A Model for Understanding Cross Currency Swaps

For those who are taking CFA exams or entering into an entry-level finance job position, swaps are likely going to be one of most challenging concepts to be faced. A swap is a kind of Over-the-Counter (OTC) agreement in which two parties exchange its financial resources during the agreed period of time. Like a forward agreement, a swap deals with future transactions of which the terms are agreed upon today. One of the most common swaps is one that exchanges one party’s floating interest rates for the other’s fixed interest rates. We call this kind a plain vanilla swap.

In this article, I’d like to share my understanding as well as methods of calculations about cross currency swaps (CCS). Although there may be dozens of financial libraries in Matlab or Python for CCS calculations, they don’t help understand the basics. In addition, I personally found explanations on the CFA textbook or Investopedia too academically to grasp. So this article is my attempt to offer my understanding and a self-explanatory Excel model for cracking a CCS problem.

In simple sentences, a CCS can be signed when two foreign companies, A and B, are entering into each other’s home country market. Each needs the domestic currency for business. Raising loans directly from local banks could be challenging for a foreign company. Alternatively, each company can borrow a loan from its home country and lends it to the other company for the latter’s local operations. Whenever receiving a loan from Company B, company A also needs to pay the interest denominated in the same currency (or vice versa). As a result, in the CCS company A’s cash inflows and outflows becomes company B’s inflows and outflows respectively.

In exchanging each other’s cash flows at every payment date, the payment amount can be either fixed or floating. “Fixed” means the same amount each time whereas “floating” means the amount is changing each time and contingent on the last forward rate. For a typical CCS problem, there are usually two puzzles to solve: 1) how to determine the fixed & floating payment amount 2) what is the market value of the CCS for a company at a certain point of time. 

A. How to use my Excel Model to compute fixed payments

Since the floating interests are based on a series of forward rates, we virtually can only determine the fixed amount of payment at t=0 (the very beginning of the swap). As one company’s cash inflows become the other’s outflows, it’s easier to solve this puzzle by doing the math for each of the following scenarios that involve USD ($) and Euro (€):

In order to calculate the fixed payments, you need to know the swap’s term structure for both currencies:

1) The life of the swap

2) The frequency of payment (i.e. how many payments) over the course of the swap

3) The spot rates corresponding to each payment period (e.g. at t=0, what is the spot LIBOR rate for 90-day period, 180-day period, 270-day period and 360-day period).

Scenario 1: Company A pays fixed $ and company B pays floating €

Fixed Payment Rate (% of notional $ payment) = (1 – $Bn)/ ($B1+$B2+…+$Bn)

Where $Bn are the present value discount factors for USD; it’s calculated by dividing 1 by (1+the corresponding spot LIBOR rate for USD loans)

Scenario 2: Company B pays fixed € and company A pays floating $.

Similarly, the Fixed Payment Rate (% of notional € payment) = (1 – €Bn)/ (€B1+€B2+…+€Bn)

Where €Bn are the present value discount factors for Euro; it’s calculated by dividing 1 by (1+the corresponding spot LIBOR rate for Euro loans)

Scenario 3: Company A pays fixed $ and company B pays fixed €

In this scenario, each fixed amount of payment has already been determined in the scenario 1 and 2. We can simply steal the figures from the above.  

Scenario 4: Company A pays floating $ and company B pays floating €

Unless all the forward rates are given for each payment date, we are UNABLE to calculate the payment amount when both parties are paying floating interests. Although we can derive the implied forward interest rates by using covered interest rate parity, yet the actual forward rates can turn out to be very different.

Fill in all the given ‘Spot LIBOR’ and ‘days” cells shaded in grey. Each rate represents the annualized rate for the corresponding length. Then, fill in the ‘payment frequency’ cell. Eventually, you’ll end up getting the PV discount factors for each LIBOR rate, as well as the ANNUALIZED payment rate for the USD-denominated and Euro-denominated notional payments respectively.  

B. Calculating the market value of the swap at a specific point of time    

Before delving right into the question, let’s ask the broader question: what is the market value of ANYTHING? It should be the net benefit free of any additional cost (i.e. total benefits – total costs). The same law applies to cross currency swaps too. At any point of time over the course of a swap, its market value should equal to its present value of benefit minus its present value of cost: PV (benefit) – PV (cost). The benefits and costs in the case of a swap represent money received and paid respectively.

In this sense, the market value of a CCS really depends the forms of benefits and costs—if the swap payments to receive and pay are floating or fixed. To make this problem easier to solve, let’s break down the question by drawing this metrics based on one company’s perspective.  

In the metric above, there are 4 scenarios: 2 ‘If to Receive’s matching 2 ‘If to Pay’s; however, we can pass a shortcut by calculating only MV1 and MV2 to get all the key results. The rest will be simply filling in the blanks. 

Again, to compute the values for each blank, we need to know the following:

1) The USD and Euro LIBOR rates starting at the new time point till each payment dates (for example: if 120 days have passed and the next payment is on day 180, we need to know the 60-LIBOR rate on day 120).

2) The forward USD-denominated and Euro-denominated interest rates determined at the very last payment.   

3) The new exchange rate

Now, let’s go through the calculations for the key present values in scenario A

1) PV(benefit) = PV(floating USD received) = (1 + the last forward interest rate for $)* (the corresponding PV discount factor)

(Recall that: the latest PV discount factor = 1/ (1+ the corresponding spot rate %)

2) PV(cost) = PV(fixed Euro paid) = (€fixed payment*sum of all the following discount factors for €payments) + the discount factor for the final € payment

Now, you have present values of both of the benefit and cost. But they are denominated in different currency! Thus, we’ll have to translate Euro-denominated PV (Cost) into USD amount. As we will always need to pay the equivalent amount of fixed Euro payments, the corresponding USD-denominated fixed payment can be different as the exchange rate (FX) changes. We’ll have to reflect the exchange rate changes when translating PV (cost) : €PV(fixed Euro paid) = $ PV (fixed Euro paid) * initial FX rates/ current FX rates.

Finally, you can compute the market value by subtracting $ PV (cost) from $ PV (benefit)

We’re done with Scenario A! Are you still hanging there? Let’s march to Scenario B

With the training of scenario A computation, scenario B becomes much easier. Simply, we can retain the same methodology and simply swap the currency denomination:

1) For PV(benefit: receive fixed USD), the formulae = (fixed $ amount * sum of discount factors) + the final discount factor. All the spot rates used here correspond to USD

2) For PV(cost: pay floating Euro), the formulae = (1+the last forward interest rate)*the corresponding discount factor

Again, don’t forget to translate PV(cost) from Euro into USD when calculating the market value ( present value of benefit – present value of cost).

So far, we have acquired the following figures:

1) Floating $ benefit

2) Fixed $ benefit

3) Floating €cost

4) Fixed €cost

Fortunately, there are all we need for computing the market value for any given scenario.

In summary, it’s quite challenging to deal with cross currency swaps when you just ran into this concept. The purpose of this article is NOT FOR PROFESIONAL USE; instead, it’s solely for academic illustration by going through the basics and some key procedures. Although you could easily acquire the results by using various financial libraries, they don’t necessarily illustrate how the CCS works; thus, they don’t provide a solid foundation for understanding swaps.

Despite its simplicity in formatting and discreteness, I hope you will find this article helpful. Free free to download the Excel attachment (personal use only). Let me know if you have any question. Thanks for reading!  

An Excel Model for Understanding Cross Currency Swaps.docx