Economics · Reading 19

Exchange Rate Calculations

CFA Level I · Economics · Reading 19 · about 26 min

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What you'll learn

LOS 19.a Calculate and interpret currency cross rates from two (or more) quotes that share a common currency.
LOS 19.b Explain the no-arbitrage link between spot rates, forward rates and interest rates; calculate forward rates from points, percentages or interest rates; interpret forward premiums and discounts.

Module 19.1

Foreign Exchange Rates

This reading calculates currency cross rates from two quotes against a common third currency. It then derives the no-arbitrage forward rate from the spot rate and the two interest rates, converts forward quotes given in points or percentages into outright rates, and interprets forward premiums and discounts.

LOS 19.a — Currency cross-rates

A cross rate is the exchange rate between two currencies derived from the rate of each against a third, common currency (usually the USD or EUR). Cross rates are needed when a currency pair has no active market of its own.

Method: make the units cancel. Write every quote as price/base and arrange the quotes so that the common currency cancels, leaving the required pair:

Key concept

\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}

If a quote has the wrong orientation, invert it first: $B / A = 1/ (A / B)$ .
Two quotes with the same base (e.g. A/USD and C/USD): divide, $A / C = (A / USD) \div (C / USD)$ .
Two quotes with the same price currency (e.g. USD/A and USD/C): divide the other way, $A / C = (USD / C) \div (USD / A)$ .
With three quotes, chain them: $A / D = A / B \times B / C \times C / D$ .

Example. Given 1.3620 CAD/USD and 0.7940 EUR/USD, find CAD/EUR:

\frac{CAD}{EUR} = \frac{CAD}{USD} \times \frac{USD}{EUR} = 1.3620 \times \frac{1}{0.7940} = 1.7154

One euro costs about 1.7154 Canadian dollars. The inverse, EUR/CAD, is $1/1.7154 = 0.5830$ .

Sanity check. Ask "how many units of the price currency does one unit of the base currency buy?" If the euro buys more dollars than the Canadian dollar does, the CAD/EUR rate must be above 1.

Common exam traps

Multiplying where the quotes should be divided, so the units do not cancel. Writing the units next to each number prevents this. In the CAD/EUR example, $1.3620 \times 0.7940 = 1.0814$ instead of 1.7154.
Forgetting to invert at the end when the answer is asked in the opposite orientation (e.g. the chain gives X/Z but the question asks for Z/X). In the CAD/EUR example, the EUR/CAD rate is 0.5830, not 1.7154.
"Direct" and "indirect" depend on the investor's home currency; the arithmetic of a cross rate does not change.

LOS 19.b — Spot, forward and interest rates: no-arbitrage, points, percentages, premiums and discounts

The no-arbitrage (interest rate parity) relation

If currencies trade freely and forward contracts are available, the percentage difference between forward and spot rates must approximately equal the difference between the two currencies' risk-free interest rates. Otherwise an investor could borrow one currency, convert it at spot, invest at the other currency's rate, and sell the proceeds forward for a riskless profit. The no-arbitrage condition rules this out: the round trip cannot earn more than the domestic riskless rate.

The idea behind it: investing at home, and investing abroad with the currency sold forward, are both riskless, so they must end with the same amount of domestic currency. With quotes as domestic/foreign (d/f, price/base):

(1 + r_{d}) = \frac{1}{S _{d / f}} (1 + r_{f}) F_{d / f}

Key concept

\frac{F _{d / f}}{S _{d / f}} = \frac{1 + r _{d}}{1 + r _{f}} F_{d / f} = S_{d / f} \frac{1 + r _{d}}{1 + r _{f}} S_{d / f} = F_{d / f} \frac{1 + r _{f}}{1 + r _{d}}

The price currency's rate is in the numerator and the base currency's rate in the denominator. In percentage terms the same relation reads

\frac{F _{d / f}}{S _{d / f}} - 1 = \frac{r _{d} - r _{f}}{1 + r _{f}} \approx r_{d} - r_{f}

so the forward premium or discount on the base currency is roughly the interest rate differential for the period. Exam convention: the percentage relation is printed as $%Δ s = (r_{f} - r_{d}) / (1 + r_{d})$ , which holds only when the spot rate is quoted as foreign currency per unit of domestic currency. Current practice: with domestic/foreign (price/base) quotes, as used throughout this module, the forward premium is $(r_{d} - r_{f}) / (1 + r_{f})$ .

Key concept

Interest rates	Forward vs spot (P/B quote)	Base currency	Price currency
$r_{price} > r_{base}$	Forward > spot	Forward premium	Forward discount
$r_{price} < r_{base}$	Forward < spot	Forward discount	Forward premium
Equal	Forward = spot	Neither	Neither

The currency with the higher interest rate trades at a forward discount (it "depreciates" in the forward market by roughly the rate differential).

Periods shorter than a year. Use the interest rate for the period rather than the annual rate. Money market rates are annualized, usually on a 360-day basis, so for a 90-day forward each rate becomes $r \times 90/360$ :

F = S \times \frac{1 + r _{price} ( days /360 )}{1 + r _{base} ( days /360 )}

Example. Spot is 1.4800 AUD/GBP, the one-year AUD rate is 4.5% and the one-year GBP rate is 5.0%.

F = 1.4800 \times \frac{1.045}{1.050} = 1.4730 AUD/GBP

The GBP (higher rate, base) is at a forward discount of $1.47295/1.4800 - 1 = - 0.48%$ , close to the 0.5% rate gap.

Example. Spot is 1.4650 CHF/GBP. The 180-day CHF rate is 1.20% and the 180-day GBP rate is 4.80%, both quoted as annualized rates on a 360-day basis. Find the 180-day forward rate in points.

De-annualize each rate: CHF $1.20% \times 180/360 = 0.60%$ ; GBP $4.80% \times 180/360 = 2.40%$ .
Apply parity with the price currency (CHF) in the numerator: $F = 1.4650 \times 1.0060/1.0240 = 1.43925$ .
Convert to points using this quote’s 0.0001 point size (four decimal places): $(1.43925 - 1.4650) /0.0001 \approx - 257.5$ points.

The GBP, the higher-yielding base currency, trades at a forward discount of about $1.76%$ for the half year.

If the quoted forward differs from the no-arbitrage forward, arbitrage is possible: buy the currency that is cheap in the forward market relative to parity and sell the one that is dear. If the forward price of the base currency is too high, borrow the price currency, buy the base currency at spot, invest it at the base-currency rate and sell the proceeds forward. The profit is what is left after repaying the loan with interest. Arbitrage trading pushes the spot and forward rates back to parity.

Forward rates are not forecasts. Reading a forward rate as the expected future spot rate would mean the spot rate is expected to move by roughly the interest rate differential, with the higher-yielding currency expected to depreciate. Capital-market reasoning points the other way: higher domestic rates attract capital and strengthen the currency. Historically, interest rate differentials have forecast future spot rates poorly, though they may be unbiased and usually get the direction right. Treat the no-arbitrage forward as the rate that rules out arbitrage at a point in time. It is not the expected future spot rate.

Forward quotes in points or percentages

Forward points are quoted in units of the last decimal place of the spot quote. For a four-decimal quote, one point = 0.0001:

F = S + (points \times point size)

The size of a point follows the number of decimals in the quote. For a two-decimal quote such as 148.25 JPY/USD, one point is 0.01, so +35 points means +0.35.

A percentage forward quote is applied multiplicatively:

F = S \times (1 + percentage)

Examples. Spot 0.6480 USD/AUD and +27.5 points gives $0.6480 + 0.00275 = 0.65075$ . Spot 0.6480 with -0.35% gives $0.6480 \times (1 - 0.0035) = 0.64573$ .

Forward premium or discount

For the base currency:

forward premium (+) or discount (-) = \frac{F}{S} - 1

A positive value means the base currency is at a forward premium (it buys more of the price currency forward than spot). The price currency is then at a forward discount.
To state the premium or discount of the price currency, invert both quotes first so that it becomes the base currency. The two percentages differ slightly in size.
A premium or discount for a period shorter than a year can be annualized roughly by multiplying by (12 / months).

Common exam traps

One point is 0.0001 on a four-decimal quote: +14.2 points = +0.00142, not +0.0142 or +0.142.
Reading negative forward points as a discount on the price currency. Forward below spot puts the base currency at a discount and the price currency at a premium.
Putting the base currency's interest rate in the numerator, which makes the higher-yielding currency appear to trade at a forward premium. In the AUD/GBP example, $1.4800 \times 1.050/1.045 = 1.4871$ instead of 1.4730.
Using annual rates directly in a 90- or 180-day parity calculation. Scale each rate to the period first (days/360 for rates quoted on a 360-day basis). In the CHF/GBP example, $1.4650 \times 1.0120/1.0480 = 1.41468$ instead of 1.43925.
When asked for the forward rate in the inverted quote, compute the forward first, then invert.

Exam shortcuts

Before computing a cross rate, check its size: if one unit of the base currency buys more of the common currency than one unit of the price currency does, the cross rate must be above 1.
The currency with the higher interest rate trades at a forward discount, so whether the forward is above or below spot is known before any calculation: with a P/B quote, forward exceeds spot when the price currency's rate is higher.
The forward premium or discount on the base currency is roughly the interest rate differential for the period, $r_{price} - r_{base}$ , which checks a computed forward rate.

Bottom line

A cross rate is found by arranging price/base quotes so the common currency cancels, $A / C = A / B \times B / C$ , inverting a quote first when it has the wrong orientation.
Under no-arbitrage, with domestic/foreign (price/base) quotes, $F / S = (1 + r_{d}) / (1 + r_{f})$ , with the price currency's rate in the numerator and the base currency's rate in the denominator.
With price/base quotes the forward premium on the base currency is $(r_{d} - r_{f}) / (1 + r_{f})$ , roughly the rate differential; the exam convention prints the relation as $(r_{f} - r_{d}) / (1 + r_{d})$ , which holds only for quotes of foreign currency per unit of domestic currency.
For a forward shorter than a year, each annualized money market rate is scaled to the period, usually $r \times days /360$ , before applying parity.
A forward quoted in points is $F = S + points \times point size$ , where one point is the last decimal place of the spot quote, and a percentage quote gives $F = S (1 + percentage)$ .
The forward premium (+) or discount (−) on the base currency is $F / S - 1$ , and the price currency's premium or discount is found by inverting both quotes first.
If the quoted forward differs from the no-arbitrage forward, borrowing one currency, converting at spot, investing and selling the proceeds forward earns a riskless profit, and arbitrage trading restores parity.
The no-arbitrage forward rate is the rate that rules out arbitrage at a point in time, not the expected future spot rate, and interest rate differentials have historically forecast future spot rates poorly.

Quick check

Question 1Core

Using the quotes EUR/USD 0.9240 and NZD/EUR 1.7960, an analyst calculates the spot cross rate between the US dollar and the New Zealand dollar. Stated as USD/NZD, it is closest to:

Show answer and explanation

Correct answer: B

Convert EUR/USD into USD/EUR by inverting, then divide by NZD/EUR (equivalently, multiply by EUR/NZD) so the euro cancels and USD/NZD remains.

USD/EUR = \frac{1}{0.9240} = 1.08225

\frac{USD}{NZD} = \frac{USD/EUR}{NZD/EUR} = \frac{1.08225}{1.7960} = 0.60259

Why the other options are wrong

A. This divides EUR/USD by NZD/EUR without first inverting EUR/USD, so the units do not cancel to USD/NZD.
C. This multiplies the two quotes: $0.9240 \times 1.7960 = 1.6595$ . The units (EUR/USD $\times$ NZD/EUR) give NZD/USD, which is the inverse of the USD/NZD rate asked for ( $1/1.6595 = 0.60259$ ).

Key takeaway Invert first, then chain the quotes so the common currency cancels.

Practice Questions

Question 2Core

The spot exchange rate is 1.3450 CAD/USD. The 180-day money market rate is 4.20% for the Canadian dollar and 5.40% for the US dollar, both quoted as annualized rates on a 360-day basis. The 180-day no-arbitrage forward CAD/USD exchange rate is closest to:

Show answer and explanation

Correct answer: B

The no-arbitrage forward rate equals spot times (1 + price-currency rate for the period) / (1 + base-currency rate for the period). Money market rates are de-annualized using days/360. Because the US dollar (base currency) has the higher rate, it trades at a forward discount, so the forward rate is below spot.

F_{C A D / U S D} = 1.3450 \times \frac{1 + 0.0420 ( 180/360 )}{1 + 0.0540 ( 180/360 )} = 1.3450 \times \frac{1.0210}{1.0270} = 1.3450 \times 0.99416 = 1.3371

Why the other options are wrong

A. This uses the full annual rates ( $1.3450 \times 1.042/1.054$ ) instead of the 180-day rates.
C. This puts the USD (base-currency) rate in the numerator; the price-currency (CAD) rate belongs on top.

Key takeaway For forwards shorter than one year, scale each money market rate by days/360 before applying the parity formula.

Question 3Core

The central bank of Selvia unexpectedly raises its policy rate, and Selvian money market rates move well above euro-area rates. Two analysts comment on the outlook for the Selvian lev (SLV) against the euro:

Ruiz: "The higher-yielding currency always trades at a forward discount, so the rate increase tells us the SLV will now weaken against the euro."
Okoye: "Higher Selvian rates should draw in foreign capital, which tends to push the SLV up against the euro."

Which evaluation of the two comments is most accurate?

Show answer and explanation

Correct answer: C

Interest rate parity is a no-arbitrage condition: at a point in time, the higher-yielding currency trades at a forward discount so that borrowing in one currency and investing in the other with a forward cover earns no riskless profit. It does not say how the spot rate responds when interest rates change, and interest rate differentials have historically been poor predictors of future spot rates. Capital-market analysis points the other way: a higher domestic interest rate attracts foreign investment, which raises demand for the currency and tends to make it appreciate. Okoye's comment follows this reasoning, while Ruiz treats the forward rate as a forecast.

Why the other options are wrong

A. Ruiz treats the no-arbitrage forward rate as the expected future spot rate. Parity only sets the forward rate that rules out arbitrage today; it does not describe how the spot rate moves after a rate change, and rate differentials have forecast future spot rates poorly.
B. A change in the policy rate does affect the exchange rate. The exchange rate is one channel of the monetary transmission mechanism: higher domestic rates attract capital and tend to strengthen the domestic currency.

Key takeaway Treat the no-arbitrage forward rate as the rate that prevents arbitrage today; it is not a forecast of the future spot rate. A higher domestic interest rate tends to attract capital and strengthen the currency.

Question 4Core

A dealer's SGD/AUD quotes are 0.8870 for spot settlement and 0.8795 for delivery in 12 months. Measured against spot, the forward quote is a:

Show answer and explanation

Correct answer: B

The forward SGD/AUD rate is below spot, so the forward quote is a discount. With four-decimal quotes the difference of 0.0075 equals 75 points. The AUD (base currency) is at a forward discount, which means the SGD (price currency) is at a forward premium to the AUD.

F - S = 0.8795 - 0.8870 = - 0.0075 = - 75 points

AUD: $0.8795/0.8870 - 1 = - 0.85%$ (discount); SGD: $0.8870/0.8795 - 1 = + 0.85%$ (premium).

Why the other options are wrong

A. The forward rate is below spot, so the quote is a discount of 75 points. The SGD, as the price currency, is at a premium.
C. Each point is 0.0001, so a 0.0075 difference is 75 points. The SGD is also at a premium, because the AUD buys fewer SGD forward than spot.

Key takeaway The sign of the points describes the base currency; the price currency moves the other way.

This reading has 29 questions in the full bank. Practice all of them.

Key Takeaways

Invert first, then chain the quotes so the common currency cancels.
For forwards shorter than one year, scale each money market rate by days/360 before applying the parity formula.
Treat the no-arbitrage forward rate as the rate that prevents arbitrage today; it is not a forecast of the future spot rate. A higher domestic interest rate tends to attract capital and strengthen the currency.
The sign of the points describes the base currency; the price currency moves the other way.