School Launch

How Lending Multiplies Deposits

A bank keeps part of each deposit as reserves and lends the rest. The borrower spends the loan, and the money lands as a deposit at another bank, which keeps a part and lends the rest again. Step through the rounds and watch the running total of deposits climb toward its ceiling.

Banking Lesson 3.1 · about 15 minutes

Each bar is the running total of deposits after that round, split into what has been kept as reserves and what has been lent on. Click or tap a bar to read it; with the chart focused, the left and right arrow keys move one round.

Rounds: 1

Deposits so far
Textbook ceiling
Where this chain ends
Newest deposit
Reserves held so far
Rounds to reach 99%

The formula and the spreadsheet

Each deposit brings back a share q of itself as the next deposit:

Adding the first n rounds is a geometric series:

The textbook multiplier 1 ÷ r is a ceiling, not a forecast. Banks often keep more than the required reserves, people keep some of the money as cash, and a loan is made only when a creditworthy borrower wants one. Each leak makes the real chain end lower.

Round by round

The first deposit

$
%

Real-world leaks

Leave both at 0% for the textbook model.

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%

Try an example

From Lesson 3.1. The U.S. reserve requirement has been 0% since March 2020; the model uses a ratio, as many textbooks still do.

Following one deposit

A family deposits $1,000. Under a 10% reserve rule the bank keeps $100 and lends $900. The borrower pays a builder, who deposits the $900 at another bank. That bank keeps $90 and lends $810, and so on. Every round is 90% of the round before, so the deposits are

$1,000 + $900 + $810 + $729 + …

The rounds shrink but never stop, and the total creeps toward $10,000. The original $1,000 has not been copied; it has been lent and re-deposited many times, and each re-deposit is a real balance someone can spend.

The ceiling

With reserve ratio r, each deposit returns a share q = 1 − r as the next deposit. The sum of the first n rounds is

D(1 − qn) / (1 − q),

and as n grows, qn shrinks to 0, leaving D / r. The number 1 / r is the simple money multiplier: 10 at a 10% ratio, 20 at 5%. When the chain is done, the reserves held by all the banks together add up to the whole first deposit, because each bank kept its share.

Why it is a ceiling

Each leak lowers q. If banks keep extra reserves e and the public keeps a share c of each loan as cash, then q = (1 − r − e)(1 − c) and the chain ends at D / (1 − q). The model also assumes every dollar a bank could lend finds a willing, creditworthy borrower; when it does not, the chain stops sooner. Since 2020 the U.S. requires no reserves at all, yet banks still hold them to settle payments and meet withdrawals, and capital and liquidity rules limit lending instead.

Check yourself

  1. Under a 10% reserve rule, $1,000 of new reserves is deposited and every loan is redeposited. Find the deposits in the first four rounds and their total.
    Answer

    $1,000, $900, $810 and $729, for a total of $3,439. The textbook maximum is $1,000 ÷ 0.10 = $10,000. Load the first example to check.

  2. If $170,000 of new reserves enters a system with an 8% reserve ratio, what is the most total deposits can grow?
    Answer

    The multiplier is 1 ÷ 0.08 = 12.5, so the most is $170,000 × 12.5 = $2,125,000.

  3. With a 10% ratio, how many rounds does it take for the total to pass 90% of the ceiling?
    Answer

    The total is 90% of the ceiling once 0.9n ≤ 0.1, so n ≥ ln 0.1 / ln 0.9 ≈ 21.9: 22 rounds. Press “+10 rounds” twice, then “Next round” twice, and read the percent.

  4. Same $1,000 at 10%, but banks keep 5% extra and people keep 10% of each loan as cash. Where does the chain end?
    Answer

    q = 0.85 × 0.90 = 0.765, so the chain ends at $1,000 ÷ 0.235 ≈ $4,255.32, well under half of the $10,000 ceiling. Of the first $1,000, about $638.30 ends up as bank reserves and $361.70 as cash in people’s hands.

A bit of history: In seventeenth-century London, goldsmiths kept merchants’ gold in their strongrooms and gave written receipts. Noticing that only a part was ever called for at once, they began lending out some of the gold, and their receipts began to pass from hand to hand like money: an early form of fractional-reserve banking.

“Wherefore then gavest not thou my money into the bank?” — Luke 19:23