A mature market and full yield curve have developed for cross-currency swaps in the past two years and individual trades can run into billions of dollars. Alex Puaca provides an outline of the market and determines some openings for profitable arbitrage trade

Cross-currency swaps can no longer be considered the poor cousin of the interest rate swap. They have developed a full yield curve and fast markets at the short end (up to a year in tenor). Further down the curve, where cross-currency swaps trade in much larger size, the relationship is reversed, with cross-currency swaps and corresponding interest rate swaps defining the market in forwards. The overlap area – the one to three-year range – provides opportunities for profitable trade. And where there are liquid forward prices, arbitrage relationships exist that determine exactly – down to capital and credit costs – the corresponding cross-currency swaps.

Cross-currency swaps denote an exchange of interest rate payments in two different currencies where the underlying index is the London interbank offered rate (Libor) or a similar interbank standard (Tokyo interbank offered rate, Tibor, for example). Exchange of principal at the start of the swap and, more importantly, at maturity is taken as given, although the analysis for structures in which there is no exchange is similar. The usual convention for quoting the basis (or margin) relative to the US dollar side of the cross-currency swap is followed.

Cross-currency swaps allow an institution to switch from one currency to another. For example, using a bond issue an institution can tap a source of fixed rate funds in the euro market, where its name may be well known and its credit well perceived, and swap into another market, dollars say, to achieve funding at a level significantly better than direct issuance offers. This type of liquidity arbitrage is a major source of supply in determining cross-currency swap levels. The converse of this, the asset swap or synthetic floater, works to counteract this price action, but as this market is smaller, the effect of new bond issuance tends to dominate.

In our analysis of cross-currency swaps, we will make the assumption that credit risk can be ignored. While this is standard practice in this type of discussion, the simplification may be dangerous. Credit is a significant factor in some currencies as the underlying interbank rate is fixed by reference to a finite, pre-determined set of banks with differing credit ratings. We will also neglect the cost of capital.

The simple approach
By breaking up a cross-currency swap into its two constituent parts, one is left with two floating rate loans in different currencies. The position can, in theory, be run and hedged as separate cash positions. Based on the prices in table A, transacting a 10-year dollar/yen swap, in which we receive dollar Libor and pay yen Libor minus 24 basis points, can look attractive. Effectively, we have lent a sum in dollars at a flat rate to Libor and created funding in an equivalent yen amount for 10 years at a significant sub-Libor margin.

Given that we may have dealt with a good quality counterparty (a very competitive market means there is little difference in pricing offered by good or lesser credits), the whole transaction can make sense when we calculate the
capital allocation and the return required. In the past, institutions were able to account for and run positions in this way.

However, with the dominance of mark-to-market revaluation (that is, calculating the immediate replacement cost of the transaction), this approach is no longer possible for longer-term swaps. Regulators no longer allow most trading institutions to run their books on an investment, or accrual, basis. An increasingly volatile market means that a trade that looks sensible on a cash basis may show a significant loss using a mark-to-market methodology.

However, the simple valuation methodology should not be ignored for the shorter end of the yield curve, where banks can hold positions for longer periods, perhaps even until maturity. And it is precisely this methodology that ensures cross-currency swaps levels do not stray too far from parity for short tenors.

As cross-currency swaps appear closely related to the spot and forward foreign exchange markets, one might expect any basis to be zero. But this is not the case. In the past, varying capital adequacy requirements in different countries gave rise to non-zero margins, although this effect is less marked today. The box arbitrage in the foreign exchange markets showing how to calculate a break even forward rate (see figure 1) is not perfect, and itself can suffer from systematic biases. Most of these can be accounted for mathematically, but they still do not explain the margin in full. That most unquantifiable of phenomena, supply and demand, is responsible, as our earlier reference to new issue swaps shows. For many European currencies the basis is small, as the swap trades close to par, but this is not the case for the Japanese yen curve.

The box arbitrage is the natural place to begin an analysis of cross-currency swaps, given the close relationship between the two products. It gives

where F is forward points (excluding the usual scaling factor, eg 10,000), S is the spot exchange rate, RT is the terms interest rate, RB is the base interest rate and t is time in years.

We use standard foreign exchange terminology to express the relationship between spot and forward exchange rates and the two underlying interest rates. In defining spot, the denominator or strong currency is referred to as the base and the numerator or weaker one as the terms currency. For example the term “euro dollar” means the number of dollars per unit of euro. The base currency is the euro and the terms currency the dollar.

This expression is not the normal one, rather it is the rearrangement that explicitly shows how the forward foreign exchange points depend on the interest rate differential.

Table B shows the arbitrage channel, defined by certain cash rates and spot, taken on December 22 1999, for the six-month euro dollar forward. It fixes the limits, outside which risk-free profits can be made (subject to our assumptions), and involves crossing the bid/offer spreads in the respective money markets. It gives a market of 133.2/143.4 as opposed to the real price of 138/139. Using mid-market interest rates, it gives a level of 138.31. That the real forward is well inside the arbitrage pricing is expected, as the foreign exchange markets tend to be more liquid in maturities of less than one year than the corresponding interest rate markets.

Moreover, there is a second arbitrage-type calculation that gives rise to a narrower channel. An institution looking to borrow money in euros can either go to the underlying cash market or raise the funds in dollars then swap them into euros on the foreign exchange market. It will choose the cheaper of the two alternatives.

While this is not arbitrage in its strict sense, it does allow money to be saved. We term it pseudo-arbitrage. The pseudo-arbitrage channel is much tighter than the true arbitrage channel, as it is calculated using the same side of the cash markets (that is, bid to bid or offer to offer). From equation (1), it is clear that the deviation from our mid-market value of the forward points so derived will depend on the difference of the bid/offer spreads in the respective cash markets (as opposed to the pure arbitrage case where it depends on the sum). Simply define:

for both interest rates and insert this into equation (1). If the spreads are similar or identical for both currencies, we will get a pseudo-arbitrage price that is very close to our mid-market calculation. In our example, it is 138.27/138.34. Taking into account parameters such as the cost of capital would widen this out to achieve something close to the true market.

Our reason for being so precise about a well-known arbitrage calculation reflects the fact that it is highly relevant to the cross-currency swaps market. The first fixing for a swap on both sides of the transaction is at the Libor (or other interbank) rate at specific times in the trading day. Once the swap has commenced, the user needs to be able hedge the rates that have been fixed – most easily achieved using a foreign exchange swap. Note that the offered side of the cash market is taken for both fixings, hence the need for the detailed analysis above to determine if there is a systematic cost in rolling the hedge.

Timing mismatch
Although similar spreads in the underlying cash markets mean the pseudo-arbitrage forward price will be close to mid-market, this will almost certainly not be true if we take into account timing mismatches. The reference rates used in cross-currency swaps are set once a day, possibly at different times for each of the two underlying currencies, whereas the foreign exchange forward hedge reflects current cash rates. Thus there may be a significant profit or loss on the first leg of a swap, which must be taken into account in determining the overall margin over the life of the transaction. The same theory applies for every subsequent fix of the swap when the hedge is rolled over. Further, using a series of forwards and the corresponding interest rate hedges, the risk may be hedged completely at the time of the transaction. In theory, therefore, an arbitrage relationship exists between the cross-currency swap and the foreign exchange market.

In practice, this relationship boils down to a question of liquidity. At the shorter end of the yield curve (up to one year), the foreign exchange markets are more liquid in terms of their bid/offer spreads. But for longer maturities, the cross-currency swap market is more liquid. Shorter-term swaps are therefore priced off the forward market, while medium and longer-term swaps, and the corresponding interest rate swaps, tend to drive the forwards for those periods.

In the latter case, one has to analyse how to build the underlying discount factor curves in the presence of a non-zero basis swap margin (for example, see Risk, October 1995). The answer is to add the margin to the non-dollar pure interest rate curve and bootstrap in the normal way, although care must be taken to compare like with like, and interpolation can be tricky. But the problem is not a difficult one.