Forward, Futures and Swaps
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Exam questions
CONTENT
1. FORWARD AND FUTURES
1.1 Forward contract on a zero coupon (June, 1994)
1.2 Forward contract on the Bel-20 (June, 1995)
1.3 Notional Government Bond Futures Contract (June, 1995)
1.4 Government Bond forward and futures (June, 1996)
1.5 Forward contract on a currency (June 1997)
2. INTEREST RATE SWAP
2.1 Interest Rate Swap (June, 1994)
2.2 Currency swap (June, 1996)
2.3 Interest Rate Swap (June 1997)
Forward, Futures and Swaps
1. FORWARD AND FUTURES
1.1 Forward contract on a zero coupon (June, 1994)
The Belgian Automobile Company plans to issue commercial paper in 3
months from now in order to finance their car import and distribution activity.
Commercial paper is short-term promissory notes issued in the open market as an
obligation of the issuing entity. Like T-bills, it is a discount instrument, in other
words, a zero coupon.
BAC will issue, in 3 months from now, commercial paper with a face value
equal to BEF 100 mio and a maturity of 6-month after the date of issuance (it will
mature in 9 months from today). The rate on this transaction will depend on the level
of the 6-month interest rate prevailing in 3 months. Mr Delco, the finance director of
BAC wishes to lock in the interest rate that he will pay. He is considering entering a
forward transaction on this zero coupon.
The BIBOR yield curve prevailing today is as follows:
Maturity
Yield
(month)
(With continuous
compounding)
3
5.00%
6
5.40%
9
5.60%
12
5.70%
(1) Calculate the 6-month forward rate (with continuous compounding) for a
transaction starting in 3 months. Briefly explain the logic underlying your
calculation
Consider next the issuance in 3 months of commercial paper as a 3-month forward
sale of a 6-month zero coupon with a face value of BEF 100 mio.
(2) Calculate the spot price of this zero coupon and the 3-month forward price.
(3) Suppose that Banque Ducoin is willing to buy or sell forward in 3 months this zero
coupon for a delivery price of 96.5 mio. What arbitrage opportunity would this
provide for BAC ?
(4) Calculate the gain that BAC could achieve by realizing the arbitrage that you have
identified in the preceding question.
1.2 Forward contract on the Bel-20 (June, 1995)
The current date is June 1, 1995. Reuter indicates the following quotations for spot
and forward transaction on the Bel-20 index
Maturity
Price
Spot
1.450
September
1.460
December
1.520
The (continuous) interest on the Belgian market is 5%. . Assume,unless stated
otherwise, that no dividend will be paid on the Bel-20 shares until December.
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(1) Calculate the theoretical forward price of a September contract.
(2) Given the quoted price for the September contract, what arbitrage would you
initiate in order to make an immediate profit ?
(3) Suppose that you take a short position on a December contract. Two month later,
the spot price of the Bel-20 is 1.500. What would be the value of your contract
at that time ?
(4) What would be the theoretical forward price of a September contract if the
continuous dividend yield on the index is 4% ?
1.3 Notional Government Bond Futures Contract (June, 1995)
The current date is June 1, 1995. Consider a futures contract on the Belgian
Government (BGO) traded on Belfox with maturity at the end of September 1995.
The underlying asset is a notional bond with a face value of BEF 2,5 millions and a
coupon of 9%. The cheapest to deliver bond is OLO 94-2000 with a coupon of 7,75
and a final maturity at the end of december 2000. The current quoted price for this
bond is 105,18. The conversion factor for this bond is 0,9568
The interest rate (with simple compounding) is 6%.
(1) Calculate the cash price of the cheapest to deliver bond
(2) Compute the quoted futures price for the September BGO contract.
(3) Suppose that you have taken a long position on one BGO contract at the futures
price calculated in (2). One month later, the futures price is 102,50. What profit
or loss would you have realized over this period.
(4) Explain why the conversion factor for the cheapest to deliver bond is less than 1.
1.4 Government Bond forward and futures (June, 1996)
The current date is June 1, 1996. Your boss has asked you to provide information on
the December 1996 BGB Futures Contract that will start trading soon. The underlying
asset is a notional bond with a face value of BEF 2,5 millions and a coupon of 9%.
The deliverable bonds are the following:
Code
Maturity
Coupon
Quoted price
265
273
275
283
29.04.2004
31.03.2005
30.10.2004
15.05.2006
7,25
6,50
7,75
7,00
106,50
102,08
109,99
105,93
Concordance factor
(December 1996)
0,9082
0,8581
0,9314
0,8761
The interest rate (with continuous compounding) is 6%
(1) A client wishes to buy bond 273 forward at the end of December 1996. Calculate
the forward price (cash price) for this transaction.
(2) Suppose that the forward price (cash price) quoted on the market for a December
1996 forward contract on bond 273 is 104,00. Set up an arbitrage to take
advantage of this market quotation.
(3) Suppose that bond 265 is the cheapest to deliver bond for the BGB Futures
contract. Calculate the quoted futures price.
(4) If the quoted futures price is 117,53, what would be the cheapest to deliver bond?
Forward, Futures and Swaps
1.5 Forward contract on a currency (June 1997)
A corporation wants a six-month forward contract to buy 1 million US dollars.
The current BEF/dollar spot exchange rate is 36.0. The price of a domestic 6-month
Treasury bill is BEF 98.0199 per BEF 100 face value and the price of the equivalent
US instrument is USD 97.0446 per USD 100 face value.
(1) Calculate the current 6-month forward exchange rate. Briefly explain the logic
underlying your calculation.
(2) Suppose that a financial institution is willing to buy or sell forward USD against
BEF in 6 month at a delivery price of 35 BEF/USD. What arbitrage opportunity
would this create? Build an arbitrage table showing the profit at maturity that
could be realised per USD.
Suppose the corporation has bought USD forward at the forward exchange rate
calculated in question (1.1). Three month later, the spot exchange rate is 35
BEF/dollar and the 3-monthly continuously compounded interest rates are 4% in BEF
and 5% in USD.
(3) Calculate the value of the forward contract of the company.
(4) How could the company lock in the result on its initial forward contract?
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Forward, Futures and Swaps
2. INTEREST RATE SWAP
2.1 Interest Rate Swap (June, 1994)
Inventive Industry (2I) wishes to short a swap in which 2I will pay cash flows
based on a floating rate and receive cash flows based on a fixed rate. They request a
quote from BBW (Banque du Brabant Wallon) on the following plain vanilla swap:
Notional principal amount : BEF 500 mio
Maturity : 3 years
Floating index : 12-month BIBOR
Fixed coupon rate : ?%
Payment frequency : Annual
The BIBOR yield curve prevailing at the origination of this swap is as follow:
Maturity
Yield
(year)
(With continuous
compounding)
1
6.00%
2
7.00%
3
7.50%
(1) Describe the future cash flows between 2I and BBW for this swap.
(2) Consider first the swap as a pair of loan contracts. Find the values of these loans
and calculate the minimum fixed rate to be offered at 2I by BBW.
(3) Explain why the swap as can viewed as a portfolio of FRAs.
(4) Explain carefully the relationship between the value of the swap and the value of
the individual FRAs.
2.2 Currency swap (June, 1996)
One year ago, Microbel, a Belgian software company entered a currency swap with
Bank Vanden Hoek in order to hedge annual dollar denominated fees to be received
for the next 4 years from USCorp, a US company. The swap has a remaining life of 3
years and the next payments will take place in one year. It involves exchanging an
outflow of $3,5 million against an inflow of BEF 90 million once a year. The term
structure of interest rates in both Belgium and the United States is currently flat and
the interest rates are 4 percent in Belgian franc and 6 percent in dollars. All interest
rates are quoted with annual compounding. The current spot exchange rate is BEF 32
per $.
(1) Explain how to decompose this swap into two bonds.
(2) Calculate the value of this swap based on this decomposition
(3) Explain the decomposition of the swap into forward contracts
(4) Compute the value of the individual forward contracts. Compare your result with
you answer to question (2.2)
2.3 Interest Rate Swap (June 1997)
A firm has bought 3 years ago a 5-years interest rate swap for which it receives
floating payments and pays fixed payments once a year on June 13. The notional
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Forward, Futures and Swaps
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amount of the swap is BEF 100 millions and the swap rate is 5.75 percent. The last
payment took place yesterday and the swap has a remaining life of 2 years.
The current term structure of interest rate (with continuous compounding) is:
Maturity
(year)
1
2
Yield on zero-coupon
(with continuous compounding)
3.00%
3.50%
(1) Calculate the value of this swap based on a decomposition into two bonds.
(2) Decompose this swap into FRAs. Calculate the value of each individual FRA.
Compare your result with your answer to question (2.1).
(3) Show how to value the swap using the current forward interest rates.
(4) Calculate the current 2-year swap rate prevailing on the market. Show how to use
this rate to value the “old” swap above.
Forward, Futures and Swaps
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Forward, Futures and Swaps
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SOLUTIONS
1. FORWARD AND FUTURES
1.1 Forward contract on a zero coupon (June, 1994)
(1) Continuous 6-month forward rate in 3 months = 5.90%
(2) Spot price of zero-coupon = 95.89
Forward price of zero-coupon = 97.09
(3) Value of contract quoted by Banque Ducoin = 0.586
Contract undervalued  Reverse cash and carry arbitrage
(4) Profit at maturity from the arbitra ge = 0.59
1.2 Forward contract on the Bel-20 (June, 1995)
(1) F = 1,468.24
(2) Reverse cash and carry Immediate profit = 8.14
(3) f = -5.12
(4) F = 1,453.63 if q=4%
1.3 Notional Government Bond Futures Contract (June, 1995)
(1) Cash price of CTD = 105.18 + 3.23 = 108.41
(2) Quoted futures price BGO = 109.60
(3) Fbp  Tick = -710.35 bp  250 = -177,588
(4) Conversion factor of CTD < 1 because coupon of CTD < 9%
1.4 Government Bond forward and futures (June, 1996)
(1) 7 month forward price for bond 273:
Cash price = Quoted price + Accrued interest = 102.08 + 6.5  (2/12) = 103.16
Forward price = 103.16  exp(.06  7/12) = 106.84
(2) 104 < 106.84  reverse cash and carry
Today: Sell spot (+103.16) and invest (-103.16) + Buy forward at 104. Total CF = 0
Dec. 96 : -ST + 106.84 + (ST - 104) = 2.84
(3) If 265 = CTD
Cash price = 106.50 + 7.25  (1/12) = 107.11
Forward price = 107.11  exp(.06  7/12) = 110.92
Clean forward price = 110.92 - 7.25  (8/12) = 106.09
Futures price = Clean forward price / Concordance factor = 116.81
(4) If quoted futures price = 117,53: CTD = 265
265
273
275
283
S-kF
1.22
0.52
2.96
-0.24
1.5 Forward contract on a currency (June 1997)
(1) Forward exchange rate calculation
Forward exchange rate = SpotXrate  DF$ / DFBEF
SpotXrate  DF$ = S = BEF value of a $ denominated zero-coupon (underlying asset)
ForwardXrate = future value of S
36  0.970446 / 0.980199 = 35.642
or : Forward exchange rate = SpotXrate  exp((rBEF-r$)(T-t))
Forward, Futures and Swaps
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0.980199 = exp(-rBEF  0.5)  rBEF = ln(1/0.980199)/0.5 = 4.00%
0.970446 = exp(-r$  0.5)  r$ = ln(1/0.970446)/0.5 = 6.00%
(2) Arbitrage : If quoted delivery price = 35 BEF/$ : reverse cash and carry arbitrage
t=0
t=0.5
Short US zero coupon
+ 34.94 = 35  0.970446
- 1  XT
(=borrow 1 $ and buy BEF)
Invest at BEF risk-free rate
- 34.94
35.94 = 34.94 /0.980199
Buy $ forward against BEF
0
XT – 35
at 35 BEF/$
Total
0
0.64
(XT = SpotXrate at maturity)
(3) Value of forward contract
Value of forward contract f = S – PV(K)
S = BEF value of US zero coupon
= 35  exp(-0.05  .25) = 34.57
PV(K) = 35.642  exp(-0.04  0.25) = 35.29
f = -0.72
(4) To lock in the result on its initial forward contract, sell USD at new forward price
New forward exchange rate = 34.91
Resultat at maturity = 34.91 – 35.642 = -0.73
PV = - 0.72
2. INTEREST RATE SWAP
2.1 Interest Rate Swap (June, 1994)
(2) Swap rate = 7.72%
(4) Swap can be decomposed into 3 FRAs having the following values
Settlement date
0
1
Value
-7.24
2.64
2
4.60
2.2 Currency swap (June, 1996)
(1) Decomposition: Long BEF bond + Short USD bond
(2) Bond value = Annual Cash Flow  Annuity factor
Long BEF bond = 90  2.775091
= 249.76 BEF
Short USD bond = (3.5  2.673012)  32 = 299.38 BEF
Swap value
= - 49.62
(3) Swap can be decomposed into 3 forward sales of USD against BEF at an exchange
rate of 90/3.5=25.71 BEF/USD.
Maturity
1
2
3
Swap exch. rates
25.71
25.71
25.71
Fwd exch. rates
31.40
30.80
30.22
Gain/loss
-19.89
-17.81
-15.78
PV
-19.12
-16.47
-14.03
Swap value = -49.62
Forward, Futures and Swaps
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2.3 Interest Rate Swap (June 1997)
(1) Value of floating rate note = 100
Value of fixed rate note = 104.18
Value of swap = - 4.18
(2) Swap can be decomposed into 2 FRA with settlement date at time 0 and 1
Value of FRA1 = 100 – 102.62 = -2.62
Value of FRA2 = 97.04 – 98.60 = -1.56
Value of swap = -2.62 – 1.56 = -4.18
(3) Valuation based on forward interest rates
1
2
1-year forward rate
3.05%
4.08%
(with annual compounding)
Swap rate
5.75%
5.75%
Gain/loss of swap
-2.70
-1.67
Present value
-2.62
-1.56
Value of swap = -4.18
(4) New swap rate = 3.55%
New swap (short, receive fix)
Old swap (long, pay fix)
Present value
Swap value - -4.18
1
3.55
-5.75
-2.13
2
3.55
-5.75
-2.05