Centre Interuniversitaire D'analyse Financière

Forward, Futures and Swaps1

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)

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:

(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

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.

(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:

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?

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?

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:

(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 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:

(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.

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 arbitrage = 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

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))

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

(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

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.

Swap value = -49.62

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

Value of swap = -4.18

(4) New swap rate = 3.55%

Swap value - -4.18