Why Open-Market Operations Matter
A central bank may announce an overnight interest-rate target, but banks still need a way to settle payments with one another. Those settlements use reserve balances, which are liabilities of the central bank and assets of commercial banks. In a scarce-reserves framework, the aggregate supply of reserves is limited relative to banks’ desired holdings. An open-market operation changes that supply and, in the simplified model used here, moves the market-clearing overnight rate.
The direction of the transaction matters. When the central bank purchases a security, it pays by creating reserve balances; when it sells a security, the buyer’s payment reduces reserve balances. The resulting rate change depends on how reserve supply compares with banks’ demand, not just on the words “purchase” or “sale.” The balance-sheet entries show the quantity change; a reserve-market model shows its effect on the rate.
Balance-Sheet Mechanics
The accounting is easiest to follow by recording both sides of each balance sheet. Assume the security is bought or sold at its stated settlement value, settlement occurs immediately, and no other transaction changes balances at the same time. A central-bank purchase increases the central bank’s securities asset. Its payment increases its reserve liability. A sale reverses both entries: securities assets and reserve liabilities fall.
The commercial-bank entries depend on who sells or buys the security. If a bank sells a security to the central bank, the bank exchanges one asset for another: its securities fall and its reserves rise. If the central bank buys from a nonbank, the nonbank receives a bank deposit. The commercial bank credits that deposit and receives reserves during settlement. For a sale to a nonbank, the central bank’s reserve liabilities still fall, while the nonbank pays from a deposit and the commercial bank reduces both its reserves and its deposit liability.
| Transaction | Central bank | Commercial bank | Nonbank, if involved |
|---|---|---|---|
| Central bank buys from a bank | Securities assets rise; reserve liabilities rise | Reserve assets rise; securities assets fall | Not involved |
| Central bank buys from a nonbank | Securities assets rise; reserve liabilities rise | Reserve assets rise; deposit liabilities rise | Securities fall; deposit assets rise |
| Central bank sells to a bank | Securities assets fall; reserve liabilities fall | Securities assets rise; reserve assets fall | Not involved |
| Central bank sells to a nonbank | Securities assets fall; reserve liabilities fall | Reserve assets fall; deposit liabilities fall | Securities rise; deposit assets fall |
Worked Example: A Purchase from a Commercial Bank
Method: Record the central bank’s purchase as an asset acquisition financed by an increase in its reserve liabilities, then record the bank’s asset swap. All amounts are in billions of dollars. The central bank initially has securities of 500 and loans of 100 on the asset side; its liabilities are reserves of 180 and currency of 320, with equity of 100. Each side totals 600. A commercial bank initially has reserves of 40, securities of 60, and loans of 300; it has deposits of 360 and equity of 40. Each side totals 400. The central bank buys 20 of securities from this bank.
Central bank. Securities increase from \(500\) to \(500+20=520\). Total assets become \(520+100=620\). Reserve liabilities increase from \(180\) to \(180+20=200\). Liabilities and equity total \(200+320+100=620\), matching assets.
Commercial bank. Reserves increase from \(40\) to \(40+20=60\), while securities decrease from \(60\) to \(60-20=40\). Loans remain 300, so total assets are \(60+40+300=400\), unchanged. Deposits and equity remain \(360+40=400\).
The operation adds 20 to aggregate reserves. The bank has exchanged securities for reserves, so its total assets do not change at settlement. Common mistake avoided: treating a purchase from a bank as a new bank deposit. In this transaction, the bank itself sells the security and receives reserves; its deposit liabilities do not change.
Worked Example: A Purchase from a Nonbank Seller
Method: Follow the security to the central bank, the deposit to the nonbank seller, and the settlement reserves to the commercial bank. All amounts are in billions of dollars. Initially, the central bank has securities of 400 and other assets of 50; its reserve liabilities are 120, currency liabilities are 280, and equity is 50. Each side totals 450. A commercial bank has reserves of 25, securities of 45, and loans of 230, funded by deposits of 270 and equity of 30. Each side totals 300. The central bank purchases securities worth 15 from a nonbank customer of the commercial bank.
Central bank. Securities rise to \(400+15=415\). Assets total \(415+50=465\). Reserve liabilities rise to \(120+15=135\); liabilities and equity total \(135+280+50=465\).
Commercial bank. Settlement increases reserves to \(25+15=40\). The bank credits the seller’s deposit, increasing deposits to \(270+15=285\). Securities and loans are unchanged, so assets total \(40+45+230=315\). Liabilities and equity total \(285+30=315\).
Nonbank seller. The seller’s securities decrease by 15 and its deposit assets increase by 15. At the stated transaction value, the exchange changes the composition, not the value, of the seller’s assets. The aggregate reserve increase is 15, even though the security was sold by a nonbank. Common mistake avoided: recording the deposit increase but forgetting that the bank also receives reserves in settlement.
From Reserve Supply to the Overnight Rate
To isolate the rate effect, use a simplified reserve market. Let \(R_d\) denote the quantity of reserves banks collectively wish to hold, measured in billions of dollars, and \(R_s\) the quantity supplied. Let \(i_{ON}\) be the quoted annualized overnight rate, measured in percentage points in the numerical example below. Assume banks’ reserve demand slopes downward with respect to that rate over the range being analyzed: a higher overnight rate makes holding an additional unit of reserves more costly relative to lending it, so desired reserve holdings are lower. Assume reserve supply is fixed during the comparison and the market clears where desired holdings equal supplied reserves.
The direction follows directly from equilibrium. At the initial rate, a purchase raises \(R_s\) while leaving the demand schedule unchanged. Supplied reserves then exceed desired holdings. The rate must fall along the demand schedule to induce banks to hold more reserves, restoring equality. A sale lowers \(R_s\); at the old rate, desired holdings exceed supply. The rate must rise to reduce desired holdings until the market clears.
This is a comparative-static result, not a universal numerical rule. It relies on a fixed demand schedule, a scarce-reserves range in which demand responds to the rate, and no offsetting changes in other reserve-supply factors. If banks’ demand shifts at the same time as supply changes, or if reserve supply is not scarce, the rate response need not match this simple prediction.
Worked Example: Calculating the Rate Effect of a Purchase and Sale
Method: Substitute reserve supply into the market-clearing condition and solve the demand schedule for the rate. For this hypothetical market, reserve demand is \(R_d=180-20i_{ON}\), where reserves are measured in billions of dollars and \(i_{ON}\) is measured in annualized percentage points. Assume this linear schedule applies over the rates calculated here, remains unchanged throughout, and all operations settle without an offsetting reserve drain. Initial reserve supply is 100.
Initial equilibrium. Set supply equal to demand: \(100=180-20i_{ON}\). Rearranging gives \(20i_{ON}=80\), so \(i_{ON}=4\%\). Checking, \(180-20(4)=100\), equal to supplied reserves.
After a purchase of 20. The purchase raises supply to \(100+20=120\). Equilibrium requires \(120=180-20i_{ON}\), so \(20i_{ON}=60\) and \(i_{ON}=3\%\). Checking, \(180-20(3)=120\).
After a subsequent sale of 10. The sale reduces supply from 120 to \(120-10=110\). Then \(110=180-20i_{ON}\), giving \(20i_{ON}=70\) and \(i_{ON}=3.5\%\). Checking, \(180-20(3.5)=110\).
The purchase lowers the rate from 4% to 3%, and the sale raises it from 3% to 3.5%. The second operation only reverses half of the purchase’s reserve addition, so it does not return supply or the rate to the initial level. Common mistake avoided: assuming any sale restores the original rate. The size of the sale must match the earlier reserve increase, and the other assumptions must also remain unchanged.
Why the Result Holds—and When to Be Careful
The linear example illustrates a general result. Suppose reserve demand over the relevant range is \(R_d=A-bi_{ON}\), where \(A\) and \(b\) are positive constants, and reserve supply is fixed at \(R_s\). Equilibrium implies \(R_s=A-bi_{ON}\). Solving for the rate gives \(i_{ON}=(A-R_s)/b\). Therefore, if supply rises by \(\Delta R_s>0\), the change in the rate is \(-\Delta R_s/b<0\), as long as the same demand schedule applies. A supply reduction has the opposite effect. The negative sign comes from the downward slope of reserve demand, not from an accounting convention.
The accounting and the equilibrium calculation answer different questions. The balance sheet establishes whether an operation adds or drains reserves. The reserve-demand model determines the direction and, if a schedule is provided, the size of the rate change. Do not infer an exact rate movement from a reserve quantity alone; that also requires information about the demand schedule.
A purchase adds securities to the central bank’s assets; a sale reduces them. Determine whether a bank or a nonbank trades the security.
A purchase increases central-bank reserve liabilities and commercial-bank reserve assets. A sale decreases both. If a nonbank participates, also record the change in its bank deposit and the bank’s deposit liability.
Add the reserve injection or subtract the reserve drain, checking for other changes that may offset the operation.
Set reserve supply equal to reserve demand. Use the demand schedule and its stated units to find the new overnight rate.
An open-market operation can be used to implement an operating target, the policy rate the central bank seeks to achieve in the market. But the mechanical result above depends on the model’s assumptions. Payment flows can redistribute reserves among banks without changing their aggregate quantity; other transactions can also alter aggregate reserves. If those factors offset an operation, the net supply change is smaller than the operation’s stated size. The analysis must therefore distinguish the operation’s gross settlement effect from the resulting net change in reserves.
A frequent exam error is to reverse the central bank’s entries when it sells securities. In a sale, the central bank’s securities assets fall and its reserve liabilities fall. The commercial bank that buys the security records securities assets rising and reserve assets falling. Reversing these sides incorrectly describes a purchase by the central bank. A separate trap is to say that a sale to a nonbank has no effect on reserves: the nonbank pays from a deposit, and its bank transfers reserves to settle the payment.
Check Your Understanding
For each problem, use the stated assumptions and keep the balance-sheet entries distinct from the rate calculation.
- A central bank buys 8 billion dollars of securities directly from a commercial bank. State the changes in central-bank securities assets and reserve liabilities, and in the bank’s securities and reserve assets. (Answer: Central-bank securities assets +8 billion and reserve liabilities +8 billion; bank securities assets −8 billion and reserve assets +8 billion.)
- A central bank sells 12 billion dollars of securities to a nonbank, paid for from its deposit at a commercial bank. State the changes in central-bank securities assets and reserve liabilities, and in the bank’s reserves and deposit liabilities. (Answer: Central-bank securities assets −12 billion and reserve liabilities −12 billion; bank reserve assets −12 billion and deposit liabilities −12 billion.)
- Reserve demand is \(R_d=150-15i_{ON}\), with reserves in billions and the rate in annualized percentage points. If reserve supply is 90 billion and the schedule is unchanged, calculate the equilibrium rate. (Answer: \(i_{ON}=4\%\), since \(150-15(4)=90\).)
- Using the demand schedule in Question 3, reserve supply initially equals 90 billion. A purchase adds 15 billion, with no other change. Calculate the initial and new rates and verify the new equilibrium. (Answer: Initial rate \(4\%\); new supply \(105\) billion and new rate \(3\%\), since \(150-15(3)=105\).)
- In a hypothetical market, reserve demand is \(R_d=210-30i_{ON}\), reserves are measured in billions, and the rate is in annualized percentage points. Supply begins at 120 billion; the central bank sells 9 billion of securities to a bank, with no offsetting change. Find the initial and final rates and state the direction of the change. (Answer: Initial rate \(3\%\); final supply \(111\) billion and final rate \(3.3\%\), since \(210-30(3.3)=111\); the rate rises by 0.3 percentage points.)