How to value a cross-currency swap

The valuation of a CCS is quite similar to the valuation of an interest-rate swap. The CCS is valued by discounting the future cash flows for both legs at the market interest rate applicable at that time. The sum of the cash flows denoted in the foreign currency (hereafter euro) is converted with the spot rate applicable at that time. One big difference with an interest-rate swap is that a CCS always has an exchange of notional.

Looking at a CCS with a fixed-fixed structure (both legs of the swap have a fixed rate), the undiscounted cash flows are already known at the start of the deal, they are simply the product of the notional, the fixed rate and the year fraction.

The discounting of the cash flows requires a more complex method. The US dollar curve is the base of everything and is, therefore, not different from valuations of plain vanilla US dollar interest-rate swaps. Looking at a euro/US dollar CCS, the eurocurve (excluding credit spreads) is made of two parts:

• The euro interest rate curve and
• The basis spread.

This basis spread curve represents a ‘compensation’ for the changes in the forward FX rates between the two currencies used in the swap. Before the global credit crisis this spread was close to zero. Nowadays, the spread ranges from 18 basis points (bp) (10-year spread) to 40bp (one-year spread), but reached 120bp as shown by figure 1.

The big peak which is visible in the last quarter of 2008 was caused by the credit crisis (the default of Lehman Brothers and Bear Stearns, and the sale of Merrill Lynch, etc). Due to the lack of liquidity in the market during the crisis, the (liquidity) spreads in the US became a lot higher than those in Europe. To make up for this window of arbitrage, the basis spread decreased at a similar pace.

Here is an example: The characteristics of our USD-EUR example swap are:
The first leg in US dollar has a notional of USD 10,000,000 and a fixed interest of 2.50%

The valuation is performed at January 31st, 2011. The FX rate at that moment was EUR/USD 1.3697. The second leg in euro has a notional of EUR 7,481,670 and a fixed interest of 3.00%. The valuation is done from the perspective of the party which pays the euro flows and receives the US dollar flows. The frequency of the payment is annual and there is no amortization of the notional.

• In columns B and E the future cash flows are calculated by multiplying the notional with the fixed rate applicable for that leg. This results in cash flows of USD 250,000 (column B) and -/- EUR 224,450 (column E).
• The market value of the cash flows is calculated by multiplying the cash flows with their discount factor (column C for the US dollar and column F for the euro).
• The euro market value (column G) is converted to US dollar by multiplying it with the spot EUR/USD, i.e. 1.3697. Adding this converted value to the US dollar market value of column D results in the net market value (column H).

To demonstrate the impact of the basis spread we will repeat step 2 and 3 without the basis spread. The euro market value excluding basis spread is shown in column J, it is calculated by multiplying column E and I. The adjusted net market values are shown in column K. The difference of the sum of column H and K is 7,5 basis points of the US dollar notional. The basis spread impact can be checked, for the first year, by calculating the variation between the value in column G (222,206) with the value in column J (221,347), the result is 39bp which is in line with figure 1.

The above calculation shows that the exclusion of the basis spread in the valuation of the cross-currency swap results in a wrong net market value.