Forwards and Futures

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, and COMMODITY SWAPS. We will also see that
a combination of hedging with futures and
swapping the basis, leads to risk-free strategies.
2
SWAPSA 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 CAPITALNotional as in
conceptual
3
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
4

• 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.

5
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
6
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
7
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)(Lt30bps) 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.
8
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
9
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.
10
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.
11
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.
12

• 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

This calculation is based on the assumption that
the payment is every 182 days.
13

• The floating side is quoted as a money market
  yield
• basis. Therefore, the first payment is

Other future payments will be determined every 6
months by the six-month LIBOR at that time.
14
FIXED 7.19
Party A
Party B
FLOATING
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.
•  

15

• 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.

16

• 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.

17
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).
18
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.
19
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 FOR FIVE YEARS
20
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.
21
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.
22
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
23

• FIXED FOR FLOATING SWAP
• A DIRECT SWAP
• FIRM FIXED RATE FLOATING RATE
• F1 15 LIBOR 2
• F2 12 LIBOR 1
• notional 10M

LIBOR
12
LIBOR2
F1
F2
12
The result of the swap F1 pays fixed 14,
better than 15. F2 pays floating LIBOR,
better than LIBOR 1
24
2. AN INDIRECT SWAP FIRM FIXED RATE FLOATING
RATE F1 15 LIBOR 2 F2
12 LIBOR 1 The notional amount 10M
I
L25bps
L
L 2
12
F2
F1
12
12,25
F1 pays 14,25 fixed Better than 15. F2
pays L25bps Better than L1. The
Intermediary gains 50 bps 50,000.
25
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
I
L5bp L
F2
F1
L2
12
12 125bp
Clearly, there exist many other possible swaps
between the two firms in this example.
26

• 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.

27
EXAMPLE A RISK MANAGEMENT SWAP
MARKET BONDS
FL1 Floating rate 1. FL2 Floating rate 2.
FL1
LOAN
10
COUNTERPARTY A
BANK

FL2
LOAN
12
FIRM A BORROWS AT A FIXED RATE FOR 5 YEARS
28
THE BANKS 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.
29
EXAMPLE A RISK MANAGEMENT SWAP
MARKET SHORT TERM BOND
FL1
10
COUNETRPARTY a
BANK

FL2
FL2
FL1
COUNTERPARTY b
12
FIRM A
30
THE BANKS 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!
31

• 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
• Eurodollar futures exist.
• 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.

32

• 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.

33

• 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
• dealers mid rate.

34

• The estimation of a fair mid rate is
  complicated a bit by the facts that
• The convention is to quote swap coupons for
  generic swaps on a semiannual bond basis, and
• 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.

35

• 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 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.

36

• 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

37

• ?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

38
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
39

• 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
• SEP94 93.95 6.05 39,42 91

40

• 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.

41

• 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.

42

• 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 f4 is

43

• Step 2 and 3

The swaps coupon is the dealer mid rate. To this
rate , the dealer will add several basis points.
44
4.33s
FIXED
Swap dealer
Client
7.19
3-M LIBOR
FLOATING
LIBOR 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.

45
4.35s
FIXED
Swap dealer
Client
6-M LIBOR
FLOATING
46

• 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
  m36 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

47
Step 2 The Fixed Rate Equivalent effective
annual rate on a bond basis is FRE
(5.17)(365/360) 5.24.
48

• 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.

49
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.
50
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 dates Days between payment Dates Treasury Bills Prices B(0,T) Euro Dollar Deposit L(0,T)
t1 182 t2 365 t3 548 t4 730 182 183 183 182 .9679 .9362 .9052 .8749 .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.
51

• The fixed side of the swap.
• At the first payment date, t1, the dollar value
  of the payment is
•  
•  
• 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
•  

52

• 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.
  
•  

53

• 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

54

• 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.

55

• 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

56

• Arriving at the value of the floating
• rate payments requires more analysis.
• If the swap is already in existence, let ? denote
  the pre specified LIBOR rate. At date T1, the
  payment is
• and a new LIBOR rate is set.
• On T1, the value of the remaining floating rate
  payments is
•  
• NP NPL(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.  

57

• The total value of the floating rate payments at
  date T1 is 
•  

The value of the floating rate payments at date 0
is the PV of
58

• 2. If the swap is initiated at date 0, then the
  above equation simplifies as follows
• Let ?(0) denote the current LIBOR rate. By
  definition

59

• 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

Notice that this value can be positive ,zero, or
negative depending upon current rates. This
conclude the analysis of plain vanilla swap
valuation.
60

• 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.

61
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 dates Days between payment Dates Treasury Bills Prices B(0,T) Euro Dollar Deposit L(0,T)
t1 182 t2 365 t3 548 t4 730 182 183 183 182 .9679 .9362 .9052 .8749 .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.
62

• 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)
• NP1 - .8684
• The equality implies
• R/100 .1316/1.8421
• R 7.14 per annum.

63
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
64

• 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.

65

• 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.

66

• 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.

67

• 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

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.
68

• 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 330,000,000.6189281
  1
• DEM301,315,273.

69

• 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.

70

• 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.

71

• The cash flows are summarized in the
• following table
• Date USD CHF DEM
• 3.30.82 20,066,667 12,375,000 30,000,000
• 3.30.83 33,600,000 12,375,000 30,000,000
• 3.30.84 33,600,000 12,375,000 30,000,000
• 3.30.85 33,600,000 12,375,000 30,000,000
• 3.30.86 243,600,000 212,375,000 330,000,000
• YTM 8 11 16.8
• NPV 205,485,000 191,367,478 301,315,273
• 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

72

• 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.

73

• 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.

74
DIRECT SWAP FIXED FOR FIXED
9.35
F1
F2
11.5
11.5
U.S.A F2 DEPOSITS 72.5M IN COUNTERPARTYF1S
ACCOUNT IN NEW YORK CITY
Great Britain F1 BORROWS 50M AND DEPOSITS IT IN
COUNTERPARTY F2s ACCOUNT IN LONDON
At maturity, the original principals are
exchanged to terminate the swap.
75

• 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 partys 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

76
DIRECT SWAP FIXED FOR FIXED
9.55
F1
F2
11.25
11.5
U.S.A F2 DEPOSITS 72.5M IN COUNTERPARTYF1S
ACCOUNT IN NEW YORK CITY
Great Britain F1 BORROWS 50M AND DEPOSITS IT IN
COUNTERPARTY F2s ACCOUNT IN LONDON
At maturity, the original principals are
exchanged to terminate the swap.
77
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 B F2 PROJECT OF F1
COUNTRY A F1 PROJECT OF F2
78
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.
79
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
80
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.
81
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 CHIL
E BRAZIL CH1 12 R16 BR2 15 R17
82
DIRECT SWAP FIXED FOR FIXED
R15
CH1
BR2
12
R17
12
BRAZIL BR2 BORROWS R10M AND DEPOSITS IT IN CH1S
ACCOUNT IN SAO PAULO
CHILE CH1 BORROWS CP2.5B AND DEPOSITS IT IN BR2S
ACCOUNT IN SANTIAGO
CH1 pays R15 BR2 pays CP12 R2
83
INDIRECT SWAP FIXED FOR FIXED
INTERMEDIARY
R15.50
12
R17
14.50
CH1
BR2
R17
12
BRAZIL BR2 BORROWS R10M AND DEPOSITS IT IN CH1S
ACCOUNT IN SAO PAULO
CHILE CH1 BORROWS CP2.5B AND DEPOSITS IT IN BR2S
ACCOUNT IN SANTIAGO
84
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 intermediarys revenue
declines. When the exchange rate reaches
CP466,67/R the intermediary gain is zero. If the
Chilean Peso continues to depreciate the
intermediary loses money on the deal.
85
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.
86
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 GB2s account in
Mexico D.F., Mexico, While GB2 borrows
5,000,000 in Great Britain and deposits it in
MX1s account in London, Great Britain.
87
DIRECT SWAP FIXED FOR FLOATING
L 1
MX1
GB2
MP15
L 1
MP15
MEXICO MX1 BORROWS MP75M AND DEPOSITS IT IN
GB2S ACCOUNT IN MEXICO D.F., MEXICO.
ENGLAND GB2 BORROWS 5,000,000 AND DEPOSITS IT IN
MX1S ACCOUNT IN LONDON, GREAT BRITAIN
MX1 pays L1 GB2 pays MP15
88
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.
89

• 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 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.

90

• 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

91

• Domestic and Foreign Term Structure
• Maturity Price of a zero coupon Bond
  Months FF
• 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 French Francs is

92

• Therefore, the present value of the
• interest rate payments in U.S Dollars plus
• principal is

The coupon payment of the semi-annual interest
payments in U.S Dollars is
93

• Therefore, the present value of the
• interest rate payments in U.S Dollars plus
• principal is

Therefore, the value of the foreign currency
swap is
94
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.
95

• 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.

96

• 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.

97
Spot oil market
Spot Price
Oil
24.20/bbl
Swap Dealer
Refinery
Average Spot Price
The commodity 30,000 bbls.
98

• 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.

99
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. MCs
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.
100
MCs cash flow is - 9.50 Index 9.55
Index 0.05/UNIT
101
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.
102
FLOATING-FOR-FLOATING NATURAL GAS SWAP
INDEX2
INDEX1
producer
MC
USER
Gas
Gas
INDEX2 - 0.08
INDEX1
Swap Dealer
In this case MCs cash flow is (Index2)
(Index1) (Index1) ( Index2 - 0.08)
0.08/UNIT.
103

• 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.
•  

104

• The value of this payment at date 0 is the
  present value of V(t1, t1)
•  
• V(0, t1) PV0V(t1, t1)
• PV0S(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.

105

• 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 PV0S(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
• PV0S(t1) - F(0, t1)B(0, t1).

106

• 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
• PV0S(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.

107

• Repeating this argument for the remaining
  payments, it can be shown that the
• value of the commodity swap
• at date 0 is

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
zero-coupon bond prices, B(0, t1), all of which
are market prices observable at date 0.
108
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.
109
April 12 1145AM From BPs 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.
110
A PARTIAL SUMMARY
DATE SPOT
FUTURES
April 12 Buy from BM. Long August
NYMEX Sell to SST. Futures. F4,12 aug
3.87. August 12 (i) Buy NG from BM Short
August NYMEX S1 IF . Futures. (ii) Sell
NG to SST for Faug aug L3D S2 F4, 12
aug X PARTIAL CASH FLOW (F4,12 aug X)
IF L3D - F4,12 aug L3D X IF.
111

• How can BM eliminate the
• spread risk?
• BP decides to enter a spread swap.
• Clearly, this is a
• floating for floating swap.
• 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

112
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.
113
SUMMARY OF CASH FLOWS MARKET CASH
FLOW Spot 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 todays NYMEX - .03. I.e., 3.87 -
.06 3.81. THE END
114
THE BP EXAMPLE
SWAP DEALER
MARKET
SWAP
IF
L3D - .09
F4,12AUG - .06
IF
BM
SPOT
BP
SST
NG
NG
LONG F4,12AUG
SHORT L3D
FUTURES
NYMEX
115
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.
116

• 1. THE FUTURES SHORT HEDGE
• TIME CASH FUTURES BASIS
• 0 S0 F0,t B0 S0 -
  F0,t
• k Sk Fk, 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
SWAP DEALER
Bk,t
117

• 1. THE FUTURES LONG HEDGE
• TIME CASH FUTURES BASIS
• 0 S0 F0,t B0 S0 -
  F0,t
• k Sk Fk, Bk,t Sk - Fk,t
• 
  
• The purchasing price for the LONG hedger
• F0,t Bk,t .
• 2. THE SWAP OF THE LONG HEDGER

B0
LONG HEDGER
SWAP DEALER
Bk,t
118
1. PRICE RISK

2. BASIS RISK
3. NO RISK AT ALL THE CASH FLOW IS
THE CURRENT CASH PRICE!

FUTURES HEDGING
BASIS SWAP
119
BASIS SWAP
NYMEX
Buy gas at Screen - 10
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.