Treating the stated loan rate as the bank’s net return. Expected credit losses, funding costs, and operating costs can reduce or outweigh interest income.
- Ignoring where a borrower spends the loan proceeds. An outward payment can reduce both deposits and reserves at the lending bank.
Check Your Understanding
Work through each problem, keeping the model assumptions separate from the bank’s actual choices.
- A bank has $24,000 in excess reserves and a required reserve ratio of 8%. It receives $220,000 of loan applications, of which $180,000 meet its credit standards. Assume loans are initially deposited at the bank and there are no outward payments. What is reserve-based deposit headroom, how much can it lend under these facts, and what excess reserves remain? (Answer: Headroom is $300,000; it lends $180,000; $9,600 of excess reserves remain.)
- A bank considers a one-year $200,000 loan at 7%. Default probability is 3%, recovery after default is 35% of principal, funding costs 4% of principal, and operating costs are 1.5% of principal. If default produces no interest and there are no other costs, what is the expected profit? (Answer: −$1,320.)
- A bank has deposits of $1,500,000, reserves of $210,000, and a 12% reserve requirement. It makes a $250,000 loan, initially deposited at the same bank. Then 30% of that new deposit is paid to a customer of another bank. What are the lending bank’s deposits and reserves after settlement, and how much must it borrow to meet its reserve requirement? (Answer: Deposits are $1,675,000; reserves are $135,000; it must borrow $66,000.)
- A 10% reserve ratio and a $50,000 reserve injection imply a $500,000 deposit expansion in the simple model. In a different scenario, qualified loan demand is only $260,000, and the bank has enough reserve headroom for that amount. If all approved loans are initially deposited at that bank, what is the initial loan-created deposit, and why is it not automatically $500,000? (Answer: $260,000; the model’s full expansion requires further rounds of lending and redepositing, which are not guaranteed when qualified demand is limited.)
- A bank has deposits of $900,000, reserves of $105,000, and a 10% reserve requirement. It makes a $100,000 loan that is initially deposited at the bank, then the borrower pays $40,000 to someone at another bank. After settlement, calculate deposits, reserves, required reserves, and any reserve shortfall. (Answer: Deposits are $960,000; reserves are $65,000; required reserves are $96,000; the shortfall is $31,000.)