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Eduzan / MODULE 03 — Economics

ECO 8 Exchange Rate Calculations

How to build cross rates, read forward points, and tie spot rates, forward rates, and interest rates together with no-arbitrage logic. Check-yourself items are study aids.

Currencies trade in a decentralized foreign exchange (FX) market where a small set of pairs carry a standard, actively quoted price. A cross rate is the exchange rate between two currencies that the market does not quote directly. It is recovered from two quotes that share a third currency, usually the US dollar. The professional market avoids the labels direct and indirect, since those depend on where the observer happens to sit, and instead names each quote by a fixed convention.

Throughout this lesson a quote written as price/base gives the number of units of the price currency needed to buy one unit of the base currency. So USD/EUR means US dollars per euro, and CAD/USD means Canadian dollars per US dollar. Reading the two symbols as a fraction is the key to combining quotes: multiply two quotes and any shared currency cancels, leaving the cross rate.

The shared US dollar cancels, leaving Canadian dollars per euro.

When the two market quotes do not share the dollar in the position that cancels, one quote is inverted first. To build a Canada-yen rate (JPY/CAD) from the two conventional quotes CAD/USD and JPY/USD, the CAD/USD quote is flipped to USD/CAD so that the dollar cancels cleanly.

Four decimal places is the norm for most quotes, while the Japanese yen carries only two.

Example 1 · Worked

A dealer quotes the two conventional pairs CAD/USD at 1.3020 and USD/EUR at 1.1701, and separately CAD/USD at 1.3020 and JPY/USD at 111.94.

1. Find the euro-Canada cross rate (CAD/EUR).

Solution. Multiply dollar-Canada by euro-dollar so the dollar cancels:

2. Find the Canada-yen cross rate (JPY/CAD).

Solution. The dollar sits in the wrong place, so invert dollar-Canada first: 1 divided by 1.3020 is 0.7680 (USD/CAD). Then multiply by dollar-yen:

Because a client can see both the cross-rate quote and the two underlying quotes, the three prices must stay consistent, or the market will trade the gap away. A cross rate that disagrees with its building blocks opens a riskless round trip through three currencies, known as triangular arbitrage.

Example 2 · Worked

The underlying quotes of CAD/USD 1.3020 and JPY/USD 111.94 imply a Canada-yen rate of 85.97, as computed above. Suppose a second dealer misquotes Canada-yen at 86.20.

Show the arbitrage and its size per Canadian dollar.

Solution. Both prices convert yen into Canadian dollars, so they must match. A trader buys one Canadian dollar where it is cheap, at JPY 85.97, and sells it where it is dear, at JPY 86.20. The riskless gain is Traders and algorithms watch for exactly this gap, so in practice these discrepancies are tiny and short lived. For study purposes a cross rate can be treated as consistent with its two underlying quotes.

Check yourself

You hold CNY/HKD at 0.8422 and CNY/ZAR at 0.9149. How do you build the ZAR/HKD cross rate, and what is it?

Invert CNY/ZAR to ZAR/CNY, then multiply by CNY/HKD so the Chinese yuan cancels: (1 divided by 0.9149) times 0.8422, which is about 0.9205 rand per Hong Kong dollar.

Why do triangular arbitrage gaps rarely persist?

Human traders and automated algorithms continuously monitor cross rates against their underlying quotes, so any mispricing is bought and sold until it closes.

Once a rate is quoted price/base, judging which currency strengthens is a matter of watching the base currency. A rise in a price/base quote means one unit of the base currency now costs more of the price currency, so the base currency has appreciated. A fall means the base currency has depreciated. The percentage move in the base currency against the price currency is the new quote divided by the old quote, minus one.

Positive means the base currency appreciated; negative means it depreciated.

To measure the move of the price currency instead, invert the quote first, or read the base-currency result the other way round. This directional discipline lets an analyst rank several currencies from strongest to weakest over a forecast horizon without ever losing track of which side of the quote is moving.

Example 3 · Worked

A research report lists spot rates and expected rates one year out.

Spot rates and expected rates in one year

Currency	Spot rate	Expected in one year
USD/EUR	1.1701	1.1619
CHF/USD	0.9900	0.9866
USD/GBP	1.3118	1.3066

1. What is the spot CHF/EUR cross rate?

Solution. The dollar cancels when USD/EUR multiplies CHF/USD:

2. What is the spot GBP/EUR cross rate?

Solution. Divide euro-dollar by pound-dollar, which is the same as inverting USD/GBP first:

3. How much is the euro expected to move against the US dollar?

Solution. The euro is the base currency in USD/EUR, and the quote is expected to fall, so the euro depreciates:

4. By how much should the US dollar shift versus the British pound?

Solution. The pound is the base currency in USD/GBP, and that quote falls, so the pound depreciates, which is the same as the dollar appreciating against the pound:

Chaining these results ranks the three currencies. The dollar strengthens against both the euro and the pound, since USD/EUR and USD/GBP are both expected to fall. Comparing the two, the euro falls by 0.7 percent while the pound falls by only 0.4 percent, so the pound is stronger than the euro. From strongest to weakest, the order is US dollar, then British pound, then euro.

Where does the Swiss franc fit?

CHF/USD is expected to fall from 0.9900 to 0.9866, so the US dollar depreciates against the franc, which is the same as the franc appreciating against the dollar. Inverting to USD/CHF, the franc gains about +0.35 percent against the dollar (1.0136 divided by 1.0101, minus one). Since the franc rises against the dollar while the dollar rises against both the euro and the pound, the franc appreciates against all three. Ranked by appreciation against the US dollar, from most to least, the order is Swiss franc, British pound, euro.

Check yourself

A USD/EUR quote falls over the year. What has happened to the euro, and to the US dollar?

The euro is the base currency, so a falling quote means one euro buys fewer dollars: the euro has depreciated and the US dollar has appreciated against it.

A forward exchange rate is the price agreed today for a currency exchange that settles on a future date. In professional markets the forward is not quoted as a full number but as forward points, also called pips or swap points. The points are simply the forward rate minus the spot rate, scaled so that one point lines up with the last decimal place of the spot quote.

Scale by 10,000 for four-decimal quotes. For the yen, quoted to two decimals, scale by 100 instead.

A positive point count, where the forward exceeds spot, marks a forward premium on the base currency. A negative count, where the forward falls short of spot, marks a forward discount. Whatever the base currency does, the price currency does the opposite. For a spot USD/EUR of 1.15885 and a one-year forward of 1.19532, the difference 0.03647 scales to +364.7 points.

Sample spot rate and forward points, USD/EUR

Maturity	Spot rate or forward points
Spot	1.15885
One week	+5.6
One month	+27.1
Three months	+80.9
Six months	+175.6
Twelve months	+364.7

Each maturity carries its own point quote, measured over the gap from spot settlement to the settlement of the forward.

The absolute number of points grows with maturity because the points are proportional to the interest rate differential between the two currencies, scaled by the term. To turn quoted points back into a forward rate, divide the points by 10,000 (by 100 for the yen) and add the result to the spot rate.

Example 4 · Worked

Using the USD/EUR spot rate of 1.15885 and the three-month quote of +80.9 points.

1. What is the three-month forward rate?

Solution. Scale the points down and add to spot:

2. State the same forward as a percentage of spot.

Solution. The percentage premium is the forward over spot, minus one: Reversing the step confirms the rate: 1.15885 times (1 + 0.00698) is approximately 1.16694. Professional markets usually quote points rather than percentages.

Check yourself

A BRL/MXN spot rate is 0.1378 and the six-month forward rate is 0.14193. What are the six-month forward points?

The forward is above spot, so the points are positive: (0.14193 − 0.1378) times 10,000, which is 0.00413 times 10,000, or +41.3 points.

A dealer lists a three-month CAD/USD forward rate of 1.0123 and quotes the forward as +6.8 percent. What is the spot rate?

Spot times (1 + 0.068) equals the forward, so spot equals 1.0123 divided by 1.068, which is about 0.9478.

Forward rates are not free-floating prices. They are pinned to spot rates and interest rates by an arbitrage relationship called covered interest rate parity. Consider an investor with one unit of domestic currency for one period. One route is to invest at home at the domestic risk-free rate, ending with (1 + r_d). The other route converts to foreign currency at the spot rate, invests abroad at the foreign risk-free rate, and locks in the conversion back using a forward rate, so no currency risk remains. Both routes carry no risk, so they must return the same amount.

Writing the spot and forward rates on a foreign-per-domestic convention (S_f/d), where the domestic currency is the base, the two equivalent investments give the parity condition, which rearranges into a formula for the forward rate.

The forward that removes all arbitrage. If the exchange rate is quoted domestic per foreign instead, invert the rates or use the mirror-image equation.

Read as a ratio, the forward over spot equals (1 + r_f) over (1 + r_d). So the base currency trades at a forward premium when its own interest rate is below the other currency’s rate. Stated without reference to any convention: the currency with the higher interest rate always trades at a forward discount, and the currency with the lower interest rate trades at a forward premium.

Example 5 · Worked

The spot rate S_f/d is 1.6535. Over a 12-month horizon the home currency earns a risk-free 3.50 percent while the foreign currency earns 5.00 percent.

1. What 12-month forward rate removes arbitrage?

Solution. Apply the forward formula: The foreign currency carries the higher rate, so it trades at a forward discount and the domestic currency at a premium (the forward 1.6775 exceeds the spot 1.6535).

2. A dealer instead misquotes the 12-month forward at 1.6900. Show the arbitrage.

Solution. Put the misquoted forward into the hedged foreign leg: The hedged foreign route returns 2.73 percent, below the domestic 3.50 percent. So an investor borrows at the higher foreign rate, sells the foreign currency spot, buys it back at the cheap forward rate, and lends at the domestic rate, banking 3.50 − 2.73, which is 77 basis points per unit, with no capital committed. Such a gap would be arbitraged away almost at once.

Forward rates as expected spot rates

If a forward rate is read as the expected future spot rate, parity implies that the spot rate is expected to move, in percentage terms, in step with the gap between the two interest rates.

Based on the foreign-per-domestic convention.

This interpretation should be handled with care. The implied direction is counterintuitive: all else equal, a higher domestic interest rate implies slower expected appreciation, or faster expected depreciation, of the domestic currency, even though a rate rise is often expected to strengthen a currency. More telling, the historical record shows forwards doing a weak job of anticipating where spot rates actually land. The margin of error is large, and the FX market reflects far more than a single yield differential. It is safest to treat a forward rate as the output of the arbitrage equation, not as a market forecast.

Check yourself

A forward premium on the base currency indicates what about interest rates?

It signals that the price currency pays a higher interest rate than the base currency. Put the other way, the base currency has the lower rate and so trades at a premium.

If the base currency trades at a forward discount, what happens to the forward points and the forward percentage?

Both are negative. A discount means buying the base currency for future delivery is cheaper than buying it spot, so the forward sits below the spot rate.

Real contracts run for a fraction of the year, so the parity formula is generalized by a time factor. Let the horizon be a fraction of the interest-rate period, so interest earned over the horizon is the annual rate times that fraction. Money-market deposits used in this arbitrage are typically quoted on an actual/360 day count. The forward rate, and the difference between forward and spot rates (the swap points), follow.

The horizon fraction is tau, for example 30/360. Swap points are proportional to the spot rate and the interest rate differential, and approximately proportional to the term.

This shows why the number of forward points grows with maturity and with the size of the interest rate gap, though not in an exactly linear way.

Example 6 · Worked

The spot rate S_f/d is 1.6555, the 30-day domestic risk-free rate is 2.00 percent per year, and the 30-day foreign risk-free rate is 3.00 percent per year, on an actual/360 basis.

1. What is the 30-day forward rate, and the premium in pips?

Solution. Insert the horizon fraction 30/360: The forward sits 14 pips above spot (1.6569 − 1.6555 = 0.0014), a forward premium.

2. Leaving everything else unchanged, extend the term to 180 days.

Solution. Use the swap-points form: The premium widens to 82 pips. The term is six times longer, but the points do not sextuple exactly: 6 times 14 would give 84, not 82.

3. Back at 30 days, double the interest rate gap by lifting the foreign rate to 4.00 percent.

Solution. The differential goes from 1.00 percent to 2.00 percent, and the points roughly double:

Hedging with forward points

A treasurer expecting a foreign-currency receipt uses these mechanics directly. Suppose a French company will receive British pounds in about one month and wants to convert them to euros. With a GBP/EUR spot rate of 0.8752 and one-month forward points of −1.4, the company sells pounds and buys euros forward at a rate found by adding the scaled points to spot.

The negative points make this a forward discount on the base currency, the euro. A discount on the base currency signals that the base-currency interest rate (the Eurozone rate here) is higher than the price-currency rate (the UK rate). If the 12-month forward rate were 0.87295, the 12-month points would be the scaled difference from spot: (0.87295 − 0.87520) times 10,000, which is −22.5 points.

Check yourself

The 270-day annualized rates are 1.370 percent for the euro and 1.325 percent for the pound, with a GBP/EUR spot rate of 0.8489. What is the 270-day forward rate and the number of forward points?

Using price over base rates: 0.8489 times [(1 + 0.01325 times 270/360) divided by (1 + 0.01370 times 270/360)] equals 0.84862. The points are (0.84862 − 0.84890) times 10,000, which is −2.8 points, a small forward discount. Omitting the 270/360 day count or mis-scaling gives the wrong answers of −3.8 or −22.8.

The JPY/AUD spot rate is 82.42, the yen rate is 0.15 percent, and the Australian dollar rate is 4.95 percent, on a 360-day basis. What are the 90-day forward points?

The forward is 82.42 times [(1 + 0.0015 times 90/360) divided by (1 + 0.0495 times 90/360)], which is 81.443. Because the yen is quoted to two decimals, scale by 100: 100 times (81.443 − 82.42) equals −97.7 points. The Australian dollar carries the higher rate, so it trades at a forward discount.

End of lesson.

Back to MODULE 03 — Economics

ECO 8 Exchange Rate Calculations

Overview

Cross Rates and Triangular Arbitrage

Reading Currency Strength from Rate Changes

Forward Points and Percentage Quotes

Covered Interest Rate Parity

Forward rates as expected spot rates

Forward Premiums, Discounts, and Maturity

Hedging with forward points