What It Means to Defend a Peg
A fixed exchange rate, or peg, is a commitment by a monetary authority to maintain the currency’s value at a stated exchange rate or within a specified range. A peg can face pressure when private buyers and sellers want to exchange currencies at a rate different from the target. To resist that pressure, the central bank can enter the foreign-exchange market and trade its reserves.
The direction of the transaction matters. A central bank that sells foreign currency receives domestic currency; one that buys foreign currency pays domestic currency. Those trades add to the market’s supply or demand, respectively, and counter the pressure on the peg. The exact interpretation of “upward” or “downward” exchange-rate pressure depends on the quotation convention, so this tutorial uses one convention throughout.
Which Way Must the Central Bank Trade?
At the peg, compare the quantity of foreign currency that private participants want to buy with the quantity they want to sell. If demand exceeds supply, there is excess demand for foreign currency. Without intervention, buyers competing for limited foreign currency would push \(E\) above the peg. The central bank counters this pressure by selling foreign currency from its reserves and buying domestic currency.
If private supply of foreign currency exceeds private demand, there is excess supply of foreign currency. Without intervention, sellers would accept fewer units of domestic currency per unit of foreign currency, pushing \(E\) below the peg. The central bank counters by buying foreign currency and paying domestic currency.
| Pressure at the peg | Likely movement in \(E\) | Central-bank trade | Reserve effect |
|---|---|---|---|
| Excess demand for foreign currency | Upward; domestic currency faces depreciation pressure | Sells foreign currency, buys domestic currency | Foreign reserves fall |
| Excess supply of foreign currency | Downward; domestic currency faces appreciation pressure | Buys foreign currency, pays domestic currency | Foreign reserves rise |
The descriptions “upward” and “downward” refer to the stated quote \(E\), not to an exchange rate quoted in the opposite direction. If a problem instead gives foreign currency per unit of domestic currency, the interpretation reverses. Always write the units alongside the rate before deciding what the intervention direction should be.
Calculating the Intervention at the Peg
For a simple market-clearing calculation, assume quantities are measured in units of foreign currency over the same period, the peg is maintained at \(E^*\), and the stated private demand and supply are the quantities at that rate. Assume also that there are no other transactions or changes in the market during the calculation. The central bank supplies the residual amount when private demand exceeds private supply; it purchases the residual when private supply exceeds private demand.
Here \(D_F\) denotes private demand for foreign currency and \(S_F\) private supply of foreign currency. If the intervention amount is \(Q_F\), its domestic-currency value at the peg is \(E^*Q_F\). Multiplying by \(E^*\) is valid because the quote is domestic currency per unit of foreign currency. These equations describe the intervention needed to close the stated gap at the peg; they do not claim that private quantities would remain unchanged in a more complete market model.
Why the Residual-Quantity Rule Works
The rule follows from adding the central bank’s trade to private market supply or demand. If private buyers demand more foreign currency than private sellers provide, a central-bank sale adds exactly the missing foreign currency to market supply. If private sellers provide more than private buyers demand, a central-bank purchase adds exactly the missing demand.
When \(D_F>S_F\), private buyers are short \(D_F-S_F\) units of foreign currency at the peg.
A sale of \(D_F-S_F\) units raises total market supply to \(S_F+(D_F-S_F)=D_F\). The stated demand and supply are then equal at the peg.
When \(S_F>D_F\), a central-bank purchase of \(S_F-D_F\) units raises total demand to \(D_F+(S_F-D_F)=S_F\), closing the gap at the peg.
This is a quantity-accounting argument, not a guarantee that the peg is sustainable indefinitely. It identifies the trade needed to clear the specified market imbalance at the target rate, subject to the available reserves and the assumptions used to describe the market.
Worked Example: Meeting Excess Demand for Foreign Currency
A hypothetical country pegs its currency at \(E^*=4.20\) domestic units per unit of foreign currency. During a trading period, private demand for foreign currency at the peg is 26 million units, while private supply is 19 million. Calculate the central bank’s intervention and the domestic currency involved. Assume no other market transactions.
Method. Compare private demand and supply, then have the central bank supply the difference. The demand for foreign currency is larger, so the domestic currency faces depreciation pressure.
Step 1: Find the private shortfall.
\(D_F-S_F=26-19=7\) million units of foreign currency.
Step 2: Identify the trade.
The central bank must sell 7 million units of foreign currency from its reserves and receive domestic currency.
Step 3: Value the sale at the peg.
\(4.20 \times 7\) million \(=29.40\) million domestic currency units. The bank receives 29.40 million domestic units at the peg.
Check. Total foreign-currency supply becomes \(19+7=26\) million, matching demand. The common mistake is to buy foreign currency because demand is high. That would add demand rather than supply and intensify the pressure.
Worked Example: Resisting Appreciation Pressure
A hypothetical central bank maintains \(E^*=0.75\) domestic units per unit of foreign currency. At that rate, private sellers offer 32 million units of foreign currency, while private buyers demand 25 million. Determine the direction and size of the intervention and the domestic-currency payment. Assume no other transactions.
Method. With excess supply of foreign currency, the central bank buys the surplus. This adds demand for foreign currency and resists a fall in \(E\).
Step 1: Find excess supply.
\(S_F-D_F=32-25=7\) million units of foreign currency.
Step 2: Identify the trade.
The central bank must buy 7 million units of foreign currency. Its foreign reserves increase by that amount.
Step 3: Calculate the payment.
At \(0.75\) domestic units per foreign unit, the payment is \(0.75 \times 7\) million \(=5.25\) million domestic currency units.
Check. Total foreign-currency demand becomes \(25+7=32\) million, matching private supply. The bank pays domestic currency; it does not receive it. Confusing the cash-flow direction can lead to the correct intervention quantity but the wrong reserve and payment changes.
Reserves, Constraints, and the Limits of a Calculation
A central bank’s capacity to sell foreign currency is constrained by its usable foreign reserves: the foreign assets it can actually deploy to meet market demand. The gross reserve total may not equal the amount available for intervention, for example if some assets are committed or otherwise unavailable. When the bank buys foreign currency, its foreign reserves increase; when it sells foreign currency, they decrease.
A reserve calculation can reveal the scale of a pressure relative to resources. Under the simplifying assumption that the monthly net demand for foreign currency remains constant and no replenishment or other reserve use occurs, the quotient of usable reserves and monthly intervention gives the number of months of coverage. This is a conditional arithmetic comparison, not a prediction of when a peg must fail: pressure, policy, private behavior, and reserve availability can change.
Worked Example: Comparing Persistent Pressure with Usable Reserves
A hypothetical central bank has 41 million units of foreign reserves, of which 6 million are committed to other near-term obligations and are unavailable for defending its peg. The market has excess demand for foreign currency of 7 million units per month. Assume that this pressure stays constant, the bank uses only the available reserves, and there are no replenishments or other reserve transactions. How long can the available reserves cover the stated intervention?
Method. First separate usable reserves from committed reserves, then divide by the intervention needed each month.
Step 1: Calculate usable reserves.
\(41-6=35\) million units of foreign currency are available for the defense.
Step 2: Calculate coverage.
\(35\div7=5\) months of intervention at the assumed constant monthly rate.
Step 3: Interpret the result.
After one month, \(35-7=28\) million units remain; after two months, \(28-7=21\) million; after three months, \(21-7=14\) million; after four months, \(14-7=7\) million; after five months, \(7-7=0\). Thus, the stated usable reserves exactly cover five months under the assumptions.
Check. The total five-month sale is \(5\times7=35\) million, equal to usable reserves, not the gross 41 million. The calculation does not establish that the peg will fail after five months; it shows only the coverage implied by fixed pressure and no offsetting reserve flows.
Reading the Transaction and Avoiding Common Traps
Intervention has two linked effects to track: the foreign-currency transaction and its domestic-currency counterpart at the peg. When the central bank sells foreign currency, foreign reserves fall and the bank receives domestic currency. When it buys foreign currency, reserves rise and it pays domestic currency. Stating both sides helps detect sign errors.
In the simple examples, the central bank’s transaction changes the supply or demand for foreign currency. It is also useful to distinguish the central bank’s ability to transact from the credibility of its commitment. A central bank might have reserves sufficient for one market imbalance but face larger or continuing pressure later. The quantity gap for one period, the reserve stock, and a judgment about whether a peg is sustainable are related but are not the same calculation.
- Do not ignore the quotation convention. With domestic currency per foreign currency, a rise in \(E\) means domestic depreciation. A different quote reverses that interpretation.
- Do not reverse the trade. Excess private demand for foreign currency calls for a central-bank sale; excess private supply calls for a central-bank purchase.
- Do not multiply by the reciprocal rate. If \(E\) is domestic units per foreign unit, the domestic-currency value is \(E\) times the foreign-currency quantity.
- Do not equate gross reserves with usable reserves. Use the reserve amount specified as available for intervention.
- Do not treat a constant-pressure estimate as a certainty. A coverage calculation depends on its stated assumptions and is not, by itself, a forecast of a break in the peg.
Check Your Understanding
For each problem, use the stated quote and assume no unmentioned transactions. Track the intervention direction as well as the quantity or value requested.
- A peg is \(E^*=3.60\) domestic units per foreign unit. Private demand is 18 million foreign units and private supply is 23 million. What intervention is needed, and how many domestic units are paid or received? (Answer: Buy 5 million foreign units and pay 18 million domestic units.)
- At a peg of \(E^*=1.25\) domestic units per foreign unit, private demand for foreign currency is 44 million units and supply is 37 million. Find the intervention and the domestic-currency amount received or paid. (Answer: Sell 7 million foreign units and receive 8.75 million domestic units.)
- A central bank has 58 million foreign units in gross reserves, of which 13 million are unavailable. Excess demand requires a sale of 9 million foreign units each month. Under constant pressure with no replenishment, how many full months can usable reserves cover, and how much remains afterward? (Answer: Usable reserves are 45 million; they cover 5 full months, leaving 0 million.)
- A country quotes its peg as 0.40 foreign units per domestic unit. Private demand is 20 million foreign units and supply is 28 million. How much must the central bank buy or sell, and what is the domestic-currency value at the peg? (Answer: Buy 8 million foreign units; since 1 domestic unit buys 0.40 foreign units, the payment is \(8\div0.40=20\) million domestic units.)
- At \(E^*=2.50\) domestic units per foreign unit, a period begins with 34 million usable foreign units. Private demand exceeds supply by 6 million foreign units in each of two consecutive periods. Assuming no other reserve changes, what is the total intervention, the ending reserve balance, and the domestic currency received? (Answer: Sell 12 million foreign units; 22 million remain; receive 30 million domestic units.)