Interest Rate Futures:

1
Chapter 13

• Interest Rate Futures
• Applications and Pricing

2
Applications and Pricing

• Hedging debt positions
• Speculative positions
• Managing asset and liability positions
• Formation of synthetic fixed-rate and
  floating-rate debt and investment positions
• Pricing of futures contracts using the
  carrying-cost model
• Use of foreign currency futures contracts to
  hedge international investment and debt position
  against exchange-rate risk

3
Hedging

• Naïve Hedge
• Cross Hedge

4
Naïve Hedging Model

• The simplest model to hedge a debt position is to
  use a naive hedging model.
• For debt positions, a naive hedge can be formed
  by hedging each dollar of the face value of the
  spot position with one market-value dollar in the
  futures contract.
• A naive hedge also can be formed by hedging each
  dollar of the market value of the spot position
  with one market-value dollar of the futures.

5
Long Hedge - Future 91-Day T-Bill Investment

• Consider the case of a treasurer of a corporation
  who is expecting a 5 million cash inflow in June
  that she is planning to invest in T-bills for 91
  days.
• If the treasurer wants to lock in the yield on
  the T-bill investment, she could do so by going
  long in June T-bill futures contracts.

6
Long Hedge - Future 91-Day T-Bill Investment

• Example If the June T-bill contract were trading
  at the index price of 95, the treasurer could
  lock in a yield (YTMf) of 5.1748 on a 91-day
  investment made at the futures' expiration date
  in June

7
Long Hedge - Future 91-Day T-Bill Investment

• To obtain the 5.1748 yield, the treasurer would
  need to form a hedge in which she bought nf
  5.063291 June T-bill futures contracts (assume
  perfect divisibility)

nf Investment in June 5,000,000 5.063291
Long Contracts f0
987,500
8
Long Hedge - Future 91-Day T-Bill Investment

• At the June expiration date, the treasurer would
  close the futures position at the price on the
  spot 91-day T-bills.
• If the cash flow, CF, from closing is positive,
  the treasurer would invest the excess cash in
  T-bills.
• If it is negative, the treasurer would cover the
  shortfall with some of the anticipated cash
  inflow earmarked for purchasing T-bills.

9
Long Hedge - Future 91-Day T-Bill Investment

• Hedge Relation

10
Long Hedge - Future 91-Day T-Bill Investment

• Suppose at the June expiration, the spot 91-day
  T-bill rate is at 4.5.
• The manager would find T-bill prices higher at
  989,086 but would realize a profit of 8,030.38
  from closing the futures position.
• Combining the profit with the 5M CF, the
  manager would be able to buy 5.063291 T-bills and
  earn a rate off the 5M investment of 5.1748.

11
Long Hedge - Future 91-Day T-Bill Investment
12
Long Hedge - Future 91-Day T-Bill Investment

• Suppose at the June expiration, the spot 91-day
  T-bill rate is at 5.5.
• The manager would find T-bill prices lower at
  986,740 but would realize a loss of 3,848 from
  closing the futures position.
• With the inflow of 5 million, the treasurer
  would need to use 3,848 to settle the futures
  position, leaving her only 4,996,152 to invest
  in T-bills.
• However, with the price of the T-bill lower in
  this case, the treasurer would again be able to
  buy 5.063291 T-bills, and therefore realize a
  5.1748 rate of return from the 5 million
  investment.

13
Long Hedge - Future 91-Day T-Bill Investment
14
Long Hedge - Future 91-Day T-Bill Investment

• Note, the hedge rate of 5.1748 occurs for any
  rate scenario.

15
Long Hedge - Future 182-Day T-Bill Investment

• Case
• Money market manager is expecting a 5M CF in
  June that she plans to invest in a 182-day
  T-bill.
• Since the T-bill underlying a futures contract
  has a maturity of 91 days, the manager would need
  to go long in both a June T-bill futures and a
  September T-bill futures (note there is
  approximately 91 days between the contract) in
  order to lock in a return on a 182-day T-bill
  investment.

16
Long Hedge - Future 182-Day T-Bill Investment

• If June T-bill futures were trading at IMM of 91
  and September futures were trading at IMM of
  91.4, then the manager could lock in a 9.3 rate
  on an investment in 182-day T-bills by going
  long in 5.115 June T-bill futures and 5.11
  September contracts.

17
Long Hedge - Future 182-Day T-Bill Investment
18
Long Hedge - Future 182-Day T-Bill Investment

• Suppose in June, the spot 91-day T-bill rate is
  at 8 and the spot 182-day T-bill rate is at
  8.25.
• At these rates, the price on the 91-day spot
  T-bill would be 980,995, the price on the
  182-day spot would be 961,245, and if the
  carrying-cost model holds, the price on the
  September futures would be 979,865.
• At these prices, the manager would be able to
  earn a profit of 24,852 from closing both
  futures contract (which offsets the higher T-bill
  futures prices) and would be able to buy 5.227
  182-day T-bills, yielding a rate of 9.3 from a
  5M investment.

19
Long Hedge - Future 182-Day T-Bill Investment
20
Long Hedge - Future 182-Day T-Bill Investment

• Suppose in June, the spot 91-day T-bill rate is
  at 10 and the spot 182- day T-bill rate is at
  10.25.
• At these rates, the price on the 91-day spot
  T-bill would be 976,518, the price on the
  182-day spot would be 952,508, and if the
  carrying-cost model holds, the price on the
  September futures would be 975,413.
• At these prices, the manager would incur a loss
  of 20,798 from closing both futures contracts.
  However, with lower T-bill futures prices, the
  manager would still be able to buy 5.227 182-day
  T-bills, yielding a rate of 9.3 from a 5M
  investment.

21
Long Hedge - Future 182-Day T-Bill Investment
22
Short Hedge Managing the Maturity Gap

• In June, a bank makes a 1M loan for 180 days
  that it plans to finance by selling a 90-day CD
  now at the LIBOR of 8.258 and a 90-day CD
  ninety days later (in September) at the LIBOR
  prevailing at that time.
• To minimize its exposure to market risk, the bank
  goes short in 1.03951 September Eurodollar
  futures at 92.4 (IMM).
• By doing this, the bank is able to lock in a rate
  on its CD financing for 180 days of 8.17.

23
Managing the Maturity Gap

• Bank sells 1M of CD now (June) at 8.258. At
  the September
• maturity, the bank would owe 1,019,758.
• To hedge this liability, the bank would go short
  in 1.03951 Eurodollar
• futures at 981,000.

24
Managing the Maturity Gap

• In September, the bank will sell a new 90-day CD
  at the prevailing LIBOR to finance its
  1.019758M debt on the maturing CD plus (minus)
  any debt (profit) from closing its short
  September Eurodollar futures position.
• If the LIBOR rate is higher, the bank will have
  to pay greater interest on the new CD, but it
  will realize a profit on its futures that, in
  turn, will lower the amount of funds it needs to
  finance.
• On the other hand, if the LIBOR is lower, then
  the bank will have lower interest payment on its
  new CD, but it will also incur a loss on its
  futures position and therefore have more funds
  that need to be financed.

25
Managing the Maturity Gap

• As shown in the exhibit on the next slide, at a
  September LIBORs of 7.5 or 8.7, the banks
  total debt at the end of the 180-day period will
  be 1,039,509, which equates to a rate of 8.17.
• Note This is true for any rate.

26
Managing the Maturity Gap
27
Cross Hedge Price-Sensitivity Model

• Cross Hedging is hedging a position with a
  futures contract in which the asset underlying
  the futures is different than the asset to be
  hedged.
• Example
• Future CP sale hedged with T-bill futures
• AA Bond portfolio hedged with T-bond futures

28
Cross Hedge Price-Sensitivity Model

• One model used for cross hedging is the
  price-sensitivity model developed by Kolb and
  Chiang (1981) and Toers and Jacobs (1986)).
• This model has been shown to be relatively
  effective in reducing the variability of debt
  positions.
• The model determines the number of futures
  contracts that will make the value of a portfolio
  consisting of a fixed-income security and an
  interest rate futures contract invariant to small
  changes in interest rates.

29
Cross Hedge Price-Sensitivity Model
30
Cross Hedging Example Hedging a Future CP Issue
with T-bill Futures

• A company plans to sell a 182-day CP issue with a
  10M principal in June to finance its anticipated
  accounts receivable.
• The company would like to lock in the current CP
  rate of 6, ensuring it of funds from the CP sale
  of 9.713635M.
• Using the price-sensitivity model, the company
  locks in a rate by going short in 20 June T-bill
  futures contracts at IMM index 95.

31
Cross-Hedging Example Hedging a Future CP Issue
with T-Bill Futures
32
Cross Hedging Example Hedging a Future CP Issue
with T-bill Futures

• If CP sold at a discount yield that was 25 BP
  greater than the discount yield on T-bills, then
  the company would be able to lock in a rate on
  its CP of 5.48 when it sold its CP and closed
  its futures position (assuming the time of the CP
  sale and T-bill futures expiration are the same).

33
Cross Hedging Example Hedging a Future CP Issue
with T-bill Futures
34
Cross Hedging Example Hedging a Future AAA Bond
Sale Issue with T-bill Futures

• Bond portfolio manager plans to sell AA bond
  portfolio in June. Currently, the fund has the
  following features
• Current Value 1.02M,
• YTM 11.75
• Duration 7.66 years
• Weighted Average Maturity 15 years.

35
Cross Hedging Example Hedging a Future AAA Bond
Sale Issue with T-bill Futures

• Suppose the manager is considering hedging the
  portfolio against interest rate changes by going
  short in June T-bond futures contracts currently
  trading at f0 72 16/32 with the T-bond most
  likely to be delivered on the contract having the
  following features
• YTM 9,
• Maturity 18 years
• Duration of 7 years
• Using the Price-Sensitivity Model, the portfolio
  manager could hedge the bond portfolio by selling
  14 futures contracts.

36
Cross Hedging Example Hedging a Future AAA Bond
Sale Issue with T-Bond Futures
37
Cross Hedging Example Hedging a Future AAA Bond
Sale Issue with T-bill Futures

• If the manager hedges the bond portfolio with 14
  June T-bond short contracts, she will be able to
  offset changes in the bond portfolio's value
  resulting from interest rate changes.

38
Cross Hedging Example Hedging a Future AAA Bond
Sale Issue with T-bill Futures

• Example, suppose interest rates increased from
  January to mid-May causing the price of the bond
  portfolio to decrease from 102 to 95 and the
  futures price on the June T-bond contract to
  decrease from 72 16/32 to 68 22/32.
• In this case, the fixed-income portfolio would
  lose 70,000 in value (decrease in value from
  1,020,000 to 950,000).
• This loss, though, would be partially offset by a
  profit of 53,375 on the T-bond futures position
  Futures Profit 1472,500 - 68,687.50
  53,375.
• Thus, by using T-bond futures the manager is able
  to reduce some of the potential losses in her
  portfolio value that would result if interest
  rates increase.

39
Speculating with Interest Rate Futures

• While interest rate futures are extensively used
  for hedging, they are also frequently used to
  speculate on expected interest rate changes.
• A long futures position is taken when interest
  rates are expected to fall.
• A short position is taken when rates are expected
  to rise.

40
Speculating with Interest Rate Futures

• Speculating on interest rate changes by taking
  such outright or naked futures positions
  represents an alternative to buying or short
  selling a bond on the spot market.
• Because of the risk inherent in such outright
  futures positions, though, some speculators form
  spreads instead of taking a naked position.
• A futures spread is formed by taking long and
  short positions on different futures contracts
  simultaneously.

41
Speculating with Interest Rate Futures

• Outright Positions
• Long Expect rates to decrease
• ST Rates use T-bills or Eurodollar futures
• LT Rates use T-bonds or T-note futures
• Short Expect rates to increase
• ST Rates use T-bills or Eurodollars futures
• LT Rates use T-bonds or T-note futures

42
Speculating with Interest Rate Futures

• Spread
• Intracommodity Spread long and short in futures
  on the same underlying asset but with different
  expirations.
• Intercommodity Spread Long and short in futures
  with different underlying assets but the same
  expiration.

43
Intracommodity Spread

• More distant futures contracts (T2) are more
  price-sensitive to changes in the spot price than
  near-term futures contracts (T1)

44
Intracommodity Spread

• A speculator who expected the interest rate on
  long-term bonds to decrease in the future could
  form an intracommodity spread by going
• long in a longer-term T-bond futures contract and
  
• short in a shorter-term one.
• This type of spread will be profitable if the
  expectation of long-term rates decreasing occurs.

45
Intracommodity Spread

• That is, the increase in the T-bond price
  resulting from a decrease in long-term rates,
  will cause the price on the longer-term T-bond
  futures to increase more than the shorter-term
  one. As a result, a speculators gains from his
  long position in the longer-term futures will
  exceed his losses from his short position.
• If rates rise, though, losses will occur on the
  long position these losses will be offset
  partially by profits realized from the short
  position on the longer-term contract

46
Intracommodity Spread

• If a bond speculator believed rates would
  increase but did not want to assume the risk
  inherent in an outright short position, he could
  form a spread with
• a short position in a longer term contract and
• a long position in the shorter term one.

47
Intracommodity Spread

• Note that in forming a spread, the speculator
  does not have to keep the ratio of long- to-short
  positions one-to-one, but instead could use any
  ratio (2-to-1, 3-to-2, etc.) to give him his
  desired return-risk combination.

48
Intercommodity SpreadRate-Anticipation Swap

• Consider the case of a spreader who is
  forecasting a general decline in interest rates
  across all maturities (i.e., a downward parallel
  shift in the yield curve).
• Since bonds with greater maturities are more
  price sensitive to interest rate changes than
  those with shorter maturities, a speculator could
  set up a rate-anticipation swap by going long in
  the longer-term bond with the position partially
  hedged by going short in the shorter-term one.

49
Intercommodity SpreadRate-Anticipation Swap

• Instead of using spot securities, the specualtor
  alternatively could form an intercommodity spread
  by going long in a T-bond futures contract that
  is partially hedge by a short position in a
  T-note (or T-bill) futures contract.
• On the other hand, if an investor were
  forecasting an increase in rates across all
  maturities, instead of forming a
  rate-anticipation swap with spot positions, she
  could go short in the T-bond futures contract and
  long in the T-note.
• Forming spreads with T-note and T-bond futures is
  one of the more popular intercommodity spread
  strategies it is referred to as the NOB strategy
  (Notes over Bonds).

50
Intercommodity SpreadQuality Swap

• Another type of intercommodity spreads involves
  contracts on bonds with different default risk
  characteristics it is an alternative to a
  quality swap.
• For example, a spread formed with futures
  contracts on a T-bond and a Municipal Bond Index
  (MBI) or contracts on T-bills and Eurodollar
  deposits.
• Like quality swaps, profits from these spreads
  are based on the ability to forecast a narrowing
  or a widening of the spread between the yields on
  the underlying bonds.

51
Intercommodity SpreadQuality Swap

• For example, in an economic recession the demand
  for lower default-risk bonds often increases
  relative to the demand for higher default-risk
  bonds.
• If this occurs, then the spot yield spread for
  lower grade bonds over higher grade would tend to
  widen.
• A speculator forecasting an economic recession
  could, in turn, profit from an anticipated
  widening in the risk premium by forming an
  intercommodity spread consisting of a long
  position in a T-bond futures contract (no default
  risk) and short position in a MBI contract (some
  degree of default risk).

52
Intercommodity SpreadQuality Swap

• Similarly, since Eurodollar deposits are not
  completely riskless, while T-bills are, a
  spreader forecasting riskier times (and the
  resulting widening of the spread between
  Eurodollar rates and T-bill rates) could go long
  in the T-bill contract and short in the
  Eurodollar contract.
• A spread with T-bills and Eurodollars contracts
  is known as a TED spread.

53
Managing Asset and Liability Positions

• Interest rate futures can also be used by
  financial and non-financial corporations to alter
  the exposure of their balance sheets to interest
  rate changes. The change can be done for
• Speculative purposes increasing the firms
  exposure to interest rate changes
• Hedging purposes reducing the firms exposure to
  interest rate changes.

54
Managing Asset and Liability Positions

• Example
• Consider an insurance company that as a matter of
  policy maintains an immunized position in which
  the duration of its bond portfolio is equal to
  the duration of its liabilities DA DL.
• With a duration gap of zero, DA - DL 0, the
  companys economic surplus is invariant to
  interest rate changes.
• Suppose, though, that the managers expect rates
  will fall across all maturities in the future and
  would like to change the insurance companys
  interest rate exposure to a moderately
  speculative one in which the company has a
  positive duration gap DA - DL gt 0.
• As noted in Chapter 8, one way for the company to
  do this would be to increase the duration of its
  bond portfolio by changing the allocation sell
  short-term bonds and buy long-term ones.

55
Managing Asset and Liability Positions

• An alternative to this expensive strategy would
  be to take a long position in T-bond futures.
• If rates decrease as expected, then the value of
  the companys bond portfolio would increase and
  it would also profit from its long futures
  position.
• If rates were to increase, then the company would
  see not only a decline in the value of its bond
  portfolio but also losses on it futures position.
  
• Thus, by adding futures the company has
  effectively increased its balance sheets
  interest rate exposure by creating a positive
  duration gap.

56
Managing Asset and Liability Positions

• Instead of increasing its balance sheets
  exposure to interest rate changes, a company may
  choose to reduce it.
• For example, a company with a positive duration
  gap and a concern over futures interest rate
  increases could reduce the gap by taking a short
  position in an interest rate futures contract.

57
Managing Asset and Liability Positions

• This method of hedging or speculating in which
  the original composition of assets and
  liabilities is not changed is referred to as
  off-balance sheet restructuring.

58
Synthetic Debt and Investment Positions

• There are some cases in which the rate on debt
  and investment positions can be improved by
  creating synthetic positions with futures and
  other derivative securities such as swaps.
• These cases involve
• Creating a synthetic fixed-rate loan by combining
  a floating-rate loan with short positions in a
  series of Eurodollar futures contracts
• Creating a synthetic floating-rate loan by
  combining a fixed-rate loan with long positions
  in a series of Eurodollar futures contracts
• Creating synthetic fixed-rate investment by
  combining an investment in a floating-rate note
  with a long position in a series of Eurodollar
  futures
• Creating floating-rate investment by combining an
  investment in a fixed-rate note with a short
  position in a series of Eurodollar futures.

59
Synthetic Debt and Investment Positions

• Note
• In practice, exchange-traded interest rate
  futures contracts are usually priced so that such
  opportunities dont exist.
• That is, if the equilibrium carrying-cost model
  governing interest rate futures prices holds,
  then the rate on synthetic positions will be
  equal to the rate on the spot.

60
Synthetic Fixed-Rate Loan

• A corporation wanting to finance its operations
  or capital expenditures with fixed-rate debt has
  a choice of either a direct fixed-rate loan or a
  synthetic fixed-rate loan formed with a
  floating-rate loan and short positions in
  Eurodollar futures contracts, whichever is
  cheaper.
• Consider the case of a corporation that can
  obtain a one-year, 1M fixed-rate loan from a
  bank at 11 or alternatively can obtain a
  one-year, floating-rate loan from a bank.

61
Synthetic Fixed-Rate Loan

• In the floating-rate loan, suppose the loan
  starts on 9/20 with the rate at 11.25 and is
  then reset on 12/20, 3/20, and 6/20 at the
  prevailing LIBOR plus 150 BP.
• The company fixes the floating rate by going
  short in a series of Eurodollar futures
  (Eurodollar strip).

62
Synthetic Fixed-Rate Loan

• Suppose the company goes short in Eurodollar
  contracts with expirations of 12/20, 3/20, and
  6/20 and the following prices

63
Synthetic Fixed-Rate Loan

• By doing this, the company is able to lock in a
  fixed rate of 10.12

64
Synthetic Fixed-Rate Loan

• For example, if the LIBOR is at 9 on date 12/20,
  the company will have to pay 26,250 on its loan
  the next quarter but it will also have a profit
  on its 12/20 Eurodollar futures of 1,250 that it
  can use to defray part of the interest expenses,
  yielding an effective hedged rate of 10.

65
Synthetic Fixed-Rate Loan
66
Synthetic Fixed-Rate Loan

• If the LIBOR is at 6 on date 12/20, the company
  will have to pay only 18,750 on its loan the
  next quarter but it will also have to cover a
  loss on its 12/20 Eurodollar futures of 6,250.
  The payment of interest and the loss on the
  futures yields an effective hedged rate of 10.

67
Synthetic Fixed-Rate Loan
68
Synthetic Floating-Rate Loan

• A synthetic floating-rate loan is formed by
  borrowing at a fixed rate and taking a long
  position in a Eurodollar or T-bill futures
  contract.

69
Synthetic Floating-Rate Loan

• For example, suppose the corporation in the
  preceding example had a floating-rate asset and
  wanted a floating-rate loan instead of a fixed
  one.
• Suppose the corporation could take a
  floating-rate loan at LIBOR plus 200 BP or it
  could form a synthetic floating-rate loan by
  borrowing at a fixed rate for one year and going
  long in a series of Eurodollar futures expiring
  at 12/20, 3/20, and 6/20

70
Synthetic Floating-Rate Loan

• The synthetic loan will provide a lower rate than
  the direct floating-rate loan if the fixed rate
  is less than 10.5.
• For example, suppose the corporation borrows at a
  fixed rate of 10 for one year with interest
  payments made quarterly at dates 12/20, 3/20, and
  6/20 and then goes long in the series of
  Eurodollar futures to form a synthetic
  floating-rate loan.

71
Synthetic Floating-Rate Loan

• On date 12/20, if the settlement LIBOR were 9
  (settlement index price of 97.75 and a closing
  futures price of 977,500), the corporation would
  lose 1,250 ( (977,500 - 978,750)) from its
  long position on the 12/20 futures contracts and
  would pay 25,000 on its fixed-rate loan
  ((.10/4)(1M) 25,000).
• The companys hedged annualize rate would be
  10.5 (4(25,000 1,250)/1,000,000 .105),
  which is .5 less than the rate paid on the
  floating-rate loan (LIBOR 200BP 9 2.0
  11).

72
Synthetic Floating-Rate Loan

• If the settlement LIBOR were 6 (settlement
  index price of 94 and a closing futures price of
  985,000), the corporation would realize a profit
  of 6,250 ( (985,000 - 978,750) from the long
  position on the 12/20 futures contracts and would
  pay 25,000 on its fixed-rate loan.
• Its hedged annualize rate would be 7.5
  ((4)(25,000 - 6,250))/1,000,000 .075), which
  again is .5 less than the rate on the floating-
  rate loan (LIBOR 200BP 6 2.0 8.0).

73
Synthetic Investment

• Futures can also be used on the asset side to
  create synthetic fixed and floating rate
  investments.
• An investment company setting up a three-year
  unit investment trust offering a fixed rate could
  invest funds either in three-year fixed-rate
  securities or a synthetic one formed with a
  three-year floating-rate note tied to the LIBOR
  and long positions in a series of Eurodollar
  futures, which ever yields the higher rate.

74
Synthetic Investment

• An investor looking for a floating-rate security
  could alternatively consider a synthetic
  floating-rate investment consisting of fixed-rate
  security and a short Eurodollar strip.

75
Futures Pricing

• The underlying asset price on a futures contract
  primarily depends on the spot price of the
  underlying asset.
• The difference between the futures (or forward
  price) and the spot price is called the basis
  (Bt)

76
Futures Pricing

• Note By definition a normal futures market is
  defined as one with a positive basis, while an
  inverted futures market is defined as one with a
  negative basis

77
Futures Pricing

• For most futures (and forward) contracts, the
  futures price exceeds the spot price before
  expiration and approaches the spot price as
  expiration nears.
• Thus, the basis usually is positive and
  decreasing over time, equaling or nearing zero at
  expiration (BT 0).
• Futures and spot prices also tend to be highly
  correlated with each other, increasing and
  decreasing together their correlation, though,
  is not perfect.
• As a result, the basis tends to be relatively
  stable along its declining trend, even when
  futures and spot prices vacillate.

78
Carrying-Cost Model

• The relationship between the spot price and the
  futures or forward price can be explained by the
  carrying-cost model (or cost of carry model).
• In this model, arbitrageurs ensure that the
  equilibrium forward price is equal to the net
  costs of carrying the underlying asset to
  expiration.
• The model is used to explain what determines the
  equilibrium price on a forward contract. However,
  if short-term interest rates are constant, the
  carrying-cost model can be extended to pricing
  futures contracts.

79
Carrying-Cost Model

• In terms of the carrying-cost model, the price
  difference between futures and spot prices can be
  explained by the costs and benefits of carrying
  the underlying asset to expiration.
• For futures on debt securities
• The carrying costs include the financing costs of
  holding the underlying asset to expiration.
• The benefits include the coupon interest earned
  from holding the security.

80
Carrying-Cost Model

• To illustrate the carrying-cost model consider
  the pricing of a T-bill futures contract.
• With no coupon interest, the underlying T-bill
  does not generate any benefits during the holding
  period and the financing costs are the only
  carrying costs.

81
Pricing T-Bill FuturesCarrying Cost Model
82
Pricing T-Bill Futures

• Example
• If the rate on a 161-day spot T-bill is 5.7 and
  the repo rate (or RF rate ) for 70 days is 6.38,
  then the price on a T-bill futures contract with
  an expiration of 70 days would be 98.74875

83
Pricing T-Bill Futures

• The futures price is governed by arbitrage. If
  the market price does not equal f, then
  arbitrageurs would take a position in the futures
  and an opposite position in the spot.
• This arbitrage strategy is referred to as a cash
  and carry arbitrage.

84
Pricing T-Bill Futures

• Example Suppose f M 99
• An arbitrageur would go short in the futures,
  agreeing to sell a 91-day T-bill for 99 seventy
  days later and would go long in the spot,
  borrowing 97.5844 at 6.38 for 70 days to finance
  the purchase of the 161-day T-bill that is
  trading at 97.5844.
• Seventy days later (expiration), the arbitrageur
  would sell the bill (which now would have a
  maturity of 91 days) on the futures for 99 (fM)
  and pay off his financing debt of 98.74875 (f),
  realizing a cash flow of 2,512.50.

85
Pricing T-Bill Futures
86
Pricing T-Bill Futures

• Note at f M 99, a money market manager planning
  to invest for 70 days in a T-bill at 6.38 could
  earn a greater return by buying a 161-day bill
  and going short in the 70-day T-bill futures to
  lock in the selling price.

87
Pricing T-Bill Futures

• For example, using the above numbers, if a money
  market manager were planning to invest 97.5844
  for 70 days, she could buy a 161-day bill for
  that amount and go short in the futures at 99.
• Her return would be 7.8, compared to only 6.38
  from the 70-day T-bill

88
Pricing T-Bill Futures

• Example Suppose f M 98
• An arbitrageur would go long in the futures,
  agreeing to buy a 91-day T-bill for 98 seventy
  days later and would go short in the spot,
  borrowing the 161-day T-bill, selling it for
  97.5844 and investing the proceeds at 6.38 for
  70 days.
• Seventy days later (expiration), the arbitrageur
  would buy the bill (which now would have a
  maturity of 91 days) on the futures for 98 (fM),
  use the bill to close his short position, and
  collect 98.74875 (f) from his investment,
  realizing a cash flow of 7,487.50

89
Pricing T-Bill Futures
90
Pricing T-Bill Futures

• Note at f M 98, a money market manager with a
  161-day T-bill could earn an arbitrage by selling
  the bill for 97.5844 and investing the proceed at
  6.38 for 70 days, then going long in the 70-day
  T-bill futures.
• Seventy days later, the money market manager
  would receive 98.74875 from the investment and
  would pay 98 on the futures to reacquire the bill
  for a CF of .74875 (per 100 face value).

91
Pricing T-Bill Futures Implications

• Implication 1
• If the carrying-cost model holds, then the spot
  rate on a 70-day bill (repo rate) will be equal
  to the synthetic rate (implied repo rate) formed
  by buying the 161-day bill and going short in the
  70-day futures.

92
Pricing T-Bill Futures Implications
93
Pricing T-Bill Futures Implications

• Formally, the implied repo rate is defined as the
  rate in which the arbitrage profit from
  implementing the cash and carry arbitrage
  strategy is zero

94
Pricing T-Bill Futures Implications

• The actual repo rate is the one we use in solving
  for the equilibrium futures price in the
  carrying-cost model in our example, this was the
  rate on the 70-day T-bill (6.38).
• Thus, the equilibrium condition that the
  synthetic and spot T-bill be equal can be stated
  equivalently as an equality between the actual
  and the implied repo rates.

95
Pricing T-Bill Futures Implications

• Implication 2
• If the carrying-cost model holds, then the YTM of
  the futures will be equal to the implied forward
  rate (RI)

96
Pricing T-Bill Futures Implications

• In terms of our example, if f0M f0 98.74875,
  then the implied futures rate will be 5.18

97
Pricing T-Bill Futures Implications

• The implied forward rate on a 91-day T-bill
  investment to be made 70 days from the present,
  RI(91,70), is obtained by
• Selling short the 70-day T-bill at 98.821
  (100/(1.0638)70/365 (or equivalently borrowing
  98.821 at 6.38)
• Buying S0(T)/S0(T91) S0(70)/S0(161)
  98.821/97.5844 1.01267 issues of the 161-day
  T-bill
• Paying 100 at the end of 70 days to cover the
  short position on the maturing bond (or the loan)
• Collecting 1.01267(100) at the end of 161 days
  from the long position.

98
Pricing T-Bill Futures Implications

• This locking-in strategy would earn an investor a
  return of 101.267, 91 days after the investor
  expends 100 to cover the short sale thus, the
  implied forward rate on a 91-day investment made
  70 days from the present is 1.267, or
  annualized, 5.18

99
Pricing T-Bill Futures Implications

• Thus, if the carrying-cost model holds, then the
  implied yield on the futures is equal to the
  implied forward rate.

100
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• The T-bond futures contract gives the party with
  the short position the right to deliver, at any
  time during the delivery month, any bond with a
  maturity of at least 15 years.

101
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• When a particular bond is delivered, the price
  received by the seller is equal to the quoted
  futures price on the futures contract times a
  conversion factor, CFA, applicable to the
  delivered bond.
• The invoice price, in turn, is equal to that
  price plus any accrued interest on the delivered
  bond.

Invoice Price (f0) (CFA) Accrued Interest
102
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• The CBOT uses a conversion factor based on
  discounting the deliverable bond by a 6 YTM. The
  CBOTs rules for calculating the CFA on the
  deliverable bond are as follows
• The bonds maturity and time to the next coupon
  date are rounded down to the closest three
  months.
• After rounding, if the bond has an exact number
  of six-month periods, then the first coupon is
  assumed to be paid in six months.
• After rounding, if the bond does not have an
  exact number of six-month periods, then the first
  coupon is assumed to be paid in three months and
  the accrued interest is subtracted.

103
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• Example A 5.5 T-bond maturing in 18 years and 1
  month would be
• Rounded down to 18 years
• The first coupon would be assumed to be paid in
  six months
• The CFA would be determined using a discount rate
  of 6 and face value of 100

104
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• The CFA for the bond would be .945419

105
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• If the bond matured in 18 years and four months,
  the bond would be assume to have a maturity of 18
  years and three months.
• Its CFA would be found by determining the value
  of the bond three months from the present,
  discounting that value to the current period, and
  subtracting the accrued interest ((3/6)(2.75)
  1.375).

106
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• During the delivery month, there are a number of
  possible bonds that can be delivered.
• The party with the short position will select
  that bond that is cheapest to deliver.
• The CBOT maintains tables with possible
  deliverable bonds.

107
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• The tables show the bonds current quoted price
  and its CFA.
• For example, suppose three possible bonds are

108
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• If the current quoted futures price were 90 16
  (90.5), the costs of buying and delivering each
  bond would be

Thus, the cheapest bond to deliver would be
number 2.
109
Pricing T-Bond Futures Cheapest-to-Deliver Bond

• Over time and as rates change, the
  cheapest-to-deliver bond can change.
• In general
• If rates exceed 6, the CBOTs conversion system
  favors bonds with higher maturities and lower
  coupons.
• If rates are less than 6, the system tends to
  favor higher coupon bonds with shorter maturities.

110
Pricing T-Bond Futures Wild-Card Play

• Under the CBOT's procedures, a T-bond futures
  trader with a short position who wants to deliver
  on the contract has the right to determine during
  the expiration month not only the eligible bond
  to deliver, but also the day of the delivery.

111
Pricing T-Bond Futures Wild-Card Play

• The delivery process encompasses the following
  three business days
• Business Day 1, Position Day The short position
  holder notifies the clearinghouse that she will
  deliver.
• Business Day 2, Notice of Intention Day The
  clearinghouse assigns a long position holder the
  contract (typically the holder with the longest
  outstanding contract).
• Business Day 3, Delivery Day The short holder
  delivers an eligible T-bond to the assigned long
  position holder who pays the short holder an
  invoice price determined by the futures price and
  a conversion factor.

112
Pricing T-Bond Futures Wild-Card Play

• Since a short holder can notify the
  clearinghouse of her intention to deliver a bond
  by 8 p.m. (Chicago time) at the end of the
  position day (not necessarily at the end of the
  futures' trading day), an arbitrage opportunity
  has arisen because of the futures exchange's
  closing time being 200 (Chicago time) and the
  closing time on spot T-bond trading being 400.

113
Pricing T-Bond Futures Wild-Card Play

• Thus, a short holder knowing the settlement price
  at 200 p.m., could find the price of an eligible
  T-bond decreasing in the next two hours on the
  spot market.
• If this occurred, she could buy the bond at the
  end of the day at the lower price, then notify
  the clearinghouse of her intention to deliver
  that bond on the futures contract.
• If the bond price does not decline, the short
  holder can keep her position and wait another
  day.
• This feature of the T-bond futures contract is
  known as the wild-card option. This option tends
  to lower the futures price.

114
Pricing T-Bond Futures Equilibrium Price

• Like T-bill futures, the price on a T-bond
  futures contract depends on the spot price on the
  underlying T-bond (S0) and the risk-free rate.
• Note The pricing of a T-bond futures contract is
  more complex than the pricing of T-bill or
  Eurodollar futures because of the uncertainty
  over the bond to be delivered and the time of the
  delivery.

115
Pricing T-Bond Futures Equilibrium Price

• If we assume that we know the cheapest-to-deliver
  bond and the time of delivery, the equilibrium
  futures price is
• where
• S0 current spot price of the cheapest-to-deliver
  T-bond (clean price plus accrued interest)
• PV(C) present value of coupons paid on the bond
  during the life of the futures contract

116
Pricing T-Bond Futures Equilibrium Price

• Example, suppose the following
• The cheapest-to-deliver T-bond underlying a
  futures contract has the following features
• Coupon 10
• CFA 1.2
• Currently Price 110 (clean price)
• The cheapest-to-deliver T-bonds last coupon date
  was 50 days ago, its next coupon is 132 days from
  now, and the coupon after that comes 182 day
  later.
• The yield curve is flat at 6.
• The T-bond futures estimated expiration is T
  270 days.

117
Pricing T-Bond Futures Equilibrium Price

• The current T-bond spot price is 111.37 and the
  present value of the 5 coupon received in 132
  days is 4.8957

118
Pricing T-Bond Futures Equilibrium Price

• The equilibrium futures price based on a 10
  deliverable bond is therefore 111.16 per 100
  face value

119
Pricing T-Bond Futures Equilibrium Price

• The quoted price on a futures contract written on
  the 10 delivered bond would be stated net of
  accrued interest at the delivery date.
• The delivery date occurs 138 days after the last
  coupon payment (270-132).
• Thus, at delivery, there would be 138 days of
  accrued interest. Given the 182-day period
  between coupon payments, accrued interest would
  therefore be 3.791

Accrued Interest (138/182)(5) 3.791
120
Pricing T-Bond Futures Equilibrium Price

• The quoted futures price on the delivered bond
  would be 107.369
• With a CFA of 1.2, the equilibrium quoted futures
  price would be 89.47

Quoted Futures Price 111.16 3.791 107.369
Quoted Futures Price on Bond 111.16 -
(138/182)5 107.369 Quoted Futures Price
107.369/1.2 89.47
121
Pricing T-Bond Futures Equilibrium Price

• Like T-bill futures, cash-and-carry arbitrage
  opportunities will exist if the T-bond futures
  were not equal to 111.16 (or its quoted price of
  89.47).

122
Pricing T-Bond Futures Equilibrium Price

• Example, if futures were priced at f M 113, an
  arbitrageur could
• Go short in the futures at 113
• Buy the underlying cheapest-to-deliver bond for
  111.37
• Finance the bond purchase by
• Borrowing 106.4743 ( S0 PV(C) 111.37
  4.8957) at 6 for 270 days
• Borrowing 4.8957 at 6 for 132 days

123
Pricing T-Bond Futures Equilibrium Price

• 132 days later, the arbitrageur would receive a
  5 coupon that he would use to pay off the
  132-day loan of 5 ( 4.8957(1.06)132/365).
• At expiration, the arbitrageur would
• Sell the bond on the futures contract at 113
• Pay off his financing cost on the 270-day loan of
  111.16 ( 106.4743(1.06)270/365).

124
Pricing T-Bond Futures Equilibrium Price

• At expiration, the arbitrageur realized a profit
  of 1.84 per 100 face value

fM f0 113 111.16 1.84 per 100 face
value
125
Pricing T-Bond Futures Equilibrium Price

• This risk-free return would result in
  arbitrageurs pursuing this strategy of going
  short in the futures and long in the T-bond,
  causing the futures price to decrease to 111.16
  where the arbitrage disappears.
• If the futures price were below 111.16,
  arbitrageurs would reverse the strategy, shorting
  the bond, investing the proceeds, and going long
  in the T-bond futures contract.

126
Notes on Futures Pricing

• Note
• For many assets the costs of carrying the asset
  for a period of time exceeds the benefits.
• As a result, the futures price on such assets
  exceeds the spot price prior to expiration and
  the basis (ft-St) on such assets is positive.

127
Notes on Futures Pricing

• A market in which the futures price exceeds the
  spot price is referred to as a contango or normal
  market.
• If the futures price is less then the spot price
  (a negative basis), the costs of carrying the
  asset is said to have a convenience yield in
  which the benefits from holding the asset exceed
  the costs.
• A market in which the basis is negative is
  referred to as backwardation or an inverted
  market.
• For futures on debt securities, an inverted
  market could occur if large coupon payments are
  to be paid during the period.

128
Notes on Futures Pricing

• Note
• The same arbitrage arguments governing the
  futures and spot price relation also can be
  extended to establish the equilibrium
  relationship between futures prices with
  different expirations.

129
Notes on Futures Pricing

• Note
• The futures price is related to an unknown
  expected spot price.
• Several expectation theories have been advanced
  to explain the relationship between the futures
  and expected spot prices.

130
Notes on Futures Pricing

• One of the first theories was broached by the
  famous British economists John Maynard Keynes and
  J.R. Hicks.
• They argued that if a spot market were dominated
  by hedgers who, on balance, wanted a short
  forward position, then for the market to clear
  (supply to equal demand) the price of the futures
  contract would have to be less than the expected
  price on the spot commodity at expiration
  (E(ST)) f0 lt E(ST).
• According to Keynes and Hicks, the difference
  between E(ST) and f0 represents a risk-premium
  that speculators in the market require in order
  to take a long futures position. Keynes and
  Hicks called this market situation normal
  backwardation.

131
Notes on Futures Pricing

• C.O. Hardy argued for the case of f0 gt E(ST),
  even in a market of short hedgers.
• His argument, though, is based on investor's risk
  behavior.
• He maintained that since speculators were akin to
  gamblers, they were willing to pay for the
  opportunity to gamble (risk-loving behavior).
• Thus, a gambler's fee, referred to as a contango
  or forwardation, would result in a negative risk
  premium.

132
Notes on Futures Pricing

• Finally, there is a risk-neutral pricing
  argument.
• In this argument, the futures price represents an
  unbiased estimator of the expected spot price (f0
  E(ST)) and, with risk-neutral pricing,
  investors purchasing an asset (bond) for S0 and
  expecting an asset value at T of E(ST) f0
  require an expected rate of return equal to the
  risk-free rate.
• As a result, in a risk-neutral market, the
  futures price is equal to the expected spot
  price

133
Hedging International Positions with Currency
Futures

• When investors purchase and hold foreign
  securities or when corporations and governments
  sell debt securities in external markets or incur
  foreign debt positions, they are subject to
  exchange-rate risk.
• As noted in Chapter 7, major banks provide
  exchange-rate protection by offering forward
  contracts to financial and non-financial
  corporations to hedge their international
  positions.
• In addition to contracts offered in this
  interbank forward market, hedging exchange rate
  risk can also be done using foreign currency
  futures contracts listed on the Chicago
  Mercantile Exchange (CME), as well as a number of
  exchanges outside the U.S.

134
Hedging International Positions with Currency
Futures

• Short Hedging Case
• Consider a U.S. fund that has a sizable
  investment in Eurobonds that will pay a principal
  in British pounds of 10M next September.
• Suppose the current spot exchange rate is
  1.425/, making the dollar value of the
  principal worth 14.25M.
• Suppose the fund is concerned that the /
  exchange rate could decrease by September,
  reducing the amount of dollars they would receive
  when they convert 10M.

135
Hedging International Positions with Currency
Futures

• To minimize its exchange-rate exposure, the fund
  could go short in an interbank forward contract
  in which it agrees to sell 10M at the September
  principal payment date at a specified forward
  exchange rate.
• Alternatively, the fund could take a short
  position in a CME September futures contract.
• Given the contract size on CMEs British pound
  contract of 62,500, the fund would need go short
  in 160 CME British pound contracts in order to
  hedge its 10M September receipt

nf 10,000,000/62,500 160
136
Hedging International Positions with Currency
Futures

• If the futures price on the September contract
  were equal to f0 1.425/, and the September
  principal payment occurred at the same time as
  the futures expiration, then the fund would be
  able realize a 14.25M cash inflow when it
  converted its 10M principal to dollars at the
  spot / exchange rate at the September principal
  payment date and closed its 160 British pound
  futures contracts at an expiring futures price
  equal to the spot exchange rate.

137
Hedging International Positions with Currency
Futures

• The hedge is illustrated in the exhibit on the
  next slide.
• The exhibit shows the funds hedged revenue of
  14.25M at expiration from converting the 10M at
  spot exchange rates of 1.47/ and 1.39/ and
  from closing its 160 short futures contracts by
  going long at expiring futures prices equal to
  the different spot exchange rates.

138
Hedging International Positions with Currency
Futures
139
Hedging International Positions with Currency
Futures

• Long Hedging Case
• Consider the case of a U.S. corporation that has
  issued a Eurobond denominated in British pounds.
• Suppose the company has to make a September
  principal payment in pounds of 5M and that the
  September CME British pound futures is trading at
  f0 1.425/ and expires at the same time the
  principal payment is due.

140
Hedging International Positions with Currency
Futures

• In this case, the U.S. company could hedge the
  dollar cost on its principal payment against
  exchange-rate changes by going long in 80
  September futures contracts
• At expiration, the company would realize a hedged
  dollar cost of 7.125M when it purchased 5M at
  the spot exchange rate and closed its 80 long
  futures contracts at expiring futures prices
  equal to the spot exchange. This hedge is
  illustrated in the exhibit on the next slide.

nf 5,000,000/62,500 80
141
Hedging International Positions with Currency
Futures
142
Pricing Currency Futures and Forward Exchange
Rates

• The carrying cost model can be used to determine
  the equilibrium price of a currency forward or
  futures exchange rate.
• In international finance, the carrying cost model
  governing the relationship between spot and
  forward exchange rates is referred to as the
  interest rate parity theorem (IRPT).
• In terms of IRPT, the forward price of a currency
  or forward exchange rate (f0) is equal to the
  cost of carrying the spot currency (priced at the
  spot exchange rate of E0) for the contracts
  expiration period.

143
Pricing Currency Futures and Forward Exchange
Rates

• In terms of IRPT, the equilibrium forward price
  or exchange rate is
• where
• RUS U.S. risk-free rate
• RF foreign risk-free rate

144
Pricing Currency Futures and Forward Exchange
Rates

• If the interest rate parity condition does not
  hold, an arbitrage opportunity will exist.
• The arbitrage strategy to apply in such
  situations is known as covered interest arbitrage
  (CIA).

145
Pricing Currency Futures and Forward Exchange
Rates

• To illustrate, suppose the annualized U.S. and
  foreign interest rates are RUS 4 and RF 6,
  respectively, and the spot exchange rate is E0
  0.40/FC.
• By IRPT, a one-year forward contract would be
  equal to 0.39245283/FC

146
Pricing Currency Futures and Forward Exchange
Rates

• If the actual forward rate, f0M, exceeds
  0.39245283/FC, an arbitrage profit would exist
  by
• Borrowing dollars at RUS
• Converting the dollar to FC at E0
• Investing the fund in a foreign risk-free rate of
  RF
• Entering a short forward contract to sell the FC
  at the end of the period at f0M

147
Pricing Currency Futures and Forward Exchange
Rates

• Example If f0M 0.40/FC, an arbitrageur could
• Borrow 40,000 at RUS 4 creating a loan
  obligation at the end of the period of 41,600
  (40,000)(1.04).
• Convert the dollars at the spot exchange rate of
  E0 0.40/FC to 100,000 FC (
  (2.5FC/)(40,000)).
• Invest the 100,000 FC in the foreign risk-free
  security at RF 6 creating a return of
  principal and interest of 106,000 FC one year
  later.
• Enter a forward contract to sell 106,000 FC at
  the end of the year at f0M 0.40/FC.

148
Pricing Currency Futures and Forward Exchange
Rates

• One year later, the arbitrageur would
• receive 42,400 when she sells the 106,000 FC on
  the forward contract and
• owe 41,600 on her debt obligation,
• for an arbitrage return of 800.

149
Pricing Currency Futures and Forward Exchange
Rates

• Such risk-free profit opportunities, in turn,
  would lead arbitrageurs to try to implement the
  CIA strategy.
• This would cause the price on the forward
  contract to fall until the riskless opportunity
  disappears. The zero arbitrage profit would
  occur when the interest rate parity condition is
  satisfied.

150
Pricing Currency Futures and Forward Exchange
Rates

• If the forward rate is below the equilibrium
  value, then the CIA is reversed. In the example,
  if f0M 0.38/FC, an arbitrageur could
• Borrow 100,000 FC at RF 6 creating a 106,000
  FC debt.
• Convert the 100,000FC at the spot exchange rate
  to 40,000.
• Invest the 40,000 in the U.S. risk-free security
  at RUS 4.
• Enter a forward contract to buy 106,000 FC at the
  end of the year at f0M 0.38/FC.

151
Pricing Currency Futures and Forward Exchange
Rates

• At the end of the period, the arbitrageurs
  profit would be 1,320
• As arbitrageurs attempt to implement this
  strategy, they will push up the price on the
  forward contract until the arbitrage profit is
  zero this occurs when the interest rate parity
  condition is satisfied.

40,000(1.04) (0.38/FC)(106,000) 1,320
152
Websites

• For information on forward exchange rates go to
  www.fxstreet.com
• For information on CME currency futures contracts
  go to www.cme.com and click on Delayed Quotes
  in Market Data, and then Currency Products.