Covered Interest Parity Lost: Understanding the Cross-Currency Basis: formula sheet

Key formulas

CIP, direct quote (domestic per 1 foreign)

F = S × (1 + r_d × T) ÷ (1 + r_f × T)

S and F are units of domestic currency per 1 unit of foreign currency. Simple interest for money-market maturities. T in years.

CIP with compounding

F = S × (1 + r_d)^T ÷ (1 + r_f)^T

Use for longer maturities or when rates are annual compounded. Continuous form: F = S × e^((r_d − r_f)T).

Forward points

Forward points = F − S

Positive means the foreign currency is at a forward premium against the domestic one. Quoted in pips, so check the scaling.

Approximate forward premium

(F − S) ÷ S ≈ (r_d − r_f) × T

A quick check only. Do not use it when exact answers are options.

Arbitrage rule

If F_market > F_CIP, sell the forward, buy foreign currency spot, and lend abroad.

If F_market < F_CIP, do the reverse. Direction is set by which route returns more domestic currency.

CIP condition (no basis)

F ÷ S = (1 + r_domestic) ÷ (1 + r_foreign), with rates for the same horizon and exchange rate quoted as domestic per unit foreign

Forward-to-spot ratio equals the interest rate ratio. Basis is zero when this holds.

Cross-currency basis (simple, annualised, one period)

x ≈ r_non-USD, implied − r_non-USD, direct

The implied rate is the non-dollar rate the FX swap market gives you when you hand over non-dollar currency and receive dollars, which is the synthetic dollar route. The direct rate is what you would earn lending the same currency in its own money market. A negative x means the swap market pays less on the non-dollar currency than the direct rate, so obtaining dollars through the swap costs more. Sign conventions vary by source, so check how the question defines it.

Forward points

Forward points = F − S

The forward points carry the interest differential plus any basis in an FX swap.

Basis swap quoting convention

Non-USD leg pays: floating rate + basis (bp). USD leg pays: floating rate flat

Negative basis means the non-USD leg pays below its reference rate. It signals a premium to obtain dollars.

Covered interest parity (no-arbitrage)

F ÷ S = (1 + i_d) ÷ (1 + i_f)

S and F are spot and forward in domestic currency per unit of foreign currency. i_d is the domestic rate, i_f the foreign rate, over the same period. The forward premium offsets the interest differential.

Implied USD rate from FX swap (quote: USD per EUR)

(1 + i_USD,implied) = (1 + i_EUR) × F ÷ S

Use S and F as USD per EUR. Gives the dollar rate you pay by borrowing euros and swapping into dollars. Annualise for periods under a year.

CIP deviation (basis)

x = i_USD,cash − i_USD,implied

Sign conventions differ. In the usual quoting, x is negative when swap-implied dollar borrowing costs more than cash USD rates. Always check which rate is compared with which.

Annualised forward premium or discount

(F − S) ÷ S × (360 ÷ days)

Positive means the foreign currency is at a forward premium. Use the day-count the question gives.

CIP condition

F ÷ S = (1 + r_USD) ÷ (1 + r_foreign), with F and S in USD per unit of foreign currency

Holds with no frictions. F is the forward rate and S is the spot rate.

Cross-currency basis (sign convention)

r_foreign implied via FX swap from USD = r_foreign direct + Basis

This is the market convention: the basis is a spread added to the foreign-currency rate (for example, EUR rate + basis = the EUR rate implied by borrowing USD and swapping). A negative basis means the implied foreign rate is lower, so USD borrowed through the swap costs more than USD borrowed directly. The supplier of dollars earns a premium of about −Basis. Use this convention throughout and check the sign convention in the question.

Arbitrage profitability rule

For a negative basis, arbitrage pays only if (−Basis) > cost of balance sheet use + credit and funding costs

−Basis is the premium earned for supplying dollars through the swap. If the cost exceeds that premium, the deviation persists without being exploited.

Quick revision

• CIP: a covered (hedged) foreign investment should earn the same return as the domestic one.
• Forward-rate form: F = S × (1 + i domestic) ÷ (1 + i foreign), with rates quoted for the same horizon and the exchange rate as domestic per foreign.
• The higher-interest-rate currency trades at a forward discount if CIP holds.
• An FX swap exchanges currencies at spot and reverses at a forward rate; it is a collateralised-style funding tool.
• A cross-currency basis swap exchanges floating-rate payments in two currencies, with the basis spread added to the non-USD leg.
• A negative basis against USD means the implied cost of borrowing USD through swaps exceeds the direct USD rate.
• CIP deviations became large and persistent after the 2008 global financial crisis.
• A key driver is strong demand for USD funding from non-US banks and investors.
• Limits to arbitrage include balance-sheet costs and regulatory constraints, so arbitrageurs do not fully close the gap.
• A wider basis signals USD funding stress and raises hedged funding costs.
• Treasury responses include diversifying funding sources, managing USD liquidity buffers and rollover risk.

Common mistakes

• Inverting the rate ratio Fix: Put the rate of the numerator currency on top. If the quote is USD per EUR, USD is on top.
• Treating the forward as a forecast of future spot Fix: CIP forward comes from no-arbitrage and interest rates, not from views. Expected spot belongs to uncovered parity.
• Treating an FX swap and a currency swap as the same product Fix: FX swap: short-dated, principal exchange at spot and forward, interest embedded in forward points. Cross-currency basis swap: longer-dated, floating interest paid in each currency, with a spread.
• Reading a negative basis as a gain for the dollar borrower Fix: A negative basis on the non-dollar leg means the dollar borrower pays a premium. Dollars obtained via the swap cost more than direct dollars.
• Saying a negative basis means dollars are cheap to borrow. Fix: Remember that the basis is added to the non-dollar leg. A negative basis lowers what a euro or yen lender receives, so dollars obtained via swaps cost more.
• Using S and F in the wrong direction. Fix: Write out the quote, for example USD per EUR, before inserting numbers. Check that the answer is sensible.
• Saying the basis proves a free arbitrage exists. Fix: Always net the balance sheet and credit costs against the basis. If costs are higher, there is no profitable arbitrage.
• Blaming only demand for dollars. Fix: Give two sides: demand imbalance and constrained supply from banks.

Exam tips

• Write the quote convention beside the numbers before you calculate. Most lost marks come from inversion.
• When four options are given, use the direction test first. It often removes two options.
• Know that CIP is a no-arbitrage identity and uncovered parity is not. Expect conceptual options on this.
• Link CIP to the basis. A negative basis means the market forward implies a higher cost of raising USD through FX swaps than CIP does.
• Check whether the question uses simple or compounded rates. Use the one stated.
• Always find which leg carries the spread and read its sign before choosing an answer.
• Questions often link a negative basis to dollar shortage, so be ready to name the drivers: funding demand and limits to arbitrage.
• Know the difference between FX swaps (short, embedded in forward points) and basis swaps (long, floating legs with a spread).