Futures Hedging Strategies and Interest Rate Futures
Read Chapters 3 and 6 in Hull
Outline
 Hedges
 Basis risk
 Optimal hedge ratio
 Hedge using index futures
 Stack and roll
 Day count conventions
 Treasury bills and bonds
 Eurodollar futures
 Convexity adjustment
 Duration
 Durationbased hedge ratio
Hedges
 You should use a long futures hedge when you know you will purchase an asset in the future and want to lock in the price
 Futures and asset are different things
 Long futures position – makes money if asset price increases and loses money if it falls
 A spot price increase will make asset more expensive to buy
 You should use a short futures hedge when you know you will sell an asset in the future & want to lock in the price
 Short futures position – makes money if asset price drops and loses money if it increases
 A spot price decrease will make a loss when you sell it
 Benefit of Hedging
 Companies should focus on their main business and take steps to minimize risks from interest rates, exchange rates, and other market variables
 Problems oft Hedging
 Shareholders can diversify thier investments and can hedge to reduce risk
 Shareholds may increase their risk to hedge when competitors do not
 You could have trouble explaining a situation to your manager if the company gained on the underlying asset but lost on the hedge
 Futures prices converge to spot market
Basis Risk
 Basis risk  the value of a futures contract will not move in line with that of the underlying exposure.
 Basis is the difference between spot & futures
 Basis = Spot price (S) – Future price (F)
 Basis risk arises from the uncertainty when the hedge is closed out
 Note – you are not buying the futures to lock in a future price.
 You are buying (shorting) the future as a means to offset gains/losses on the asset's price changes
 Long hedge to purchase an asset
 Define
 F_{1} : You buy futures price when you set up hedge
 F_{2} : Futures price when you sell futures to purchase asset
 S_{2} : Asset price at time of purchase
 b_{2} : Basis at time of purchase
Cost of asset 
S_{2} 
Gain on Futures 
F_{2} − F_{1} 
Net amount paid 
S_{2} − (F_{2} − F_{1}) = F_{1} + (S_{2} − F_{2}) = F_{1} + b_{2} 
Net amount received 
−S_{2} + (F_{2} − F_{1}) 
 Company uses long hedge
 Company hedges at t_{1} and buys asset at t_{2} from spot market
 If basis rises (↑ S_{2} or ↓ F_{2} ), company's position worsens
 ↑ S_{2} − ( ↓ F_{2} − F_{1}) = ↑ S_{2} − ↓ hedge = F_{1} + b_{2} ↑
 Company must buy asset at higher price for S_{2} and/or takes a loss on the hedge
 Not a complete hedge
 Short hedge to sale an asset
 Define
 F_{1} : Futures price when you set up hedge
 F_{2} : Futures price when you sell asset at time 2
 S_{2} : Asset price at time of sale
 b_{2} : Basis at time of sale
Price of asset 
S_{2} 
Gain on Futures 
F_{1} − F_{2} 
Net amount received 
S_{2} + (F_{1} − F_{2}) = F_{1} + (S_{2} − F_{2}) = F_{1} + b_{2} 
 Company uses short hedge
 Company hedges at t_{1} and sells asset at _{t}2 on spot market
 If basis rises (↑ S_{2} or ↓ F_{2} ), company's position improves
 ↑ S_{2} + ( F_{1} −↓ F_{2} ) = ↑ S_{2} + ↑ hedge = F_{1} + b_{2} ↑
 Company can sell asset at higher price for S_{2} and/or takes a gain on the hedge
 Not a complete hedge
 Choice of Contract
 Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge
 If asset has no futures contract for a hedge, choose the contract whose futures price is most highly correlated with the asset price.
 Basis has two components
Optimal Hedge Ratio
 Proportion of the exposure that should optimally be hedged equals
 h is the minimum hedge ratio
 σ_{S} is the standard deviation of ΔS, the change in the spot price during the hedging period
 σ_{F} is the standard deviation of ΔF, the change in the futures price during the hedging period
 ρ is the coefficient of correlation between ΔS and ΔF.
 Optimal Number of Contracts, N*
 * means optimal number of contracts
 h* means optimal hedge ratio
 Q_{A} Size of position being hedged (units)
 Q_{F} Size of one futures contract (units)
 Note: Tailing adjustment is accounting for the daily settlement of futures.
 That is why is uses value and not quantity
 Optimal number of contracts, N*, if no tailing adjustment
 Optimal number of contracts, N*, after tailing adjustment to allow daily settlement of futures
 V_{A} Value of position being hedged, V_{A} = S_{0} x Q_{A}
 V_{F} Value of one futures contract, V_{F} = F_{0} x Q_{F}
 Example
 Airline will purchase 2 million gallons of jet fuel in one month and hedges using heating oil futures
 From historical data σ_{F} = 0.0313, σ_{S} = 0.0263, and ρ = 0.928
 h* = 0.928 x 0.0263 / 0.0313 = 0.7777
 Optimal number of contracts assuming no tailing adjustment
 The size of one heating oil contract is 42,000 gallons
 N* = 0.7777 x 2,000,000 / 42,000 = 37.03
 Optimal number of contracts after tailing adjustment
 The spot price is 1.94 for jet fuel and the futures price is 1.99 for heating oil both dollars per gallon
 V_{A} = 2,000,000 x 1.94 = 3,880,000
 V_{F} = 42,000 x 1.99 = 83.580
 N* = 0.7777 x 2=3,880,000 / 83,580 = 36.10
Hedging Using Index Futures
 To hedge the risk in a portfolio the number (N*) of contracts that should be shorted is
 A stock portfolio loses value when stock prices fall
 A short earns money when prices fall
 Define
 V_{A} is the current value of the portfolio
 β is its beta
 V_{F} is the current value of one futures
 V_{F} = futures price times contract size
 Example
 Futures price of S&P 500 is 1,000
 Size of portfolio is $5 million
 Beta of portfolio is 1.5
 One contract is on $250 times the index
 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?
 V_{A} = $5 million
 V_{F} = 1,000 ($250) = $250,000
 N* = 1.5 x $5 million / $250,000 = 30
 You short the contracts in case the S&P 500 Index falls
 You are happy if the value of your portfolio increases
 You are not happy if the value of the portfolio falls
 The short gains when the portfolio falls in value
 What if beta changes?
 What position is necessary to reduce the β of the portfolio to 0.75?
 V_{A} = $5 million
 V_{F} = 1,000 ($250) = $250,000
 β = 0.75
 N* = 0.75 x $5 million / $250,000 = 15
 Sell 15 contracts from the original 30
 Portfolio less sensitive to market
 What position is necessary to increase the β of the portfolio to 2.0?
 β = 2.0
 N* = 2.0 x $5 million / $250,000 = 40
 Buy 10 additional contracts with the original 30
 Portfolio more sensitive to market
 Why Hedge Equity Returns
 May want to be out of the market for a while.
 Hedging avoids the costs of selling and repurchasing the portfolio
 Suppose stocks in your portfolio have an average beta of 1.0, but you feel they have been chosen well and will outperform the market in both good and bad times.
 Hedging ensures that the return you earn is the riskfree return plus the excess return of your portfolio over the market.
Stack and Roll
 We can roll futures contracts forward to hedge future exposures
 Initially we enter into futures contracts to hedge exposures up to a time horizon
 Just before maturity we close them out and replace them with new contract to reflect the new exposure
 Liquidity Issues
 In any hedging situation, the hedger may experience losses on the hedge while he/she does not realize gains on the underlying exposure
 This can create liquidity problems
 For example
 Metallgesellschaft sold long term fixedprice contracts to businesses on heating oil and gasoline
 Company hedged using stack and roll
 Company went long on futures to lock in price for oil and gasoline
 The price of oil fell
 Company realized huge losses ($1.33 billion) to cover future contracts for margin calls
 Stock Picking
 If you think you can pick stocks that will outperform the market, futures contract can be used to hedge the market risk
 If you are right, you will make money whether the market goes up or down
 Rolling The Hedge Forward
 We can use a series of futures contracts to increase the life of a hedge
 Each time we switch from 1 futures contract to another we incur a type of basis risk
Day Count Convention
 Day Count Conventions
 Defines the period of time to which the interest rate applies
 The period of time used to calculate accrued interest relevant when the instrument is bought or sold Day Count Conventions in the U.S.
Treasury Bonds: 
Actual/Actual (in period) 
Corporate Bonds: 
30/360 
Money Market Instruments: 
Actual/360 
 Examples
 Bond: 8% Actual/ Actual in period.
 4% is earned between coupon payment dates.
 Accruals on an Actual basis.
 When coupons are paid on March 1 and Sept 1, how much interest is earned between March 1 and April 1?
 interest = 0.08 x 31 / 184 = 0.0135
 Count days in the month: March 31, April 30, May 31, June 30, July 31, August 31, Septermber 1
 Bond: 8% 30/360
 Assumes 30 days per month and 360 days per year.
 When coupons are paid on March 1 and Sept 1, how much interest is earned between March 1 and April 1?
 Total days = 6 months * 30 = 180 days
 30 days interest
 interest = 0.08 x 30 / 180 = 0.0133
 TBill: 8% Actual/360:
 8% is earned in 360 days.
 Accrual calculated by dividing the actual number of days in the period by 360.
 How much interest is earned between March 1 and April 1?
 Days = 31
 Months = 180
 interest = 0.08 x 31 / 180 = 0.0138
 The February Effect
 How many days of interest are earned between February 28, 2013 and March 1, 2013 when
 Day count is Actual/Actual in period?
 Day count is 30/360?
 Solution
 Between Feb 28 and March 1 Day = 1 for actual / 365
 Under 30/360, you get 3 days
 Note – check for leap year, February 29, 2016
Treasury Bills and Bonds
 Treasury Bill Prices in the US
 P is quoted price
 Y is cash price per $100
 Face value = $100
 n: number of days
 U.S. Treasury Bond Price Quotes
 Cash price = Quoted price + Accrued Interest
 Called dirty price
 Bond has not paid interest yet
 Buyer collects the whole coupon payment
 Prorate the accrued interest
 Note: Bond price = 9005 = 90 + 05/32
 Remember second number is divided 32
 Treasury Bond Futures
 The conversion factor for a bond is approximately equal to the value of the bond assuming the yield curve is flat at 6% with semiannual compounding
 Writer of futures must deliver bond
 Treasury bonds have different coupon rates and face values
 Conversion factor puts all bonds on equal footing
Cash price received by party with short position = Most Recent Settlement Price × Conversion factor + Accrued interest
 Example
 Most recent settlement price = 90.00
 Conversion factor of bond delivered = 1.3800
 Accrued interest on bond =3.00
 Price received for bond is
 price = 1.3800×90.00+3.00 = $127.20 per $100 of principal
 Chicago Board of Trade (CBOT) TBonds & TNotes
 Factors that affect the futures price
 Delivery can be made any time during the delivery month
 Any range of eligible bonds can be delivered
 The wild card play
 If bond prices fall after 2:00 pm on first delivery day of month
 Party with short position issues intent to deliver at 3:45 pm
 The party closes the short by buying at the cheaper price at 3:45 pm.
 If price rises, the party keeps the short open.
Eurodollar Futures
 A Eurodollar is a dollar deposited in a bank outside the United States
 Eurodollar futures are futures on the 3month Eurodollar deposit rate
 same as 3month LIBOR rate
 One contract is on the rate earned on $1 million
 A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $25
contract price = 10,000 × [100 − 0.25 × (100 − Q)]
 A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $25
 Show
 Quote (Q)
 Interest = 100 – Q
 A one basis point corresponds to a Q = 99.99
 contract price = 10,000 × [100 − 0.25 × (100 − 99.99)] = 999,975
 Contract price = 999,975
 Difference = 1,000,000 – 999,975 = 25
 A Eurodollar futures contract is settled in cash
 When it expires (on the third Wednesday of the delivery month) the final settlement price is 100 minus the actual three month deposit rate
 Table below shows Eurodollar futures quotes
Date 
Quote 
Nov 1 
97.12 
Nov 2 
97.23 
Nov 3 
96.98 
……. 
…… 
Dec 21 
97.42 
 Example
 Suppose you buy (take a long position in) a contract on November 1
 The contract expires on December 21
 The table above shows the prices
 a) How much do you gain or lose on the first day?
 On Nov. 1 you invest $1 million for three months on Dec 21
 The contract locks in an interest rate
 interest rate = 100  97.12 = 2.88%
 b) How much do you gain or lose on the second day?
 interest = 97.23 – 97.12 = 0.11
 11 basis points
 Gain = $25 (11) = $275
 c) How much do you gain or lose over the whole time until expiration?
 You earn 100 – 97.42 = 2.58% on $1 million for three months
 Basis points = 97.42 (Dec. 21) – 97.12 (Nov. 1) = 0.3
 30 basis points
 You locked in a rate of 2.88%, which equals $7,200
 Gain = 7,200 – 6,450 = 750
Convexity Adjustment
 Forward Rates and Eurodollar Futures
 Eurodollar futures contracts last as long as 10 years
 For Eurodollar futures lasting beyond two years we cannot assume that the forward rate equals the futures rate
 Two reasons explain why
 Futures is settled daily where forward is settled once
 Futures is settled at the beginning of the underlying threemonth period;
 Forward Rate Agreement (FRA) is settled at the end of the underlying threemonth period
 FRA – two parties switch interest rate payments on some amount of money
 For example, parties switch fixed for variable interest
 Parties only settle the difference in interest rates
 A "convexity adjustment" often made is
 T_{1} is the start of period covered by the forward/futures rate
 T_{2} is the end of period covered by the forward/futures rate (90 days later that T_{1})
 σ is the standard deviation of the change in the short rate per year
 Forward Rate = Futures Rate − 0.5 σ^{2} T_{1} T_{2}
 Table below shows convexity adjustment for σ=0.012
Maturity of Futures 
Convexity Adjustment (bps) 
2 
3.2 
4 
12.2 
6 
27.0 
8 
47.5 
10 
73.8 
 Example 1
 σ = 0.012
 T_{1} = 2
 T_{2} = 2.25, Eurodollar contract is 3 months
 convexity adj. = 0.5(0.012)^{2}(2)(2.25) = 0.000324
 Example 2
 σ = 0.012
 T_{1} = 10
 T_{2} = 10.25, Eurodollar contract is 3 months
 convexity adj. = 0.5(0.012)^{2}(10)(10.25) = 0.00738
Duration
 Duration of a bond that provides cash flow c_{i} at time t_{i} is
 where B is its price and y is its yield (continuously compounded)
 This leads to
 Similar to elasticity
 If the yield, y, increases by 1% or (100 bps), then the bond price falls by approximately ΔB / B percent.
 Below shows how to derive duration
 When the yield y is expressed with compounding m times per year
 The expression is referred to as the "modified duration"
 Duration Matching
 This involves hedging against interest rate risk by matching the durations of assets and liabilities
 It provides protection against small parallel shifts in the zero curve
 Using Eurodollar Futures
 One contract locks in an interest rate on $1 million for a future 3month period
 How many contracts are necessary to lock in an interest rate on $1 million for a future sixmonth period?
 Two contracts
 One contract for the first 3 months
 One contract for the second 3 months
DurationBased Hedge Ratio
 Define
 V_{F}: Contract Price for Interest Rate Futures
 D_{F}: Duration of Asset Underlying Futures at Maturity
 P: Value of portfolio being Hedged
 D_{P}: Duration of Portfolio at Hedge Maturity
 N*: number of contracts
 Example
 Three month hedge is required for a $10 million portfolio.
 Duration of the portfolio in 3 months will be 6.8 years.
 3month Tbond futures price is 9302 so that contract price is $93,062.50
 Contracts are based on $100,000
 Note: 9302 = 93 + 2/32 = 93.0625
 Duration of cheapest to deliver bond in 3 months is 9.2 years
 Number of contracts for a 3month hedge is
 Limitations of DurationBased Hedging
 Assumes that only parallel shift in yield curve take place
 Assumes that yield curve changes are small
 When TBond futures is used assumes there will be no change in the cheapesttodeliver bond
 GAP Management
 Banks use a more sophisticated approach to hedge interest rate.
 They bucket the zero curve
 Divide loans and sources into buckets by maturity
 First bucket – between 0 and 1 month
 Second bucket – between 1 and 3 months
 Match the assets and liabilities for each bucket
 Hedging exposure to situation where rates corresponding to one bucket change and all other rates stay the same
