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Chapter 4 FRA, FLOATERS AND SWAPS 4.1

Introduction

• One of the most active fixed income markets is the interest rate swap market. It is the largest derivatives market in the world. The notional amount of interest rate swaps (IRS) at the end of 2013 was $457 trillion, which represented 64% of the global OTC derivative market. The gross market value of outstanding IRS contracts - that is, the cost of replacing all outstanding contracts at market prices - was about $13 trillion by the end of 2013. The swap market is an over-the-counter market, with end users dealing directly with major financial institutions. It is also a global market, with trade taking place around the world, 24 hours a day, in instruments denominated in all of the major currencies and lots of the minor ones, too. For many users, swaps are now the instrument of choice for interest rate and currency risk management. • In essence, a swap is an agreement between two parties, called counterparties, to exchange a sequence of cash flows in the same or different currencies. The most important kinds of swaps are interest rate swaps and foreign currency swaps. We examine in this chapter standard forms of two of the most popular swaps: interest rate swaps and cross-currency swaps. —

—

In a standard interest rate swap, one party pays the other the difference between a fixed rate and a floating rate of interest for several periods (commonly between two and ten years). If you think of the fixed rate as analogous to the coupon on a long bond, this allows the buyer to transform long debt into short debt, or the reverse. In a standard cross-currency swap, one party pays the other the difference between interest on a loan denominated in one currency and interest on a loan denominated in a second currency. Interest payments in both cases can be either fixed or floating. In this case the swap allows the buyer to transform debt in one currency into debt in another, thus altering its exposure to movements in currency prices.

2

FRA, Floaters and Swaps

•

So-called “plain vanilla” interest rate swap, in which one counterparty pays the other the difference between fixed and floating rates is often represented visually as

Floating Rate Counterparty

Counterparty

A

B Fixed Rate

•

As illustrated, the counterparty , the fixed rate payer in this case, pays a fixed rate to the counterparty  , and receives a floating rate in return. In fact they would net the two payments: the counterparty with the larger payment sends the difference to the other one.

—

Both the

fixed

and

floating

rates are applied to a

notional principal.

We

refer to the principal as “notional” to emphasize that the two parties do not exchange principal payments (which are typically the same anyway).

— •

Since the notional principal is never exchanged, credit risk is limited to the value of the swap at any point in time.

In the jargon of the trade, the buyer (fixed rate payer) pays fixed interest payments in exchange for receiving floating ones, and the seller (floating rate payer) pays variable interest payments in exchange for receiving fixed ones. The buyer is often called “long a swap” or “shorting the bond” and the seller is often called “shorting a swap” or “long a bond”.

•

Although the parties do not exchange principal, it’s helpful to think of an in-

terest rate swap as an exchange of a floating rate note for a fixed rate note. For notes of equal value at the settlement date, this arrangement has the same net cash flows as the interest rate swap above. The principal payments at the end, for example, net to zero, so for this purpose it’s immaterial whether we include them or not. The advantage is that we can apply what we know about

the pricing of bonds and FRNs. The value of a swap to a fixed rate payer is then the difference between the value of an FRN and a fixed rate note with the stated coupon.

Forward Rate Agreement (FRA)

•

3

In a plain vanilla cross-currency swap, one party pays the other the difference between interest and principal in two currencies, illustrated here for the US dollar and the Japanese Yen:

Yen Interest & Principal

Counterparty A

•

Counterparty Dollar Interest & Principal

B

The interest payments can be fixed or floating in either direction. Thus we see that a cross-currency swap differs from an interest rate swap in denominating the interest and principal in different currencies and in calling for an exchange of principals at maturity. The principals generally have the same value when the swap is arranged, but can be substantially different at maturity - a factor that plays an important role in the swap’s risk characteristics.

•

The origin of the swap market can be traced to the late 1970s, when currency traders developed currency swaps as a technique to evade British controls on

the movement of foreign currency. Since then, a significant industry has arisen to facilitate swap transactions. The swap facilitators who help counterparties consummate swap transactions may serve as brokers and dealers. In the early days of the swap market, brokers identified and brought prospective counterparties into contact with each other without participating in the swap itself. Swap dealers are in the business of making a swap market in swaps and serving as counterparties to those seeking to complete swaps. In today’s swap market, swap dealers predominate.

•

4.2 4.2.1

•

Before we deal with swaps, we need some background on forward rate agreement and floating rate notes.

Forward Rate Agreement (FRA)

fi

De nition and examples

There is an explicit market for forward rates. A contract based on the forward rate is known as a forward rate agreement (FRA).

4

FRA, Floaters and Swaps

•

•

Like any forward contract, an FRA is a legally binding agreement to buy or sell some commodity at a specific future time, place, and price.

—

With FRAs, the “commodity” bought or sold is money and the “price” is, therefore, an interest rate.

—

FRAs involve the exchange of a specified period of time.

— —

An FRA can be tailored to meet the exact requirement of the user. Because of its simplicity, flexibility and lack of up-front fees, an FRA offers customers effective tools for managing interest rate risk.

FRA’s are the basic building block for all other interest rate risk management products.

— — — •

fixed rate for a floating one over a single,

The Eurodollar futures contracts are simply exchange traded FRAs. An interest rate swap is a series of FRAs out to the maturity of swaps. All interest rate risk management products are closely linked to the simplest product, the FRA.

FRAs involve the exchange of a fixed rate for a floating rate over a single period.

—

fixed rate is set when the contract is bought or sold, but the floating rate (and therefore the settlement value) is not known until the first of the The

exposure period.

— —

The

floating rate is usually LIBOR.

If the contract is for a three-month period in the future, then the

floating

rate is the then current three-month LIBOR rate.

•

FRAs are traded by both banks and corporates and between banks. The FRA market is very liquid in all major currencies and rates are readily quoted on screens by both banks and brokers. Dealing is over the telephone or over a dealing system such as Reuters.

•

When an FRA is traded, the buyer is borrowing (and the seller is lending) a

specified notional sum at a fixed rate of interest for a specified period, the “loan” to commence at an agreed date in the future.

—

We use the term “notional” because with an FRA no borrowing or lending of cash actually takes place.

Forward Rate Agreement (FRA)

— —

5

There is no exchange of cash at the time of the trade. The cash payment that does arise is the difference in interest rates between that at which the FRA was traded and the actual rate prevailing when the FRA matures. The buyer is the notional borrower, and so if there is a rise in interest rates between the date that the FRA is traded and the date that the FRA comes into effect, she will be protected.

—

The buyer may be using the FRA to hedge an actual exposure, that is an actual borrowing of money, or simply speculating on a rise in interest rates.

—

The counterparty to the transaction, the seller of the FRA, is the notional lender of funds, and has fixed the rate for lending funds. If there is a fall in interest rates the seller will gain, and if there is a rise in rates the seller will pay.

—

•

Thus, the buyer of the FRA pays the receives

—

•

Again, the seller may have an actual loan of cash to hedge or be a speculator.

fixed and pays floating.

fixed rate and receives floating; the seller

The company with the LIBOR payment on the loan can hedge the risk of rising rates by buying the FRA, but for one period only.

If a buyer of an FRA wished to hedge against a rise in rates to cover a threemonth loan starting in three months’ time, she would transact a “three-againstsix month” FRA, or more usually a 3 6 or 3-v-6 FRA. This is referred to in

×

the market as a “threes-sixes” FRA, and means a three-month loan beginning in three months’ time. So a “ones-fours” FRA (1-v-4) is a three-month loan in one month’s time, and a “threes-nines” FRA (3-v-9) is six-month money in three months’ time.

•

Formally, an FRA is an agreement to exchange fixed rate payment at a rate  for LIBOR on an underlying loan with principal  for the period  (settlement date) to

 +

(maturity date), which implies that contract period is  .

—

The settlement of the FRA contract is made at time manner.

—

The difference between paying fixed and receiving floating interest payments at the time  +  on a loan of  from time  to  +  is

 ( − )  (  + )



in a discounted

(4.1)

6

FRA, Floaters and Swaps

where  is the principal amount of notional loan,  is the LIBOR rate at  for an loan maturing at  +  ,

— Both the floating rate  and the fixed rate are on a LIBOR basis. —  is called as the reference rate which is usually the LIBOR rate on the fixing date for the contract period in question. — The settlement amount at  for the FRA is the discounted amount   =

 ( − )  (  + ) 1 +   (  + )

(4.2)

where   is the value of the FRA at time  .

•

We denote  (  + ) as the time difference in years between the dates  and  +  , which is usually referred to as year fraction between the dates  and  +  . The particular choice that is made to measure the time between two dates reflects what is known as the day count convention. According the market conventions, for FRA contracts,

 (  + ) =

days 

where money market day count ( ) is 360 for dollars and ‘days’ is the number of actual days between interest reset days.

•

Example) On the 14th Jan., 2007 a firm buys a 12 month FRA, on 3 month LIBOR, on a principal of $5 million at a rate of 5.5%. We have the following contracts details: FRA Principal Amount () Dealing Date (Trading Date) Spot Date () Settlement date of contract ( ) Interest rate index Contract rate Contract position

12 × 15 $ d/m/y d/m/y d/m/y $LIBOR (months) % long/short

5,000,000 14-Jan-2007 16-Jan-2007 16-Jan-2008 3 5.5 long

— In this example the underlying loan principal is $5 million. — The contract has a spot date of 16 January, 2007, and has a maturity of 12 months.

7

Forward Rate Agreement (FRA)

— The underlying interest rate index on which payment will be based is 3 month $LIBOR.

— The fixed rate to be paid 5.5%. — The contract position is LONG, indicating that the client who buys the FRA will receive LIBOR and pay fixed. — The payoff of this FRA is positive if $LIBOR in 12 months time exceeds 5.5%. Suppose, for example, that 3 month LIBOR is 6% on that date. The payoff is Principal Amount × (LIBOR rate - Fixed rate) ×

= 5 000 000 × (006 − 0055) × = 6 15764

90 360

× 09852

 

× discount factor

where the discount factor is calculated as discount factor 1 = 1 +    1 = 90 1 + 006 × 360 = 09852 4.2.2

•

No Arbitrage Relationship of FRA’s

Consider a CD with a coupon of % at time  with the maturity date of  +  . The price at time  of a CD with a coupon % is 

 (  + ) =

1 +    1 +   

(4.3)



Hence, at time  , a forward contract to sell such a CD at par has a payoff of

1 −   (  + ) =

•

( − )    1 +   

Thus, a forward contract to sell at par a CD with coupon period is equivalent to a long FRA on



period LIBOR.

(4.4)



and maturity



8

FRA, Floaters and Swaps

•

Example) Suppose for example that a forward contract is made to sell a 90 day $5,000,000 CD with a coupon of 5.5% at par. The payoff, if LIBOR turns out to be 6% at time  , is

"

 1−

1 + fixed rate ×

   

1 + LIBOR × ∙ 1 + 0055 × = 5 000 000 1 − 1 + 0060 × = 615764

•

#

90 360 90 360

¸

A similar relationship exists between FRA’s and zero coupon bond forwards. The price of a zero coupon bond with maturity value of  at  maturing at  +  is related to the LIBOR rate by

 (  + ) =

  1 +   

(4.5)

Consider now a forward contract to sell a zero coupon bond at a contract price

=

 1+×

(4.6)

  

The payoff on the contract is

 −  (  + )   − =  1 +   1+×   =

1+×

Ã

= 

•

Ã



 

1−

( − )   1 +   

!

1+× 1+

×

    

!

1

1+×

 



Thus, a short (long) zero bond forward contract at a contract price

= is equivalent to rate

.

1 1+×  

 1+×

 

long (short) FRAs on a notional principal of

(4.7)

$ at a

9

Forward Rate Agreement (FRA)

• Caplet and floorlet are closely related to FRA through the put-call parity. Specifically, a caplet (floorlet) is an option to enter a long (short) FRA at a fixed rate . The payoff on a caplet is

max [ −  0] days 

(4.8)

1 +  days 

and the payoff on a floorlet is

max [ −   0] days  1 +  days 

(4.9)

• Thus, taking long (short) position in a caplet and short (long) position in a floorlet at the same time on  period LIBOR at a strike rate  on a loan with principal , we have the exactly the same payoff of a long (short) FRA. 4.2.3

Valuation and Pricing of FRA’s

• Consider a long FRA, on a principal amount  for the loan period  to  +  , at a fixed rate . Its value at time  is

µ

  ( ) =  (  ) −  1 +  

¶

 (  + ) 

(4.10)

where  (  ) is the price at time  of a zero-coupon bond that pays $1 at the maturity date  , and  (  + ) is the price at time  of a zero-coupon bond that pays $1 at the maturity date  +  . • The value of a FRA can be obtained from the following logical steps; 1. Consider the actual cash flows of the FRA on the settlement date,  . In the figure, the symbol ˜ emphasizes the fact that the cash flows are stochastic as of time ; ~ days ~  A iT B T , T    M

t

T

 Ak

days ~ B T , T    M

T+ 

10

FRA, Floaters and Swaps

2. Assuming that the institution can borrow or lend at the LIBOR rate existing at time  , the FRA has the equivalent cash flows at time  +  shown in the figure;

~ days  A iT M

t

T+

T

 Ak

days M

3. If we add and subtract the notional principal of cash flows with the same value.

$,

this again leaves the

¢

¡

at time  +  is on  1 +    value of $ at time  , which is now

4. Note that the LIBOR loan payment of

a principal of $. Hence it has a non-stochastic cash flows. Thus, the value of the FRA is the same as that of the non-stochastic cash

flows in the following figure.

5. Now, the non-stochastic cash

flows

bond prices to obtain the time



can be discounted at the time



zero

value of the FRA.

A

t

T

T+

days    A1 k M  

•

Example) Consider an FRA with the following details. We would like the value of position on the current date, i.e., on 16-Jan-2007 when the FRA turns into a

3 × 6:

11

Forward Rate Agreement (FRA)

FRA 4×7 Principal Amount () £ 5,000,000 Spot Date () d/m/y 16-Dec-2006 Settlement date of contract ( ) d/m/y 16-Apr-2007 Interest rate index £ LIBOR (months) 3 Contract rate % 6 Contract position long/short long 1. The relevant time periods are 16-Jan-2007 16-Apr-2007 16-Jul-2007 . 2. Suppose that the interest rate for 90 days is 625% and the rate for 181 days is 65%. 3. Now, the price of the 3 month zero coupon bond is 



 +

1 1 +    1 = 1 + 00625 × = 098482

 ( 90) =

90 365

and the price of the 6 month zero is  ( 181) 1 = 1 + 00650 × = 096877

181 365

4. Thus, the value of the FRA is Value of floating µ leg - Value¶ of fixed leg =  (  ) −  1 + 

 

 (  + )

µ

= 5000000 × 098482 − 5000000 × 1 + 006 × = 4 924 115 − 4 916 327 = 778752

91 365

¶

× 096877

12

FRA, Floaters and Swaps

•

The value at time



of a long FRA with an arbitrary rate

µ

 (  ) − 1 + 

¶  



for

=1

is<...