Deriving the Simple Deposit Multiplier

Why Successive Redeposits Form a Multiplier

When a bank receives a deposit, it holds the required fraction as reserves and can lend the rest under the simple deposit-expansion model. If the loan proceeds are deposited at another bank, that bank can in turn hold a fraction in reserves and lend the remainder. Repeating this process produces a sequence of deposits. The simple deposit multiplier summarizes the total of that sequence relative to the initial reserve injection.

The result is a model, not a claim that every reserve increase automatically produces a fixed amount of lending in practice. Its usefulness is that it makes the mechanics of deposit expansion precise when a specific set of simplifying assumptions holds.

Definition: Let $r$ be a fixed required reserve ratio, with $0<r\leq 1$, and let $C$ be an initial cash deposit that adds $C$ to the banking system’s reserves. Under the simple model, the simple deposit multiplier is the ratio of total deposits generated to the initial reserve injection. It equals $1/r$ when banks lend all funds not required as reserves and all loan proceeds are redeposited.

This definition uses a cash deposit as the initial injection: the bank initially receives $C$ in cash, which is added to its reserves. The first bank holds the required share of the deposit and lends the remainder. In subsequent rounds, loan proceeds are deposited rather than withdrawn as currency.

Model Assumptions

Before applying the multiplier, specify the assumptions. The simple model assumes the same reserve ratio applies to every bank and every round; each bank holds exactly the required reserves against its new deposits; banks lend all funds beyond those required reserves; and every loan is redeposited in the banking system. It also treats the rounds as having no other changes to deposits or reserves.

These conditions make each new round a constant fraction of the previous one. If $r=1$, banks lend none of the deposit and the multiplier is $1$. A zero reserve ratio is excluded: the reciprocal $1/r$ is undefined, and the model does not give a finite multiplier.

Deriving the Geometric Series

Start with an initial cash deposit of $C$. The first bank holds $rC$ as reserves and lends $(1-r)C$. If that loan is redeposited, the next bank holds $r(1-r)C$ in reserves and lends $(1-r)^2C$. The second-round deposit is therefore $(1-r)C$. Each subsequent deposit is $1-r$ times the preceding deposit.

The complete sequence of deposit increases is $C$, $(1-r)C$, $(1-r)^2C$, and so on. The initial deposit is part of the total: it must not be added again after summing the series.

\begin{aligned} \Delta D_{\text{total}} &= C+(1-r)C+(1-r)^2C+\cdots\\ &= C\left[1+(1-r)+(1-r)^2+\cdots\right]\\ &= \frac{C}{1-(1-r)}\\ &= \frac{C}{r}. \end{aligned}

The infinite geometric series converges because its common ratio is $1-r$, which lies in the interval $[0,1)$ for $0<r\leq1$. For a geometric series with first term $C$ and common ratio $1-r$, its sum is $C/[1-(1-r)]$. Simplifying the denominator gives $r$, so total deposits are $C/r$. Dividing total deposits by the initial injection $C$ gives the multiplier $1/r$.

Result: Under the stated assumptions, total deposits are $C/r$, and the simple deposit multiplier is $1/r$. Since total required reserves equal $r$ times total deposits, the model also gives $r(C/r)=C$: the total reserves required against all the deposits equal the initial reserve injection.

Total loans are not the same as total deposits. In this model, total loans equal the deposit increases after the initial deposit. Thus total loans are $C/r-C$, or $C(1-r)/r$. This distinction is useful when checking whether a question asks for the deposit total or the amount lent.

Worked Example: Finding Total Deposits and Loans

Method. Use the geometric series with the initial cash deposit as the first deposit increase. A hypothetical banking system receives a $ \$12{,}000 $ cash deposit. Its fixed reserve ratio is $25\%$, and every loan is redeposited. Find total deposits and total loans under the model.

Step 1: Convert the percentage to a decimal. The reserve ratio is $r=25/100=0.25$, so the fraction lent in each round is $1-r=1-0.25=0.75$.

Step 2: Write the first deposit increases. The initial deposit is $ \$12{,}000 $. The next deposit is $0.75(\$12{,}000)=\$9{,}000$. The following one is $0.75(\$9{,}000)=\$6{,}750$. The sequence continues by multiplying each term by $0.75$.

Step 3: Sum the series. The total is $C/r=\$12{,}000/0.25=\$48{,}000$. The multiplier is $1/0.25=4$, and $4(\$12{,}000)=\$48{,}000$.

Step 4: Find total loans and check reserves. Total loans are total deposits minus the initial deposit: $\$48{,}000-\$12{,}000=\$36{,}000$. Required reserves against the total deposits are $0.25(\$48{,}000)=\$12{,}000$, equal to the initial reserve injection.

Common mistake avoided: The multiplier gives total deposits, including the initial deposit. Adding another $ \$12{,}000 $ to $ \$48{,}000 $ would count the first term twice.

Worked Example: Applying a Fractional Reserve Ratio

Method. Express the reserve ratio as a decimal, then apply the reciprocal to the initial deposit. A hypothetical bank system receives a $ \$6{,}400 $ cash deposit and has a fixed reserve ratio of $12.5\%$. Assume all excess funds are lent and loan proceeds are fully redeposited. Find total deposits, total reserves, and total loans.

Step 1: Convert the reserve ratio. Since $12.5\%=12.5/100=0.125=1/8$, the fraction available to lend in each round is $1-0.125=0.875=7/8$.

Step 2: Calculate total deposits. The multiplier is $1/r=1/0.125=8$. Therefore, total deposits are $8(\$6{,}400)=\$51{,}200$.

Step 3: Calculate reserves. Required reserves are $0.125(\$51{,}200)=\$6{,}400$. Equivalently, because $0.125=1/8$, one-eighth of $ \$51{,}200 $ is $ \$6{,}400 $.

Step 4: Calculate loans and check. Total loans are $\$51{,}200-\$6{,}400=\$44{,}800$. The first deposit rounds begin $ \$6{,}400 $, $ \$5{,}600 $, $ \$4{,}900 $, and $ \$4{,}287.50 $, because each term is $7/8$ of the previous one. The series total is consistent with the multiplier, and the reserve check returns the original $ \$6{,}400 $.

Common mistake avoided: Do not treat $12.5$ as the decimal reserve ratio. The ratio is $0.125$, so the multiplier is $8$, not $1/12.5$.

Finite Rounds and the Remaining Expansion

The multiplier gives the limit of the whole sequence, not the amount created after a specified number of rounds. If a problem asks for only the first few rounds, sum those deposit increases separately. The difference between that partial sum and the full total is the expansion still represented by the later terms.

Worked Example: Calculating a Partial Expansion

Method. Add the specified initial and redeposit rounds, then compare the partial sum with the infinite-series total. A hypothetical system receives an initial cash deposit of $ \$7{,}500 $. The fixed reserve ratio is $25\%$. Find the total deposits after the initial deposit and the next three redeposit rounds, and find how much deposit expansion remains under the model.

Step 1: Find the repeated fraction. With $r=0.25$, the fraction lent and redeposited is $1-r=0.75$.

Step 2: Calculate the four deposit increases. The initial deposit is $ \$7{,}500 $. The next is $0.75(\$7{,}500)=\$5{,}625$. The third is $0.75(\$5{,}625)=\$4{,}218.75$. The fourth is $0.75(\$4{,}218.75)=\$3{,}164.0625$.

Step 3: Sum these rounds. Their total is $\$7{,}500+\$5{,}625+\$4{,}218.75+\$3{,}164.0625=\$20{,}507.8125$, or $ \$20{,}507.81 $ to the nearest cent.

Step 4: Find the full total and remaining expansion. The multiplier is $1/0.25=4$, so total deposits under the full model are $4(\$7{,}500)=\$30{,}000$. The remaining expansion is $\$30{,}000-\$20{,}507.8125=\$9{,}492.1875$, or $ \$9{,}492.19 $ to the nearest cent. As a check, the next deposit would be $0.75(\$3{,}164.0625)=\$2{,}373.046875$; that term and all later terms sum to $\$2{,}373.046875/0.25=\$9{,}492.1875$.

Common mistake avoided: Do not label a partial sum as the full multiplier result. The rounds specified in the question determine which terms to add.

Interpreting and Checking the Result

A multiplier of $4$, for example, means that the model’s total deposits are four times the initial reserve injection. It does not mean that each bank lends four times its own deposit, nor that total loans equal four times the initial injection. The multiplier summarizes deposits across all rounds and all banks.

A reliable check is to multiply the calculated total deposits by the reserve ratio. If the assumptions hold, this product should equal the initial reserve injection. A second check is to subtract the initial deposit from total deposits to obtain total loans in this setup. These checks catch errors in the reserve ratio, in the number of rounds, and in whether the initial deposit has been counted once or twice.

The simple multiplier is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus as a model of bank deposit expansion. It is a benchmark that depends on fixed conditions. If banks hold reserves beyond the required amount or funds leave the banking system instead of being redeposited, the geometric sequence described here no longer follows as stated. In applying the result, first confirm that the problem specifies the simple model’s conditions; do not assume the multiplier describes observed outcomes without them.

Key takeaway: With a fixed reserve ratio $r$, full lending of excess funds, and complete redepositing, each deposit round is $1-r$ times the preceding round. The resulting geometric series sums to $C/r$, so the simple deposit multiplier is $1/r$.

Check Your Understanding

For each problem, assume the fixed reserve ratio and complete-redeposit conditions of the simple model unless the question says otherwise.

A hypothetical bank system receives an initial cash deposit of $ \$18{,}000 $. The required reserve ratio is $25\%$. Find the multiplier, total deposits, and total loans. (Answer: multiplier $4$; total deposits $ \$72{,}000 $; total loans $ \$54{,}000 $.)
An initial cash deposit of $ \$4{,}800 $ enters a system with a $20\%$ reserve ratio. Find the deposit increases in the initial round and the next two redeposit rounds, their partial total, and the remaining expansion to the model’s full total. (Answer: $ \$4{,}800 $, $ \$3{,}840 $, and $ \$3{,}072 $; partial total $ \$11{,}712 $; remaining expansion $ \$12{,}288 $.)
A hypothetical system has an initial reserve injection of $ \$12{,}000 $ from a cash deposit and a reserve ratio of $12.5\%$. Find total deposits and total loans, and verify the reserve total. (Answer: total deposits $ \$96{,}000 $; total loans $ \$84{,}000 $; required reserves $ \$12{,}000 $.)
A reserve ratio is $4\%$, and the initial cash deposit is $ \$2{,}500 $. Find the multiplier, total deposits, total reserves, and total loans. (Answer: multiplier $25$; total deposits $ \$62{,}500 $; total reserves $ \$2{,}500 $; total loans $ \$60{,}000 $.)
A hypothetical bank system receives $ \$9{,}000 $ at a fixed reserve ratio of $15\%$. Count the initial deposit and the next two redeposit rounds, then find the full model total and the deposit expansion remaining after those three terms. (Answer: the three terms are $ \$9{,}000 $, $ \$7{,}650 $, and $ \$6{,}502.50 $; partial total $ \$23{,}152.50 $; full total $ \$60{,}000 $; remaining expansion $ \$36{,}847.50 $.)

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