Transcript Forwards and Futures - Bauer College of Business

Swaps
Chapter 6
1
SWAPS
Swaps are a form of derivative instruments.
Out of the variety of assets underlying
swaps we will cover:
INTEREST RATES SWAPS,
CURRENCY SWAPS,
COMMODITY SWAPS,
EQUITY SWAPS and
BASIS SWAPS.
2
SWAPS
A SWAP is a contract between two
parties for an
exchange of cash flows during
some time period.
The cash flows are determined
based on the UNDERLYING ASSET
3
1.
2.
3.
4.
5.
It follows that a swap involves
Two parties
An underlying asset
Cash flows
A payment schedule
An agreement as to how to
resolve problems
4
1. Two parties:
The two parties in a swap are 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 swap dealer.
5
2. The Underlying
asset is
the basis for the determination of the
cash flows. It is almost never
exchanged by the parties.
Examples:
USD100,000,000,
GBP50,000,000,
50,000 barrels of crude oil
An equity index
6
2. The Underlying
asset is called the
NOTIONAL AMOUNT
Or
The PRINCIPAL
Because it only serves to determine the cash
flows. Neither party needs to own it and it almost
never changes hands.
7
3. The cash flows
may be of two types:
a fixed or a floating cash flow.
Fixed interest rate
vs.
Floating interest rate
Fixed price
Vs.
Market price
8
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.
If the underlying asset are 100,000 barrels
of oil:
Ex: (100,000)($24,75) = $2,475,000; Fixed.
(100,000)(St );
Floating.
9
4. The payments
are always net.
The contract determines the cash
flows timing as annual,
semiannual or monthly, etc. Every
payment is the net of the two cash
flows
10
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.
11
Typical Uses of an
Interest Rate Swap
• Converting a
• Converting an
liability from
investment from
– fixed rate to
– fixed rate to
floating rate
floating rate
– floating rate to
– floating rate to
fixed rate
fixed rate
12
Why SWAPS?
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.
13
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 EUR35million.
The swap is for five years.
Two weeks later, the six-month LIBOR
rate is 6.45% per annum.
14
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:
 Not ionalDays in  Fixed Rat e




100

amount P eriod 
(182) 7.19
 EUR35,000,000
365 100
 EUR1,254,802.74.
This calculation is based on the assumption
that the payment is every 182 days. 15
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)
 EUR35,000,000
360
100
 EUR1,194,375.
Other future payments will be determined
every 6 months by the six-month LIBOR at
that time.
16
7.19%
Party A
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:
EUR1,254,802.74 - EUR1,194,375.00
= EUR60,427.74.
17
Another Example of a “Plain
Vanilla” Interest Rate Swap
• An agreement by Microsoft to
receive 6-month LIBOR & pay a
fixed rate of 5% per annum every 6
months for 3 years on a notional
principal of USD100 million
• Next slide illustrates cash flows
18
The principal amount …………… USD100.000.000.
The cash flows are………………...semiannual
5% FIXED
SWAP
DEALER
MICROSOFT
6-month LIBOR
19
Cash Flows to Microsoft
(See Table 6.1, page 127)
---------Millions of USD--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
20
Intel and Microsoft (MS)
Transform a Liability
(Figure 6.2, page 128)
5%
5.2%
MS
Intel
LIBOR+0.1%
LIBOR
21
SWAP DEALER is Involved
(Figure 6.4, page 129)
4.985%
5.015%
5.2%
Intel
SD
MS
LIBOR+0.1%
LIBOR
LIBOR
22
Intel and Microsoft (MS)
Transform an Asset
(Figure 6.3, page 128)
5%
4.7%
Intel
MS
LIBOR-0.25%
LIBOR
23
SWAP DEALER is Involved
(See Figure 6.5, page 129)
4.985%
5.015%
4.7%
SD
Inte
LIBOR-0.25% l
LIBOR
MS
LIBOR
24
These examples illustrate 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, the principal
amount 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.
25
4. On the fixed payment side, a 365-day
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 Six-month LIBOR.
Estimates of future LIBOR values are
obtained from LIBOR yield curves which
are based on Euro Strip of Euro dollar
futures strips.
26
The Comparative Advantage Argument
A firm has an ABSOLUTE ADVANTAGE if it
can obtain better rates in both the fixed and
the floating rate markets.
Firm A has a RELATIVE ADVANTAGE in one
market if:
the difference between what firm A pays
more than firm B in the floating rate (fixed
rate) market is less than the difference
between what firm A pays more than firm B in
the fixed rates
(floating rate) market.
27
Example:
A FIXED FOR FLOATING SWAP
Two firms need EUR10M financing for projects. They
face 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%).
28
Now, suppose that the firms decide to enter
a FIXED for FLOATING swap based on
the notional of EUR10.000.000.
The cash flows: Annual payments to be
made on the first business day in March for
the next five years. The SWAP always begins with
each party borrowing capital in the market in which it
has a RELATIVE ADVANTAGE. Thus:
F1 borrows S EUR10,000,000 in the market for floating
rates, I.e., for LIBOR + 2% for 5 years.
F2 borrows EUR10,000,000 in the market for fixed rates,
I.e., for 12%.
NOW THE TWO PARTIES EXCHANGE THE TYPE OF CASH FLOWS BY
ENTERING THE SWAP FOR FIVE YEARS.
29
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.
30
FIXED FOR FLOATING SWAP
1. A DIRECT SWAP:
FIRM FIXED RATE
FLOATING RATE
F1
15%
LIBOR + 2%
F2
12%
LIBOR + 1%
12%
LIBOR
F2
LIBOR+2%
F1
12%
The result of the swap:
F1 pays fixed 14%,
better than 15%.
F2 pays floating LIBOR, better than
LIBOR + 1%
31
2. AN INDIRECT SWAP with a SWAP DEALER:
FIRM
FIXED RATE
FLOATING RATE
F1
15%
LIBOR + 2%
F2
12%
LIBOR + 1%
12%
L+25bps
F2
L
F1
SD
12%
L + 2%
12,25%
F1 pays 14,25% fixed: Better than 15%. F2 pays
L+25bps : Better than L+1%. The swap dealer gains
50 bps = $50,000.
32
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 swap
dealers to be happy with 10 basis points. In the present
example, another possible swap arrangement is:
12%
L+5bp
F2
L
F1
SD
12%
L+2%
12%+5bp
Clearly, there exist many other possible
swaps between the two firms in this example.
33
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.
34
EXAMPLE: A RISK MANAGEMENT SWAP
BONDS MARKET
FL1 = 6-MONTH BANK RATE.
FL2 = 6-MONTH LIBOR.
LOAN
FL1
10%
SWAP DEALER A
BANK
LOAN
FL2
12%
FIRM A
BORROWS AT A
FIXED RATE FOR 5
YEARS
35
THE BANK’S CASH FLOW:
12% - FLOATING1 + FLOATING2 – 10% = 2% + SPREAD
Where the
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.
36
EXAMPLE: A RISK MANAGEMENT SWAP
BOND MARKET
FL1
10%
SWAP DEALER A
BANK
FL2
FL2
12%
FL1
SWAP DEALER B
FIRM A
37
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
FIXED RATE!
38
PRICING SWAPS
The swap coupons (payments) for shortdated 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
they are liquid out to at least four years.
39
The Euro strip is a series of successive threemonth 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.
40
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.
41
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.
42
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 threemonth LIBOR rates from the Euro strip
to obtain the implied annual effective
LIBOR for the full-tenor of the swap.
The second step: Convert this full-tenor
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.
43
NOTATIONS:
The swap is an m-months or
m/12 years swap. The swap is to be priced off threemonth Eurodollar futures, thus, pricing requires n
sequential futures series. n = m/3; m = 3n.
Step 1: Use the futures Euro strip to Calculate the
implied effective annual LIBOR for the full tenor of
k
the swap:
n

N(t) 
r0,3n    [1 (r3(t -1),3(t)
)]  1, where
360 
 t 1
360
k
; N(t)denotestheact ualnumber
 N(t)
of days coveredby thet - th Eurodollarfutures.
44
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
45
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:
1
f
SC  {[1  FRE(0,3n)] - 1}(f),
which, upon subst it ut ion of
FRE(0,3n),can be rewrit t enas :
1
f
365
SC  {[1  r0,3n
]  1}(f).
360
46
Example:
For illustration purposes let us observe Eurodollar
futures settlement prices on April 24, 2001.
Eurodollar Futures Settlement Prices April 24,2001.
CONTRACT
JUN01
SEP01
DEC01
MAR02
JUN02
SEP02
DEC02
MAR03
JUN03
SEP03
DEC03
MAR04
JUN04
SEP04
PRICE
95.88
95.94
95.69
95.49
95.18
94.92
94.64
94.52
94.36
94.26
94.11
94.10
94.02
93.95
LIBOR
4.12
4.06
4.31
4.51
4.82
5.08
5.36
5.48
5.64
5.74
5.89
5.90
5.98
6.05
FORWARD
0,3
3,6
6,9
9,12
12,15
15,18
18,21
21,24
24,27
27,30
30,33
33,36
36,39
39,42
DAYS
92
91
90
92
92
91
91
92
92
91
90
92
92
91
47
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 01 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.
48
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.
49
Suppose that we want to price a one-year
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:
50
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  .0431 )(1  .0451 ) 
360
360 

 4.34%,on moneymarketbasis.
360
365
1
51
Step 2 and 3:
1
f
SC  {[1  FRE(0,3n)] - 1}(f),which, upon
substitution of FRE(0,3n),can be rewrittenas :
1
f
365
SC  {[1  r0,3n
]  1}(f)
360
1
365 4
 {[1  .0434
]  1}(4)  4.33% on a
360
quarterlybond basis.
The swap’s coupon is the dealer mid rate. To this
rate , the dealer will add several basis points. 52
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.
53
1
2
365
SC  [1  (.0434
)  1)](2);
360
SC  4.35%,on a semiannual
bond basis.
4.35% FIXED
Swap
dealer
Client
6-M LIBOR FLOATING
54
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-paysfixed-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:
55
N(t) 

r0,36    [1  (r3(t -1),3(t)
)]
360 
 t 1
12
360
1096
1
92
91
90 

)(1 .0406
)(1 .0431 ) 
 (1  .0412
360
360
360 

92
92
91 

 (1  .0451360)(1 .0482360)(1 .0508360) 


 (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
56
Step 2: The Fixed Rate Equivalent effective annual
rate on a bond basis is:
FRE = (5.17%)(365/360) = 5.24%.
Finally,
Step 3:
The equivalent semiannual Swap Coupon
is calculated:
SC = [(1.0524).5 – 1](2) = 5.17%.
57
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.
58
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
59
amortizing.
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.
Let B(0,T)=PV of USD1.00 paid at T.
Let L(0,T)=PV of 1EuroUSD paid at T.
These prices are derived from the Treasury and
Eurodollar term structures, respectively.
The data are:
60
Payments
dates
t1 = 182
t2 = 365
t3 = 548
t4 = 730
Days
between
payment
Dates
182
183
183
182
Treasury
Bills Prices
B(0,T)
Eurodollar
Deposit
L(0,T)
.9679
.9362
.9052
.8749
.9669
.9338
.9010
.8684
61
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 ).
62
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.
63
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)NP .
64
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.
65
SWAP VALUATION: The general formula
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
66
The value of the floating rate payments:
1. If the swap is already in existence, let λ denote
the pre specified LIBOR rate. At date T1, the
payment is:
1
P
k
N [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.
67
The value of the floating rate payments at date T1 is:
k1
VFLOATING (T1 )  N P λ
360
 N P  N P L(T1 , Tn ).
The value of the floating payments at date 0 is the present
value of:
VFLOATING (T1 ) :
 k1

VFLOATING (0)  N P  λ
 1L(0,T1 ) - N P L(0,Tn ).
 360 
T hisholds true because L(T1 , Tn )L(0,T1 )  L(0,Tn ).
68
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 , thevalue of thefloating
rate paymentsis :
69
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 )].
70
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:
T HE SWAP VALUE for the party
receiving fixed and paying 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


71
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

72
PAR SWAP Valuation
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
dates
t1 = 182
t2 = 365
t3 = 548
t4 = 730
Days
between
payment
Dates
Treasury
Bills Prices
B(0,T)
182
183
183
182
.9679
.9362
.9052
.8749
Euro
Dollar
Deposit
L(0,T)
.9669
.9338
.9010
.8684 73
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.
74
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 Europe, exchange it to USD and repay it
later, exchanging USD into EUR.
Currency swaps are basically, interest rate swaps
75
accross international borders.
Case Study of a currency swap:
IBM and The World Bank(1982)
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.
76
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 6.1785%
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
USD.5181/DEM in March 1980 to USD.3968/DEM in
August 1981. Thus, coupon payments of DEM100 had
fallen in USD cost from USD51.81 to USD39.68. 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.
77
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 starting date.
78
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 Factor 
1
n
360
,
(1  y)
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.
79
The discount factors were calculated:
Date
3.30.82
3.30.83
3.30.84
3.30.85
3.30.86
Days
215
575
935
1295
1655
CHF
.9550775
.8843310
.8188250
.7581813
.7020104
DEM
.9395764
.8464652
.7625813
.6870102
.6189281
80
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.
81
The terms of the swap were agreed upon on AUG 11,
1981. Thus, The World Bank would have been left
exposed to currency risk for two weeks until AUG 25.
The World Bank decided to hedge the above derived
NPV amounts with 14-days currency forwards. Assuming
that these forwards were at USD.45872/CHF and
USD.390625/DEM, The World Bank needed a total
amount of:
USD87,783,247 to buy the CHF
+
USD117,701,753 to buy the DEM;
a total of
USD205,485,000. 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.
82
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:
USD205,485,000/0.9785 = USD210,000,000.
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:
USD210,000,000(.16)[215/360]
= USD20,066,667.
83
The cash flows are summarized in the following table:
Date
USD
CHF
DEM
3.30.82
3.30.83
3.30.84
3.30.85
3.30.86
YTM
NPV
20,066,667
33,600,000
33,600,000
33,600,000
243,600,000
8%
205,485,000
12,375,000
12,375,000
12,375,000
12,375,000
212,375,000
11%
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 CHF and GEM cheaper than it would had
it gone to the currency markets directly.
84
THE SWAP
CHF200M
SWITZ
DEM300M
IBM
GERMANY
CHF CUPON
DEM CUPON
USD
CHF
CUPON CUPON
DEM
CUPON
WORLD
BANK
USD
CUPON
IBM
PAY
RECEIVE
Date
USD
CHF
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
RECEIVE
DEM
30,000,000
30,000,000
30,000,000
30,000,000
330,000,000
USD
CAPITAL
USA
85
THE ANALYSIS OF CURRENCY SWAPS
F1 in country A looks for financing a project in
country B
AT THE SAME TIME
F2 in country B, looks for financing a project in
country A
COUNTRY A
COUNTRY B
F1
F2
PROJECT OF F2
PROJECT OF F1
86
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.
87
CURRENCY SWAP FIXED FOR FIXED
CLP = Chilean Peso
BR = 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:
88
FIRM
CHILE
BRAZIL
CH1
CLP12%
BR16%
BR2
CLP15%
BR17%
CH1 has comparative advantage in Chile
only. CH1 borrows in Chile in CLP and BR2
borrows in Brazil in BR. The swap begins
with the interchange of the principal
amounts 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
89
exchange hands once more.
ASSUME THAT THE CURRENT EXCHANGE
RATE IS:
BR1 = CLP250
ASSUME THAT CH1 NEEDS BR10.000.000 FOR
ITS PROJECT IN BRAZIL AND THAT BR2
NEEDS EXACTLY CLP2,5B FOR ITS PROJECT
IN CHILE.
Again:
FIRM
CHILE
BRAZIL
CH1
$12%
R16%
BR2
$15%
R17%
90
DIRECT SWAP FIXED FOR FIXED
BR15%
CH1
BR2
CLP12%
CLP12%
BR17%
CHILE
BRAZIL
CH1 BORROWS
CLP2.5B AND
DEPOSITS IT IN BR2’S
ACCOUNT IN
SANTIAGO
BR2 BORROWS
BR10M AND DEPOSITS
IT IN CH1’S ACCOUNT
IN SAO PAULO
CH1 pays BR15%; BR2 pays CLP12% + BR2%
91
INDIRECT SWAP FIXED FOR FIXED
SWAP DEALER
BR15.50%
CLP12%
CH1
CLP12%
CLP14.50%
BR17%
BR2
BR17%
CHILE
BRAZIL
CH1 BORROWS
CLP2.5B AND
DEPOSITS IT IN BR2’S
ACCOUNT IN
SANTIAGO
BR2 BORROWS
BR10M AND DEPOSITS
IT IN CH1’S ACCOUNT
IN SAO PAULO
92
THE CASH FLOWS:
CH1:
PAYS BR15.50%
BR2:
PAYS CLP14.50%
SWAP DELER: CLP2.50 – BR1.50%
EXAMPLE:
CLP2,5B(0.025) – BR10M(0.015)(250)
= CLP62,500,000 - CLP37,500,000 = CLP25,000,000
Notice: In this case, CH1 saves 0.25% and BR2 saves
0.25%, while the SWAP DEALER bears the exchange rate
risk. If the CLP depreciates against the BR the
intermediary’s revenue declines. When the exchange rate
reaches CLP466,67/BR the intermediary gain is zero. If the
Chilean Peso continues to depreciate the intermediary loses
money on the deal.
93
Foreign Currency Swaps
EXAMPLE:
a “plain vanilla”
foreign currency swap.
F1, an Italian firm has issued bonds with face
value of EUR50M with a annual coupon of
11.5%, paid semi-annually and maturity of
seven years.
F1 would prefer to have USD and to be making
interest payments in USD. Thus, F1 enters into a
foreign currency swap with F2 - usually a SWAP
DEALER. In the first phase of the swap, F1
exchanges the principal amount of EUR50M with
party F2 and, in return, receives principal worth
USD60M. Usually, this exchange is done at the
current exchange rate, i.e.,
S = USD1.20/EUR in this case.
94
The swap contract is as follows: F1 agrees to
make semi – annual interest rate payments
to F2 at the rate of 9.35% per annum based
on the USD denominated principal for a
Seven Year period. In return, F1 receives from
F2 a semi- annual interest rate at the annual
rate of 11.5%, based on the EURO
denominated principal for a seven years.
The swap terminates seven years later,
when the principals are again exchanged:
F1 receives the principal amount of EUR50M
F2 receives the principal amount of USD60M.
95
DIRECT SWAP FIXED FOR FIXED
USD9.35%
F1
F2
EUR11.5%
EUR11.5%
USD9.35%
ITALY
U.S.A
F1 BORROWS
EUR50M AND
DEPOSITS IT IN F2’s
ACCOUNT IN MILAN
F2 DEPOSITS USD60M
IN F1’S ACCOUNT IN
NEW YORK CITY
At maturity, the original principals are exchanged
96 to
By entering into the foreign currency swap,
F1 has successfully transferred its EUR
liability into a USD liability.
In this case, F2 payments to F1 were based
on the the same rate of party’s F1 payments
in Italy EUR11.5%. Thus, F1 was able to
exactly offset the EUR interest rate
payments. This is not necessarily always the
case. It is quite possible that the interest
rate payments F1 receives from SWAP
DEALER F2 only partially offset the EUR
expense. In the same example, the situation
may change to:
97
DIRECT SWAP FIXED FOR FIXED
USD9.55%
F1
EUR11.5%
F2
EUR11.25%
USD9.55%
ITALY
U.S.A
F1 BORROWS EUR50M
AND DEPOSITS IT IN
F2’s ACCOUNT IN
MILAN
F2 DEPOSITS USD60M
IN F1’S ACCOUNT IN
NEW YORK CITY
At maturity, the original principals are exchanged.
98
EXAMPLE: FIXED FOR FLOATING
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
99
vice – versa.
A DIRECT SWAP FIXED FOR FLOATING INTEREST RATES
MEXICO
GREAT BRITAIN
MX1
MXP15%
GBPLIBOR + 3%
GB2
MXP18%
GBPLIBOR + 1%
ASSUME: The current exchange rate is: GBP1 =
MXP15.
MX1 needs GBP5.000.000 in England GB2
needs MXP75.000.000 in Mexico.
THUS: MX1 borrows MXP75M in Mexico and deposits
it in GB2’s account in Mexico D.F. While GB2
borrows GBP5,000,000 in Great Britain and deposits
it in MX1’s account in London.
100
DIRECT SWAP FIXED FOR FLOATING
GBP L + 1%
MX1
GB2
MXP15%
GBP L + 1%
MXP15%
MEXICO
ENGLAND
MX1 BORROWS
MXP75M AND
DEPOSITS IT IN
GB2’S ACCOUNT IN
MEXICO D.F.
GB2 BORROWS
GBP5,000,000 AND
DEPOSITS IT IN MX1’S
ACCOUNT IN LONDON,
MX1 pays GBP L+1%; GB2 pays MXP15%
101
DIRECT SWAP FIXED FOR FLOATING
AGAIN:
MX1 pays GBP L+ 1%; GB2 pays
MXP15%.
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 GBP L+ 1%, a better rate than
GBP LIBOR + 3%, the rate it would have paid had it
borrowed directly in the floating rate market in
Great Britain.
GB2 pays MXP15% fixed, which is better than the
MXP18% it would have paid had it borrowed
directly in Mexico.
102
A CURRENCY SWAPS VALUATION
Under the terms of a swap, party A receives EUR
interest rate payments and making USD interest
payments.
BEUR = PV of the payments in EUR from party B,
including the principal payment at maturity.
BUSD = PV of the payments in USD from party A,
including the principal payment at maturity.
S0(EUR/USD) = the current exchange rate.
Then, the value of the swap to counterparty A in
terms of Euros is:
V
VEUR = BEUR - S0(EUR/UED)BEUR.
103
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 EUR and pays 3.75%
per annum semi-annually in USD.
104
The principals in the two currencies are
EUR12M; USD10M, reflecting the Current
exchange rate: S0(EUR/USD) = 1.20
Information about the US and ITALIAN term
structures of interest rates is given in
following table:
105
Domestic and Foreign Term Structure*
Maturity Price of a zero coupon Bond Months
USD
EUR
6
.0840 (3.22%) .9699 (6.09%)
12
.9667 (3.38%) .9456 (5.59%)
18
.9467 (3.65%) .9190 (5.63%)
24
.9249 (3.90%) .8922 (5.70%)
*Figures in parenthesis are continuously compounded yields.
The coupon payment of the semi-annual interest
payments in EUR is:
5.875 1
[EUR12M]
( )
100 2
 EUR352,500.
106
Therefore, the present value of the Interest rate
payments in USD plus principal is:
P V(Cash Flows)USD  [BEUR ]S(USD/EUR)
 EUR352,500[.9699 .9456 



 .9190 .8922][USD.8333/EUR]
  EUR12,000,000(.8922)


 USD10,016,718.
The coupon payment of the semi-annual interest
payments in USD is:
3.75 1
[USD10M]
( )  USD187,500 .
100 2
107
Therefore, the present value of the interest rate
payments in USD plus principal is:
BUSD
 USD187,500[.9840 .9667



 .9467 .8249]
  USD10,000,000(.9249)


 USD9,946,931.
108
The value of the foreign currency swap is:
PV(Cash Flow)USD - BUSD
 USD10,014,364- USD9,965,681
 USD48,683.
109
3.COMMODITY SWAPS
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.
110
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.
B pays A an agreed upon floating price
for the same 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.
111
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 USD24.20 per barrel.
The refinery agrees to make monthly payments to
the swap dealer at a fixed price of USD24.20 per
barrel.
The swap dealer agrees to pay the refinery the
average daily price for oil during the CURRENT
month.
The notional principal is 30,000 barrels.
The swap is for 36 months.
112
Spot oil
market
Oil
Daily Spot
Price
USD24.20/bbl = USD726,000
Swap
Dealer
Refinery
Average Spot Price
The commodity: 30,000 Barrels(1,000/day).
113
SPOT OIL
MARKET
Daily spot
price
USD726,000
OIL
Swap
Dealer A
Italian
Refinery
USD726,000/1.2
=EUR648,214
Average spot price
USD726,000
Swap
Dealer B
114
Valuation of Commodity of Swaps
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.
115
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.
116
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).
117
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.
118
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 )Np .
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.
119
FINANCIAL ENGINEERING 1
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
Employing a swap of the remaining
spread.
120
April 12 – 11:45AM
From BP’s derivatives trading room
1. The 1st call: BP agrees to buy NG from BM in
August at the market price on AUG 12.
2. The 2nd call: BP hedges the NG purchase going
long NYMEX’ SEP NG futures.
3. The 3rd call: BP finds a buyer for the gas - SST.
But, SST negotiates the purchase price, P, to be
at some discount, X, off the current SEP NYMEX
NG futures. X is left unknown for now.
P = F4.12,SEP - X
121
A PARTIAL SUMMARY of BP POSITION
DATE
SPOT
FUTURES
April 12 CONTRACTS:
Buy from BM.
SELL TO SST
August 12
(i) Buy NG from BM
for S1
Long SEP NYMEX
Futures.
F4,12; SEP = $6.87.
Short SEP NYMEX Futures.
F8,12,SEP
(ii) Sell NG to SST for
P = F4, 12; SEP – X
PARTIAL CASH FLOW ON AUG 12:
F4,12; SEP – X – S1 + F 8,12; SEP - F4,12; SEP
= F 8,12; SEP – S1 - X
122
How can BM eliminate the BASIS risk?
BP decides to enter a swap.
Clearly, this is a floating for floating swap.
4. The 4th call: BP enters a swap whereby
BP pays the Swap dealer
F8,12,SEP – USD.09
and receives
S1
from the Swap dealer.
The swap is described as follows:
123
A FLOATING FOR FLOATING SWAP
F8,12,SEP – USD.09
SWAP DEALER
BP
S1
The principal amount underlying the swap is
the same amount of NG that BP buys from
BM and sells to SST.
124
SUMMARY OF CASH FLOWS ON AUG 12
MARKET
Spot:
CASH FLOW
F4, 12; SEP - X - S1
Futures:
+ F 8, 12; SEP - F4, 12; SEP
Swap:
- F 8,12; SEP + USD.09 + S1
TOTAL
= USD.09 - X.
125
BP decides to make 3 cents per unit of NG.
Solving USD.03 = USD.09 - X
yields: X = USD.06.
5. The 5th call BP calls SST and agree on
the purchase price.
On AUG 12, SST buys the NG from BP for
P = USD6.87 - USD.06 = USD6.81.
126
THE BP EXAMPLE
MARKET
SWAP DEALER
SWAP:
S1
F8.12,SEP - .09
F4,12;SEP - .06
S1
SPOT:
BM
NG
FUTURES
LONG
F4,12;SEP
BP
SST
NG
SHORT
F8.12,SEP
NYMEX
127
4. BASIS SWAPS
A basis swap is a risk management tool that allows a
hedger to eliminate the BASIS RISK associated with
a 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 receives 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
CASH
FUTURES
0
S0
F0,T
k
Sk
Fk,T
BASIS
B0,T = 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 HEDGER:
B0,T
SHORT
HEDGER
SWAP DEALER
Bk,T
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
CASH
FUTURES
0
S0
F0,T
k
Sk
Fk,T
BASIS
B0,T = S0 - F0,T
Bk,T = Sk - Fk,T
The purchasing price for the LONG hedger is: F0,T + Bk,T .
2. THE SWAP OF THE LONG HEDGER:
B0,T
LONG
HEDGER
SWAP DEALER
Bk,T
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 NG at
“Screen - 10”
NYMEX
USD6.60
USD6.50
GAS
PRODUCER
POWER
PLANT
GAS
F
- 10
S-F
SWAP
DEALER
Power plant is a long hedger. B0 = – .10. BK = S– F.
Power plant may swap the bases and the final
purchasing price is:
USD6.60 + S – F – [S - F – (-.10)] = USD6.50.
FINANCIAL ENGINEERING 2.
A Mexican firm. All its costs in MXP are fixed for the next 5 years. All its
revenews in MXP are fixed for the next 5 years. The only floating cost is the
cost of oil it buys. The firm buys 150,000 barrels of crude oil every 3 months
in the market price and pays in USD. It wishes to change this floating USD
payment into a fixed MXP payment.
RECIEVE
PAY
SWAP DEALER 1. FLOATING
FIXED
S + 0,50
USD26/bbl
AVERAGE (S)
USD25/bbl
SWAP DEALER 2. FLOATING
FIXED
USD LIBOR
11%
USD LIBOR – 25pbs
10% (=USD3.9M)
SWAP DEALER 3. FLOATING
FIXED
USD LIBOR+25pbs
8%(=MXP37.181.898)
LIBOR
MXP35M
THE FIRM
AVERAGE (S)
S
USD26/bbl = USD3.9M
FLOATING
FIXED
All payments are quartely payments.
The swap is for 20 quarterly cash flows.
133
The swap: Quarterly payments for the next 5 years
OIL PRODUCERS
USD156M =
USD3.9M/0.025
150,000bbls
150,000bbls
USD S
USD26/bbl Total USD3.9M
USD3.9M
SWAP DEALER
INTEREST RATES
MEXICAN FIRM
USD ave(S)
USD LIBOR
MXP37,181,898
NPV=3.9/(1.025)t
= USD60,797,733
EXCHANGE:
SWAP DEALER
COMMODITY
USD LIBOR
SWAP DEALER
FORX
USD1=MXP10
=MXP607,977,330
MXP607,977,330=C/(1.02)t
MXP1,859,094,907=
C = MXP37,181,898
MXP37,181,898/0.02
USD156M
134