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 enter 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 refer to different things
    • Long futures hedge – makes money if asset price increases and loses money if it falls
    • A spot price increase will make asset more expensive to buy
    • The gain of the hedge offsets the higher spot price
  • You enter a short futures hedge when you know you will sell an asset in the future and want to lock in the price
    • Short futures hedge – makes money if asset price drops and loses money if it increases
    • A spot price decrease makes a loss when you sell it
    • Gain on hedge offsets loss on a lower spot price
  • 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 of Hedging
    • Shareholders can diversify their investments and 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 and futures prices
      • 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 change
  • 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 an 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 an 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 t2 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 be hedged optimally
    • 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 hedge ratio

  • 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 hedge ratio with 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

optimal hedge ratio with tailing adjustment

  • Example
    • Bus company will purchase 1 million gallons of diesel fuel in one month and hedges using heating oil futures
    • From historical data σ F = 0.05 , σ S = 0.02 , and ρ = 0.85
      • h* = 0.850 x 0.020 / 0.050 = 0.340
  • Optimal number of contracts assuming no tailing adjustment
    • The size of one heating oil contract is 42,000 gallons
      • N* = 0.340 x 1,000,000 / 42,000 = 8.095
  • Optimal number of contracts after tailing adjustment
    • The spot price is 2.20 for diesel fuel and the futures price is 1.90 for heating oil both dollars per gallon
    • V A = 1,000,000 x 2.20 = 2,200,000
    • V F = 42,000 x 1.90 = 79,800
      • N* = 0.340 x 2,200,000 / 79,800 = 9.37

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 the beta
    • V F is the current value of one futures
      • V F = futures price times contract size

hedging with index futures

  • Example
    • Futures price of S&P 500 is 1,500
    • Size of portfolio is $10 million
    • Beta of portfolio is 1.2
    • 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 = $10 million
      • V F = 1,500 ($250) = $375,000
        • N* = 1.2 x $10 million / $375,000 = 32
      • 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 0.85?
      • V A = $10 million
      • V F = 1,500 ($250) = $375,000
      • β = 0.85
        • N* = 0.85 x $10 million / $375,000 = 22.667 (or 23 contracts)
      • Long 9 contracts from the original 32
      • Remember - a long unwinds a short, and vice versa
      • Portfolio less sensitive to market
    • What position is necessary to increase the β of the portfolio to 2.5?
      • β = 2.5
        • N* = 2.5 x $10 million / $375,000 = 66.667 (or 67 contracts)
      • Short 35 additional contracts with the original 32
      • 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 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 = 70-15 = 70 + 15/32
      • Remember to divide second number by 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 = 85.00
    • Conversion factor of bond delivered = 1.45
    • Accrued interest on bond =5.50
    • Price received for bond is
      • price = 1.45×85.00+5.50 = $128.75 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
  • Use equation to calculate price

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 equates 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
Oct 1 96.55
Oct 2 97.11
Oct 3 95.95
……. ……
Nov 21 98.65
  • Example
    • Suppose you buy (take a long position in) a contract on November 1
    • The contract expires on November 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 - 96.55 = 3.45%
      • b) How much do you gain or lose on the second day?
        • interest = 97.11 – 96.55 = 0.56
        • 56 basis points
        • Gain = $25 (56) = $1,400
      • c) How much do you gain or lose over the whole time until expiration?
        • You earn 100 – 98.65 = 1.35% on $1 million for three months
        • Basis points = 98.65 (Nov. 21) – 96.55 (Oct. 1) = 2.1
        • 210 basis points
          • Gain = 210 ($25)=$5,250
        • You locked in a rate of 3.45%, which equals $8,625
        • The final rate is 1.35% or 135 x $25 = 3,375
          • Gain = 8,625 – 3,375 = 5,250

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 beginning of the underlying three-month period
        • FRA – two parties switch interest rate payments on some amount of money
        • For example, the long position party pays a fixed interest and receives a variable interest rate
        • Parties 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 10.89
4 41.14
6 90.75
8 159.72
10 248.05
  • Example 1
    • σ = 0.022
    • T 1 = 2
    • T 2 = 2.25 , Eurodollar contract is 3 months
      • convexity adj. = 0.5(0.022) 2(2)(2.25) = 0.001089
      • Multiply by 10,000 to get number in table
  • Example 2
    • σ = 0.022
    • T 1 = 10
    • T 2 = 10.25 , Eurodollar contract is 3 months
    • convexity adj. = 0.5(0.022) 2(10)(10.25) = 0.024805

Duration

  • Duration of a bond that provides cash flow c i at time t i 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 $5 million for a future 3-month period
      • How many contracts are necessary to lock in an interest rate on $5 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
    • 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

duration based hedge ratio

  • Example
    • Three month hedge is required for a $25 million portfolio.
    • Duration of the portfolio in 3 months will be 5.5 years.
    • 3-month T-bond futures price is 97-13 so that contract price is $97,406.25
      • Contracts with zero discount equal $100,000
      • Note: 97-13 = 97 + 13/32 = 97.40625
    • Duration of cheapest to deliver bond in 3 months is 8.7 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 we use T-Bond futures, we assume no changes 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