### Swaps

After completing this reading, you should be able to:

• Explain the mechanics of a plain vanilla interest rate swap and compute its cash flows.
• Explain how a plain vanilla interest rate swap can be used to transform an asset or a liability and calculate the resulting cash flows.
• Explain the role of financial intermediaries in the swaps market.
• Describe the role of the confirmation in a swap transaction.
• Describe the comparative advantage argument for the existence of interest rate swaps and evaluate some of the criticisms of this argument.
• Explain how the discount rates in a plain vanilla interest rate swap are computed.
• Calculate the value of a plain vanilla interest rate swap based on two simultaneous bond positions.
• Calculate the value of a plain vanilla interest rate swap from a sequence of forward rate agreements (FRAs).
• Explain the mechanics of a currency swap and compute its cash flows.
• Explain how a currency swap can be used to transform an asset or liability and calculate the resulting cash flows.
• Calculate the value of a currency swap based on two simultaneous bond positions.
• Calculate the value of a currency swap based on a sequence of FRAs.
• Describe the credit risk exposure in a swap position.
• Identify and describe other types of swaps, including commodity, volatility, and exotic swaps.

## What’s an Interest Rate Swap?

An interest rate swap is an agreement to exchange one stream of interest payments for another, based on a specified principal amount, over a specified period of time. Here is an example of a plain vanilla interest rate swap with Bank A paying the LIBOR + 1.1% and Bank B paying a fixed 4.7%. As in most financial transactions, a swap dealer is in between the two parties taking a commission on the trade.

## Cash Flows of a Plain Vanilla Interest Rate Swap

In a plain vanilla IRS, company $$A$$ agrees to pay Company $$B$$ a periodic fixed rate on a notional principal over the term of the swap. In return, Company $$B$$ agrees to pay Company $$A$$ a periodic floating rate on the same notional principal.

Both payments are in the same currency, and only the net payment is exchanged. That the principal is notional implies it’s not exchanged either. The floating leg of the swap uses LIBOR as the reference rate. For example, the rate could be set at the three-month LIBOR + 1.2%.

### Example:

Let’s assume companies $$A$$ and $$B$$ have just entered into a two-year plain vanilla IRS with semiannual payments and the 6-month LIBOR as the reference. We assume further that the fixed leg is pegged at $$2.75\%$$, and the notional principal is $$10 \quad million$$.

$$Table\quad 1-Six-month\quad LIBOR$$

$$\begin{array}{|c|c|} \hline beginning\quad of\quad the\quad period & LIBOR \\ \hline 1 & 2.0\% \\ \hline 2 & 2.5\% \\ \hline 3 & 3.0\% \\ \hline 4 & 3.5\% \\ \hline 5 & 4.0\% \\ \hline \end{array}$$

Table 2 shows the swap’s cash flows from the beginning to the end:

$$Table\quad 2-Swap\quad Cash\quad Flows$$

$$\begin{array}{|c|c|c|c|c|c|} \hline End \quad of \quad period & LIBOR \quad at & Fixed \quad cash & Floating \quad cash & Net \quad cash \quad flow & Paid \quad by \\ {} & beginning & flow & flow & {} & {} \\ {} & of \quad period & {} & {} & {} & {} \\ \hline 1 & 2.0\% & 137,500 & 100,000 & 37,500 & A \\ \hline 2 & 2.5\% & 137,500 & 125,000 & 12,500 & A \\ \hline 3 & 3.0\% & 137,500 & 150,000 & 12,500 & B \\ \hline 4 & 3.5\% & 137,500 & 175,000 & 37,500 & B \\ \hline \end{array}$$

For example,at the end of the second period,

$$Fixed\quad leg=0.0275\times 0.5\times 10,000,000=137,500$$

$$Floating\quad leg=0.025\times 0.5\times 10,000,000=125,000$$

There are two important points to note:

• Cash flows are made at the end of the period but use the rate (LIBOR) at the beginning of the period.
• 0.5 is the day count.

Interest rate swaps can help entities/investors to transform a liability or even an asset. For example, assume that the fixed rate payer, company $$A$$, has a 2-year floating rate liability, and the floating rate payer, company $$B$$, has a fixed rate liability. By entering into the swap arrangement, $$A$$ transforms its floating rate liability into a fixed rate liability ($$A$$ pays fixed and receives floating). On the other hand, $$B$$ transforms its fixed rate liability into a floating rate liability ($$B$$ pays floating and receives fixed).

Similarly, assume that company $$A$$ has a fixed rate asset while $$B$$ has a floating rate asset tied to LIBOR. By entering into the swap arrangement, $$A$$ is able to transform its fixed rate asset into a floating rate asset. $$B$$, on the other hand, is able to transform its floating rate asset into a fixed rate asset.

## The Role of Financial Intermediaries and Confirmation In a Swap Transaction

Just like in other OTC instruments, parties to a swap do not interact one on one. A broker intertwines themselves between the parties such that all transactions occur through them. In most cases, therefore, a swap party stays unaware of the identity of the party in the offsetting position. The broker effectively serves as an intermediary.

The details of each swap agreement are contained in a document called the confirmation. Such documents are drafted by the International Swaps and Derivatives Association (ISDA). Each party must append their signature on the confirmation to show their commitment to the agreement.

Let’s look at an example of two firms, $$A$$ and $$B$$.

• $$A$$ wants to borrow floating
• $$B$$ wants to borrow fixed

$$Table\quad 3-Borrowing\quad costs$$

$$\begin{array}{|c|c|c|} \hline Firm & Fixed \quad borrowing & Floating \quad borrowing \\ \hline A & 6\% & LIBOR \\ \hline B & 8\% & LIBOR + 100bps \\ \hline \end{array}$$

From the table, we can see that $$A$$ can borrow fixed at $$6\%$$, and $$B$$ can borrow fixed at $$8\%$$. Also, $$A$$ can borrow floating at LIBOR, and $$B$$ can borrow floating at LIBOR + 100bps. However, the difference in borrowing rates for $$A$$ and $$B$$ is higher in the fixed market than in the floating market (200bps vs. 100bps). Therefore, $$A$$ has an absolute advantage in both markets but a comparative advantage in the fixed market. $$B$$, on the other hand, has a comparative advantage in the floating market.

When a comparative advantage exists, the implication is that the parties involved can reduce their borrowing costs by entering into a swap agreement. The net borrowing savings by entering into a swap is the difference between the differences, i.e., $$\Delta fixed\quad -\quad \Delta floating$$.

If we assume that the net borrowing savings are split evenly between the parties, we will divide the total borrowing savings by 2,i.e.,

$$Borrowing\quad savings\quad per\quad party=\frac { \Delta fixed\quad -\quad \Delta floating }{ 2 } =\frac { 200bps-100bps }{ 2 } =50bps$$

A problem with the comparative advantage argument is that it assumes the floating rates will remain in force in the long-term. In practice, the floating rate is reviewed at 6-month intervals and may increase or decrease to reflect the credit risk of the borrower. It also assumes zero transaction costs even when an intermediary is involved in the swap (which is standard practice).

## Computing The Discount Rate In A Plain Vanilla Interest Rate Swap

In essence, a swap is a series of cash flows, and therefore its value is determined by discounting all those cash flows to the present (valuation date). The cash flows are discounted using spot rates developed using the swap curve. The curve makes use of the following relationship between forward rates and spot rates, assuming continuous compounding:

$${ R }_{ forward }={ R }_{ 2 }+\left( { R }_{ 2 }-{ R }_{ 1 } \right) \frac { { T }_{ 1 } }{ { T }_{ 2 }-{ T }_{ 1 } }$$

Where:

$${ R }_{ i }$$=spot rate corresponding with $${ T }_{ i }$$ years

$${ R }_{ forward }$$=forward rate between $${ T }_{ 1 }$$ and $${ T }_{ 2 }$$

## Value of a Plain Vanilla Interest Rate swap Using the Bond Methodology

In essence, the pay fixed, receive floating party has a long position in a floating rate (since it’s an inflow) and a short position in the fixed rate (since it’s an outflow). The pay floating, receive fixed party has a short position in the floating rate (since it’s an outflow) and a long position in the fixed rate (since it’s an inflow).

If we denote the value of the swap as $${ V }_{ swap }$$, the present value of fixed-leg payments as $${ P }_{ fix }$$, and the present value of floating-leg payments as $${ P }_{ flt }$$, then:

To the pay fixed, receive floating,

$${ V }_{ swap }={ P }_{ flt }-{ P }_{ fix }$$

To the pay floating, receive fixed,

$${ V }_{ swap }={ { P }_{ fix }-P }_{ flt }$$

The important thing to note here is that the two positions are mirror images of each other.

## Currency Swap

A currency swap works much like an interest rate sap, but there are several key differences:

• A currency swap involves the exchange of both principal and interest rate payments, in different currencies.
• Currency swaps use the spot exchange rate
• Because the principals in a currency swap are in different currencies, they are exchanged at the inception of the swap. This ensures the principals have equal value using the spot exchange rate.
• In a currency swap, there’s no netting of payments, again because the payments are not in the same currency.

Currency swaps can be used to:

• Transform a liability in one currency into a liability in a different currency
• Transform an investment in one currency into an asset in another currency

Two companies can also get into a currency swap to exploit their comparative advantages regarding borrowing in different currencies. For example,

• Firm $$X$$ can borrow in $at 6%, or in £ at 4% • Firm $$Y$$ can borrow in$ at 4.5%, or in £ at 3.2%

If $$X$$ wants to borrow $$£$$, and $$Y$$ wants to borrow , the two may be able to able to save on their borrowing costs. That could happen if each borrows in the market in which they have a comparative advantage, and then swapping into their preferred currencies for their liabilities.

Swaps have counterparty credit risk. If the floating rate is above the fixed rate, then the pay floating, receive fixed party will make a payment to the pay fixed, receive floating party based on the difference between the two rates. In the event that the pay floating, receive fixed party defaults, the pay fixed, receive floating party will suffer a credit loss. Assuming the recovery rate is zero, the total loss will be equal to the present value of the net interest payments still to be made. This is known as the replacement cost of the swap.

## Other Types of Swaps

• Equity swap: In an equity swap, one of the parties commits themselves to make payments reflecting the return on a stock, portfolio, or stock index. In turn, the counterparty commits themselves to make payments based on either a floating rate or a fixed rate.
• Swaption: A swaption gives the holder the right to enter into an interest rate swap. It’s purchased for a premium whose value is determined by the strike rate specified in the swaption. Swaptions can either be American or European
• Commodity swap: A floating (or market or spot) price based on an underlying commodity is traded for a fixed price over a specified period.

## Question

A steel manufacturing firm recently issued a $$500 \quad million$$ fixed rate debt at $$3\%$$ per annum to fund an ambitious expansion plan. The chief risk manager at the firm has advised that the firm convert this debt into a floating rate obligation by tapping into the interest rate swap market. In this regard, he has identified four other firms interested in swapping their debt from floating to a fixed rate. The table below provided the various rates at which all the five firms can borrow:

$$\begin{array}{|c|c|c|} \hline Firm & Fixed-rate\quad \left( \% \right) & Floating \quad rate \\ {} & {} & 6-month\quad LIBOR\quad + \\ \hline Steel & 5.0 & 2.5 \\ \hline Firm \quad X & 4.5 & 1.0 \\ \hline Firm \quad Y & 7.0 & 4.0 \\ \hline Firm \quad Z & 6.5 & 3.0 \\ \hline Firm \quad T & 5.5 & 3.5 \\ \hline \end{array}$$

Identify the firm with which the manufacturer stands to yield the greatest possible combined benefit.

1. Firm T
2. Firm Y
3. Firm X
4. Firm Z

$$\begin{array}{|c|c|c|c|c|c|} \hline Firm & Fixed \quad rate & Floating \quad rate & Fixed \quad spread & Floating & Possible \\ {} & {} & LIBOR \quad + & {} & spread & benefit \\ \hline Steel & 5.0 & 2.5 & {} & {} & {} \\ \hline Firm \quad X & 4.5 & 1.0 & -0.5 & -1.5 & 1.0 \\ \hline Firm \quad Y & 7.0 & 4.0 & 2.0 & 1.5 & 0.5 \\ \hline Firm \quad Z & 6.5 & 3.0 & 1.5 & 0.0 & 1.5 \\ \hline Firm \quad T & 5.5 & 3.5 & 1.0 & 1.0 & 0.0 \\ \hline \end{array}$$
The net borrowing savings by entering into a swap is the difference between the spreads, i.e., $$\Delta fixed\quad -\quad \Delta floating$$.