What is the deposit multiplier?
The deposit multiplier (also called the deposit expansion multiplier or simple deposit multiplier) is a theoretical factor that shows how much bank deposits can increase for each unit of reserves a banking system holds. It comes from fractional reserve banking, where banks keep only a fraction of deposits as reserves and lend out the remainder.
Key definitions
– Reserve requirement (or required reserves): the minimum share of deposits a central bank requires commercial banks to hold and not lend out.
– Fractional reserve banking: a system in which banks keep only part of deposits on hand and lend out the rest.
– Deposit multiplier: the maximum potential increase in total deposits in the banking system per dollar of reserves, assuming certain ideal conditions.
Basic formula
Deposit multiplier = 1 / reserve ratio
where the reserve ratio is expressed
as a decimal.
Worked numeric example — simple model
– Reserve ratio (rr) = 0.10 (10%).
– Deposit multiplier = 1 / rr = 1 / 0.10 = 10.
– If the banking system receives $1,000 of new reserves and all assumptions hold, maximum possible increase in deposits = 10 × $1,000 = $10,000.
You can also show the same process as successive rounds:
1. Bank A keeps $100 (10%) and lends $900.
2. Borrower spends the $900 and it gets redeposited at Bank B.
3. Bank B keeps $90 (10% of $900) and lends $810.
4. Repeat indefinitely; the geometric series sum equals $1,000 + $900 + $810 + … = $10,000 total deposits.
Key assumptions behind the simple deposit multiplier
– Banks hold only required reserves; they do not hold excess reserves.
– Borrowers redeposit all loan proceeds back into the banking system (no currency drain).
– No taxes, fees, or regulatory capital constraints that reduce lending.
– Banks are willing to make loans and there is loan demand.
Why real-world multipliers are smaller
In practice, several “leakages” and frictions reduce the theoretical maximum:
– Currency drain: households/firms hold some cash outside banks, lowering redeposits.
– Excess reserves: banks may hold reserves beyond the requirement (especially if interest is paid on reserves).
– Regulatory constraints: capital requirements and liquidity rules limit how much banks can expand assets.
– Weak loan demand or credit standards: banks may not find creditworthy borrowers.
– Central bank policy tools (e.g., interest on reserves, reserve exemptions) that change banks’ incentives.
A more realistic money-multiplier formula
To account for currency holdings and excess reserves, use:
Money multiplier = (1 + c) / (rr + c + e)
where
– c = currency-to-deposit ratio (currency held by public / checkable deposits),
– rr = required reserve ratio,
– e = excess-reserve-to-deposit ratio (excess reserves / checkable deposits).
Worked numeric example — realistic case
– Suppose rr = 0.10 (10%), c = 0.20 (public holds 20% as cash relative to deposits), e = 0.02 (banks hold 2% of deposits as excess reserves).
– Multiplier = (1 + 0.20) / (0.10 + 0.20 + 0.02) = 1.20 / 0.32 = 3.75.
– So $1,000 of new reserves would be associated with about $3,750 of total deposits, not $10,000.
Practical checklist — how to estimate deposit/money expansion
1. Obtain or estimate parameters:
– required reserve ratio (rr) from the central bank or law,
– currency-to-deposit ratio (c) from recent monetary statistics,
– observed excess-reserve ratio (e) from central bank balance-sheet data.
2. Compute multiplier = (1 + c) / (rr + c + e).
3. Multiply the multiplier by the change in monetary base (new reserves) to estimate change in deposits (or broader money measure).
4. Adjust for time lags and changing behavior (e.g., crisis periods raise e and c).
How central banks influence the multiplier
– Changing reserve requirements (rr) directly alters the theoretical multiplier.
– Paying interest on reserves tends to increase e (excess reserves), reducing the multiplier.
– Open-market operations change the monetary base (reserves), but the ultimate effect on deposits depends on the multiplier at that time.
– During large-scale asset purchases (quantitative easing), central banks increased reserves, but large e and changes in c meant the money supply did not expand proportionally.
Limitations and modern context
– Many advanced economies rely less on simple reserve ratios; some central banks target interest rates and pay interest on reserves, which weakens the link between reserves and broad money.
– The deposit/money multiplier is a useful conceptual tool for teaching money creation, but empirical multipliers vary over time and across countries. Use current central-bank data when doing quantitative estimates.
Sources for further reading
– Investopedia — Deposit Multiplier: https://www.investopedia.com/terms/d/deposit_multiplier.asp
– Federal Reserve Education — Money creation in the banking system: https://www.federalreserveeducation.org/about-the-fed/structure-and-functions/money-creation
– Bank of England — What is the money multiplier?: https://www.bankofengland.co.uk/knowledgebank/what-is-the-money-multiplier
– European Central Bank — How is money created?: https://www.ecb.europa.eu/explainers/tell-me/html/money_creation.en.html
Practical steps to estimate a deposit (money) multiplier in a simple way
1) Decide which concept you mean
– Deposit multiplier (bank-lending view): uses the reserve-to-deposit relationship and currency leakages. Formula used below.
– Empirical money multiplier: often defined as a broad money aggregate (M2, M3) divided by the monetary base; this is what data series report.
2) Choose a formula and collect inputs
– Standard textbook formula (with currency ratio c and required reserve ratio r):
m = (1 + c) / (r + c)
where
– c = currency held by the public ÷ demand deposits (currency-to-deposit ratio)
– r = fraction of deposits banks hold as reserves (required reserve ratio; can include desired reserves if you want a behavioral r)
– Data to collect:
– Recent currency-to-deposit ratio from your central bank or national accounts.
– Reserve requirement (or average reserve holdings) from the central bank.
– If relevant, measure excess reserves separately and treat them as part of r for practical estimates.
3) Worked numeric example
– Suppose:
– Currency held by public = $200 billion
– Demand deposits = $800 billion → c = 200/800 = 0.25
– Required reserve ratio r = 0.10 (10%)
– Compute multiplier:
m = (1 + 0.25) / (0.10 + 0.25) = 1.25 / 0.35 = 3.571…
– Interpretation:
– If the monetary base (reserves + currency) increases by $100 billion and the relationships above remain stable, the simple-model estimate of the increase in deposits/broad money would be roughly $357.1 billion (100 × 3.571).
– Check assumptions (below) before using this result for policy or forecasting.
4) Quick checklist before you trust the number
– Are banks holding large excess reserves? If yes, add them to r or use empirical base data.
– Has the central bank started paying interest on reserves? That raises banks’ desired r and weakens the multiplier.
– Is the public’s preference for cash changing (e.g., more electronic payments)? That changes c.
– Time horizon: short-run multipliers differ from long-run ones because behavior adjusts.
– Use observed series (monetary aggregate ÷ monetary base) for a reality check.
Key assumptions and limitations (be explicit)
– The simple formula assumes banks lend out all non-reserve deposits and that c and r are stable. Real-world behavior often violates both assumptions.
– It ignores off-balance-sheet funding, shadow-banking, and central-bank policies (e.g., quantitative easing, interest on
reserves). Those policies can decouple the monetary base from broad money by changing banks’ incentives or by expanding the base with assets that do not immediately translate into deposit creation.
Other important limitations
– Lending capacity and demand: The formula assumes banks have capital, willingness, and creditworthy borrowers. In a recession banks may hold excess reserves or tighten lending standards, lowering real-world creation of deposits independent of r and c.
– Regulatory/liquidity rules: Capital requirements, liquidity-coverage ratios, and macroprudential buffers constrain how much banks can lend relative to deposits.
– Shadow banking and wholesale funding: Non-deposit funding (money-market mutual funds, repo markets, securitization) can create money-like liabilities outside the traditional deposit–reserve channel.
– Open-economy flows: Cross-border capital flows and foreign-currency liabilities change the domestic deposit base in ways the simple multiplier does not capture.
– Time variation and expectations: c and r can vary with interest rates, payment technology, and expectations about inflation or crises; short-run multipliers can differ materially from long-run averages.
Practical steps to estimate a working money multiplier
1. Choose definitions
– Pick the monetary aggregate you want to explain (e.g., M1: currency + demand deposits + traveler’s cheques; M2: M1 + savings deposits, small time deposits, retail MMFs).
– Pick a base measure (monetary base or high-powered money = currency in circulation + reserves at the central bank).
2. Compute the empirical multiplier
– Simple approach: multiplier = (chosen monetary aggregate) ÷ (monetary base).
– Use level or rolling averages to smooth volatility (e.g., 3- or 12-month moving averages).
3. Compare to theoretical m = (1 + c) / (c + r)
– Estimate c (currency/deposit ratio) and r (reserve/deposit ratio) from data:
c = Currency in circulation ÷ Bank deposits
r = Total reserves ÷ Bank deposits
– Plug into the formula and compare to the empirical multiplier for consistency checks.
4. Adjust for excess reserves and interest on reserves
– If central banks pay interest on reserves or excess reserves are large, treat excess reserves as part of r (it increases r and lowers m).
5. Check robustness
– Repeat for alternative aggregates (M1 vs M2), sample periods (pre- vs post-crisis), and after accounting for sterilized central-bank asset purchases.
6. Interpret results
– A stable empirical multiplier close to the theoretical one suggests the simple model has explanatory power.
– Large, persistent gaps imply important roles for excess reserves, off-balance-sheet flows, or structural changes in payments behavior.
Worked numeric examples
Example A — textbook case
– Suppose currency/deposit ratio c = 0.20 (people hold $0.20 in cash for every $1 in deposits).
– Reserve ratio r = 0.10 (banks hold $0.10 in reserves per $1 of deposits).
– The theoretical multiplier m = (1 + c) / (c + r) = (1 + 0.20) / (0.20 + 0.10) = 1.20 / 0.30 = 4.0.
– Interpretation: $1,000 increase in the monetary base could support $4,000 in deposits (and a larger increase in the chosen money aggregate, depending on definitions).
Example B — excess reserves matter
– Same c = 0.20. Now required reserves = 0.10 but banks hold excess reserves = 0.05, so effective r = 0.15.
– m = (1 + 0.20) / (0.20 + 0.15) = 1.20 / 0.35 ≈ 3.4286.
– Interpretation: The same $1,000 base expansion now supports about $3,429 in deposits because excess reserves reduce lending.
Quick checklist before using the multiplier
– Define which money aggregate and base you are using.
– Check for large excess reserves or interest-on-reserves policies.
– Confirm whether reserve requirements are meaningful in your jurisdiction (some countries have low or zero reserve requirements).
– Review recent payment-technology trends and currency usage (affects c).
– Use data from reputable sources and report the sample period and any smoothing.
Where to find data (select sources)
– U.S. Federal Reserve — H.6 Monetary Aggregates: https://www.federalreserve.gov/releases/h6/
– Federal Reserve Economic Data (FRED), St. Louis Fed (search and download series for monetary base, M1, M2, reserves): https://fred.stlouisfed.org/
– Bank for International Settlements (statistics on bank balance sheets and cross
-border banking and bank-level series): https://www.bis.org/statistics.htm
Worked numeric example (step-by-step)
Assumptions
– You want M2 as the money aggregate. – Choose a sample day when the monetary base MB = $1,000 billion (currency plus reserves). – Reserve requirement r = 10% (0.10). – Currency-to-deposit ratio c = 20% (0.20). This is currency outstanding divided by deposit currency components consistent with M2. – Excess-reserve ratio e = 2% (0.02) (excess reserves divided by deposits).
1) Compute the multiplier formula
m = (1 + c) / (r + e + c)
2) Plug numbers in
m = (1 + 0.20) / (0.10 + 0.02 + 0.20) = 1.20 / 0.32 = 3.75
3) Compute money aggregate
M2 = m × MB = 3.75 × $1,000 billion = $3,750 billion
Contrast: if excess reserves fall to zero and currency is negligible (c ≈ 0), m = 1 / r = 1 / 0.10 = 10, so M2 = 10 × $1,000b = $10,000b. This shows how currency preference and excess reserves materially reduce the multiplier.
Quick checklist: computing an empirical multiplier from data
– Choose the money aggregate (M1, M2, etc.) and define which deposit components are included. – Decide the base definition: usually MB = currency in circulation + bank reserves at the central bank. – Obtain time-consistent series for currency outstanding (C), deposits included in your aggregate (D), required reserves (RR), and total reserves (TR). – Compute c = C / D. – Compute r = RR / D (use statutory reserve requirements or effective requirements if available). – Compute e = (TR − RR) / D (excess reserves divided by deposits). – Calculate m = (1 + c) / (r + e + c). – Multiply m by MB to estimate the money aggregate; compare to reported M to check model fit. – Report sample period, frequency, and any smoothing (e.g., 3-month moving average) and test sensitivity to assumptions.
Common pitfalls and caveats
– Reserve requirements may be zero or not binding. If r = 0, the simple multiplier formula degenerates and deposit creation is governed by other constraints (funding markets, capital, liquidity). – Interest on reserves (IOR) or interest on excess reserves (IOER) changes incentives: when IOER is positive and attractive, banks hold more excess reserves, raising e and lowering m. – Payments innovations (e.g., mobile wallets) and currency-sharing habits change c over time; use consistent, contemporaneous series. – Central-bank liabilities besides reserves and currency (e.g., term deposits, reverse repos, central bank bills) complicate the “base” notion; these instruments can effectively sterilize or re-lever the base. – Cross-border flows and foreign-currency deposits mean domestic MB changes don’t map cleanly to domestic deposits. – The multiplier is a reduced-form relationship that assumes proportional responses; actual bank lending depends on credit demand, capital rules, risk appetite, and macroprudential policy.
How to test the model empirically (simple steps)
1) Gather MB and your chosen M series (monthly or weekly). 2) Compute m_observed = M / MB for each date. 3) Compute model m_model = (1 + c) / (r + e + c) using contemporaneous c, r, e. 4) Plot m_observed vs. m_model and compute correlation and mean absolute error. 5) Run a simple regression m_observed = α + β × m_model + ε to test correspondence (remember endogeneity and omitted-variable issues). 6) Check subsamples (pre/post crisis, pre/post IOER introduction) to see structural breaks.
Practical uses and realistic expectations
– Use the multiplier as a diagnostic: it helps explain broad swings in money-growth given base changes, and flags when bank behavior deviates from textbook proportionality. – Do not use it as a precise forecasting engine; structural shifts (policy changes, payments
changes (payments-system changes, regulatory shifts, or a sudden rise in nonbank intermediation) can break historical relationships and make the simple multiplier model misleading.
Suggested monitoring checklist
– Watch the three behavioral ratios. Track contemporaneous:
– c = currency held by the public / deposits,
– r = required reserves / deposits (use the legal reserve schedule or an effective ratio if requirements are temporary),
– e = excess reserves / deposits.
– Monitor MB and M series separately. Use consistent definitions (e.g., monetary base vs. M1 or M2).
– Flag large policy changes: IOER (interest on excess reserves), changes in reserve requirements, large-scale asset purchases (QE), or structural payments innovations.
– Compare m_observed = M / MB to m_model = (1 + c) / (r + e + c) regularly (monthly or quarterly).
– Run simple diagnostics: correlation, MAE (mean absolute error), and the OLS regression m_observed = α + β m_model + ε as a quick fit check. Look for structural breaks (pre/post crisis, pre/post-IOER).
Step-by-step reproducible procedure (practical)
1. Choose M (M1 or M2) and MB definitions and data frequency (monthly recommended).
2. Compute deposits D consistent with your M definition (for M2 use demand and time deposits as appropriate).
3. Compute ratios:
– c = Currency_in_public / D
– r = Required_reserves / D (or use total reserves required by regulation)
– e = Excess_reserves / D
4. Compute m_model = (1 + c) / (r + e + c).
5. Compute m_observed = M / MB.
6. Plot the two series and compute:
– Pearson correlation between m_model and m_observed,
– Mean absolute error: MAE = mean(|m_observed − m_model|),
– Run OLS m_observed = α + β m_model + ε; inspect α and β, and standard errors.
7. Test robustness: use lags, subsamples, and add control variables (short-term interest rate, a crisis dummy).
Worked numeric example
Assume:
– Deposits D = $10,000 (units: billions for clarity),
– Currency held by public = $2,000 → c = 2,000 / 10,000 = 0.20,
– Required reserves = $1,000 → r = 1,000 / 10,000 = 0.10,
– Excess reserves = $500 → e = 500 / 10,000 = 0.05,
– Monetary base MB = $4,000,
– Broad money M = $13,700.
Compute:
– m_model = (1 + c) / (r + e + c) = (1 + 0.20) / (0.10 + 0.05 + 0.20) = 1.20 / 0.35 = 3.4286.
– m_observed = M / MB = 13,700 / 4,000 = 3.425.
Interpretation: m_observed ≈ m_model, suggesting the textbook proportions roughly hold for this snapshot. If repeated over time, you would expect some scatter; large sustained divergence signals behavioral or structural change.
Common pitfalls and caveats
–