Calculating Cross-Currency Basis with Quarterly Rates and Correct Quotes

Summary

The note addresses why a cross-currency basis calculation may appear far from zero before the financial crisis. The key correction is to match the interest-rate convention to the contract horizon: for a three-month period, convert quoted annualized rates to quarterly returns before comparing them with the forward-to-spot exchange-rate ratio. Using annualized rates directly makes the comparison inconsistent.

The example also highlights quotation direction. When spot and forward rates are quoted as domestic currency per unit of foreign currency, invert both before applying the stated formula if it requires foreign currency prices. A yen example uses spot and forward quotes and a foreign interest rate to illustrate how the adjusted exchange-rate ratio can align with the dollar return, producing a near-zero basis. The response suggests checking a crisis-period date to observe a basis. This is an illustrative calculation rather than a comprehensive treatment of day-count conventions, compounding, or market-specific basis definitions.

Key ideas

• Match the interest-rate period to the forward contract horizon before calculating basis.
• Annualized rate quotes must be converted to returns over the contract period.
• Confirm the exchange-rate quotation direction and invert spot and forward quotes when required.
• A worked yen example illustrates a near-zero pre-crisis basis after convention corrections.
• Market-specific conventions and compounding details may affect the calculation.

Tags

# Calculation cross-currency basis


# Calculation cross-currency basis












I am trying to calculate cross-currency basis on the 3-month horizon for a certain set of currencies. The formula should be $ccb = F/S (1+y_{foreign currency}) - (1+y_{USD})$ where $y_{USD}$ is Libor 3months in USD, S is the spot rate and F is the forward rate (both expressed as the price of foreign currency) and y_foreign currency is the 3-months Libor in the foreign currency. Now, before 2008 I should obtain cross-currency basis close to 0 for almost all the currencies, but that is not the case. For example Japan in 26/11/2004 had an interest rate differential equal to -.0239 (-2.3%). The ratio between spot and forward, however, is not giving me the expected result. What am I doing wrong?

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/49207

Since we are dealing with quarterly returns we have to use the returns over one quarter (one period) not the annualized returns that are commonly quoted and that you used in your formula. So the formula is

$$ccb/4 =\frac{F}{S}(1+y_f/4)-(1+y_{usd}/4)$$

Now the calculation. If the conventional quotation (i.e. in USDJPY terms) S is 103.00 and the forward points are -61, then F is 102.39. But in this formula we need the rates in the opposite direction S=1/103 and F=1/102.39. So F/S is 1.0059576. Since yf is 0.00053, 1+yf/4 is 1.0001325. So (F/S)*(1+yf/4) is 1.0060909. This is not too different from 1+y/4 which is 1.006. So unless I made a mistake (entirely possible) we are on target for essentially no currency basis.

Now try it for a date during the Global Financial Crisis and there should be cross currency basis.

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