###### The Black-Scholes-Merton Model

After completing this reading you should be able to: Explain the lognormal property... **Read More**

**After completing this reading, you should be able to:**

- Explain the mechanics of a plain vanilla interest rate swap and compute its cash flows.
- Describe the role of the confirmation in a swap transaction.
- 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.
- Identify and describe other types of swaps, including commodity, volatility, credit defaults, and exotic swaps.
- Describe the credit risk exposure in a swap position.

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. The principal in an interest rate swap is known as a notional principal because it is not exchanged. Only interest rates calculated with respect to the notional principal can be exchanged.

Assume two parties, A and B. Party A, agree to pay Party B a fixed interest rate at 4% per annum, compounded semiannually, on a principal of USD 100,000. Party B, in return, agrees to pay party A interest at the six-month LIBOR (London Inter-Bank Offered Rate). Exchanges occur every six months over a period of three years. The tables below summarize the possible scenarios of parties A and B.

Scenarios for party B: Pays LIBOR and receives a fixed rate.

$$ \begin{array}{c|c|c|c|c} \textbf{Time } & \textbf{6-month} & \textbf{Floating } & \textbf{Fixed} & \textbf{Net}\\ \textbf{in Years} & \textbf{LIBOR} & \textbf{amount} & \textbf{amount} & \textbf{cashflow} \\ \textbf{ } & \textbf{(% per year)} & \textbf{paid (USD)} & \textbf{received (USD)} & \textbf{(USD)} \\ \hline 0.0 & 3.00 & & & \\ 0.5 & 3.20 & 1500 & 2000 & +500 \\ 1.0 & 3.44 & 1600 & 2000 & +400 \\ 1.5 & 4.00 & 1720 & 2000 & +280 \\ 2.0 & 4.30 & 2000 & 2000 & –\\ 2.5 & 4.44 & 2150 & 2000 & -150 \\ 3.0 & 4.70 & 2220 & 2000 & -220 \end{array} $$

Note: The exchange takes place one period after the LIBOR is observed. Therefore, the first exchange occurring after six months will occur using the LIBOR rates observed at time zero. The exchange after one year will take place after the LIBOR rates observed at time 0.5 into the contract, and so forth.

At time 0.5, the floating rate amount will be \(3\%×0.5×100,000 = 1,500\).

The fixed rate amount will be \(4\%×0.5×100,000= 2,000\).

As such, the cash flows exchanged will be the netted amount, \(2,000 – 1,500 = 500\).

The above calculations are just approximations as they do not consider day counts. For more accurate results, it is important to consider day count conventions.

Suppose that the exchanges in the above example take place on 1^{st} January and 1^{st} July. The first cash flow exchange is on 1^{st} July of that year, and the floating rate exchanged will be:

$$ \frac{183}{360} × 3\% × 100,000 = 1,525 $$

Note: 183 was obtained by adding the total number of days between 1^{st} January and 1^{st} July, and the day count convention for LIBOR is actual/360.

It is important to see that 1,525 is 25 more than the approximate value shown in the table (1,500).

Similarly, the fixed rate will be expressed with day count conventions as shown below:

$$ \frac{183}{360} × 4\% × 100,000 = 2,033.33 $$

Again, 2033.33 is USD 33.33 more than the approximate value shown in the table (2,000).

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.

The contents of confirmation include the dates when payments will be exchanged, the day count conventions to be used in calculating payments, and the way payments will be calculated.

**Ask quote**: This is the rate a firm is willing to receive to pay LIBOR.**Bid quote**: This is the rate a firm is willing to pay to receive LIBOR.**Swap rate**: The average of the bid and the ask quotes.

When using an overnight rate (which will replace swaps as discussed in the chapter on Properties of Interest Rates), the floating rate can be obtained using the formula:

$$ R = (1+d_1 r_1) (1+d_2 r_2) … (1+d_n r_n) – 1 $$

Where:

R is the floating rate,

d is the number of days, and

r is the overnight rate.

Apart from weekends and holidays, d = 1. Weekends and holidays lead to the overnight rate being applied more than once. On a Friday, for example, d=3.

Interest rate swaps can be used to transform assets into liabilities, or vice-versa, by converting fixed (floating) rates loans and liabilities into floating (fixed) rates.

Assume that the 3-year bid and ask quotes are 3.06% and 3.09%, respectively. A company borrows a bank USD 5,000 at a fixed interest rate of 4%, compounded quarterly. To convert this fixed-rate loan into a floating rate liability, the company can use the three-year bid and ask quotes to enter into a three-year swap with another company. It will then have three sets of cash flows:

- It will pay 4% on the borrowed USD 5,000 to the bank;
- It will pay LIBOR on USD 5,000 under the terms of the swap to the swap dealer; and
- It will receive a fixed rate of 3.06% from the swap dealer on USD 5,000 under the terms of the swap.

The net out interest payment will therefore be:

$$ 4\% + LIBOR – 3.06\% =0.94\%+ LIBOR $$

The company will then have converted a 4% fixed interest rate into a 0.94% + LIBOR floating interest rate.

Suppose the above company borrowed the USD 5,000 on a floating interest rate of three months LIBOR plus 0.5% (50 basis points). To convert the floating rate to a fixed rate, the company can use the ask quote of 3.09%. It will then have the below set of cash flows:

- It will pay LIBOR + 0.5 % on it USD 5,000 borrowings;
- It will receive LIBOR under the terms of the swap; and
- It will pay a fixed rate of 3.09% under the terms of the contract.

The net out interest payment will therefore be:

$$ LIBOR + 0.5\% – LIBOR + 3.09\% =3.59\% $$

The company will then have converted a floating interest rate of LIBOR + 50 basis points into a fixed interest rate of 3.59%.

Assets with floating (fixed) interest rates can be converted to fixed (floating) interest rates using the same concept.

Just like in other OTC instruments, parties to a swap do not interact one on one. A swap dealer intertwines themselves between the parties taking a commission on the trade.

In most cases, therefore, a swap party stays unaware of the identity of the party in the offsetting position. The swap dealer effectively serves as an **intermediary**.

Let’s look at an example of two firms, \(A\) and \(B\).

- \(A\) wants to borrow floating
- \(B\) wants to borrow fixed

$$ \textbf{Borrowing costs} $$

$$

\begin{array}{c|c|c}

\textbf{Firm} & \textbf{Fixed borrowing} & \textbf{Floating borrowing} \\ \hline

\text{A} & 6\% & \text{LIBOR} \\ \hline

\text{B} & 8\% & \text{LIBOR + 100bps} \\

\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., \(\Delta \text{fixed} -\Delta \text{floating}\).

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.,

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

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 borrower’s credit risk. It also **assumes zero transaction costs** even when an intermediary is involved in the swap (which is standard practice).

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 \({ T }_{ 2 }\)

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.

A currency swap works much like an interest rate swap, 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.
- There’s no netting of payments in a currency swap again because the payments are not in the same currency.
- The two sets of cash flows in a swap are known as legs.

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.

Assume that USD 5,000 at a fixed rate of 3% is being received in exchange for 4,000 Euros at a fixed rate of 2.5%. Payments are exchanged every year for three years, and interest rates are annually compounded.

There are two legs present in this example, a USD leg and a EURO leg.

Interest for the USD leg \(= 0.03 × 5,000 = 150\).

Interest for the Euro leg \(= 0.025 × 4,000 = 100\).

$$\begin{array}{l|c|c} \textbf{Time} & \textbf{USD} & \textbf{Euro} \\ \textbf{(years)} & \textbf{Cash flow} & \textbf{Euro Cash flow} \\ \hline 0 & -5,000 & +4,000 \\ 1 & +150 & -100 \\ 2 & +150 & -100 \\ 3 & +150 & -100 \end{array}$$

Assume that a year after the first exchange occurs, the risk-free rate for all maturities in USD is 4.5% and 3.5% in Euros. Assume also that 1 Euro = 1.15 USD.

To value this swap, we follow the below steps;

- Value the remaining currency X cash flows in currency X terms (time is measured from the valuation date and not from the start of the swap).
- Value the remaining currency Y cash flows in currency Y terms (time is measured from the valuation date and not from the start of the swap).
- Convert the value of the currency Y cash flows to currency X at the current exchange rate.

$$\begin{align*} \text{Value in USD} &= 150(1.045^{-1})+150(1.045^{-2}+150(1.045^{-3}) = 412.3447\\ \text{Value in Euros}& = 100(1.035^{-1})+100(1.035^{-2}+100(1.035^{-3}) = 280.1637\\ \end{align*} $$

The value of the swap in USD is therefore:

$$ \text{Value of the Swap} = 412.3447 – 280.1637×1.15 = 90.156445 $$

**Floating for fixed**: A floating rate in one currency is exchanged for a fixed rate in another currency.**Floating for floating**: A floating rate in one currency is exchanged for a floating rate in another currency.

These two currency swaps are valued by valuing each leg in their respective currencies. In valuing the floating rate, we assume that the forward rate will be realized.

In an equity swap, one of the parties commits to making 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.

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

A floating (or market or spot) price based on an underlying commodity is traded for a fixed price over a specified period.

Historical volatility observed over a certain period of time is applied to the notional principal in exchange for pre-determined fixed volatility applied on the same notional principal.

This insures against default by a company or bond. Similar to a car insurance contract, the buyer of the protection pays the seller fixed payments over a specified period. In case there is no default, the seller does not pay anything and simply receives the protection fixed payments. In case of default, the seller then pays the buyer the notional amount.

Swaps can give rise to credit risk, especially when no collateral has been posted. The initial pricing of a transaction should, therefore, consider expected credit losses by each party.

Credit risk is greatly eliminated from transactions done through a central counterparty (CCP), and this is because the CCP requires both initial and variation margins to be posted. The margins can then be transferred to the affected party in case of a default.

## Practice 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}

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

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

A. Firm T

B. Firm Y

C. Firm X

D. Firm Z

The correct answer is

D.$$

\begin{array}{c|c|c|c|c|c}

\textbf{Firm}& \textbf{Fixed rate} & \textbf{Floating rate} & \textbf{Fixed spread} & \textbf{Floating} & \textbf{Possible} \\ {} & {} & \textbf{LIBOR +} & {} & \textbf{spread} & \textbf{benefit} \\ \hline \text{Steel} & 5.0 & 2.5 & {} & {} & {} \\ \hline \text{Firm X} & 4.5 & 1.0 & -0.5 & -1.5 & 1.0 \\ \hline \text{Firm Y} & 7.0 & 4.0 & 2.0 & 1.5 & 0.5 \\ \hline \text{Firm Z} & 6.5 & 2.5 & 1.5 & 0.0 & 1.5 \\ \hline \text{Firm T} & 5.5 & 3.5 & 1.0 & 1.0 & 0.0 \\ \end{array}

$$The net borrowing savings by entering into a swap is the

difference between the spreads, i.e., \(\Delta \text{fixed} -\Delta \text{floating}\).