SWAPS

Definition

“A swap is an over-the-counter agreement between two companies to exchange cash flows in the future. The agreement defines the dates when the cash flows are to be paid and the way in which they are to be calculated. Usually the calculation of the cash flows involves the future value of an interest rate, an exchange rate, or other market variable.”

Source: J. Hull, Options, Futures, and Other Derivatives, 9th Edition, Pearson Education Limited 2018

Cash flows whose exchange is the subject of the contract are usually associated with debt instruments or foreign currencies. For this reason, we distinguish two basic types of swap transactions:

interest
currency

Characteristics of interest rate swap

Interest rate swap is a contract in which one company agrees to pay to the other cash flows based on predefined fixed interest rate on an agreed notional amount for a certain period (usually from 2 to 10 years). The second counterparty, in return pays to the first company cash flows based on a floating rate on the same notional principal and the same period.

The main parameters of interest rate swap are:

fixed rate
floating rate
notional principal
period
the number of payments during the year

Usually, one of the parties has deposited its cash into a fixed-rate deposit, while the other has a floating-rate deposit.

Characteristics of currency swap

A currency swap (fixed-for-fixed) is an agreement where counterparties exchange the defined amount denominated in two currencies for specific period (usually up to 5 years). They agree for certain fixed interest rates for each currency respectively. On the top of the regular interest payments there is a capital flow after the time specified in the contract. After this time, regardless of the currency exchange rate at the time of settlement, the counterparties transfer back the original principal amounts in each currency.

There are also other types of currency swaps, where floating rates for currencies are involved:

fixed-for-floating
floating-for-floating

Liability and asset transfer

Interest rate swaps can be split into two groups, based on the purpose of the contract:

swaps that transform a liability: where the cashflow is driven by difference between fixed and floating interest rate (liability swaps). It transforms a floating-rate loan into a fixed-rate loan (or vice versa)
swaps that transform an assets: swaps in which the parties transfer all the benefits from specific assets (assets swaps)

Exapmples of the both types will be provided in further readings.

Market participants

Swaps are transactions made in banking sector and there are not precisely standardized (like in case of futures) - they are therefore not traded on the regulated market. Special institutions have been set up to facilitate investors’ swap transactions. These are the organizers of the swap market - intermediary agents (usually brokers or dealers), facilitating matching partners and enabling transactions.

Intermediaries can focus on the search of both parties of the transaction, acting on their own account or act as one of the parties to the contract. Undertaking by the intermediaries the role of one of the parties to the transaction exposes them to financial risk. This risk can be significant.

Intermediaries acting as one part of the contract are in fact market makers. First of all, they have to price a swap contract in such a way that the valuation includes their margin for the risk taken. On the other hand, as a result of many transactions concluded, market makers face the necessity of managing their portfolios i.e. hedging all the risks undertaken.

The main motivation for swap transaction

trading needs - routine trade operations of some companies may lead in a natural way to the situation in which these companies are exposed to the risk related to the interest rate or a certain currency risk

comparative imbalance - companies can sometimes have some advantage in terms of the conditions under which they can access a certain type of financing. It means that some companies can borrow capital on exceptionally favorable terms, and then use a swap to change the loan characteristics to suit the specific needs of the company.

The history of the swap market

The beginning of the swap market can be traced back to the late seventies of XX century, when currency market participants used swap transactions to circumvent the British control over flows of foreign currency. The first interest swap appeared in 1981 and was between IBM and the World Bank. Since then, we have been witnessing the rapid development of this market. Swap contracts constitute a majority of OTC derivatives.

As of 2018 June month end there where 350 trillion USD notional amounts outstanding Interest Rate Swaps out of which: 35% denominated in USD, 25% in EUR and 10% in Yen.

Interest Rate Swap constitutes 72% of all OTC interest rate derivatives (17% are FRA contracts) in terms of notional amounts outstanding.

Currency swap sums up to 26 trillion USD of notional amount and is responsible for 26% of OTC foreign exchange derivatives.

Source: https://www.bis.org/statistics/derstats.htm

Regulations of the swap market

The swap market was created to a large extent because these instruments allow to avoid many restrictions related to options or futures contracts listed on stock exchanges. Of course, there are also limitations specific to swap contracts.

Swaps are tailored to the individual needs of the contract partners. Partners can create a completely new contract on a piece of paper that would satisfy their needs better than stock exchange instruments. Partners can set any amount they want to exchange because they are not limited by any fixed contractual terms they would face in the case of instruments listed on the stock exchange.

Swap parties can also choose the most convenient expiration date, without having to adjust their needs to the dates ffered by the stock exchange. This is a very important feature of the swap market, because this flexibility allows participants to take a much longer time horizon than is possible with exchange-rate instruments.

Another advantage of the swap market is that it provides privacy that is impossible to achieve in the stock market. Only contract partners know about the transaction and its details. It is impossible to achieve that on the stock market, where players are able to recognize the moves of certain companies because they know who represents them.

The swap market, unlike listed futures and options, are the subject of less severe government regulation. Swap market participants were afraid that Commodity Futures Trading Commission (CFTC) would try to impose their own rules on them. However, the CFTC has officially announced that it is not intending to interfere in the swap transactions market, either now or in the future.

However, the swap market has its own limitations

to enter into transactions, the potential partner must find someone who wants to act as the opposite party. If the investor wants a specific delivery date or precisely defined terms for exchanging cash flows, finding the other party can be a serious problem unless there is no market maker for the instrument
since a swap contract is an agreement between two parties, a change of its terms or a previous expiration requires the consent of both partners
in the case of swap contracts, there is no institution guaranteeing that both parties fulfill the terms of the contract. This means that swap deal partners need to be sure about their mutual solvency

The swap market has developed mechanisms to deal with all three types of restrictions. The most serious problem is the possibility of not complying with the terms of the contract. Estimating the financial capabilities of the other party can be difficult and expensive. That is why only companies and institutions participate in the swap market, which often engage in transactions or have access to the most serious intermediaries that can certify their solvency / creditworthiness.

Interest rate swap (IRS)

In the case of a simple interest rate swap, the starting position of one party is a debt instrument with a fixed interest rate, whereas the other has a position in floating interest rate. The investor who initially take the position on the floating rate instrument is exposed to the risk related to the change of interest rates. IRS allows the investor to eliminate this risk. For the partner originally holding a fixed-rate instrument, the interest rate swap increases the interest rate risk that he takes. The simple example is presented below. It shows the nature of a swap transaction closer.

IRS example
Let us assume that the swap covers four years and involves annual interest payments based on 1 million EURO notional outstanding. Let’s assume that company A has agreed to pay the B company a fixed interest rate of 10 percent. The B company has agreed to pay to company A floating interest rate set as LIBOR + 2 percent. Below figure presents the basic elements of the transaction.

Company A pays company B 10 percent each year for one million EURO (i.e. 100,000 EURO). Company B makes payments to company A, but their actual cash flow depends on LIBOR movements. Theoretically, the parties also exchange one million euro between them. However, actually replacing one million EURO would not make any sense. As a result, the nominal amounts that are the basis for the interest rate are not exchanged. However, they decide about interest payments. Because capitals are not actually exchanged, they are called reference capital, which is used only for calculations, but they are not subject to a real transfer. Knowing the amount of capital, we can calcualte the actual annual cash flows between the two parties of the transaction. Suppose that at the time of the first payment LIBOR is 9 percent. This means that company A will be required to pay the company B the sum of 100,000 EURO. Company B will, on the other hand, owe 110,000 EURO to A. In total, after agreeing on the commitments, the company B must pay 10,000 EURO company A. In fact, only the net payment is made, i.e. the payment of the difference between the liabilities.

Currency swap

In the case of currency swaps, one of the parties has a position in certain currency, but it wants to have exposure to a different currency. A swap transaction is made if one party offers the second one the specified notional principal amount in one currency in exchange for its equivalent in another currency (it may also be an exchange of credit payments). For example, company X may have USD that it wants to exchange for EUR. However, the company Y may have EUR and wants to exchange it for dollars. In such a situation, a swap agreement may be made. A simple currency swap involves three different types of cash flows:

First, the parties actually exchange cash when the transaction is initiated. The motive for making a foreign exchange swap is the actual need to have assets denominated in a particular currency. We see here the differences between the currency swap and the interest rate swap. In the latter case, there is only one currency, thanks to which it is possible to limit only to the net payment.
Secondly, the currency swap partners are making mutual periodic payments related to the interest rates on currencies at all times.
Third, at the end of the period determined for by the swap, the parties exchange the nominal amount again.

Currency swap example

Let us assume that the current EUR and USD currency exchange rate is 0.9 USD for EURO, the dollar interest rate is 5%, and EUR interest rate is 6%. Company X has 9 million USD and wants to exchange them for EUR. In return for USD, the company Y, when the swap is initiated, will pay to X 10 million EUR. Moreover, let us assume that the swap will last for 5 years, and the parties will transfer cash flows related to the interest rate on currencies annually. The company Y will pay 5 percent of the 9 million USD received, i.e. annual amount due to X will be $ 450,000. However, company X, which received EUR 10 million, will pay to the company Y 6 percent of this sum, i.e. EUR 600,000.

In practice, the parties will only offset net liabilities. Assuming that after the first year the FX rate will be 0.95 USD for EURO, the dollar will be worth 1.052 EUR. With this rate, company X will pay 570,000 USD (600,000 EUR times for 0.95), while the company Y side will pay 450,000 USD. So company X will pay a 120,000 USD difference. In another period, the exchange rate may be different, which will of course also affect the net payment. After five years, the parties again exchange nominal amounts, which completes the swap transactions.

Pricing of interest rate swaps (IRS)

Swap can be priced as a long position in one bond and a short position in another or as a portfolio of FRA contracts. We assume that during the duration of the swap agreement, one party of the contract will receive a fixed payment of $k$ dollars at moments $t_i$ (1 <$i$ <$n$) and will make floating payments at the same times. With such assumptions, swap can be valued using the following formula:

\[V_I = B_{fix} - B_{fl}\]
and
\[V_{II} = B_{fl} - B_{fix}\] where:
$V_I$ - the value of the swap for the party receiving the fixed payment and making the floating,
$V_{II}$ - value of the swap for the party receiving floating payments and making fixed payments,
$B_{fix}$ - the value of a fixed rate bond being part of the swap,
$B_{fl}$ - the value of bonds with floating interest as part of the swap,
$Q$ - the notional principal amount.
Typically, the Libor rate is used to discount cash flows when valuing swaps. The $B_{fix}$ and $B_{fl}$ values are determined from the following formulas:

\[B_{fix} = \sum_{i=1}^n\frac k {e^{(r_i*t_i)}} + \frac Q {e ^{(r_n*t_n)}}\] and
\[B_{fl} = \frac {k^*} {e^{(r_1*t_1)}} + \frac Q {e ^{(r_1*t_1)}}\], where:

$r_i$ - discount rates corresponding to the period remaining until $t_i$,
$k$ - payment resulting from a fixed interest rate,
$k^*$ - payment resulting from a floating interest rate,
$Q$ - face value of the swap (notional).

Note that formula (3) is an equivalent of formula (7) from the INTEREST RATES materials. The only difference is usage of continous compounding of interest rates.
The floating-rate bond is worth the notional principal immediately after a payment. Therefore we can discount only the nearest payment and the notional to calculate present value of the bond. This is expressed by the formula (4).

Relationship between interest rate swap and FRA

Swap defines obligation to transfer cash flows based on the interest rate specified in the contract for $n$ periods, regardless of current market conditions. The contract for each period can be treated as a forward contract and the entire swap as a forward contracts portfolio. The valuation procedure is then as follows:

Setting the forward rates for all Libor rates affecting the amount of payments related to the swap,
Calculate the cash flows resulting from the contract assuming that Libor rates will be equal to forward rates,
Calcualte the value of the swap contract as the current value of this cash flows.

Valuation of currency swaps

Currency swaps can also be priced as a long position in one bond and short in the other. The swap value can be calculated using the data for foreign and domestic bonds and the foreign exchange (FX) rate:
\[V_{III} = S * B_f - B_d\] where: $V_{III}$ - the value of the swap for the party receiving payments denominated in a currency other than the domestic one and making the payment of the domestic currency,
$S$ - exchange rate,
$B_f$ - value of the bond being the basis of the contract, denominated in a currency other than domestic,
$B_f$ - value of the bond denominated in domestic currency.

Decomposition of currency swaps on forward contracts

Currency swaps (foreign exchange swaps) can also be valued as a portfolio of forward contracts. The swap value can be calculated based on the time structure of the foreign exchange rate and the time structure of domestic interest rates. The valuation procedure is as follows:

Determination of forward rates:

\[FER = S * e ^ {-(r_{d} - r_{f})* T}\] $S$ - FX rate
$FER$ - Forward exchange rate rate
$r_{d}$ - domestic interest rate
$r_{f}$ - foreign interest rate
$T$ - time to maturity

determination of the value of the forward contract related to the exchange of interest payments at the moment $t_i$

\[(CF_f * FER_i - CF_d)* e^{-(r_i * t_i)}, \ \ \ \ \ \ for 1< i < n\] $CF_f$ - foreign interest payments cash flow
$CF_d$ - domestic interest payments cash flow

determining the forward contract value corresponding to the exchange of nominal amounts at time $t_n$

\[(N_f * FER_n - N_d) * e ^ {-(r_n * t_n)}\] where:
$N_f$ - notional amount in the foreign currency,
$N_d$ - notional amount in the domestic currency,

calculate the value of the swap contract as the sum of the values determined in points 1 and 2.

Excercise 1

Interest rate swap

Assume that company A has agreed to pay a 6-month Libor and receive a fixed interest rate of 8% per annum (with interest payable every six months) from the face value of $ 100 million. Swap is 1.25 years to expire. The interest rates for 3, 9 and 15 months are: 10%, 10.5% and 11% respectively. Assume that interest rates are continously compounded. The 6-month Libor is currently 10.2%. Calculate the value of this swap for company A. compounding.

Excercise 2

Interest rate swap
Determine the value of the swap from exercise 1 in the way as described in the Relationship between interest rate swap and FRA part.

Excercise 3

Currency swap

Assume that yield curves in Japan and in the US are flat. The interest rate in Japan is equal to 4% per annum, and in the US to 9% per annum (with continuous compounding). The financial institution takes position in the swap contract, under which it receives 5% on an annual basis of the amount denominated in yen and pays 8% per annum of the amount denominated in dollars. These amounts are respectively 10 million USD and 1200 million yen. The contract is valid for 3 years and the current exchange rate is 110 USDJPY. What is the value of this currency swap?

Excercise 4

Currency swap
Determine the value of the swap from exercise 3 as the sum of forward contracts.

Excercise 5

Currency swap
Company X intends to borrow US dollars based on a fixed interest rate, while company Y intends to borrow in Japanese yens also based on a fixed interest rate. The loan amounts are approximately the same, taking into account the current exchange rate. The following loan rates have been proposed to companies to reflect the risks associated with their current situation:

Company X gets offers: JPY 5.0%, USD 9.6%
Company Y gets offers: JPY 6.5%, USD 10.0%

Design a swap that will allow the bank as an intermediary institution to achieve a profit of 50 bp per year (assuming that the entire currency risk is taken over by the bank) and at the same time will bring the same profit to X and Y, compared to the initial situation, without a swap contract.
What will the swap contract look like if the currency risk is taken over by company X?
What will change in the contract if the risk is taken over by Y? Calculate the value using forward contracts approach.
## Excercise 6
Currency swap

The currency swap has 3 years to expire. Agreement assumes the exchange of the following payments: A pays to B 14% from GBP 20 million, while B pays to A 10% from USD 30 million. The term structure of interest rates is flat in both the United Kingdom and the United States. At present, US rates are 8%, and Great Britain rates are 11% (continous compounding). Interest is paid once a year. The current exchange rate is 1.65 GBPUSD.

What is the value of the swap for the company A and how much for the company B?
How will the answer to the above question change if we assume a decreasing structure of interest rates, both in GBP and USD?
USD: 8%, 7%, 6% for Y1, Y2, Y3 respectively
GBP: 11%, 10%, 9% for Y1, Y2, Y3 respectively
Will the value of the swap change if we assume that the parties pay floating payments based on market rates? Use the forwards portfolio to calculate swap value.