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
  • Duration-based 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
    • Refer to graph below

Futures price converges to spot price

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
      • F1 : You buy futures price when you set up hedge
      • F2 : Futures price when you sell futures to purchase asset
      • S2 : Asset price at time of purchase
      • b2 : Basis at time of purchase
Cost of asset S2
Gain on Futures F2 − F1
Net amount paid S2 − (F2 − F1) = F1 + (S2 − F2) = F1 + b2
Net amount received −S2 + (F2 − F1)
  • Company uses long hedge
    • Company hedges at t1 and buys asset at t2 from spot market
    • If basis rises (↑ S2 or ↓ F2 ), company's position worsens
    • ↑ S2 − ( ↓ F2 − F1) = ↑ S2 − ↓ hedge = F1 + b2
    • Company must buy asset at higher price for S2 and/or takes a loss on the hedge
    • Not a complete hedge
  • Short hedge to sale an asset
    • Define
      • F1 : Futures price when you set up hedge
      • F2 : Futures price when you sell asset at time 2
      • S2 : Asset price at time of sale
      • b2 : Basis at time of sale
Price of asset S2
Gain on Futures F1 − F2
Net amount received S2 + (F1 − F2) = F1 + (S2 − F2) = F1 + b2
  • Company uses short hedge
    • Company hedges at t1 and sells asset at t2 on spot market
    • If basis rises (↑ S2 or ↓ F2 ), company's position improves
    • ↑ S2 + ( F1 −↓ F2 ) = ↑ S2 + ↑ hedge = F1 + b2
    • Company can sell asset at higher price for S2 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
    • σS is the standard deviation of ΔS, the change in the spot price during the hedging period
    • σF 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 hedge ratio

  • Optimal Number of Contracts, N*
    • * means optimal number of contracts
    • h* means optimal hedge ratio
    • QA Size of position being hedged (units)
    • QF 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 hedge ratio with no tailing adjustment

    • Optimal number of contracts, N*, after tailing adjustment to allow daily settlement of futures
      • VA Value of position being hedged, VA = S0 x QA
      • VF Value of one futures contract, VF = F0 x QF

optimal hedge ratio with tailing adjustment

  • Example
    • Airline will purchase 2 million gallons of jet fuel in one month and hedges using heating oil futures
    • From historical data σF = 0.0313, σS = 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
    • VA = 2,000,000 x 1.94 = 3,880,000
    • VF = 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
    • VA is the current value of the portfolio
    • β is its beta
    • VF is the current value of one futures
      • VF = futures price times contract size

hedging with index futures

  • 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?
      • VA = $5 million
      • VF = 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?
      • VA = $5 million
      • VF = 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 risk-free 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
    • Similar to a credit card
  • 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 fixed-price 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
    • T-Bill: 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

treasury bill price

  • 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 = 90-05 = 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) T-Bonds & T-Notes
    • 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 3-month Eurodollar deposit rate
    • same as 3-month 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
          • Gain = 30 ($25)=$750
        • 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 three-month period;
        • Forward Rate Agreement (FRA) is settled at the end of the underlying three-month 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
    • T1 is the start of period covered by the forward/futures rate
    • T2 is the end of period covered by the forward/futures rate (90 days later that T1)
    • σ is the standard deviation of the change in the short rate per year
      • Forward Rate = Futures Rate − 0.5 σ2 T1 T2
    • 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
    • T1 = 2
    • T2 = 2.25, Eurodollar contract is 3 months
      • convexity adj. = 0.5(0.012)2(2)(2.25) = 0.000324
  • Example 2
    • σ = 0.012
    • T1 = 10
    • T2 = 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 ci at time ti is

duration

  • where B is its price and y is its yield (continuously compounded)
  • This leads to

relating duration to interest rate changes

  • 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

relating duration to interest rate changes

  • When the yield y is expressed with compounding m times per year

modified duration

  • The expression is referred to as the "modified duration"

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 3-month period
      • How many contracts are necessary to lock in an interest rate on $1 million for a future six-month period?
        • Two contracts
        • One contract for the first 3 months
        • One contract for the second 3 months

Duration-Based Hedge Ratio

  • Define
    • VF: Contract Price for Interest Rate Futures
    • DF: Duration of Asset Underlying Futures at Maturity
    • P: Value of portfolio being Hedged
    • DP: Duration of Portfolio at Hedge Maturity
    • N*: number of contracts

duration based hedge ratio

  • Example
    • Three month hedge is required for a $10 million portfolio.
    • Duration of the portfolio in 3 months will be 6.8 years.
    • 3-month T-bond futures price is 93-02 so that contract price is $93,062.50
      • Contracts are based on $100,000
      • Note: 93-02 = 93 + 2/32 = 93.0625
    • Duration of cheapest to deliver bond in 3 months is 9.2 years
    • Number of contracts for a 3-month hedge is

example duration based hedge ratio

  • Limitations of Duration-Based Hedging
    • Assumes that only parallel shift in yield curve take place
    • Assumes that yield curve changes are small
    • When T-Bond futures is used assumes there will be no change in the cheapest-to-deliver 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
 

FOLLOW ME