SWAPS
Swaps are a form of derivative
instruments. Out of the variety of
assets underlying swaps we will
cover:
INTEREST RATES SWAPS,
CURRENCY SWAPS, and
COMMODITY SWAPS.
We will also see that a combination of
hedging with futures and swapping
the basis, leads to risk-free
strategies.
1
SWAPS
A SWAP is a contractual
arrangement between two
parties for an
exchange of cash flows.
The amounts of money
involved are based on a
NOTIONAL AMOUNT OF
CAPITAL
Notional as in conceptual
2
It follows that in a swap
we have:
1. Two parties
2. A notional amount
3. Cash flows
4. A payment schedule
5. An agreement as to
how to resolve
problems
3
1. Two parties:
The two parties in a swap
are sometimes labeled as
party
and counterparty.
They may arrange the swap
directly or indirectly.
In the latter case, there are
two swaps, each between
one of the parties and the
intermediary.
4
2. The NOTIONAL AMOUNT
is
the basis for the
determination of the cash
flows. It is almost never
exchanged by the parties.
For example:
$100,000,000
£50,000,000
50,000 barrels of crude oil
5
3. The cash flows
may be of two types:
a fixed cash flow
or
a floating cash flow.
Fixed interest rate
vs.
Floating interest rate
Fixed price
Vs.
Market price
6
3. The cash flows
The interest rates, fixed or
floating, multiply the
notional amount in order to
determine the cash flows.
Ex: ($10M)(.07)=$700,000; Fixed.
($10M)(Lt+30bps); Floating.
The price, fixed or market,
multiply the commodity
notional amount in order to
determine the cash flows.
Ex: (100,000bbls)($24,75) =
$2,475,000; Fixed.
(100,000bbls)(St ); Floating.
7
4. The payments
are always net.
The agreement determines
the cash flows timing as
annual, semiannual or
monthly, etc. Every
payment is the net of the
two cash flows
8
5. How to resolve problems:
Swaps are Over The Counter
(OTC) agreements. Therefore,
the two parties always face
credit risk
operational risk, etc.
Moreover, liquidity issues such
as getting out of the
agreement, default
possiblilities, selling one side
of the contract, etc., are
frequently encountered
problems.
9
The goals of entering
a swap are:
1. Cost saving.
2. Changing the nature of
cash flow each party
receives or pays from fixed
to floating and vice versa.
10
1. INTEREST RATE
SWAPS
Example: Plain Vanilla
Fixed for Floating rates swap
A swap is to begin in two weeks.
Party A will pay a fixed rate 7.19% per
annum on a semi-annual basis, and will
receive the floating rate: six-month
LIBOR + 30bps from from Party B. The
notional principal is $35million. The
swap is for five years.
Two weeks later, the six-month LIBOR
rate is 6.45% per annum.
11
The fixed rate in a swap is usually quoted
on a
semi-annual bond equivalent yield
basis. Therefore, the amount that is paid
every six months is:
 Notional   Days in   Fixed Rate 




100

amount   Period  
(182) 7.19
 $35,000,000
365 100
 $1,254,802.74.
This calculation is based on the
assumption that the payment is every
182 days.
12
The floating side is quoted as a
money market yield
basis. Therefore, the first payment
is:
 Notional   Days in   Floating Rate 




100

amount   Period  
(182) (6.45  .30)
 $35,000,000
360
100
 $1,194,375.
Other future payments will be
determined every 6 months by the
six-month LIBOR at that time.
13
FIXED 7.19%
Party
A
FLOATING
Party
B
LIBOR 30 bps
As in any SWAP, the payments
are
netted.
In this case, the first payment is:
Party A pays Party B the net difference:
$1,254,802.74 - $1,194,375.00
= $60,427.74.
14
This example illustrates five points:
1. In interest rate swaps, payments
are netted. In the example, Party
A sent Party B a payment for the
net amount.
2. In an interest rate swap, principal
is not exchanged. This is why the
term “notional principal” is used.
3. Party A is exposed to the risk that
Party B might default. Conversely,
Party B is exposed to the risk of
Party A defaulting. If one party
defaults, the swap usually
terminates.
15
4. On the fixed payment side, a 365day year is used, while on the
floating payment side, a 360-day
year is used. The number of days
in the year is one of the issues
specified in the swap contract.
5. Future payments are not known in
advance, because they depend on
future realizations of the Sixmonth LIBOR.
Estimates of future LIBOR values
are obtained from LIBOR yield
curves which are based on Euro
Strip of Euro dollar futures strips.
16
Example:
A FIXED FOR FLOATING SWAP
Two firms need financing for projects and
are facing the following interest rates:
PARTY
FIXED RATE FLOATING RATE
F1 :
15%
LIBOR + 2%
F2 :
12%
LIBOR + 1%
F2 HAS ABSOLUTE ADVANTAGE
in both markets, but F2 has
RELATIVE ADVANTAGE only in the
market for fixed rates. WHY?
The difference between what F1
pays more than F2 in floating rates,
(1%), is less than the difference
between what F1 pays more than F2
in fixed rates, (3%).
17
Now, suppose that the firms decide
to enter a FIXED for FLOATING
swap based on the notional of
$10.000.000.
The payments:
Annual payments to be made on
the first business day in March for
the next five years.
18
The SWAP always begins
with each party borrowing capital in the
market in which it has a
RELATIVE ADVANTAGE.
Thus, F1 borrows S $10,000,000
in the market for floating rates, I.e., for
LIBOR + 2% for 5 years.
F2 borrows $10,000,000
in the market for fixed rates, I.e.,
for 12%.
NOW THE TWO PARTIES
EXCHANGE THE TYPE OF CASH
FLOWS BY ENTERING THE SWAP
19
FOR FIVE YEARS
A fundamental implicit
assumption:
The swap will take place
only if
F1 wishes to borrow capital
for a FIXED RATE, While
F2 wishes to borrow capital
for a FLOATING RATE.
That is, both firms want to change the
nature of their payments.
20
Two ways to negotiate the
contract:
1. Direct negotiations
between the two parties.
2. Indirect negotiations
between the two parties.
In this case each party
separately negotiates with
an intermediary party.
21
Usually,
The intermediary is a
financial institution – a
swap dealer - who
possesses a portfolio of
swaps.
The intermediary charges
both parties commission
for its services and also as
a compensation for the risk
it assumes by entering the
two swaps
22
FIXED FOR FLOATING SWAP
1. A DIRECT SWAP:
FIRM FIXED RATE
FLOATING RATE
F1
15%
LIBOR + 2%
F2
12%
LIBOR + 1%
notional: $10M
LIBOR
12%
F1
F2
LIBOR+2%
12%
The result of the swap:
F1 pays fixed 14%,
better than 15%.
F2 pays floating LIBOR, better than
LIBOR + 1%
23
2. AN INDIRECT SWAP
FIRM
FIXED RATE FLOATING RATE
F1
15%
LIBOR + 2%
F2
12%
LIBOR + 1%
The notional amount: $10M
L+25bps
12%
F2
L
F1
I
12%
L + 2%
12,25%
F1 pays 14,25% fixed: Better than 15%.
F2 pays L+25bps : Better than L+1%.
The Intermediary gains 50 bps = $50,000.
24
Notice that the two swaps presented above
are two possible contractual agreements.
The direct, as well as the indirect swaps,
may end up differently, depending on the
negotiation power of the parties involved.
Nowadays, it is very probable for
intermediaries to be happy with 10 basis
points. In the present example, another
possible swap arrangement is:
12%
L+5bp
L
I
F2
12%
F1
L+2%
12%+5bp
Clearly, there exist many other possible
swaps between the two firms in this
example.
25
Warehousing
In practice, a swap dealer
intermediating (making a market in)
swaps may not be able to find an
immediate off-setting swap. Most
dealers will warehouse the swap and
use interest rate derivatives to hedge
their risk exposure until they can find
an off-setting swap. In practice, it is
not always possible to find a second
swap with the same maturity and
notional principal as the first swap,
implying that the institution making a
market in swaps has a residual
exposure. The relatively narrow bid/ask
spread in the interest rate swap market
implies that to make a profit, effective
interest rate risk management is
essential.
26
EXAMPLE: A RISK MANAGEMENT
SWAP
MARKET
FL1 = Floating rate 1.
BONDS
LOAN
FL2 = Floating rate 2.
FL1
10%
COUNTERPARTY
BANK
A
FL2
LOAN
12%
FIRM A
BORROWS AT
A FIXED RATE
FOR 5 YEARS
27
THE BANK’S CASH FLOW:
12% - FLOATING1 + FLOATING2 – 10%
= 2% + SPREAD
SPREAD = FLOATING2 - FLOATING1
RESULTS
THE BANK EXCHANGES THE RISK
ASSOCIATED WITH THE DIFFERENCE
BETWEEN FLOATING1 and 12% WITH
THE RISK ASSOCIATED WITH THE
SPREAD
= FLOATING2 - FLOATING1.
The bank may decide to swap the
SPREAD for fixed, risk-free cash flows.
28
EXAMPLE: A RISK MANAGEMENT
SWAP
MARKET
SHORT
TERM BOND
FL1
10%
BANK
FL2
COUNETRPARTY
a
FL2
FL1
12%
FIRM A
COUNTERPARTY
b
29
THE BANK’S CASH FLOW:
12% - FL1 + FL2 – 10% + (FL1 - FL2 )
= 2%
RESULTS
THE BANK EXCHANGES THE RISK
ASSOCIATED WITH THE
SPREAD = FL2 - FL1
WITH A FIXED RATE OF 2%.
THIS RATE IS A
RISK-FREE RATE!
30
VALUATION OF SWAPS
The swap coupons (payments) for
short-dated fixed-for-floating
interest rate swaps are routinely
priced off the Eurodollar futures
strip (Euro strip). This pricing
method works provided that:
(1) Eurodollar futures exist.
(2) The futures are liquid.
As of June 1992, three-month
Eurodollar futures are traded in
quarterly cycles - March, June,
September, and December - with
delivery (final settlement) dates
as far forward as five years. Most
times, however, they are only
liquid out to about four years,
thereby somewhat limiting the
use of this method.
31
The Euro strip is a series of
successive three-month Eurodollar
futures contracts.
While identical contracts trade on
different futures exchanges, the
International Monetary Market (IMM)
is the most widely used. It is worth
mentioning that the Eurodollar
futures are the most heavily traded
futures anywhere in the world. This
is partly as a consequence of swap
dealers' transactions in these
markets. Swap dealers synthesize
short-dated swaps to hedge
unmatched swap books and/or to
arbitrage between real and synthetic
swaps.
32
Eurodollar futures provide a way to
do that. The prices of these
futures imply unbiased estimates
of three-month LIBOR expected to
prevail at various points in the
future. Thus, they are conveniently
used as estimated rates for the
floating cash flows of the swap.
The swap fixed coupon that
equates the present value of the
fixed leg with the present value of
the floating leg based on these
unbiased estimates of future
values of LIBOR is then the
dealer’s
mid rate.
33
The estimation of a “fair” mid rate is
complicated a bit by the facts
that:
(1) The convention is to quote swap
coupons for generic swaps on a
semiannual bond basis, and
(2) The floating leg, if pegged to
LIBOR, is usually quoted on a
money market basis.
Note that on very short-dated swaps
the swap coupon is often quoted
on a money market basis. For
consistency, however, we
assume throughout that the
swap coupon is quoted on a
bond basis.
34
The procedure by which the dealer
would obtain an unbiased mid rate
for pricing the swap coupon
involves three steps.
The first step: Use the implied
three-month LIBOR rates from the
Euro strip to obtain the implied
annual effective LIBOR for the fulltenor of the swap.
The second step: Convert this fulltenor LIBOR to an effective rate
quoted on an annual bond basis.
The third step: Restate this
effective bond basis rate on the
actual payment frequency of the
swap.
35
NOTATIONS:
Let the swap have a
tenor of m months (m/12 years). The
swap is to be priced off three-month
Eurodollar futures, thus, pricing requires n
sequential futures series; n = m/3 or,
equivalently, m = 3n.
Step 1: Use the futures Euro strip to
Calculate the implied effective annual
LIBOR for the full tenor of the swap:
k
N(t) 

r0,3n    [1  (r3(t-1),3(t)
)]   1,
360 
 t 1
360
where : k 
; N(t) denotes
 N(t)
n
the actual number of days covered
by the t - th Eurodolla r futures.
36
N(t) is the total number of days covered
by the swap, which is equal to the sum of
the actual number of days in the
succession of Eurodollar futures.
Step 2: Convert the full-tenor LIBOR,
which is quoted on a money market basis,
to its fixed-rate equivalent FRE(0,3n),
which is stated as an effective annual rate
on an annual bond basis. This simply
reflects the different number of days
underlying bond basis and money market
basis:
365
FRE(0,3n)  r0,3n
.
360
37
Step 3: Restate the fixed-rate on the
same payment frequency as the floating
leg of the swap. The result is the swap
coupon, SC.
Let f denote the payment frequency, then
the coupon swap is given by:
SC  {[1  FRE(0,3n)]
1
f
- 1}(f),
which, upon substituti on of
FRE(0,3n), can be rewritten as :
1
f
365
SC  {[1  r0,3n
]  1}(f).
360
38
Example: For illustration purposes let
us observe Eurodollar futures settlement
prices on April 24, 2001.
Eurodollar Futures Settlement Prices
April 24,2001.
CONTRACT PRICE LIBOR FORWARD DAYS
JUN01
95.88
4.12
0,3
92
SEP91
95.94
4.06
3,6
91
DEC91
95.69
4.31
6,9
90
MAR92
95.49
4.51
9,12
92
JUN92
95.18
4.82
12,15
92
SEP92
94.92
5.08
15,18
91
DEC92
94.64
5.36
18,21
91
MAR93
94.52
5.48
21,24
92
JUN93
94.36
5.64
24,27
92
SEP93
94.26
5.74
27,30
91
DEC93
94.11
5.89
30,33
90
MAR94
94.10
5.90
33,36
92
JUN94
94.02
5.98
36,39
92
39
SEP94
93.95
6.05
39,42
91
These contracts imply the three-month
LIBOR (3-M LIBOR) rates expected to
prevail at the time of the Eurodollar
futures contracts’ final settlement, which
is the third Wednesday of the contract
month. By convention, the implied rate
for three-month LIBOR is found by
deducting the price of the contract from
100. Three-month LIBOR for JUN 91 is a
spot rate, but all the others are forward
rates implied by the Eurodollar futures
price. Thus, the contracts imply the 3-M
LIBOR expected to prevail three months
forward, (3,6) the 3-M LIBOR expected to
prevail six months forward, (6,9), and so
on. The first number indicates the month
of commencement (i.e., the month that
the underlying Eurodollar deposit is lent)
and the second number indicates the
month of maturity (i.e., the month that
the underlying Eurodollar deposit is
repaid). Both dates are measured in
months forward.
40
In summary, the spot 3-M LIBOR is
denoted r 0,3 , the corresponding forward
rates are denoted r3,6, r6,9, and so on.
Under the FORWARD column, the first
month represents the starting month and
the second month represents the ending
month, both referenced from the current
month, JUNE, which is treated as month
zero.
Eurodollar futures contracts assume a
deposit of 91 days even though any
actual three-month period may have as
few as 90 days and as many as 92 days.
For purposes of pricing swaps, the actual
number of days in a three-month period
is used in lieu of the 91 days assumed by
the futures. This may introduce a very
small discrepancy between the
performance of a real swap and the
performance of a synthetic swap created
from a Euro strip.
41
Suppose that we want to price a oneyear fixed-for-floating interest rate
swap against 3-M LIBOR. The fixed rate
will be paid quarterly and, therefore, is
quoted quarterly on bond basis. We need
to find the fixed rate that has the same
present value (in an expected value sense)
as four successive 3-M LIBOR payments.
Step 1: The one-year implied LIBOR rate,
based on k =360/365, m = 12, n = 4
and f=4 is:
r0,3n
k
N(t) 

   [1  (r3(t-1),3(t)
)]   1
360 
 t 1
n
92
91 

)(1  .0406
)
 (1  .0412
360
360 


90
92 

)(1  .0451
)
 (1  .0431
360
360 

 4.34%, on money market basis.
360
365
1
42
Step 2 and 3:
1
f
SC  {[1  FRE(0,3n)] - 1}(f),
which, upon substituti on of
FRE(0,3n), can be rewritten as :
1
f
365
SC  {[1  r0,3n
]  1}(f)
360
1
365 4
 {[1  .0434
]  1}(4)
360
 4.33% on a quarterly bond basis.
The swap’s coupon is the dealer mid rate.
To this rate , the dealer will add several basis
points.
43
4.33%+s
FIXED
Client
7.19
3-M %
LIBOR
Swap
dealer
LIBOR
FLOATING
+ 30
In this swap, four net payments will take
place during the one year tenure of the
swap depending the three-month
LIBOR realizations.
This completes the example.
Next, suppose that the swap is for
semiannual payments against 6-month
LIBOR.
The first two steps are the same as in the
previous example. Step 3 is different
because f = 2, instead of 4.
44
1
2
365
SC  [1  (.0434
)  1)]( 2);
360
SC  4.35%, on a semiannual
bond basis.
4.35%+s
FIXED
Swap
dealer
Client
6-M LIBOR
FLOATING
45
The procedure above allows a dealer to
quote swaps having tenors out to the
limit of the liquidity of Eurodollar futures
on any payment frequency desired and to
fully hedge those swaps in the Euro Strip.
The latter is accomplished by purchasing
the components of the Euro Strip to
hedge a dealer-pays-fixed-rate swap or,
selling the components of the Euro Strip
to hedge a dealer-pays-floating-rate
swap.
Example: Suppose that a dealer wants
to price a three-year swap with a
semiannual coupon when the floating leg
is six-month LIBOR. Three years: m=36
months requiring 12 separate Eurodollar
futures; n = 12. Further, f = 2 and the
actual number of days covered by the
swap is N(t) = 1096.
Step 1: The implied LIBOR rate for the
entire period of the swap:
46
N(t) 
 12
r0,36    [1  (r3(t-1),3(t)
)] 
360 
 t 1
360
1096
1
92
91
90 

(1

.0412
)(1

.0406
)(1

.0431
)

360
360
360 

92
92
91 

(1

.0451
)(1

.0482
)(1

.0508
)

360
360
360


 (1  .0536 91 )(1  .0548 92 )(1  .0564 92 ) 

360
360
360 


91
90
92
 (1  .0574
)(1  .0589
)(1  .0590
)
360
360
360 

 5.17%, on money market basis.
360
1096
1
Step 2: The Fixed Rate Equivalent
effective annual rate on a bond basis is:
FRE = (5.17%)(365/360) = 5.24%.
47
Finally,
Step 3: The equivalent semiannual
Swap Coupon is calculated:
SC = [(1.0524).5 – 1](2) = 5.17%.
The dealer can hedge the swap by buying
or selling, as appropriate, the 12 futures in
the Euro Strip.
The full set of fixed-rate for 6-M LIBOR
swap tenors out to three and one-half
years, having semiannual payments, that
can be created from the Euro Strip are
listed in the table below. The swap fixed
coupon represents the dealer's mid
rate. To this mid rate, the dealer can be
expected to add several basis points if
fixed-rate receiver, and deduct several
basis points if fixed-rate payer. The par
swap yield curve out to three and one-half
years still needs more points.
48
Implied Swap Pricing Schedule
Out To Three and One-half Years
as of April 24,2001*
Tenor of swap Swap coupon mid rate
6
12
4.35%
18
24
30
36
5.17%
42
* All swaps above are priced against 6-month
LIBOR flat and assume that the notional principal
is non amortizing.
49
Swap Valuation
The example below illustrates the valuation
of an interest rate swap, given the coupon
payments are known. Consider a financial
institution that receives fixed payments at
the annual rate 7.15% and pays floating
payments in a two-year swap. Payments are
made every six months. The data are:
Payments
Days
Treasury
dates
between
Bills
payment
Prices
Dates
B(0,T)
t1 = 182
t2 = 365
t3 = 548
t4 = 730
182
183
183
182
.9679
.9362
.9052
.8749
Euro
Dollar
Deposit
L(0,T)
.9669
.9338
.9010
.8684
B(0,T)=PV of $1.00 paid at T.L(0,T)=PV of 1Euro$
paid at T. These prices are respectively, derived from
the Treasury and Eurodollar term structures. 50
The fixed side of the swap.
At the first payment date, t1, the
dollar value of the payment is:
182
VFIXED (t 1 , t1 )  N P (.0715)
,
365
where NP denotes the notional
principal.
The present value of receiving one
dollar for sure at date t1, is
0.9679. Therefore, the present
value of the first fixed swap
payment is:
VFIXED (0, t1 )  [.9679]VR (t1, t1 ).
51
By repeating, this analysis, the
present value of all fixed
payments is:
VFIXED(0)
= NP[(.9679)(.0715)(182/365)
+ (.9362)(.0715)(183/365)
+ (.9052)(.0715)(183/365)
+ (.8749)(.0715)(182/365)]
= NP[.1317].
This completes the fixed payment of the
swap.
52
On the floating side of the swap, the
pattern of payments is similar to that of a
floating rate bond, with the important
proviso that there is no principal
payment in a swap. Thus, when the
interest rate is set, the bond sells at par
value. Given that there is no principal
payment, we must subtract the present
value of principal from the principal itself.
The present value of the floating rate
payments depends on L(0, t4) - the
present value of receiving one Eurodollar
at date t4:
VFLOATING (0)  N P  N P [L(0, t 4 )]
 N P [1  .8684]
 (.1316)N P .
53
The value of the swap to the
financial institution is:
Value of Swap
= VFIXED(0) - VFLOATING(0)
= NP[.1317 - .1316]
= (.0001)NP.
If the notional principal is $45M, the value
of the swap is $4,500.
In this example, the Treasury bond prices
are used to discount the cash flows based
on the Treasury note rate. The Eurodollar
discount factors are used to measure the
present value of the LIBOR cash flows. This
practice incorporates the different risks
implicit in these different cash flow streams.
This completes the example.
54
SWAP VALUATION:
The general formula
To generalize the above example, we
replace algebraic symbols for the
numbers.
Consider a swap in which there are n
payments occurring on dates Tj, where
the number of days between payments is
kj, j = 1,…, n. Let R be the swap rate,
expressed as a percent; NP represents the
notional principal; and B(0,Tj) is the
present value of receiving one dollar for
sure at date Tj.
The value of the fixed payments is:
n
R kj
VFIXED (0)  N P {B(0, Tj )[ ][ ]}.
100 365
j1
55
Arriving at the value of the floating
rate payments requires more analysis.
1. If the swap is already in existence,
let λ denote the pre specified LIBOR
rate. At date T1, the payment is:
k1
N P [λ
]
360
and a new LIBOR rate is set.
On T1, the value of the remaining
floating rate payments is:
NP – NP{L(T1, TN)}.
where L(T1, TN) is the present value at
date T1 of a Eurodollar deposit that
pays one dollar at date Tn.
We are now ready to calculate the total
value of the floating rate payments at
date T1.
56
The total value of the floating rate
payments at date T1 is:
k1
VFLOATING (T1 )  N Pλ
360
 N P  N P L(T1 , Tn ).
The value of the floating rate payments
at date 0 is the PV of:
VFLOATING (T1 ) :
 k1

VFLOATING (0)  N P  λ
 1 L(0, T1 )
 360 
- N P L(0, Tn ).
This holds true because
L(T1 , Tn )L(0, T1 )  L(0, Tn ).
57
2. If the swap is initiated at date 0,
then the above equation simplifies as
follows:
Let λ(0) denote the current LIBOR rate.
By definition:
L(0, T1 ) 
1
and because
T1
1  λ(0)
360
k1  T1 , the value of the floating
rate payments is :
T1


VFLOATING (0)  N P  λ(0)
 1 L(0, T1 )
360 

- N P L(0, Tn ).
VFLOATING (0)  N P [1 - L(0, Tn )].
58
IN CONCLUSION: The value of the swap
for the party receiving fixed and paying
floating is the difference between the fixed
and the floating values. For example, the
value of a swap that is initiated at time 0 is:
THE
SWAP VALUE for the
receiving
fixed
and paying
party
floating
is :
VSWAP  VFIXED (0) - VFLOATING (0)
 n

R kj
N P   {B(0, Tj )[ ][
]}  N P [1  L(0, Tn ] .
100 365
 j1

Notice that this value can be positive ,zero,
or negative depending upon current rates.
This conclude the analysis of
plain vanilla
swap valuation.
59
PAR SWAPS
A par swap is a swap for which the
present value of the fixed payments equals
the present value of the floating payments,
implying that the net value of the swap is
zero. Equating the value of the fixed
payments and the value of the floating rate
payments yields the FIXED RATE, R, which
makes the swap value zero.
For PAR SWAP :
VFIXED (0)  VFLOATING (0)
 n

R kj
N P  {B(0, Tj )[ ][ ]}  N P [1  L(0, Tn ].
100 365
 j1

60
PAR SWAP Valuation
The example below illustrates the valuation
of an interest rate par swap.
Consider a financial institution that receives
fixed payments at the rate 7.15% per annum
and pays floating payments in a two-year
swap. Payments are made every six months.
The data are:
Payments
Days
Treasury
Euro
dates
between
Bills
Dollar
payment
Prices
Deposit
Dates
B(0,T)
L(0,T)
t1 = 182
182
.9679
.9669
t2 = 365
183
.9362
.9338
t3 = 548
183
.9052
.9010
t4 = 730
182
.8749
.8684
B(0,T)=PV of $1.00 paid at T.L(0,T)=PV of 1Euro$
paid at T. These prices are respectively, derived from
the Treasury and Eurodollar term structures. 61
PAR SWAP VALUATION:
Solve for R, the equation:
[
NP (R/100)(.9679)(182/365)
+ (R/100)(.9362)(183/365)
+ (R/100)(.9052)(183/365)
+ (R/100)(.8749)(182/365)
]
= NP[1 - .8684]
The equality implies:
R/100 = .1316/1.8421
R = 7.14% per annum.
62
2. CURRENCY SWAPS
Nowadays markets are global.
Firms cannot operate with disregard to
international markets trends and prices.
Capital can be transfered from one
country to another rapidly and
efficiently. Therefore, firms may take
advantage of international markets
even if their business is local. For
example, a firm in Denver CO. may find
it cheaper to borrow money in
Germany, exchange it to USD and
repay it later, exchanging USD into
German marks.
Currency swaps are basically, interest
rate swaps accross countries 63
Case Study of a currency swap:
IBM and The World Bank
A famous example of an early currency
Swap took place between IBM an the World
Bank in August 1981, with Salomon Brothers
As the intermediary.
The complete details of the swap have never
been published in full.
The following description follows a paper
published by D.R. Bock in Swap Finance,
Euromoney Publications.
64
In the mid 1970s, IBM had issued bonds in
German marks, DEM, and Swiss francs,
CHF. The bonds maturity date was March
30, 1986. The issued amount of the CHF
bond was CHF200 million, with a coupon
rate of 6 3/16% per annum. The issued
amount of the DEM bond was DEM300
million with a coupon rate of 10% per
annum.
During 1981 the USD appreciated sharply
against both currencies. The DEM, for
example, fell in value from $.5181/DEM in
March 1980 to $.3968/DEM in August 1981.
Thus, coupon payments of DEM100 had
fallen in USD cost from $51.81 to $39.68.
The situation with the Swiss francs was the
Same. Thus, IBM enjoyed a sudden,
unexpected capital gain from the reduced
USD value of its foreign debt liabilities. 65
In the beginning of 1981, The World
Bank wanted to borrow capital in
German marks and Swiss francs against
USD. Around that time, the World Bank
had issued comparatively little USD
paper and could raise funds at an
attractive rate in the U.S. market.
Both parties could benefit from USD for
DEM and CHF swap. The World Bank
would issue a USD bond and swap the $
proceeds with IBM for cash flows in CHF
and DEM.
The bond was issued by the World Bank
on August 11, 1981, settling on August
25, 1981. August 25, 1981 became the
settlement date for the swap. The first
annual payment under the swap was
determined to be on March 30, 1982 – the
next coupon date on IBM's bonds. I.e.,
215 days (rather than 360) from the swap
66
starting date.
The swap was intermediated by Solomon
Brothers.
The first step was to calculate the value of
the CHF and DEM cash flows. At that
time, the annual yields on similar bonds
were at 8% and 11%, respectively.The
initial period of 215-day meant that the
discount factors were calculated as
follows:
Discount F actor 
1
(1  y)
n
360
,
Where: y is the respective bond yield, 8%
for the CHF and 11% for the DEM and n
is the number of days till payment.
67
The discount factors were calculated:
Date
Days
CHF
DEM
3.30.82
215 .9550775
.9395764
3.30.83
575 .8843310
.8464652
3.30.84
935 .8188250
.7625813
3.30.85
1295 .7581813
.6870102
3.30.86
1655 .7020104
.6189281.
Next, the bond values were calculated:
NPV(CHF) =
12,375,000[.9550775 + .8843310
+ .8188250 + .7581813]
+ 212,375,000[.7020104]
= CHF191,367,478.
NPV(DEM)=
30,000,000[.9395764 + .8464652
+.7625813+.6870102]
+330,000,000[.61892811]
= DEM301,315,273.
68
The terms of the swap were agreed upon
on August 11, 1981. Thus, The World Bank
would have been left exposed to currency
risk for two weeks until August 25. The
World Bank decided to hedge the above
derived NPV amounts with 14-days
currency forwards.
Assuming that these forwards were at
$.45872/CHF and $.390625/DEM, The World
Bank needed a total amount of
$205,485,000;
$87,783,247 to buy the CHF and
$117,701,753 to buy the DEM.
$205,485,000.
This amount needed to be divided up to the
various payments. The only problem was
that the first coupon payment was for 215
days, while the other payments were based
on a period of 360 days.
69
Assuming that the bond carried a coupon
rate of 16% per annum with intermediary
commissions and fees totaling 2.15%, the
net proceeds of .9785 per dollar meant
that the USD amount of the bond issue
had to be:
$205,485,000/0.9785 = $210,000,000.
The YTM on the World Bank bond was
16.8%. As mentioned above, the first
coupon payment involved 215 days only.
Therefore, the first coupon payment was
equal to:
$210,000,000(.16)[215/360]
= $20,066,667.
70
The cash flows are summarized in the
following table:
Date
USD
CHF
DEM
3.30.82 20,066,667 12,375,000
3.30.83 33,600,000 12,375,000
3.30.84 33,600,000 12,375,000
3.30.85 33,600,000 12,375,000
3.30.86 243,600,000 212,375,000
YTM
8%
11%
NPV
205,485,000 191,367,478
30,000,000
30,000,000
30,000,000
30,000,000
330,000,000
16.8%
301,315,273
By swapping its foreign interest payment
obligations for USD obligations, IBM was
no longer exposed to currency risk and
could realize the capital gain from the
dollar appreciation immediately. Moreover,
The World Bank obtained Swiss francs
and German marks cheaper than it would
had it gone to the currency markets
directly.
71
Foreign Currency Swaps
EXAMPLE:
a “plain vanilla”
foreign currency swap.
Counterparty F1 has issued bonds with
face value of £50M with a annual coupon
of 11.5%, paid semi-annually and
maturity of seven years.
Counterparty F1 would prefer to have
dollars and to be making interest
payments in dollars. Thus counterparty
F1 enters into a foreign currency swap
with counterparty F2 - usually a financial
institution. In the first phase of the swap,
party F1 exchanges the principal amount
of £50M with party F2 and, in return,
receives principal worth $72.5M. Usually,
this exchange is done in the current
exchange rate, i.e., S = $1.45/£ in this
case.
72
The swap agreement is as follows:
Party F1 agrees to make to counterparty F2
semi – annual interest rate payments at the
rate of 9.35% per annum based on the
Dollar denominated principal for a seven
Year period.
In return, counterparty F1 receives from
party F2 a semi-annual interest rate at the
annual rate of 11.5%, based on the sterling
denominated principal for a seven year
period.
The swap terminates at the maturity seven
years later, when the principals are again
exchanged:
party F1 receives the principal worth £50M
and counterparty F2 receives the principal
Amount of $72.5M.
73
DIRECT SWAP FIXED FOR FIXED
$9.35%
F1
F2
£11.5%
£11.5%
Great Britain
U.S.A
F1 BORROWS
£50M AND
DEPOSITS IT IN
COUNTERPARTY
F2’s ACCOUNT IN
LONDON
F2 DEPOSITS
$72.5M IN
COUNTERPARTY
F1’S ACCOUNT
IN NEW YORK
CITY
At maturity, the original principals are
74
exchanged to terminate the swap.
By entering into the foreign currency
swap, counterparty F1 has successfully
transferred its sterling liability into a
dollar liability.
In this case, party F2 payments to party
F1 were based on the the same rate of
party’s F1 payments in Great Britain £11.5%. Thus, party F1 was able to
exactly offset the sterling interest rate
payments.
This is not necessarily always the case.
It is quite possible that the interest rate
payments counterparty F1 receives from
counterparty F2 only partially offset the
sterling expense.
In the same example, the situation may
change to:
75
DIRECT SWAP FIXED FOR FIXED
$9.55%
F1
F2
£11.25%
£11.5%
Great Britain
U.S.A
F1 BORROWS
£50M AND
DEPOSITS IT IN
COUNTERPARTY
F2’s ACCOUNT IN
LONDON
F2 DEPOSITS
$72.5M IN
COUNTERPARTY
F1’S ACCOUNT
IN NEW YORK
CITY
At maturity, the original principals are
76
exchanged to terminate the swap.
THE ANALYSIS OF
CURRENCY SWAPS
F1 IN COUNTRY A LOOKS FOR
FINANCING IN COUNTRY B
AT THE SAME TIME
F2 IN COUNTRY B, LOOKS FOR
FINANCING IN COUNTRY A
COUNTRY A
COUNTRY B
F1
F2
PROJECT OF
PROJECT OF
F2
F1
77
CURRENCY SWAP
IN TERMS OF THE BORROWING RAES,
EACH FIRM HAS
COMPARATIVE ADVANTAGE
ONLY IN ONE COUNTRY,
EVEN THOUGH IT MAY HAVE
ABSOLUTE ADVANTAGE
IN BOTH COUNTRIES.
THUS, EACH FIRM WILL BORROW IN THE
COUNTRY IN WHICH IT HAS COMPARATIVE
ADVANTAGE AND THEN, THEY EXCHANGE
THE PAYMENTS THROUGH A SWAP.
78
CURRENCY SWAP FIXED FOR FIXED
CP = Chilean Peso
R = Brazilian Real
Firm CH1, is a Chilean firm who
needs capital for a project in Brazil,
while,
A Brazilian firm, BR2, needs capital
for a project in Chile.
The market for fixed interest rates in
these countries makes a swap
beneficial for both firms as follows:
79
FIRM
CHILE
BRAZIL
CH1
$12%
R16%
BR2
$15%
R17%
With these rates, CH1 has absolute
advantage in both markets but,
comparative advantage in Chile only.
CH1 borrows in Chile in Chilean Pesos and
BR2 borrows in Brazil in Reals. The swap
begins with the interchange of the
principal amounts borrowed at the current
exchange rate.
The figures below show a direct swap
between CH1 and BR2 as well as an
indirect swap.
The swap terminates at the end of the
swap period when the original principal
amounts exchange hands once more.
80
ASSUME THAT THE CURRENT
EXCHANGE RATE IS:
R1 = CP250
ASSUME THAT CH1 NEEDS
R10.000.000 FOR ITS PROJECT IN
BRAZIL AND THAT BR2 NEEDS
EXACTLY CP2,5B FOR ITS
PROJECT IN CHILE.
AGAIN:
FIRM
CHILE
BRAZIL
CH1
$12%
R16%
BR2
$15%
R17%
81
DIRECT SWAP FIXED FOR FIXED
R15%
CH1
BR2
$12%
$12%
R17%
CHILE
BRAZIL
CH1 BORROWS
CP2.5B AND
DEPOSITS IT IN
BR2’S
ACCOUNT IN
SANTIAGO
BR2 BORROWS
R10M AND
DEPOSITS IT IN
CH1’S ACCOUNT
IN SAO PAULO
CH1 pays R15%; BR2 pays CP12% + R2%
82
INDIRECT SWAP FIXED FOR FIXED
INTERMEDIARY
R15.50%
$12%
CH1
$12%
$14.50%
R17%
BR2
R17%
CHILE
BRAZIL
CH1 BORROWS
CP2.5B AND
DEPOSITS IT IN
BR2’S
ACCOUNT IN
SANTIAGO
BR2 BORROWS
R10M AND
DEPOSITS IT IN
CH1’S ACCOUNT
IN SAO PAULO
83
THE CASH FLOWS:
CH1:
PAYS R15.50%
BR2:
PAYS CP14.50%
THE INTERMEDIARY REVENUE:
CP2.50 – R1.50%
CP2,5B(0.025) – R10M(0.015)(250)
= CP62,500,000 - CP37,500,000 =
CP25,000,000
Notice: In this case, CH1 saves 0.25% and
BR2 saves 0.25%, while the intermediary
bears the exchange rate risk. If the Chilean
Peso depreciates against the Real the
intermediary’s revenue declines. When the
exchange rate reaches CP466,67/R the
intermediary gain is zero. If the Chilean Peso
continues to depreciate the intermediary
84
loses money on the deal.
FIXED FOR FLOATING
CURRENCY SWAP
A Mexican firm needs capital for a project in
Great Britain and a British firm needs capital
for a project in Mexico. They enter a swap
because they can exchange fixed interest
rates into floating and borrow at rates that
are below the rates they could obtain had
they borrowed directly in the same markets.
In this case, the swap is
Fixed-for-Floating rates,
i.e.,
One firm borrows fixed, the other borrows
floating and they swap the cash flows therby,
changing the nature of the payments from
fixed to floating and vice – versa.
85
DIRECT SWAP FIXED FOR FLOATING
INTEREST RATES
MEXICO
GREAT BRITAIN
MX1
MP15%
£LIBOR + 3%
GB2
MP18%
£LIBOR + 1%
ASSUME: The current exchange rate is:
£1 = MP15.
MX1 needs £5.000.000 in England and
GB2 needs MP75.000.000 in Mexico.
THUS:
MX1 borrows MP75m in Mexico and deposits
it in GB2’s account in Mexico D.F., Mexico,
While GB2 borrows £5,000,000 in Great
Britain and deposits it in MX1’s account in
London, Great Britain.
86
DIRECT SWAP FIXED FOR
FLOATING
£L + 1%
MX1
GB2
MP15%
MP15%
£L + 1%
MEXICO
ENGLAND
MX1 BORROWS
MP75M AND
DEPOSITS IT IN
GB2’S
ACCOUNT IN
MEXICO D.F.,
MEXICO.
GB2 BORROWS
£5,000,000 AND
DEPOSITS IT IN
MX1’S ACCOUNT
IN LONDON,
GREAT BRITAIN
MX1 pays £L+1%; GB2 pays MP15%
87
DIRECT SWAP FIXED FOR FLOATING
AGAIN:
MX1 pays £L+ 1%;
GB2 pays MP15%.
What does this mean?
It means that both firms pay interest for the
capitals they borrowed in the markets where
each has comparative advantage.
BUT, with the swap,
MX1 pays in pounds £L+ 1%, a better rate
than £LIBOR + 3%, the rate it would have
paid had it borrowed directly in the floating
rate market in Great Britain.
GB2 pays MP15% fixed, which is better than
the MP18% it would have paid had it
borrowed directly in Mexico.
88
A plain vanilla CURRENCY
SWAPS VALUATION
Under the terms of a swap, party A
receives French francs (FF) interest rate
payments and making dollar ($) interest
payments. Let us measure the amount in
. Also, use the following notation:
BFF = PV of the payments in FF from party
B, including the principal payment at
V
maturity.
B$ = PV of the payments in $ from party
A, including the principal payment at
maturity.
S0(FF/$) = the current exchange rate.
Then, the value of the swap to
counterparty A in terms of sterling is:
VFF = BFF - S0(FF/$)B$.
89
Note that the value of the swap depends
upon the shape of the domestic term
structure of interest rates and the foreign
term structure of interest rates.
EXAMPLE: A ‘PLAIN VANILLA’
CURRENCY SWAP VALUATION
Consider a financial institution that enters
into a two-year foreign currency swap for
which the institution receives 5.875% per
annum semiannually in French francs (FF)
and pays 3.75% per annum semi-annually
in U.S. dollars ($).
The principals in the two currencies are
FF58.5M $10M, reflecting the current
exchange rate: S0(FF/$) = 5.85.
Information about the U.S. and French
term structures of interest rates is given
in following table:
90
Domestic and Foreign Term
Structure*
Maturity
Months
6
12
18
24
Price of a zero coupon Bond
$
FF
.0840 (3.22%) .9699 (6.09%)
.9667 (3.38%) .9456 (5.59%)
.9467 (3.65%) .9190 (5.63%)
.9249 (3.90%) .8922 (5.70%)
*Figures in parenthesis are continuously
compounded yields.
The coupon payment of the semi-annual
interest payments in French Francs is:
5.875 1
[FF58.5M]
( )
100 2
 FF 1,718,437.50.
91
Therefore, the present value of the
interest rate payments in U.S Dollars plus
principal is:
PV(Cash Flows) $  BFFS($/FF)
 FF1,718,437.59[.9699  .9456 



 .9190  .8922] [$.1709/FF ]



FF58,500,0
00(.8922)


 $10,014,364.
The coupon payment of the semi-annual
interest payments in U.S Dollars is:
3.75 1
[$10M]
( )  $ 187,500.
100 2
92
Therefore, the present value of the
interest rate payments in U.S Dollars plus
principal is:
 $187,500[.9840  .9667



B$  
 .9467  .8249] 



$10,000,00
0(.9249)


 $9,965,681.
Therefore, the
value of the foreign currency swap
is:
PV(Cash Flow) $ - B$
 $10,014,364 - 9,965,681
 $48,683.
93
3.COMMODITY SWAPS
The huge success of domestic interest rate
swaps and foreign currency swaps lead
investors and firms to look for other markets
for swaps. In the 1980s and the 1990s
swaps began trading on a large range of
underlying assets. Among these are:
Commodities, stocks, stock indexes, bonds
and other types of debt instruments.
The assets underlying the swaps in these
markets are agreed upon quantities of the
commodity. Here, we analyze commodity
swaps using mainly energy commodities –
natural gas and crude oil. For example,
100,000 barrels of crude oil.
94
How does a commodity swap works:
In a typical commodity swap:
party A makes periodic payments to
counterparty B at a
fixed price per unit
for a given notional quantity of some
commodity.
Party B pays party A an agreed upon
floating price
for a given notional quantity of the
commodity underlying the swap.
The commodities are usually the same.
The floating price is usually defined as the
market price or an average market price,
the average being calculated using
spot commodity prices over
some predefined period.
95
Example: A Commodity Swap
Consider a refinery that has a constant
demand for 30,000 barrels of oil per
month and is concerned about volatile oil
prices. It enters into a three-year
commodity swap with a swap dealer. The
current spot oil price is $24.20 per barrel.
The refinery agrees to make monthly
payments to the swap dealer at a fixed
rate of $24.20 per barrel.
The swap dealer agrees to pay the
refinery the average daily price for oil
during the preceding month.
The notional principal is 30,000 barrels.
96
Spot oil
market
Oil
Spot
Price
$24.20/bbl
Refinery
Swap
Average Spot Price Dealer
The commodity: 30,000 bbls.
97
Note that in the swap no exchange of the
notional commodity takes place between
the counterparties.
The refinery has
reduced its exposure to the volatile oil
prices in the markets. It still, however,
bear some risk. This is because there
may be a difference between the spot
price and the average spot price. The
refinery is still buying oil and paying the
spot price, and from the swap dealer it
receives the last month's average spot
price. It also pays to the swap dealer
$24.20 per barrel over the life of the
contract. Therefore, the spread between
the spot and last month average prices
presents some risk to the refinery.
98
A NATURAL(NG) SWAP:
FIXED FOR FLOATING.
MC – a marketing firm buys NG from a
producer for the fixed price of $9.50/UNIT
(1,000 cubic feet). At the same time MC
finds an end user and sells the NG. The end
user insists on paying a floating market price
index. The index is published daily according
NG prices in different locations.
MC’s risk is that the index falls below $9.50.
MC enters a FIXED FOR FLOATING swap in
which it pays the swap dealer the index and
recieves $9.55/tcf
The notional amount of NG is equal to the
amount purchased and sold by MC.
99
A FISED-FOR-FLOATING
NATURAL GAS SWAP
INDEX
$9.50
MC
PRODUCER
END USER
Gas
Gas
$9.55
INDEX
SWAP
DEALER
MC’s cash flow is:
- $9.50 + Index + $9.55 – Index
= $0.05/UNIT
100
FLOATING FOR FLOATING NATURAL
GAS SWAP
There are several different energy
indexes for various energy
commodities. Thus, it is very possible
that MC will buy the natural gas for one
index and sell it to the end user for
another index. In these cases, both
cash flows are based on floating rates
and MC faces the exposure of the
floating spread. MC may be able to
enter a swap and fix a positive spread
for its revenues.
101
FLOATING-FOR-FLOATING
NATURAL GAS SWAP
INDEX2
INDEX1
MC
producer
USER
Gas
Gas
INDEX1
INDEX2 - $0.08
Swap Dealer
In this case MC’s cash flow is:
(Index2) – (Index1)
+ (Index1) – ( Index2 - $0.08)
= $0.08/UNIT.
102
Valuation of Commodity of
Swaps
The value of a
“plain vanilla” commodity swap.
In a "plain vanilla" commodity swap,
counterparty A agrees to pay counterparty
B a fixed price, P(fixed, ti), per unit of the
commodity at dates t1, t2,. . ., tn.
Counterparty B agrees to pay counterparty
A the spot price, S(ti) of the commodity at
the same dates t1, t2,. . ., tn.
The notional principal is NP units of the
commodity
The net payment to counterparty A at date
t1 is:
V(t1, t1)  [S(t1) - P(fixed, t1)]NP.
103
The value of this payment at date 0 is the
present value of V(t1, t1):
V(0, t1) = PV0{V(t1, t1)}
= PV0[S(t1)] – P(fixed, t1)B(0, t1)NP ,
where B(0, t1) is the value at date 0 of
receiving one dollar for sure at date t1.
In the absence of carrying costs and
convenience yields, the present value of
the spot price S(t1) would be equal to the
current spot price. In practice, however,
there are carrying costs and convenience
yields.
104
It can be shown that the use of forward
prices incorporates these carrying costs
and convenience yields. Drawing on this
insight, an alternative expression for the
present value of the spot price PV0[S(t1)]
in terms of forward prices may be derived
as follows:
Consider a forward contract that expires
at date t1 written on this commodity with
the forward price = F(0, t1). The cash
flow to the forward contract when it
expires at date t1 is:
S(t1) - F(0, t1).
The value of the forward contract at date
0 is:
PV0[S(t1)] - F(0, t1)B(0, t1).
105
Like any forward, the forward price is
set such that no cash is exchanged when
the contract is written. This implies that
the value of the forward contract, when
initiated, is zero. That is:
PV0[S(t1)] = F(0, t1)B(0, t1).
Using this expression, the value at date 0
of the first swap payment is:
V(0,tl) = [F(0,t1) - P(fixed,t1)]B(0, tl)NP.
106
Repeating this argument for the remaining
payments, it can be shown that the
value of the commodity swap
at date 0 is:
n
V0  [F(0, t j )  P(Fixed)]B (0, t j )N p .
j1
Note that the value of the commodity
swap in this expression depends only on
the forward prices, F(0,tj), of the
underlying commodity and the zerocoupon bond prices, B(0, t1), all of which
are market prices observable at date 0.
107
FINAL EXAMPLE:
From the derivatives trading room of BP:
Hedging the sale and purchase of Natural
Gas, using NYMEX Natural Gas futures and
Creating a sure profit margin swapping the
remaining spread. First, let us define:
The following two indexes:
1. L3D
- LAST THREE DAYS
A weighted average of NYMEX
NG futures prices during the last
three trading days of the contract.
2. IF - INSIDE FERC
A weighted average of NG spot prices
at various places.
108
April 12 – 11:45AM
From BP’s derivative trading room
1. The 1st call:
BP agrees to buy NG from BM in
August for IF.
2. The 2nd call:
BP hedges the NG purchase going long
NYMEX’ August NG futures.
3. The 3rd call: BP finds a buyer for
the gas - SST. But, SST negotiates
the purchase price to be at some
discount off the current August
NYMEX NG futures. Let $X be the
discount amount. $X is left unknown
for now.
109
A PARTIAL SUMMARY
DATE
SPOT
FUTURES
Buy from BM.
Sell to SST.
Long August NYMEX
Futures.
April 12
F4,12; aug = $3.87.
August 12
(i) Buy NG from BM
S1 = IF .
(ii) Sell NG to SST for
S2 = F4, 12; aug – $X
Short August NYMEX
Futures.
Faug; aug = L3D
PARTIAL CASH FLOW:
(F4,12; aug – $X) – IF + L3D - F4,12; aug
= L3D – $X – IF.
110
How can BM eliminate the
spread risk?
BP decides to enter a spread swap.
Clearly, this is a
floating for floating swap.
4. The 4th call: BP enters a swap
whereby
BP pays the Swap dealer
L3D – $.09
and receives
IF
from the Swap dealer.
The swap is described as follows:
111
A FLOATING FOR FLOATING
SWAP
L3D - $.09
SWAP DEALER
BP
IF
The principal amount underlying the swap
is the same amount of NG that BP buys
from BM and sells to SST.
112
SUMMARY OF CASH FLOWS
MARKET
Spot:
CASH FLOW
F4, 12; AUG - $X - IF
Futures:
+ L3D - F4, 12; AUG
Swap:
+ IF - (L3D – $.09)
TOTAL
= $.09 - $X.
BP decides to make 3 cents per unit.
Solving $.03 = $.09 - $X for $X
yields:
$X = $.06.
5. The 5th call BP calls SST and both
agree that SST buys the NG from BP in
August for today’s NYMEX - $.03. I.e.,
$3.87 - $.06 = $3.81.
THE END
113
THE BP EXAMPLE
MARKET
SWAP:
SWAP DEALER
IF
L3D - .09
F4,12;AUG - $.06
IF
SPOT: BM
BP
NG
LONG
F4,12;AUG
SST
NG
SHORT
L3D
FUTURES
NYMEX
114
4. BASIS SWAPS
A basis swap is a risk management tool
that allows a hedger to eliminate the
BASIS RISK associated with the hedge.
Recall that a firm faces the CASH PRICE
RISK, opens a hedge, using futures, in
order to eliminate this risk. In most cases,
however, the hedger firm will face the
BASIS RISK when it operates in the cash
markets and closes out its futures hedging
position. We now show that if the firm
wishes to eliminate the basis risk, it may
be able to do so by entering a:
BASIS SWAP.
In a BASIS SWAP, The long hedger
pays the initial basis, I.e., a fixed payment
and pays the terminal basis, I.e., a floating
payment. The short hedger, pays the
terminal basis and receives the initial basis.
1.
THE FUTURES SHORT HEDGE:
TIME
0
k
CASH
S0
FUTURES
F0,t
Sk
Fk,
BASIS
B0 = S0 - F0,t
Bk,t = Sk - Fk,t
The selling price for the SHORT hedger is:
F0,t + Bk,t .
2. THE SWAP OF THE SHORT HEDGE:
B0
SHORT
HEDGER
Bk,t
SWAP
DEALER
3. THE SHORT HEDGER’S SELLING PRICE:
F0,t + Bk,t + B0,t - Bk,t
= F0,t + B0,t
= F0,t + S0 - F0,t
= S0 .
1.
THE FUTURES LONG HEDGE:
TIME
0
k
CASH
S0
FUTURES
F0,t
Sk
Fk,
BASIS
B0 = S0 - F0,t
Bk,t = Sk - Fk,t
The purchasing price for the LONG hedger:
F0,t + Bk,t .
2. THE SWAP OF THE LONG HEDGER
LONG
HEDGER
B0
Bk,t
SWAP
DEALER
3. THE LONG HEDGER’S PURCHASING PRICE:
F0,t + Bk,t + B0,t - Bk,t
= F0,t + B0,t
= F0,t + S0 - F0,t
= S0
1.
PRICE RISK
FUTURES HEDGING
2.
BASIS RISK
BASIS SWAP
3.
NO RISK AT ALL
THE CASH FLOW IS:
THE CURRENT CASH PRICE!
BASIS SWAP
Buy gas at
“Screen - 10”
NYMEX
$3.60
L3D
$3.50
POWER
PLANT
GAS
PRODUCER
GAS
Power plant is a long hedger.Initial basis =
–$.10. The terminal basis is S – L3D. Power
plant may swap the bases: final purchasing
price of:
$3.60 + S – L3D + (S - L3D - $.10)
= $3.50.