Chapter 14

Interest Rate Swaps and Credit Default Swaps

Swaps are a popular form of OTC derivative instruments. What is a swap?

As the name connotes, it is a contract to exchange or swap two cash flows. It
is an exchange between two parties of two payment streams that are different
from each other. In the case of an interest rate swap (IRS), the contract requires
the specification of a principal amount termed the notional principal, for it is not
meant to be exchanged, but has been specified purely to facilitate the compu-
tation of interest. Each of the two counterparties use a different benchmark for
computing the interest on the specified notional principal. For instance, one party
may use a fixed rate of interest, whereas another may use a variable rate such as
the three-month LIBOR. This is referred to as a coupon swap. The alternative is
a contract, where both the parties use variable rates to compute their respective
obligations. This is referred to as a basis swap. After the obligations have been
determined, the party owing the higher amount pays the difference between the
two computed amounts to the counterparty. This is termed as netting. Netting
is feasible because both cash flow streams are denominated in the same currency.
Had they been in two different currencies, this kind of netting would not be feasi-
ble. In this chapter, we do not consider currency swaps, which entail the exchange
of cash flows in different currencies.

Although coupon and basis swaps are possible, a contract which requires both
the parties to pay a fixed rate is not feasible. This is because the party that is
required to pay the higher rate would be paying constantly, and the counterparty
would be receiving constantly. This is clearly a manifestation of arbitrage, and
consequently the party that is required to pay a higher rate will never accept such
an arrangement. In an interest rate swap, whether it is a coupon swap or a basis
swap, we do not know a priori, which of the two counterparties will have to make
a payment, and correspondingly the identity of the receiver is unknown at the
outset.

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Interest Rate Swaps

We will now define briefly the various terms that need to be incorporated into a
swap contract.

Contract Terms

The following terms must be explicitly stated while designing an interest rate
swap contract:

• The identities of the two counterparties.

• The maturity of the swap. This is the date on which the last exchange of
cash flows takes place between the two parties.

• The interest rate used by the first party to calculate its payments. It may
be fixed or floating.

• The interest rate used by the second party to calculate its payments. It
must be floating if the first party is making payments based on a fixed rate
of interest.

• The day-count convention for the computation of interest.

• The frequency of payment

• The notional principal

Example 14.1

Consider a four-year fixed-floating swap between Scotia Bank and Dominion

Bank. Scotia will make payments based on an annual rate of 5%. Dominion
will compute its liability based on the six-month LIBOR. We assume that
every month consists of 30 days and that the year as a whole consists of 360

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days. The implication is that every semiannual period consists of 180 days, and
consequently the interest rate per semiannual period is one half of the annual
rate. Other day-count conventions such as Actual/Actual or Actual/360 are
possible. In such cases, the amount payable by the fixed rate payer varies from
period to period, even though the interest rate remains constant, because the
number of days per semiannual period varies. The exchange of payments takes
place on a semiannual basis. The notional principal amount is $8 million.

Assume that the six-month LIBOR, as observed at semiannual intervals

over the next four years, is as shown in Table 14.1.
Table 14.1
Observed values of LIBOR

Time LIBOR
0 5.25%
After 6 Months 5.40%
After 12 Months 5.10%
After 18 Months 4.75%
After 24 Months 4.30%
After 30 Months 4.00%
After 36 Months 4.80%
After 42 Months 5.20%

Using this data, let’s compute the payments to be made by the two parties
every six months. Let’s analyze the cash flows to be exchanged after six
months. Scotia has to pay $200,000 every six months. This may be calculated
as follows:
0.5 × 0.05 × 8, 000, 000 = $200, 000

This amount remains constant every period. If, however, we assume an Ac-
tual/360 day-count convention and that the actual number of days in the
period is 184 days, the amount payable becomes
184
× 0.05 × 8, 000, 000 = $204, 444.44
360

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In the following period, if the number of days is assumed to be 181, the amount
payable is
181
× 0.05 × 8, 000, 000 = $201, 111.11
360
At the end of six months, Dominion has to pay

0.5 × 0.0525 × 8, 000, 000 = $210, 000

Because Scotia owes $200,000 and Dominion owes $210,000, Dominion pays
the difference of $10,000 to Scotia. In the following period, Scotia once again
owes $200,000. But the amount owed by Dominion is

0.5 × 0.0540 × 8, 000, 000 = $216, 000

Consequently Dominion has to pay $16,000 to Scotia.

The cash flows made by the two counterparties and the net cash flow are
shown in Table 14.2.
Table 14.2
Cash flows in an interest rate swap

Time Payment by LIBOR Payment by Net

Scotia Dominion Payment
Zero - 5.25% - -
6 Month $200,000 5.40% $210,000 $(10,000)
12 Month $200,000 5.10% $216,000 $(16,000)
18 Month $200,000 4.75% $204,000 $(4,000)
24 Month $200,000 4.30% $190,000 $10,000
30 Month $200,000 4.00% $172,000 $28,000
36 Month $200,000 4.80% $160,000 $40,000
42 Month $200,000 5.20% $192,000 $8,000
48 Month $200,000 $208,000 $(8,000)
TOTAL $1,600,000 $1,552,000 $48,000

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 The payments that are made by Dominion are based on the LIBOR that
is observed at the start of the period concerned. However, the interest itself is
payable at the end of the period. For instance, the payment made by Dominion at
the end of the first period is based on a interest rate of 5.25%, which corresponds
to the LIBOR at the start of the period. This system of computation is known as
determined in advance and paid in arrears and is the most commonly used system
in practice.

The last column of Table 14.2 shows the net payment. Positive entries connote
that Scotia has to pay the counterparty, whereas negative amounts indicate that
the counterparty has to pay Scotia. The advantage of netting is that it reduces
delivery risk. For instance, if Scotia pays the gross amount it owes to Dominion
at the end of the first six months, it is exposed to the risk that the payment
of $210,000 owed by Dominion may never arrive. On the contrary, because of
netting, the risk for Scotia in this case is only the sum of $10,000 that would not
be received if Dominion were to renege.

In a swap contract, certain terms and conditions need to be specified at the very
outset to avoid ambiguities and potential future conflicts. Every swap contract
must clearly spell out the identities of the two counterparties to the deal. In our
example, the two counterparties are Scotia Bank and Dominion Bank.

Second, the tenor of the swap must be clearly stated. The tenor or maturity
of the swap refers to the length of time at the end of which the last exchange of
cash flows between the two parties takes place. In our example, the tenor is four
years. Unlike exchange-traded products like futures contracts, where the exchange
specifies a maximum maturity for contracts on an asset, swaps are OTC products
that can have any maturity that is agreed upon by bilateral discussions.

Third, the interest rates on the basis of which the two parties have to make
payments should be clearly spelled out. To avoid ambiguities, the basis on which
the cash inflow and cash outflow are arrived at for both the counterparties should
be explicitly stated. In our example, Scotia Bank is a fixed rate payer with
the rate of interest being fixed at 5.00% per annum, whereas Dominion Bank is a
floating rate payer with the amount payable being based on the six-month LIBOR
prevalent at the start of the interest computation period.

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 Fourth, the frequency with which the cash flows are to be exchanged has to
be clearly defined. In our example, we have assumed that cash flow exchanges
will take place at six-monthly intervals. In the market such swaps are referred to
as semi-semi swaps. Other contracts may entail payments on a quarterly basis or
on an annual basis. The benchmark that is chosen for the floating rate payment
is usually based on the frequency of the exchange. For instance a swap entailing
the exchange of cash flows at six-monthly intervals will specify six-month LIBOR
as the benchmark, whereas a swap entailing the exchange of payments at three-
monthly intervals will specify the three-month LIBOR as the benchmark. The
most popular benchmark in the market is the six-month LIBOR.

Fifth, the day-count convention that is used to compute the interest must be
explicitly stated. In our illustration we have assumed that every six-month period
amounts to exactly one-half of a year. The underlying convention is referred to
as 30/360. That is every month is assumed to consist of 30 days while the year
as a whole is assumed to consist of 360 days.

Finally, the principal amounts on the basis on which each party has to figure
out the payment to the counterparty have to be clearly stated. In the case of
interest rate swaps, there is obviously only one currency that is involved. However,
the magnitude of the principal has to be specified to facilitate the computation of
interest. In our example, the principal is $8 million.

In a coupon swap, the party that agrees to make payments based on a fixed
rate is referred to as the payer. The counterparty, which is committed to making
payments on a floating rate basis, is referred to as the receiver. Quite obviously
these terms cannot be used in the case of basis swaps because both the cash flow
streams are determined based on floating rates. Consequently, in order to be
explicit and avoid ambiguities, it is a good practice to describe for each of the
two parties, the rates at which they are scheduled to make and receive payments.
Thus in the preceding example, we would state that Scotia is scheduled to pay a
fixed rate of 5% and receive the six-month LIBOR, whereas Dominion is scheduled
to pay the six-month LIBOR and receive a payment based on an interest rate of
5% in return. In the case of coupon swaps, some markets refer to the fixed rate
payer as the buyer and the fixed rate receiver as the receiver.

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 The payments in a swap may be settled either on a money-market basis or
on a bond-market basis, the difference being that the first convention is based
on a 360-day year and an Actual/360 day-count convention, and the second is
based on a 365-day year and an Actual/365 day-count convention. In practice,
the fixed rates payable are quoted on a bond basis, and floating rates are quoted
on a money-market basis. To convert from a bond basis to a money-market basis,
we have to multiply the quote by 360/365; whereas to do the reverse, we multiply
by 365/360.

Key Dates in a Swap Contract

There are four important dates that have to be specified in a swap contract.
Consider the four-year swap between Scotia Bank and Dominion Bank. Assume
that the swap was negotiated on 15 June 20XX with a specification that the
first payments are for a six-month period commencing on 1 July 20XX. 15 June
is referred to as the transaction date. The date from which the interest counter
payments start to accrue is termed the effective date. In our example, the effective
date is 1 July.

Our swap, by assumption, has a tenor of four years and consequently the last
exchange of payments take place on 30 June 20XX+4. This date consequently is
referred to as the maturity date of the swap. We assume that the eight cash-flow
exchanges will occur on 31 December and 30 June of every year. The first seven
dates, on which the floating rate will be reset for the next six-monthly period, are
referred to as reset or re-fixing dates.

The Swap Rate

The fixed rate of interest that has been agreed upon in a coupon swap is referred
to as the swap rate. If the swap rate is quoted as a percentage, it is referred to as
an all-in price. However, in certain interbank markets, the fixed rate is not quoted
as a percentage. Instead, what is quoted is the difference, in basis points, between
the agreed upon fixed rate and a benchmark interest rate. The benchmark chosen

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to compute this differential is usually the government security whose remaining
term to maturity is closest to the life of the swap in question. For instance, in
the case of the Scotia-Dominion swap, the fixed rate is 5% per annum. If the
swap rate were quoted as an all-in price, it would obviously be reported as such.
However, in the second convention, the rate would be quoted as follows. Assume
that a four-year T-note has a yield to maturity of 4.80%. The swap price is then
quoted as 5% minus 4.80% or as 20 basis points.

Risk

Whether it is a coupon swap or a basis swap, an interest rate swap exposes both
the parties to interest rate risk. In the case of Scotia, which is the fixed rate payer
in this case, the risk is that the LIBOR may decline during the life of the swap.
If so, the payments due to Scotia may stand reduced, whereas the payments to
be made by it are invariant to interest rate changes. On the other hand, the risk
for Dominion, the floating rate payer, is that the LIBOR may increase over the
life of the swap. If so, Dominion’s payment amounts will increase, whereas the
payments it is due to receive will be invariant to rate changes. The same is true
in the case of a basis swap. Assume one party pays Rate-1 and the other pays
Rate-2, where both the rates are variable. The risk scenarios for the party paying
Rate-1 are that the increase in Rate-1 is more than the increase in Rate-2, the
decline in Rate-1 less than the decline in Rate-2, or Rate-1 increases and Rate-2
declines. For the counterparty, the situation is just the opposite. That is, the
increase in Rate-2 is more than the increase in Rate-1, the decrease in Rate-2 is
less than the decrease in Rate-1, or Rate-2 increases and Rate-1 declines. Also,
in every swap, both the counterparties are exposed to default risk.

Quoted Swap Rates

A swap dealer quotes two rates for a coupon swap, a bid and an ask. The bid is
the fixed rate at which the dealer is willing to do a swap that requires it to pay
the fixed rate, and the ask represents the rate at which the dealer will do a swap
that requires it to receive the fixed rate. The bid will be lower than the ask.

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 Consider the hypothetical quotes for US dollar-denominated interest rate swaps
on a given day, shown in Table 14.3. Assume that the corresponding floating rate
is the six-month LIBOR.
Table 14.3
All-in prices and spreads for swaps

Tenor All-in Prices Spread over

Treasury
Bid Ask Bid Ask
1-Year 3.70% 3.75% 20 bp 25 bp
2-Year 3.90% 3.95% 15 bp 20 bp
3-Year 3.25% 3.30% 25 bp 30 bp
5-Year 2.90% 2.95% 15 bp 20 bp
10-Year 3.45% 3.50% 20 bp 25 bp

We have assumed a spread of 5 bp for all tenors, to make matters simple. Let’s
consider the rates for a 1-year swap. The bid is 3.70%, and the ask is 3.75%. Thus
if the dealer does a swap where it has to make a fixed rate payment in return for a
cash flow based on a floating rate, it agrees to pay 3.70% per annum. However, if
the dealer is asked to do a swap wherein it receives the fixed rate in exchange for
a floating rate, it asks for a rate of 3.75% per annum. Equivalently, if the dealer
pays the fixed rate in a coupon swap, it pays 20 bp over the prevailing rate on
a one-year Treasury security, whereas if it receives the fixed rate, it demands a
spread of 25 bp over the yield on a comparable Treasury security.

Comparative Advantage and Credit Arbitrage

There could be situations where a party has a disadvantage from the standpoint
of borrowing rates with respect to another party in the markets for fixed rate
as well as for variable rate loans. However, despite this, the former may have
a comparative advantage in one of the two markets, as the following example
illustrates.

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Example 14.2

A company called Mount Holly, can borrow at a fixed rate of 5.25% and a
variable rate of LIBOR + 1% in the US debt market. On the other hand,
another company, Checkmate, can borrow at 4.90% in the fixed rate market
and LIBOR + 25 bp in the variable rate market. Thus Mount Holly has to
pay 35 bp more than Checkmate does if it borrows in the fixed rate market.
However, it has to pay 75 basis points more if it borrows on a floating rate
basis. Because Checkmate can get funds at a lower rate in both the markets,
we say that it enjoys an absolute advantage in both markets compared to
Mount Holly. However, because the spread for Mount Holly is lower in the
fixed rate market, compared to the floating rate market, we say that Mount
Holly enjoys a comparative advantage in the fixed rate market.

Assume that Mount Holly wants to borrow at a floating rate, whereas

Checkmate would like to borrow at a fixed rate. However, if Mount Holly bor-
rows in the fixed rate market, an arena where it has a comparative advantage,
and then swaps the interest payments with Checkmate, it could be a win-win
situation for both the entities. In other words, as a consequence of the swap,
both the parties can borrow at a reduced rate of interest compared to what
they would have had to pay in the absence of it.

Let’s assume that Mount Holly borrows $10 million at a fixed rate of 5.25%
per annum, and Checkmate borrows the same amount at LIBOR + 25 bp.
The two parties can then enter into a swap wherein the former agrees to pay
interest on a notional principal of $10 million at the rate of LIBOR + 40 bp
per annum in exchange for a fixed rate payment based on a rate of 4.90% from
the latter. The effective interest rate for the two parties may be computed as
follows:

Mount Holly: 5.25% + LIBOR + 40 bp - 4.90% = LIBOR + 75 bp

Checkmate: LIBOR + 25 bp, + 4.90% - LIBOR - 40 bp = 4.75%

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 Mount Holly has a saving of 25 basis points on the floating rate debt, and
Checkmate has a saving of 15 bp on the fixed rate debt. Thus, Checkmate
has an advantage of 75 bp in the market for floating rate debt and 35 bp in
the market for fixed rate debt. The difference of 40 bp manifests itself as the
savings for both parties considered together. In our illustration, Mount Holly
has saved 25 bp, and the counter-party has saved 15 bp, which adds up to 40
basis points. In practice, we can have any pair of numbers, as long as the sum
total is 40 bp.

In practice, a swap dealer such as a commercial bank plays a role in the

transaction. Assume that Mount Holly borrows at 5.25% per annum and
enters into a swap with Scotia Bank wherein it has to pay LIBOR + 55 bp in
return for a fixed-rate payment based on a rate of 5.00%. Checkmate on the
other hand borrows at LIBOR + 25 bp and enters into a swap with the same
bank wherein in receives LIBOR + 20 bp in return for payment of 4.80%.

The net result of the transaction may be summarized as follows:

Mount Holly: Effective interest paid = 5.25% + LIBOR + 55 bp - 5.00 =

LIBOR + 80 bp

Checkmate: Effective interest paid = LIBOR + 25 bp + 4.80% - LIBOR

- 20 bp = 4.85%

Scotia Bank: Profit from the transaction = LIBOR + 55 bp - 5.00 - LIBOR

- 20 bp + 4.80 = 15 bp

The difference in this case is that the comparative advantage of 40 basis

points has been split three ways. Mount Holly saves 20 basis points; Check-
mate saves 5 basis points; and the bank makes a profit of 15 basis points.

The Role of Banks in the Swap Market

When the swap market was at its nascent stage the normal practice was for invest-
ment banks to play the role of an intermediary. These banks would arrange the

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transaction by bringing together two counterparties and in return would be paid
an arrangement fee. Over a period of time, the role of an intermediary evolved
from that of an agent that facilitated a swap to that of a principal. One of the
main reasons for this was that parties to swaps did not want their identities to
be revealed to the counterparty. Second, as we have seen, swaps expose both
counterparties to default risk. For this reason, parties to a swap were more com-
fortable dealing with a bank, whose creditworthiness was easier to appraise. As
the market has evolved, such arrangement fees have become extremely rare except
perhaps for contracts that are very exotic or unusual.

In the days when the market was at its infancy, banks would primarily do
reversals. A reversal entails the offsetting of a swap with a counter agreement
with another client. For instance, they would do a fixed-floating deal with a
party only if they were hopeful of immediately concluding a floating-fixed deal
for the same tenor with a third party. Parties that carry equal and offsetting
swaps in their books are said to be running a matched book. These days banks
are less finicky about maintaining such a matched position, and in most cases are
willing to take on the inherent exposure until they eventually locate a party for
an offsetting transaction. It must be understood that a dealer that maintains a
matched book is exposed to default risk from both the parties with which it has
entered into swaps.

Valuing an Interest Rate Swap

In our illustration of a swap between Scotia Bank and Dominion Bank, we ar-
bitrarily assumed that the fixed rate or swap rate was 5% per annum. We now
demonstrate how this rate is determined in practice.

Consider a two-year swap between Bank Alpha and Bank Beta. Bank Alpha
has to pay a fixed rate of k% per annum on a semiannual basis, whereas the
counterparty has to pay the six-month LIBOR every six months. As an alter-
native, assume that instead of entering into a swap, Bank Alpha has issued a
two-year fixed rate note with a principal of $8 million on which it has to make
semiannual interest payments at the rate of k% per annum. This money has been

595
used to acquire a two-year floating rate note with the same principal, and which
pays coupons semi-annually based on the LIBOR observed at the start of the
six-monthly period.

If we look at the cash flows of this alternate arrangement, the result is equiva-
lent to that on a two-year fixed-floating swap. At the outset there is an inflow of
$8 million for Bank Alpha when the fixed rate note is issued. But this amount is
just adequate to purchase the floating-rate loan. Thus the net cash flow is zero.
Similarly, at the point of termination, that is after two years, Bank Alpha receives
$8 million when the floating rate note matures, and this amount also is just ad-
equate to retire the fixed rate note. The net result is that there is no exchange
of principal, either at the outset or at the end, which is consistent with what we
have seen for interest rate swaps.

Every six months, the floating rate note pays a coupon based on the LIBOR
at the start of the period. Bank Alpha receives this amount and is required to
pay interest at the rate of k% per annum to service the fixed rate note that it has
issued. Consequently the cash flows every six months are identical to that of the
swap. Thus a long position in a floating rate note coupled with a short position
in a fixed rate note is equivalent to a swap that requires fixed-rate payments in
return for payments based on a floating rate. Now let’s demonstrate as to how
the fixed rate of a coupon swap can be determined.

Let’s use the same vector of spot rates that we used to calibrate the Ho-Lee
and BDT models in Chapter 11.
Table 14.4
Vector of spot rates

Period Spot Rate

0 7.50%
1 7.00%
2 6.25%
3 6.75%

Because, by assumption, the current point in time is the start of the next
six-monthly period, the price of the two-year floating rate note is equal to its face
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value of $8 million. For, on a coupon reset date, the price of a default risk-free
floating rate bond reverts to its face value. The question to be answered is what
the coupon rate should be for the fixed rate note so that it too has a current
price of $8 million. Let’s first determine the discount factors corresponding to the
observed LIBOR rates.

The discount factor for a given maturity is the present value of a dollar to be
received at the end of the stated period. The convention in the LIBOR market is
that if the number of days for which the rate is quoted is N, then the corresponding
1
discount factor is given by   . For instance, the discount factor for
N
1+i×
360
1
an investment of 18 months is   where i is obviously the quoted 18-
540
1+i×
360
month LIBOR. Table 14.5 shows the vector of discount factors for our example.
Table 14.5
Discount factors

Time to Maturity Discount Factor

6M 0.9639
12M 0.9346
18M 0.9143
24M 0.8811

C
If we denote the semiannual coupon by , it must be that
2
C C C C
 
× 0.9639 + × 0.9346 + × 0.9143 + + 8, 000, 000 × 0.8811
2 2 2 2
C C
= 8, 000, 000 ⇒ 3.6939 × = 951, 200 ⇒ = 257, 505.61 ⇒ C = 515, 011.22
2 2
Thus the swap rate is:
515, 011.22
× 100 = 6.4376%
8, 000, 000

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Valuing a Swap at an Intermediate Stage

Assume that three months have elapsed since the preceding swap was initiated.
Consider the term structure in Table 14.6.
Table 14.6
Spot rates and discount factors after three months

Time to Maturity Rate Discount Factor

3M 7.75% 0.9810
9M 7.25% 0.9484
15M 6.50% 0.9249
21M 6.80 % 0.8937

The value of the fixed rate note is computed as follows:

C
× [0.9810 + 0.9484 + 0.9249 + 0.8937] + 8, 000, 000 × 0.8937
2
= 257, 505.61 × 3.748 + 8, 000, 000 × 0.8937 = 965, 131.10 + 7, 149, 600

= $8, 114, 731

The value of the floating rate bond is computed as follows. Three months hence it
pays a coupon based on the original six-month rate which is 7.50%. The amount
of this coupon is
0.5 × 0.0750 × 8, 000, 000 = $300, 000

When this coupon is paid, the value of the bond reverts to its face value of $8
million. Consequently, its value today is

8, 300, 000 × 0.9810

= $8, 142, 300

From the standpoint of the fixed-rate payer, the swap is tantamount to a long
position in a floating rate note that is combined with a short position in a fixed
rate note. Thus the value of the swap is

8, 142, 300 − 8, 114, 731 = $27, 568.95

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For the counterparty, the value is obviously -$27,568.95, for a swap is also a
zero sum game. A negative value indicates that the position holder has to pay
to assign the swap to someone else, whereas a positive value indicates that the
holder receives the value if it chooses to assign the swap to another party.

Terminating a Swap

Let’s suppose that after three months have elapsed, the fixed-rate payer in the
prceding swap decides that it no longer wants to be a party to the swap. It can get
out of the current situation in a variety of ways. One way is by way of a reversal.
That is, it can enter into a swap with 21 months to maturity, wherein it is required
to pay floating and receive fixed. The index for the floating-rate payments must
obviously be the same. In this case two swaps exist. So the party is exposed to
credit risk in both swaps. The second way to exit the swap is by selling it to a
new party. In this case, Scotia Bank has to be paid $27,568.95 by the acquirer
because the swap has a positive value. In this case, the original counterparty to
the swap, that is Dominion Bank, has to agree to the deal. Finally, Dominion
itself may buy out the swap from Scotia by paying the value. This is known as
buy-back or close-out.

Motives for the Swap

A party to a swap may enter into the contract with a speculative motive, or else
with an incentive to hedge. In addition such transactions may also be used to
undertake credit arbitrage arising due to the comparative advantage enjoyed by
the participating institutions, as we learned earlier. Let’s analyze each of these
potential uses.

Speculation

Scotia Bank and Dominion Bank are both players in the Canadian financial mar-
ket. However they have very different perspectives about the direction in which

599
interest rates are headed. Scotia believes the domestic interest rates are likely
to increase steadily over the next decade. On the contrary, Dominion is of the
opinion that domestic interest rates will decline steadily over the next 10 years.
A coupon swap with 10 years to maturity is a suitable speculative tool for both
parties. Scotia, which is bullish about interest rates, can enter the contract as the
fixed-rate payer, and Dominion, which is bearish about interest rates, can be the
counterparty as the fixed-rate receiver. Obviously both the speculators cannot
earn a profit, for one of them will be proved wrong subsequently. If interest rates
rise, Scotia, the fixed-rate payer, stands to benefit. On the contrary, if yields
decline, Dominion Bank, the floating-rate payer, gets positive cash inflows.

Hedging a Liability

Swaps can be used as a hedge against anticipated interest rate movements. Parties
may choose to hedge an asset or a liability, depending on their prior position.
Carpenters Inc., a company based in Kansas City, has taken a floating-rate loan
on which it has to pay an interest rate equal to the six-month LIBOR + 1%. Its
apprehension is that rates may increase, and consequently it would have to pay
more. It can use an interest rate swap to convert its existing liability into an
effective fixed-rate loan. One alternative way to do so in practice is to renegotiate
the loan and have it converted to a loan carrying a fixed rate of interest. This
may not be easy in real life, for there are a lot of administrative and legal issues
and related costs. However, it is relatively easier to enter into a swap with a
bank, wherein the company has to pay a fixed rate in return for a LIBOR-based
payment.

Assume that Prudential Bank agrees to enter into a swap with Carpenters
wherein it pays LIBOR in return for a fixed interest stream based on a rate of
4.25% per annum. One possibility is that the bank is bearish about interest rates
and wants to speculate. The other possibility is that the bank has an asset on
which it is earning a floating rate of interest. Being bearish, it seeks to hedge by
entering into a coupon swap as a fixed-rate receiver.

The net result from the standpoint of Carpenters may be analyzed as follows:

600
 • Outflow-1(interest on the original loan): LIBOR + 1%

• Inflow-1(receipt from Prudential Bank): LIBOR

• Outflow-2 (payment to Prudential Bank): 4.25%

• Net Outflow: 4.25% + LIBOR + 1% - LIBOR = 5.25% per annum

Thus the company has converted its variable rate liability, to an effective fixed-rate
liability carrying interest at the rate of 5.25% per annum.

Yet another reason for using swaps is to change the mix of fixed-rate and
floating-rate debt on a company’s balance sheet. Riviera Corporation has a liabil-
ity profile consisting of $50 million in fixed-rate debt and another $50 million in
floating-rate debt. It is now taking over Rodeo Corporation, and after the amalga-
mation it will have $100 million of fixed-rate debt and $70 million of floating-rate
debt. The firm is eager to maintain the 50:50 ratio between fixed and floating
debt after the merger. The total debt post-merger will be $170 million, and a
50:50 ratio implies a fixed-rate debt of $85 million and a floating-rate debt of an
equivalent amount. One way of restructuring is to borrow another $15 million at
a floating rate and repay $15 million worth of fixed-rate debt. Another alternative
is the use of an interest rate swap.

Riviera can enter into a swap wherein it receives interest at a fixed rate on a
notional principal of $15 million and pays interest on a floating rate basis on the
same amount. The portfolio of the existing liabilities plus the swap is effectively
a fixed-rate liability of $85 million and a floating-rate liability of $85 million.

Yet another reason for using a swap is the inability of a party to obtain loans
at a fixed rate of interest. In practice a small company, a new company, or even
an existing company that is large but has a weak credit rating, may be unable to
borrow at a fixed rate of interest. Such a company has to raise capital in the bond
market, and the bond markets are very particular about the credit rating of the
potential borrower. One possibility is to issue junk bonds carrying high coupons.
An alternative is to borrow at a floating rate of interest and then do a coupon
swap, wherein the entity pays a fixed rate and receives a floating rate. The net
result is the conversion of a floating-rate liability to a fixed-rate liability.

601
Hedging an Asset

A swap can be used by a borrower to convert its liability from a fixed-rate loan
to a floating-rate loan or vice versa. Such contracts, however, may also be used
by entities that seek to transform the income from their assets from a fixed rate
cash inflow to a floating rate cash inflow or vice versa. For instance, a company
that has invested in fixed rate bonds may use an interest rate swap to convert it
into a synthetic floating-rate asset.

Assume that a company has bought 100,000 bonds with a face value of $1,000
and a coupon of 6% per annum paid semiannually. Let’s assume that we are on
the issue date and that the value of the bonds is equal to the par value. On the
issue date, or any coupon date, the accrued interest will be zero. A conventional
coupon swap, wherein the company pays fixed and receives LIBOR, is appropriate
under these circumstances.

However, consider a situation where we are three months into the coupon
period. The bond has 21 months to maturity. Table 14.7 shows the prevailing
LIBOR values.
Table 14.7
Spot rates and discount factors after three months

Time to Maturity Rate Discount Factor

3M 7.75% 0.9810
9M 7.25% 0.9484
15M 6.50% 0.9249
21M 6.80 % 0.8937

In this case there is accrued interest for three months. In practice, investors
prefer an arrangement where the assets do not have any accrued interest, and
where the swap rate is equal to the coupon rate on the bonds. If we assume that
the bonds are quoting at 97-08, the clean price of the bond is
8
97 +
1, 000 × 32 = $972.50
100
602
The accrued interest, assuming a 30/360 day-count convention, is
30
1, 000 × = $15
2
and consequently the dirty price is $987.50. In practice, investors entering into
asset swaps prefer contracts where the notional principal is equal to the par value
of the bonds. In our case the difference between the par value and the dirty
price per bond is $12.50, and for 100,000 bonds, it is $1,250,000. This amount
is payable by the bond holder to the counterparty. Thus the cash flows over the
next 21 months are those in Table 14.8.
Table 14.8
Cash flows from the asset holder’s perspective

Time Amount
0 1,250,000
3M 3,000,000
9M 3,000,000
15M 3,000,000
21M 3,000,000

Each of the subsequent cash flows are $3 million, because the semiannual coupon
payment is $30 per bond.

The value of these cash flows given the LIBOR rates and using the market
method for computing the discount factors is
3, 000, 000 3, 000, 000
1, 250, 000 +  +
0.0775 0.0725 × 3

1+ 1+
4 4
3, 000, 000 3, 000, 000
+
0.0650 × 5 0.0680 × 7
 
1+ 1+
4 4
= $12, 493, 800

The corresponding fixed rate is given by

k k
12, 493, 800 = ×0.9810×100, 000, 000+ [0.9484+0.9249+0.8937]×100, 000, 000
4 2
603
 ⇒ k = 7.6709%
Had the price of the bond been equal to par, and the counterparty had paid
LIBOR, the value of the cash flows would have been
100, 000, 000 × 0.0775 × 0.25 100, 000, 000 × 0.0725 × 0.50
+
0.0775 0.0725 × 3
   
1+ 1+
4 4
100, 000, 000 × 0.065 × 0.5 100, 000, 000 × 0.068 × 0.5
+
0.0650 × 5 0.0680 × 7
   
1+ 1+
4 4
= $11, 382, 938
The corresponding fixed rate is given by
k k
11, 382, 938 = ×0.9810×100, 000, 000+ [0.9484+0.9249+0.8937]×100, 000, 000
4 2
⇒ k = 6.9889%
Thus the fixed rate is higher by 68.20 basis points because the bond is trading at
a discount. Consequently the counterparty pays LIBOR + 68.20 basis points.

Had the bond been at a premium, the analysis would be as follows. Assuming
that the bonds are quoting at 100-24, the clean price of the bond is
24
100 +
1, 000 × 32 = $1, 007.50
100
The accrued interest is $15, and consequently the dirty price is $1,022.50. Thus
the difference between the par value and the dirty price per bond is $22.50, and
for 100,000 bonds, it is $2,250,000. This amount is payable by the counterparty
to the bond holder.

Each of the subsequent cash flows are $3 million, because the semiannual
coupon payment is $30 per bond. The value of these cash flows given the LIBOR
rates and using the market method for computing the discount factors is
3, 000, 000 3, 000, 000
−2, 250, 000 +  +
0.0775 0.0725 × 3

1+ 1+
4 4
3, 000, 000 3, 000, 000
+
0.0650 × 5 0.0680 × 7
 
1+ 1+
4 4
604
 = $8, 993, 800

The corresponding fixed rate is given by

k k
8, 993, 800 = ×0.9810×100, 000, 000+ [0.9484+0.9249+0.8937]×100, 000, 000
4 2
⇒ k = 5.5220%

Had the bond been at trading at par, the corresponding fixed rate would have
been 6.9889%. Thus the fixed rate is lower by 1.4669%, and the counterparty
pays LIBOR - 1.4669%.

Equivalence with FRAs

We now show that a position in an interest rate swap is equivalent to a series of

forward rate agreements (FRAs). Consider a portfolio of one-period FRAs paying
after one, two, three, and four periods. Consider the case of an investor who has
taken a long position in the portfolio with a notional principal of $8 million. Let’s
assume that the FRAs are delayed settlement agreements that pay six months after
the LIBOR is determined. This is to be consistent with the payment convention
of an interest rate swap. Figures 14.1 and 14.2 show the interest rate, and state
price trees, for the Ho-Lee model that we derived in Chapter 11.

605
 Figure 14.1
No-arbitrage interest rate tree
t0 t1 t2 t3 t4

11.2863%




HH
 H
 H

6.7725% 11.2863%
H
HH

7.5063%HHH 

 H 
 6.7725%HHH 
9.2863%

HH
7.5000% HH
 H  H
 7.5063%
H 
4.7725% 9.2863%
H
 H
H  H
HH
 H
H H 
HH  H H 
H 
5.5063% 4.7725%
H 
7.2863%
H  H 
HH
7.5000% HH
H H
HH  HH
5.5063%
H 
2.7725% 7.2863%
H
H  H
HH HH
H 
HH 
2.7725%
H 
5.2863%
H 
HH
H
HH
5.2863%
H
H
HH

Figure 14.2
State price tree
t0 t1 t2 t3 t4
0.053160
0.112320
0.232247 0.215741
0.481928 0.340258
1.0 0.466755 0.328336
0.481928 0.343589
0.234508 0.222092
0.115651
0.056336

606
Determining the Fixed Rate

Let’s denote the unknown fixed rate by k. The value of the cash flow at time 1 is

8, 000, 000 × 0.5 × (0.075 − k)

0.075
 
1+
2
The value of the payoff from the FRA at time 2 is

8, 000, 000 × 0.5 × (0.075063 − k) 8, 000, 000 × 0.5 × (0.055063 − k)

0.481928× +0.481928×
0.075063 0.055063
   
1+ 1+
2 2
The value of the payoff from the FRA at time 3 is

8, 000, 000 × 0.5 × (0.067725 − k) 8, 000, 000 × 0.5 × (0.047725 − k)

0.232247× +0.466755×
0.067725 0.047725
   
1+ 1+
2 2
8, 000, 000 × 0.5 × (0.027725 − k)
+0.234508 ×
0.027725
 
1+
2
Finally, the value of the payoff at time 4 is

8, 000, 000 × 0.5 × (0.112863 − k) 8, 000, 000 × 0.5 × (0.092863 − k)

0.112320× +0.340258×
0.112863 0.092863
   
1+ 1+
2 2
8, 000, 000 × 0.5 × (0.072863 − k) 8, 000, 000 × 0.5 × (0.052863 − k)
+0.343589× +0.115651×
0.072863 0.052863
   
1+ 1+
2 2
The unknown fixed rate k should be set such that the value of the portfolio is
zero. Using SOLVER in Excel, we get a value of 6.7485% per annum.

An identical solution can be obtained by computing the coupon rate corre-

sponding to a par bond:
1, 000 × 0.5 × k 1, 000 × 0.5 × k
1, 000 =  + 
0.075 0.07 2
 
1+ 1+
2 2
1, 000 × 0.5 × k [1, 000 + 1, 000 × 0.5 × k]
+  +
0.0625 3 0.0675 4
 
1+ 1+
2 2
k = 6.7485%
607
The equivalence of swaps and FRAs may be demonstrated as follows. The first
payment due from Scotia Bank is known at the very outset because the applicable
LIBOR is determined at the start of the period. Hence the first transaction may
be viewed as a spot transaction in which Scotia Bank receives $300,600 from
Dominion six months hence. This may be computed as

8, 000, 000 × 0.5 × (0.075 − 0.067485) = $300, 600

The remaining three transactions are FRAs in which Scotia Bank agrees to make
payments based on a rate of 6.7485% per annum and receive payments based on
the LIBOR that prevails at the start of the corresponding period.

The par bond can also be valued as the present value of a series of cash flows
based on the implied one-period forward rates. These are
2
(1.035)
= 1.032506 ≡ 3.2506%
1.0375
3
(1.03125)
= 1.023791 ≡ 2.3791%
(1.035)2
(1.03375)4
= 1.041286 ≡ 4.1286%
(1.03125)3
The forward rate for the first period is equal to the spot rate, which is equal to
7.50% per annum. If we assume a face value of 1,000, the cash flows are $37.50
after six months, $32.506 after 12 months, $23.791 after 18 months, and $41.286
after 24 months. If we denote the unknown fixed rate by k
1, 000 × 0.5 × k 1, 000 × 0.5 × k
 + 
0.075 0.07 2
 
1+ 1+
2 2
1, 000 × 0.5 × k [1, 000 + 1, 000 × 0.5 × k]
+  +
0.0625 3 0.0675 4
 
1+ 1+
2 2
37.50 32.506
= +
0.075 0.07 2

1+ 1+
2 2
23.791 [1, 000 + 41.286]
+  +   = 1, 000
0.0625 3 0.0675 4
1+ 1+
2 2
⇒ k = 6.7485%
608
Forward-Start Swaps

A plain vanilla interest rate swap starts at time zero. That is, the first cash
flows arise one period after inception. However, we can design contracts that
are scheduled to start at a later date. Such contracts are termed forward-start
swaps. The contract rate for such a swap will obviously be different from that
for a corresponding plain vanilla swap. However, it can be determined using the
same vector of spot rates.

For instance, let’s consider a three-period (18 months) swap that is sched-
uled to come into existence six months from now. The cash flows based on the
one-period spot rates are $32.506 after 12 months, $23.791 after 18 months, and
$41.286 after 24 months. If we denote the unknown contract rate by k
1, 000 × 0.5 × k 1, 000 × 0.5 × k [1, 000 + 1, 000 × 0.5 × k]
+   +
0.07 2 0.0625 3 0.0675 4
   
1+ 1+ 1+
2 2 2
32.506 23.791 [1, 000 + 41.286]
= 2 +  3 +   ⇒ k = 6.4822%
0.07 0.0625 0.0675 4
1+ 1+ 1+
2 2 2
The same answer can be derived by computing the coupon rate corresponding
to a par bond. For this we need to compute the one-period, two-period, and
three-period forward rates one period from now.

The one-period forward rate, per annum, for a loan after one period is

3.2506 × 2 = 6.5012%

The two-period forward rate, per annum, for a loan after one period is
" 
 (1.03125) 3 1/2
#

−1 ×2
 (1.0375) 

= 5.6278%

The three-period forward rate, per annum, for a loan after one period is
" 
 (1.03375) 4 1/3
#

−1 ×2
 (1.0375) 

609
 = 6.5006%

The coupon rate of the par bond may be calculated as

1, 000 × 0.5 × c 1, 000 × 0.5 × c [1, 000 + 1, 000 × 0.5 × c]
1, 000 = + +
(1.032506) (1.028139) 2 (1.032503) 3
⇒ c = 6.4823%

Amortizing Swaps

In an amortizing swap, the notional principal declines steadily over the life of
the swap. Let’s reconsider the swap between Scotia Bank and Dominion Bank.
Assume that the initial notional principal is $8 million, and that it declines by $2
million at the end of every six-monthly period. The fixed rate may be determined
as follows
[250 + 1, 000 × 0.5 × k] [250 + 750 × 0.5 × k]
1, 000 = +
0.075 0.07 2
   
1+ 1+
2 2
[250 + 500 × 0.5 × k] [250 + 250 × 0.5 × k]
+ +
0.0625 3 0.0675 4
   
1+ 1+
2 2
k = 6.7375%

In-Arrears Swaps

In the case of an in-arrears swap, as soon as the LIBOR is determined at the end
of every period, the interest is paid out immediately. We can determine the fixed
rate for such a swap by viewing it as a portfolio of in-arrears FRAs.

Let’s denote the unknown fixed rate by k. The cash flow at time 1 is

0.481928×[8, 000, 000×0.5×(0.075063−k)]+0.481928×[8, 000, 000×0.5×(0.055063−k)]

The payoff from the FRA at time 2 is

0.232247×[8, 000, 000×0.5×(0.067725−k)]+0.466755×[8, 000, 000×0.5×(0.047725−k)]

610
 +0.234508 × [8, 000, 000 × 0.5 × (0.027725 − k)]

Finally, the payoff at time 3 is

0.112320×[8, 000, 000×0.5×(0.112863−k)]+0.340258×[8, 000, 000×0.5×(0.092863−k)]

+0.343589×[8, 000, 000×0.5×(0.072863−k)]+0.115651×[8, 000, 000×0.5×(0.052863−k)]

The unknown fixed rate k should be set such that the value of the portfolio is
zero. Using SOLVER in Excel, we get a value of 6.5016% per annum.

Extendable and Cancelable Swaps

An extendable swap gives one of the counterparties, usually the fixed-rate payer,
the option to extend the maturity date beyond the scheduled date. From the
perspective of a fixed-rate payer, the option to extend is likely to be invoked in
an economic environment in which interest rates are increasing, in which the cash
inflows due to floating-rate payments are likely to be higher than the counter
payments based on the fixed rate. The extension option needs to be priced and
has a higher fixed rate compared to a plain vanilla interest rate swap.

A cancelable swap gives one of the counterparties the option to terminate the
swap prior to the scheduled maturity date. From the perspective of a fixed-rate
payer, the option to cancel is likely to be invoked in an economic environment in
which interest rates are declining, in which the cash inflows due to floating-rate
payments are likely to be lower than the counter payments based on the fixed
rate. A swap with this option has a higher fixed rate compared to a plain vanilla
swap. A swap that is cancelable by the fixed-rate payer swap is also termed a
callable swap.

Another type of a cancelable swap is a putable swap. This gives the floating-
rate payer the right to terminate the swap prior to the original maturity date.
Obviously a payer chooses to do so when interest rates are rising. Consequently
the fixed rate for a swap with a put option is lower than that of a plain vanilla
swap.

611
Swaptions

A swaption is an option on a swap. Such an option requires the buyer to pay

an up-front premium, for the right to enter into a coupon swap. There are two
types of swaptions. The holder of the option may enter into a coupon swap as a
fixed-rate payer or as a fixed-rate receiver. In the case of a payer swaption, the
option holder can exercise it to enter into the swap as a fixed-rate payer. On the
other hand, a receiver swaption gives the holder the right to enter into a swap as
a fixed-rate receiver.

The exercise price specified in the swaption is an interest rate. The underlying
asset is a swap with a specified term to maturity. A payer swaption is exercised
only if the prevailing rate for a swap with the specified maturity is higher than the
exercise price of the swaption. Quite obviously, a receiver swaption is exercised
only if the prevailing swap rate is lower than the exercise price. Swaptions may
be European or American in nature.

Swaptions may be useful for entities that are not sure whether they will face
interest rate exposure in the future. Consider the case of a corporation that
decides to borrow in the future. It obviously worries about the specter of a rising
interest rate. It can protect itself by buying a payer swaption. If rates go up,
it borrows at the market rate and exercises the option, which results in receipt
of a series of positive cash flows because the market rates are higher than the
swap rate. However, if rates decline, it refrains from exercising the option and
borrows at the market rate, which by assumption has declined and consequently
is favorable to it.

An asset holder, such as a bank that holds a portfolio of mortgages, will be

perturbed about falling interest rates, which are likely to lead to substantial pre-
payments. Such an entity can protect itself by buying a receiver option. If rates go
up, the protection is not required, and the option is allowed to expire. However, if
rates decline, the bank exercises the swaption and receives a series of positive cash
flows that help mitigate the effect of prepayments from the mortgage holders.

Issuers of callable bonds can also use swaptions. Assume that a company has
issued 10-year deferred callable bonds with a 5-year call protection period. Let’s

612
assume that the coupon rate is 7.50% per annum. At the end of two years, rates
decline substantially, to say 4% per annum. Given a choice, the issuer wants to call
back the bonds. However, it is constrained by the call protection feature. In such
circumstances, it can sell a receiver option with a strike price of 7.50%, for which
it obviously receives a premium. Three years hence if interest rates are more than
7.50%, the swaption is not exercised, and the bonds are not recalled. However,
if the rate falls to say 5%, the counterparty does exercise the option. The issuer
of callables receives LIBOR in return for a fixed-rate payment of 7.50%. It can
then call back the bonds and issue fresh floating rate bonds carrying an interest
of LIBOR. The funds required to pay the coupon come from the counterparty
to the swaption. Thus the consequence of the transaction is that the company
continues to pay 7.50% for the remaining life of the bond. At the outset, however,
it receives a premium due to the sale of the option. It is as if it has sold the call
option inherent in the callable bond, although the inherent option per se is not a
tradable asset. Consequently, this strategy is termed call monetization.

It is not necessary that the company sell a swaption with an exercise price
equal to the coupon of 7.50%. It can choose to sell a swaption with a lower or a
higher exercise price. If it chooses a lower exercise price of, say, 6.00%, and the
rate after two years is 4%, the counterparty will exercise. The effective rate on its
debt is 6%. The premium received for the receiver swaption is lower in this case
because the exercise price of 6% is lower than the exercise price of 7.50% that was
assumed earlier. If the company sells a swaption with a higher exercise price of,
say, 9%, it has to pay a higher rate on its debt, but receives a higher premium at
the outset.

In practice, the swap rate and the bond rate need not be perfectly correlated.
Assume that the company has sold a receiver swaption with an exercise price of
7.50%. The LIBOR is assumed to be 4%. The holder pays the LIBOR in return
for a fixed rate of 7.50%. However, the credit quality of the bond issuer may have
deteriorated, and as a consequence, it has to pay LIBOR plus 1.50% if it issues
floating rate bonds to refinance its debt. The net result is that the rate for the
issuer is 9% per annum.

613
Credit Default Swaps

A credit default swap (CDS) is a contract between two parties, known as the
protection buyer and the protection seller, respectively. At the outset, the buyer
of protection has to pay a premium to the counterparty. In return it gets protection
against credit risk or default risk pertaining to an issuing entity, or to a specific
security that has been issued by an entity. The security or entity for which
protection has been sought, is known as the reference asset. In practice it may
be a standalone asset, or a basket or portfolio of assets. The protection seller
is required to make a payment if a specified credit event occurs. If the event
does not happen, the seller makes no payment. In the case of the credit event
happening, the seller pays the loss due to the event to the buyer. The contract
then terminates. The purpose of a CDS is to transfer the credit risk pertaining
to an asset without transferring its ownership.

The specified credit event may be one of the following:

• Bankruptcy or insolvency

• Default on a payment

• Decline in the price of a specified asset by more than a certain prespecified

amount

• Credit downgrade of a security or an issuing entity

If a credit event occurs, it leads to one of the following situations. The pro-
tection seller determines the post-default value, or in other words, the percentage
of the distressed asset’s value that can be recovered. The par value minus this
post-default value is then paid to the protection buyer. There also is something
called a digital swap. In this case, the post-default price or recovery value is a
prespecified amount. This mode of settlement is termed cash settlement. Finally,
the seller may pay the entire par value to the buyer and take over the security
in return. This is referred to as delivery settlement. Here is an example of cash
settlement.

614
 Example 14.3

Arizona Bank has bought a CDS from Xylo Bank. The notional principal is
$25 million. Every year the buyer has to pay a premium of 80 basis points to
the seller. Thus the premium in this case is

25, 000, 000 × 0.008 = $200, 000

If there is a credit event, and assuming that the recovery rate is 60% of par,
the seller pays

(100 − 60)
25, 000, 000 × = $10, 000, 000
100
In the absence of the event, nothing is payable, and the seller retains the
premiums.

In practice the premiums are paid quarterly or semiannually in arrears, and it

is not always necessary that the credit event conveniently occurs at the beginning
of a period. If a credit event occurs in the middle of a period, the accrued premium
has to be paid. A CDS is a tool that facilitates the trading of credit risk, just
the way other tools facilitate the trading of market risk. It should be noted that
the maturity of the swap contract does not need to match the maturity of the
reference asset, and in practice, it does not in most cases.

If the contract is on an issuing entity and not on a specific issue, the buyer may
have a choice of deliverable assets or what are termed deliverable obligations. In
such a situation, the buyer delivers the cheapest of the assets that are eligible for
delivery. In principle, there is no difference between cash settlement and delivery
settlement from the standpoint of protection against a credit event. However, in
practice, delivery settlement offers a benefit to the seller. If the status of the issuer
or the security improves after the seller has taken possession of the underlying
asset, it may be able to derive a higher recovery value than what was anticipated at
the time of default. In the market, buyers prefer cash settlement. This is because,

615
if they are not in prior possession of the underlying asset, they have to acquire it
in the market. In this case, if there are liquidity issues, the price of acquisition
of the asset may be higher than what the recovery rate warrants. In practice,
physical settlement is commonly used because it facilitates the determination of
the security’s market value. In practice, if there is a credit event, a cooling period
such as a fortnight is provided so that the market stabilizes, before the post default
value is determined. The contracts usually stipulate that if the value cannot be
ascertained after a credit event, the value of a similar asset in terms of maturity
and credit quality can be used.

In the case of a basket CDS, which is based on a portfolio of assets, one

possibility is a first to default swap. As the name suggests, the swap terminates
the moment one of the constituent assets suffers a credit event, unless of course the
CDS itself terminates before any of the component securities experience a credit
event. In the case of such swaps, if the buyer seeks protection for the remaining
constituents of the portfolio, it has to enter into a fresh contract, for the original
contract becomes void the moment a credit event occurs. An alternative is an all
to default swap, which terminates when all the assets in the basket have defaulted,
unless of course the contract itself terminates prior to that. In this case, the buyer
receives compensation for all the defaulting securities.

Valuation of a CDS

A combination of a long position in a risky bond and a long position in a CDS is

equivalent to a long position in a synthetic T-bill. Thus the premium for the CDS
should be approximately equal to the default risk premium inherent in the bond’s
yield. Take the case of a party that short sells a riskless bond with a coupon of
r% and buys a risky bond with a coupon of c%, where c = r + p, where p is the
default risk premium. To keep the issue simple, let’s assume that p is a constant.
If we assume that both bonds are trading at par, the proceeds from the short
sale are adequate to buy the risky bond, and consequently there is neither a net
inflow nor a net outflow. Assume that the investor goes long in a CDS and has
to pay a premium of s% every period. At the end of every coupon period, there
is an inflow of c = r + p from the risky bond and a outflow of r. There also is an

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outflow on account of the swap premium. The net cash flow is c-(r+s).

On the coupon date, if the risky bond does not default, the holder can sell
it and retire the short position in the riskless bond. However, if the risky bond
defaults, the holder can deliver it under the swap in return for the par value if it
is delivery settled, or sell it at the prevailing price, receive the deficit under the
swap, and use the proceeds to retire the short position. In either case the net
cash flow is zero. Because the cash flows at the outset and at the end are zero,
all intermediate cash flows should be zero to rule out arbitrage. Thus

c − (r + s) = 0 ⇒ c = r + s = r + p ⇒ s = p

This confirms our claim that the swap premium should be equal to the default
risk premium. There are some inherent assumptions in this argument. First, the
credit spread p is a constant. Second, the bonds are floating rate issues, which
reset to par on every coupon date. And as usual, we assume that there are no
market imperfections such as transaction costs, and if an asset is short-sold, the
full proceeds are immediately available for use.

Using Default Probabilities to Determine the Swap Rate

Consider an eight-year swap. Every period has a 2.50% probability of default,

assuming that there is no earlier default. We assume that the swap premium is
paid in arrears at the end of each year. If a default occurs, it happens at the middle
of the period. The recovery rate is assumed to be 60%. We denote the unknown
swap premium by s per dollar of notional principal. The survival probabilities are
given in Table 14.9.

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 Table 14.9
Default and survival probabilities

Time Default Survival

Probability Probability
1 0.025000 0.975000
2 0.024375 0.950625
3 0.023766 0.926859
4 0.023171 0.903688
5 0.022592 0.881096
6 0.022027 0.859069
7 0.021477 0.837592
8 0.020940 0.816652

The survival probability at the end of the first year is one minus the default
probability, which is 1.00 - 0.025 = 0.975. The default probability for the second
year is 0.025 × 0.975000 = 0.024375. Thus the probability of survival after two
years is 0.975000 − 0.024375 = 0.950625. In general the survival probability at
the end of period t is (1.000 − 0.025)t . The default probability for period t is the
survival probability at the end of the previous period multiplied by the default
intensity of 2.50%.

Assume that the discount rate is 6% per annum. Thus the discount factor for
a cash flow occurring at time t is
1
(1.06)t

At the end of every period, if there has been no prior default, a premium of s
is payable per dollar. Hence, the expected payoff at the end of the year is the
corresponding survival probability multiplied by s.

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 Table 14.10
Expected premium payments and their present values

Time Survival Expected Discount PV of Expected

Probability Premium Pmt. Factor Premium Pmt.
1 0.975000 0.975000s 0.943396 0.919811s
2 0.950625 0.950625s 0.889996 0.846052s
3 0.926859 0.926859s 0.839619 0.778208s
4 0.903688 0.903688s 0.792093 0.715805s
5 0.881096 0.881096s 0.747258 0.658406s
6 0.859069 0.859069s 0.704960 0.605609s
7 0.837592 0.837592s 0.665057 0.557046s
8 0.816652 0.816652s 0.627412 0.512377s
Total 5.593314s
Pmt. stands for Payment.

Table 14.11
Expected accrued premium payments and expected payoffs and their present values

Time Default Expected Discount PV of Expected PV of Exp.

Probability Payoff Factor Expected Accrual Accrual
Payment Payoff Payment Payment
0.5 0.025000 0.010000 0.971286 0.009713 0.012500s 0.012141s
1.5 0.024375 0.009750 0.916307 0.008934 0.012188s 0.011168s
2.5 0.023766 0.009506 0.864441 0.008217 0.011883s 0.010272s
3.5 0.023171 0.009268 0.815510 0.007558 0.011586s 0.009448s
4.5 0.022592 0.009037 0.769349 0.006953 0.011296s 0.008691s
5.5 0.022027 0.008811 0.725801 0.006395 0.011014s 0.007994s
6.5 0.021477 0.008591 0.684718 0.005882 0.010739s 0.007353s
7.5 0.020940 0.008376 0.645960 0.005411 0.010470s 0.006763s
Total 0.059063 0.073830s
Exp. stands for Expected.

The present value of the expected swap payment is 5.593314s + 0.073830s =

619
5.667144s. The break-even CDS spread is given by

5.667144s = 0.059063 ⇒ s = 0.010422 ≡ 104 basis points

The expected accrual premium at time t is the default probability multiplied by

half the premium, as default is assumed to occur in the middle of the year. The
expected payoff from the swap is the default probability multiplied by 0.40 because
the recovery rate is assumed to be 60%.

Chapter Summary

This chapter looked at two important OTC derivatives, namely interest rate swaps
and credit default swaps. We demonstrated the valuation of an interest rate swap
given a vector of spot rates. We showed that it is equivalent to a combination of a
fixed rate bond and a floating rate bond. We also illustrated that an interest rate
swap is equivalent to a portfolio of FRAs and showed how to compute the swap
rate given a model for determining the short rates such as the Ho-Lee model. The
alternatives to a plain vanilla interest rate swap, such as a forward-start swap,
an amortizing swap, and an in-arrears swap, were also discussed. It was shown
that swaps could be used for both hedging and speculation, in conjunction with
both assets and liabilities. While discussing the possible motives for a swap, the
concept of credit arbitrage was discussed in detail. The chapter concluded with
a study of credit default swaps. We showed how to determine the swap premium
for such contracts by assuming survival probabilities and a recovery rate.

With this final chapter, the book is concluded. This book seeks to provide
a comprehensive treatment of bonds, bond valuation, and yield computation.
Derivatives based on interest rate products are also extensively covered. To fa-
cilitate the comprehension, two detailed chapters, on time value of money, and
derivatives, are presented. At every appropriate step, the use of Excel for solving
problems is demonstrated in adequate detail.

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