© 2002 South-Western Publishing 1

Chapter 14

Swap Pricing

2

Outline

Swap pricing Solving for the swap price Valuing an off-market swap Hedging the swap Pricing a currency swap

3

Swap Pricing

Swaps as a pair of bonds Swaps as a series of forward contracts Swaps as a pair of option contracts

4

Swaps as A Pair of Bonds

If you buy a bond, you receive interest If you issue a bond you pay interest

In a plain vanilla swap, you do both– You pay a fixed rate– You receive a floating rate– Or vice versa

5

Swaps as A Pair of Bonds (cont’d)

A bond with a fixed rate of 7% will sell at a premium if this is above the current market rate

A bond with a fixed rate of 7% will sell at a discount if this is below the current market rate

Swap values are impacted in a similar manner to bond values

6

Swaps as A Pair of Bonds (cont’d)

If a firm is involved in a swap and pays a fixed rate of 7% at a time when it would otherwise have to pay a higher rate, the swap is saving the firm money– swap has value– unwinding the swap captures the value

If because of the swap you are obliged to pay more than the current rate, the swap is beneficial to the other party– cost to the firm to unwind the swap

7

Swaps as A Series of Forward Contracts

A forward contract is an agreement to exchange assets at a particular date in the future, without marking-to-market

An interest rate swap has known payment dates evenly spaced throughout the tenor of the swap

8

Swaps as A Series of Forward Contracts (cont’d)

A six month swap with a single payment date six months hence is no different than an ordinary six-month forward contract

– At that date, the party owing the greater amount remits a difference check

9

Swaps as A Pair of Option Contracts

Assume a firm buys a cap and writes a floor, both with a 5% striking price

At the next payment date, the firm will – Receive a check if the benchmark rate is above

5%– Remit a check if the benchmark rate is below 5%

10

Swaps as A Pair of Option Contracts (cont’d)

The cash flows of the two options are identical to the cash flows associated with a 5% fixed rate swap

– If the floating rate is above the fixed rate, the party paying the fixed rate receives a check

– If the floating rate is below the fixed rate, the party paying the floating rate receives a check

11

Swaps as A Pair of Option Contracts (cont’d)

Cap-floor-swap parity

5%+ =

5% 5%

Write floor Buy cap Long swap

12

Solving for the Swap Price

The role of the forward curve for LIBOR Implied forward rates Initial condition pricing Quoting the swap price Counterparty risk implications

13

Swap Pricing

The swap price is determined by fundamental arbitrage arguments

– All swap dealers are in close agreement on what this rate should be

14

The Role of the Forward Curve for LIBOR

LIBOR depends on when you want to begin a loan and how long it will last

Similar to forward rates:– A 3 x 6 Forward Rate Agreement (FRA) begins in

three months and lasts three months (denoted by )

– A 6 x 12 FRA begins in six months and lasts six months (denoted by )

63 f

126 f

15

The Role of the Forward Curve for LIBOR (cont’d)

Assume the following LIBOR interest rates:

Spot (0f3) 5.42%

Six Month (0f6) 5.50%

Nine Month (0f9) 5.57%

Twelve Month (0f12) 5.62%

16

The Role of the Forward Curve for LIBOR (cont’d)

LIBOR yield curve

Months0 3 6 9

5.42

5.50

5.575.62

spot

0 x 6

0 x 9

0 x 12

%

17

Implied Forward Rates

We can use these LIBOR rates to solve for the implied forward rates– The rate expected to prevail in three months, 3f6

– The rate expected to prevail in six months, 6f9

– The rate expected to prevail in nine months, 9f12

The technique to obtain the implied forward rates is called bootstrapping

18

Implied Forward Rates (cont’d)

An investor can– Invest in six-month LIBOR and earn 5.50%– Invest in spot, three-month LIBOR at 5.42% and

re-invest for another three months at maturity

If the market expects both choices to provide the same return, then we can solve for the implied forward rate on the 3 x 6 FRA

19

Implied Forward Rates (cont’d)

The following relationship is true if both alternatives are expected to provide the same return:

2

606330

41

41

41

fff

20

Implied Forward Rates (cont’d)

Using the available data:

%56.5

4

0550.1

41

4

0542.1

63

2

63

f

f

21

Implied Forward Rates (cont’d)

Applying bootstrapping to obtain the other implied forward rates:

– 6f9 = 5.71%

– 9f12 = 5.75%

22

Implied Forward Rates (cont’d)

LIBOR Implied forward rate curve

Months0 3 6 9

5.42

5.58

5.715.77

spot

3 x 6

6 x 9

9 x 12

%

23

Initial Condition Pricing

An at-the-market swap is one in which the swap price is set such that the present value of the floating rate side of the swap equals the present value of the fixed rate side (equilibrium)– The floating rate payments are uncertain

Use the implied forward rate curve (derived from the spot yield curve) as a proxy for the floating rate payments

24

Initial Condition Pricing (cont’d)

At-the-Market Swap Example

A one-year, quarterly payment swap exists based on actual days in the quarter and a 360-day year on both the fixed and floating sides. Days in the next 4 quarters are 91, 90, 92, and 92, respectively. The notional principal of the swap is $1.

Convert the future values of the swap into present values by discounting at the appropriate zero coupon rate contained in the forward rate curve.

25

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)

First obtain the discount factors:

013701.10542.360

9111 3

R

027653.10550.360

909111 6

R

26

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)

First obtain the discount factors:

042239.10557.360

92909111 9

R

056981.10562.360

9292909111 12

R

27

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)

Next, apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

055042.0

013951.014001.013575.013515.056981.1

36092

%77.5

042239.136092

%71.5

027653.136090

%58.5

013701.136091

%42.5

floating

PV

28

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)

Apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

X

XXXX

XXXXPV

979612.0

241779.245199.243273.249361.056981.1

36092

%

042239.136092

%

027653.136090

%

013701.136091

%

fixed

29

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)

Solving the two equations simultaneously for X gives X = 5.62%. This is the equilibrium swap fixed rate, or swap price.

...this is the starting point for a swap dealer

30

Quoting the Swap Price

Common practice to quote the swap price relative to the U.S. Treasury yield curve– interest rates are constantly changing so dealers will

quote relative to or off the U.S. treasury or Govt. Of Canada yield curve

– Maturity should match the tenor of the swap

There is both a bid and an ask associated with the swap price– The dealer adds a swap spread to the appropriate

Treasury yield

31

Counterparty Risk Implications

From the perspective of the party paying the fixed rate– Higher when the floating rate is above the fixed

rate

From the perspective of the party paying the floating rate– Higher when the fixed rate is above the floating

rate

32

Valuing an Off-Market Swap

The swap value reflects the difference between the swap price (fixed rate) and the interest rate that would make the swap have zero value

– As soon as market interest rates change after a swap is entered, the swap has value to one party or the other

33

Valuing an Off-Market Swap (cont’d)

An off-market swap is one in which the fixed rate is such that the fixed rate and floating rate sides of the swap do not have equal value - the present value of the two cash flow streams is different.

– Thus, the swap has value to one of the counterparties

34

Valuing an Off-Market Swap (cont’d)

If the fixed rate in our at-the-market swap example was 5.75% instead of 5.62%– The value of the floating rate side would not

change– The value of the fixed rate side would be lower

than the floating rate side– The swap has value to the floating rate payer

and fixed rate receiver ...in a stand alone situation

35

Valuing an Interest Rate Swap During Its Life

If fixed rates changed from 5.62 % to 5.75 % : to arrive at the value of the swap - in effect

discounting the fixed rate payments at a higher discount rate resulting in a lower PV (akin to bond pricing)

for the fixed rate payor - this swap would have value ( cash value if it were closed out) and a cost to the floating rate payor/fixed rate receiver)

36

Hedging the Swap

If interest is predominantly in one direction (e.g., everyone wants to pay a fixed rate), then the dealer stands to suffer a considerable loss

– E.g., the dealer is a counterparty to a one-year, $10 million swap with quarterly payments and pays floating

The dealer is hurt by rising interest rates

37

Hedging the Swap

The dealer can hedge this risk in the futures market

– If the deal is Libor based - then use Euro-dollar futures

– If the dealer faces the risk of rising rates, he could sell Eurodollar futures and benefit from the decline in value associated with rising interest rates

38

Pricing A Currency Swap

To value a currency swap:– Solve for the equilibrium fixed rate on a plain

vanilla interest rate swap for each of the two countries

Determine the relevant spot rates over the tenor of the swap

Determine the relevant implied forward rates

– Find the equilibrium swap price for an interest rate swap in both currencies - in effect there are two swap prices in a currency swap