Foreign exchange derivatives and arbitrage

Covered Interest Arbitrage Formula Lab

Covered interest arbitrage is the CFA idea that a forward contract should remove exchange-rate risk from an international money-market investment. Once FX risk is covered, the domestic return and the covered foreign return should be equal.

This page explains only Covered Interest Arbitrage. It uses the notation from your formula image, builds the rule from cash flows, then applies it through CFA-style examples and questions.

Start with the formula Jump to practice

Quote convention: \(S_{f/d}\), \(F_{f/d}\) foreign per domestic

Core no-arbitrage equality

\[(1+i_d)=\frac{S_{f/d}(1+i_f)}{F_{f/d}}\]

One unit of domestic currency should grow to the same maturity value whether it is invested at home or converted, invested abroad, and converted back using a forward.

The Formula, Built From Cash Flows

Use one consistent quote convention. Here, \(S_{f/d}\) and \(F_{f/d}\) mean units of foreign currency \(f\) per one unit of domestic currency \(d\). The domestic currency \(d\) is the base currency in the quote.

Step 1: Domestic investment

\[1\ d \longrightarrow (1+i_d)\ d\]

If you keep the money at home, one unit of domestic currency becomes \(1+i_d\) units of domestic currency at maturity.

Step 2: Covered foreign investment

\[1\ d \xrightarrow{S_{f/d}} S_{f/d}\ f \xrightarrow{1+i_f} S_{f/d}(1+i_f)\ f\]

Convert domestic currency into foreign currency at the spot rate and invest at the foreign interest rate for the same maturity as the forward contract.

Step 3: Use forward to return to domestic

\[\frac{S_{f/d}(1+i_f)}{F_{f/d}}\ d\]

Since \(F_{f/d}\) is foreign currency per domestic currency, divide the future foreign currency by the forward rate to get domestic currency.

No-arbitrage condition

\[(1+i_d)=\frac{S_{f/d}(1+i_f)}{F_{f/d}}\]

If the two maturity values are unequal, the higher one is the asset side and the lower one is the funding side. That is the covered interest arbitrage opportunity.

Important rearrangements

Use	Expression
No-arbitrage forward	\(F_{f/d}^{*}=S_{f/d}\frac{1+i_f}{1+i_d}\)
Covered domestic return from foreign investment	\(R_d^{covered}=\frac{S_{f/d}(1+i_f)}{F_{f/d}}-1\)
Approximate forward premium on \(d\)	\(\frac{F-S}{S}\approx i_f-i_d\)

Exam language

A CFA-style question often asks if the forward is overvalued or undervalued relative to interest rate parity. Compute \(F^*\), compare it with the market forward, then write the exact borrow-convert-invest-forward cash flows.

Decision Rules and the Meaning of Buy Low, Sell High

The phrase is correct, but only after adjusting for interest-rate carry. The forward rate that is "fair" is not usually the same as the spot rate. It is the spot rate adjusted by the interest-rate ratio.

If \(F_{mkt} < F^*\)

Market forward is too low

The domestic currency \(d\) is cheap in the forward market because one unit of \(d\) costs too few units of \(f\) forward. The covered foreign investment gives a domestic return above the domestic funding cost.

1Borrow domestic currency \(d\).
2Sell \(d\) spot and buy foreign currency \(f\).
3Invest \(f\) at the foreign rate \(i_f\).
4Buy \(d\) forward, using \(f\), to repay the domestic loan.

If \(F_{mkt} > F^*\)

Market forward is too high

The domestic currency \(d\) is expensive in the forward market. Buy \(d\) in the spot market, invest it, and sell \(d\) forward at the overpriced forward rate.

1Borrow foreign currency \(f\).
2Use \(f\) to buy domestic currency \(d\) spot.
3Invest \(d\) at the domestic rate \(i_d\).
4Sell \(d\) forward to receive \(f\), then repay the foreign loan.

CFA shortcut: do not memorize only the direction. First compute the higher covered return. Borrow in the lower-return currency, invest in the higher-return currency, and cover the exchange-rate exposure with a forward contract.

Detailed CFA-Style Numerical Examples

These examples use annual rates and one-year forwards to keep the arithmetic visible. For a 90-day or 180-day CFA question, convert the annual rates to maturity-matched periodic rates before using the formula.

Example 1 No arbitrage

Find the fair one-year forward

\(S_{USD/EUR}=1.1000\), \(i_{USD}=5.00\%\), and \(i_{EUR}=3.00\%\). Here \(f=USD\) and \(d=EUR\).

Apply the no-arbitrage forward formula:

\[F_{USD/EUR}^{*}=1.1000\left(\frac{1.0500}{1.0300}\right)=1.12136\]

1 Domestic path EUR 1.0000 -> EUR 1.0300

2 Foreign covered path EUR 1.0000 -> USD 1.1000 -> USD 1.1550 -> EUR 1.0300

At \(F=1.12136\), both paths end with the same EUR amount. No arbitrage exists because neither path dominates the other after covering FX risk.

Example 2 Forward too low

Borrow EUR, invest USD, buy EUR forward

Keep \(S_{USD/EUR}=1.1000\), \(i_{USD}=5.00\%\), and \(i_{EUR}=3.00\%\), but the market forward is only \(F_{USD/EUR}=1.1100\).

First compare the market forward with the fair forward: \(1.1100 < 1.12136\). The market forward is too low. The covered USD investment return in EUR is:

\[R_{EUR}^{covered}=\frac{1.1000(1.0500)}{1.1100}-1=4.0541\%\]

Since \(4.0541\% > 3.0000\%\), borrow EUR and invest in USD while covering the future USD back into EUR.

1 Borrow EUR 10,000,000

2 Convert at spot EUR 10,000,000 x 1.1000 = USD 11,000,000

3 Invest USD USD 11,000,000 x 1.0500 = USD 11,550,000

4 Buy EUR forward USD 11,550,000 / 1.1100 = EUR 10,405,405

5 Repay EUR loan EUR 10,000,000 x 1.0300 = EUR 10,300,000

Arbitrage profit = EUR 10,405,405 - EUR 10,300,000 = EUR 105,405.

Example 3 Forward too high

Borrow USD, invest EUR, sell EUR forward

Same spot and interest rates, but now \(F_{USD/EUR}=1.1350\). The fair forward remains \(1.12136\).

Since \(1.1350 > 1.12136\), EUR is too expensive in the forward market. You want to own EUR and sell it forward high.

1 Borrow USD USD 11,000,000

2 Buy EUR spot USD 11,000,000 / 1.1000 = EUR 10,000,000

3 Invest EUR EUR 10,000,000 x 1.0300 = EUR 10,300,000

4 Sell EUR forward EUR 10,300,000 x 1.1350 = USD 11,690,500

5 Repay USD loan USD 11,000,000 x 1.0500 = USD 11,550,000

Arbitrage profit = USD 11,690,500 - USD 11,550,000 = USD 140,500.

Example 4 Bid/ask spread

Use the rates that make the trade harder

CFA questions with transaction costs require bid/ask discipline. You buy at the ask, sell at the bid, lend at the lower rate, and borrow at the higher rate.

Input	Value	Why it matters
Spot \(S_{USD/EUR}\)	1.0995 / 1.1005	Sell EUR at bid 1.0995; buy EUR at ask 1.1005.
Forward \(F_{USD/EUR}\)	1.1140 / 1.1154	Buy EUR forward at ask 1.1154; sell EUR forward at bid 1.1140.
USD deposit / borrow	4.80% / 5.20%	Use 4.80% if investing USD; use 5.20% if borrowing USD.
EUR deposit / borrow	2.85% / 3.15%	Use 2.85% if investing EUR; use 3.15% if borrowing EUR.

Test the borrow-EUR, invest-USD path:

\[\frac{S_{bid}(1+i_{USD,lend})}{F_{ask}}-1=\frac{1.0995(1.0480)}{1.1154}-1=3.3061\%\]

The covered USD return in EUR is 3.3061%, while the EUR borrowing cost is 3.15%. That leaves a small arbitrage even after transaction costs.

On EUR 5,000,000 borrowed, profit is approximately EUR 7,804 after using bid/ask spot, bid/ask forward, deposit rates, and borrowing rates.

Interactive Parity Calculator

Enter a quote \(S_{f/d}\), a market forward \(F_{f/d}\), domestic and foreign annual rates, and a domestic notional. The calculator applies the exact formula and tells you which covered arbitrage path dominates.

Inputs

Spot \(S_{f/d}\) Foreign currency per 1 domestic currency Forward \(F_{f/d}\) Same maturity as the rates Domestic rate \(i_d\) % Use the period-matched domestic rate Foreign rate \(i_f\) % Use the period-matched foreign rate Domestic notional Used only for estimated profit

Output

Run the calculator to compare the market forward with the no-arbitrage forward and identify the trade direction.

CFA Exam Traps

Most wrong answers in covered interest arbitrage come from notation, timing, or bid/ask mistakes rather than from the formula itself.

Trap 1

Using annual rates without matching the forward maturity

If the forward is 90 days, use 90-day interest factors. With simple money-market rates, this is often \(1+r(90/360)\) or \(1+r(90/365)\), depending on the convention given in the question.

Trap 2

Ignoring the quote direction

This page uses \(S_{f/d}\), foreign per domestic. If a question quotes domestic per foreign, the fraction reverses. Always write what one unit of base currency buys.

Trap 3

Using mid-rates in a bid/ask question

In real arbitrage, the trader receives the worse price. Buy at ask, sell at bid, lend at the lower rate, and borrow at the higher rate.

Trap 4

Calling it arbitrage before covering the FX risk

The word covered means the future exchange rate is locked in with a forward contract. Without the forward, it is an uncovered carry trade, not covered interest arbitrage.

Original CFA-Style Practice Questions

Select an answer and check it. Each solution focuses only on Covered Interest Arbitrage and the formula convention used on this page.

Question 1

If \(S_{f/d}\) is quoted as foreign currency per domestic currency, which expression gives the no-arbitrage forward rate?

\(F=S\frac{1+i_d}{1+i_f}\) \(F=S\frac{1+i_f}{1+i_d}\) \(F=S(i_f-i_d)\)

Correct answer: B. With \(S_{f/d}\), one domestic unit becomes \(S(1+i_f)\) foreign units after foreign investment, then is divided by \(F\) to return to domestic. Setting that equal to \(1+i_d\) gives \(F=S(1+i_f)/(1+i_d)\).

Question 2

\(S_{USD/EUR}=1.2000\), \(i_{USD}=6\%\), and \(i_{EUR}=2\%\). What is the one-year no-arbitrage forward \(F_{USD/EUR}\)?

1.2471 1.1547 1.2240

Correct answer: A. \(F=1.2000(1.06/1.02)=1.2471\). USD is the foreign currency in this notation, and EUR is the domestic currency.

Question 3

The covered foreign investment return in domestic currency is 4.20%, while the domestic borrowing rate is 3.60%. Which arbitrage direction is consistent?

Borrow foreign, invest domestic, sell domestic forward. Do nothing because 4.20% is close to 3.60%. Borrow domestic, invest foreign, buy domestic forward.

Correct answer: C. The covered foreign investment pays more than the domestic funding cost, so borrow domestic, convert to foreign, invest foreign, and use the forward to buy domestic back at maturity.

Question 4

For \(S_{USD/EUR}=1.1000\), \(i_{USD}=5\%\), and \(i_{EUR}=3\%\), the fair forward is 1.12136. If the market forward is 1.1350, what should the arbitrageur do?

Borrow EUR, invest USD, buy EUR forward. Borrow USD, invest EUR, sell EUR forward. Borrow both currencies and invest both currencies.

Correct answer: B. The market forward is too high, so EUR is overpriced forward. Buy EUR spot by borrowing USD, invest EUR, and sell EUR forward to receive USD.

Question 5

In a bid/ask quote \(S_{USD/EUR}=1.0995/1.1005\), an arbitrageur selling EUR spot receives which rate?

1.0995 1.1005 The midpoint, 1.1000

Correct answer: A. EUR is the base currency in \(USD/EUR\). If you sell the base currency, the dealer buys it from you at the bid.

Question 6

Why is the arbitrage called "covered"?

Because both currencies have the same interest rate. Because the initial spot conversion is optional. Because the future FX conversion is locked with a forward contract.

Correct answer: C. The forward contract fixes the maturity exchange rate, so the arbitrage profit is not exposed to future spot-rate uncertainty.

Question 7

A 180-day forward is used in the arbitrage. What interest rates should be used in the formula?

Always the full annual quoted rates with no adjustment. Rates converted into 180-day interest factors. Only the higher of the two annual rates.

Correct answer: B. The spot, forward, and interest-rate factors must all share the same maturity. For a 180-day forward, use 180-day interest factors.

Question 8

Suppose \(F_{mkt}=F^*\) after using the correct bid/ask and borrowing/lending rates. Which statement is most accurate?

No covered interest arbitrage is available. Borrow the low interest-rate currency automatically. A riskless profit exists because two countries have different rates.

Correct answer: A. Different interest rates alone do not create arbitrage. Arbitrage requires a mismatch between the market forward and the interest-rate parity forward after transaction costs.