Transcript
4.1
Hedging Strategies Using Futures Chapter 4 Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.2
Long & Short Hedges • A long futures hedge is appropriate when – you know you will purchase an asset in the future and – You want to lock in the price
• A short futures hedge is appropriate when – you know you will sell an asset in the future & – you want to lock in the price
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.3
Arguments in Favor of Hedging • Companies should – focus on the main business they are in and – take steps to minimize risks arising from fluctuations in • interest rates • exchange rates • other market variables Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.4
Arguments against Hedging • Shareholders are usually well diversified and can make their own hedging decisions • It may increase risk to hedge when competitors do not • Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.5
Convergence of Futures to Spot Figure (a)
S T = FT
Figure (b)
Futures Price Spot Price
Time
Spot Price Futures Price
T
Time
T
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.6
Exercise 1:
Convergence of futures to spot • Explain using an arbitrage argument why the futures price FT converges on the stock price S T at the delivery date T.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.7
Example: short hedge • You are a dealer in Treasury bonds (T-bonds) • Between the time you buy and the time you sell the your inventory may lose value – E.g. if interest rates rise
• You know that – (long) T-bill futures and – (long) T-bill cash prices
are strongly positively correlated. Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.8
• Therefore – short T-bill futures and – long T-bill cash prices are strongly negatively correlated.
• One way of hedging risks is therefore to have a portfolio consisting of – both these instruments • (see table following) Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.9
Anatomy of the short hedge
Cash market
Short hedge in T-bond futures Date
Futures market
Buy cash bonds @ Sell cash bonds @ Loss
105-07 104-18 0-21
Sell T-bond futures @ Buy T-bond futures @ Gain
Now Later Net gain/loss=0
Note: the unhedged position would have resulted in a loss of $0-21/bond.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
105-17 104-28 0-21
4.10
Example: Long hedge • You are an exporter of grains. • You have sold to China – 1m bushels of corn – Delivery date 3 months hence – Price agreed is today’s cash price in Chicago, $2.85/bushel
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.11
• Alternative strategies: – Buy corn today and store • Advantages – No price risk – Current cash price locked in
• Disadvantages – Cost of storage – Interest cost of tying money up
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.12
– Buy corn futures • Advantages – No price risk (in spot market) – Locks in current futures price – No costs of storage
• Disadvantages – Margin ( downpayment ) is required » Some small interest cost – Hedge may not be perfect » Hence some price risk may remain Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.13
Anatomy of the Long hedge Long hedge in corn Cash market Sell cash corn @ Buy cash corn @ Loss
$2.85/bu $3.10/bu $0.25/bu
Date
Futures market
Now Later
Buy corn futures @ Sell corn futures @ Gain
$2.96/bu $3.21/bu $0.25/bu
Net gain/loss=0
Note: the unhedged position would have resulted in a loss of $0.25/bu vs. $0 with the hedge.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.14
What if your forecast is wrong? Long hedge in corn Cash market Sell cash corn @ Buy cash corn @ Gain
$2.85/bu $2.10/bu $0.75/bu
Date
Futures market
Now Later
Buy corn futures @ Sell corn futures @ Loss
$2.96/bu $2.21/bu $0.75/bu
Net gain/loss=0
Thus if instead of rising the cash price falls, the hedging strategy still works: instead of a loss in the spot market being cancelled by a gain in the futures market now we have a gain in the spot cancelled by a loss in the futures. The variance of profits is still zero. Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.15
• Thus if the forecast is wrong it doesn’t invalidate the hedging strategy – The purpose of this strategy is to reduce the riskiness of profits. – This it still does.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.16
Example: Basis and risk • In the previous two examples we assumed a perfect correlation of spot and futures prices of the form: ∆F = ∆S
where
S = spot price F = futures price Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.17
• This meant that a perfect hedge was possible: – All risk could be eliminated
• Consider the following example, however.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.18
Imperfect hedge Cash market
Short hedge in T-bond futures Date Futures market
Buy cash bonds @ Sell cash bobds @ Loss
105-07 104-18 0-21
Now Later
Sell T-bond futures @ Buy T-bond futures @ Gain Net loss=0-03
Basis 105-17 104-31 0-18
Note: the unhedged position would have resulted in a loss of $0-21 vs. $0-03 with the hedge.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
-10 -13 -3
4.19
Exercise 2:
Perfect correlation • Assume futures and spot prices are linearly related: – show that the correlation coefficient between futures and spot price changes is then unity.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.20
Graphics of perfect correlation Prices
Ft
St
Ft = α + βSt
time Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.21
Basis (b) • Definition – The difference between the cash price and the futures price of a commodity
b = S − F (short hedge) b = F − S (long hedge)
• Characteristics – Can be positive or negative – Represents the net asset position of the trader Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.22
Basis and imperfect hedging • We know that the spot and futures prices of a given underlying asset converge at the delivery date – Hence the basis should be zero at delivery
• However, in practice we will find that – The hedger may be uncertain when the asset will be bought or sold – The hedge may require the futures contract to be closed out before its expiration date. – The asset whose price is hedged may not be exactly the same as the asset underlying the futures contract Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.23
Long Hedge: Theory • Suppose that F1 : Initial Futures Price (fixed) F2 : Final Futures Price S 2 : Final Asset Price • You hedge the future purchase of an asset by entering into a long futures contract • Cost of Asset=S2 –(F2 – F1) = F1 + Basis – Where Basis = S2 – F 2 Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.24
Short Hedge: Theory • Suppose that F1 : Initial Futures Price (fixed) F2 : Final Futures Price S2 : Final Asset Price • You hedge the future sale of an asset by entering into a short futures contract • Price Realized=S 2+ (F1 –F2 ) = F1 + Basis Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.25
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 • When there is no futures contract on the asset being hedged, – choose the contract whose futures price is most highly correlated with the asset price. – There are then 2 components to basis Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.26
Basis with incomplete hedging • A trader’s net worth is the difference between her assets and liabilities – assets are what she owns; – Liabilities what she owes (to someone else)
• Suppose a trader is – Long in the asset • she owns corn
– Short in the future • she owes corn which must be delivered in 1 year Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.27
• In value terms her net asset position is b = S - hF where b = value of net assets (‘basis’) S = price of corn currently owned F = price of corn owed future h = proportion of current position hedged (see later) Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.28
Example • Suppose that the spot price of a corn is $2.95/bu and the futures price is $3.05/bu. • I decide to hedge half of the value of my spot position • This means that – for a given • contract size and • contract volume
– I choose h=0.5 or 50%
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.29
– Then the initial basis is b = 2.95 − (0.5) 3. 05 = $1.425/bu
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.30
• From the definition of basis the net assets per unit of the underlying, b, will increase with – An increase in the price of current corn – A decrease in the futures price of corn
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.31
Profits and losses • The change in b is the trader’s profit/loss: ∆b = ∆S − h ∆F
• In practice of course changes in S and F are both uncertain
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.32
Basis (profits) risk • The variance of profits is given by s b2 = s 2S + h2 s F2 − 2hs S s F ?S F
where σ i2 = var( ∆xi ) ρ SF = cor (∆S , ∆F )
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.33
Minimum-variance hedge ratio • To minimise risk we can choose the hedge ratio h such that h* =
σS ρ, σF
ρ = σ SF / σ S σ F
• Note that h*=1 (complete hedging) requires at all times that ∆F = ∆S Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.34
Exercise 3:
Optimal hedge ratio formula • Derive the optimal hedge ratio given above.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.35
Variance of basis change
Graphics: Optimal hedge ratio
h*
h
Hedge ratio Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.36
Complete hedging cont’d • This means that the change in – the current price of corn – the futures price of corn
be always § in the same direction and § of exactly the same magnitude
• Empirically, spot and futures price changes are positively but imperfectly correlated Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.37
Exercise 4:
Complete hedging • Prove that h*=1 (complete hedging) requires at all times that
∆F = ∆S
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.38
Exercise 5:
Estimating the optimal hedge ratio • Given the data in the following table – calculate the optimal monthly hedge ratio, h* – Interpret the result.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.39
Data: Monthly price changes Spot price change Futures price change
0.5 0.56
0.61 0.63
-0.22 -0.12
-0.35 -0.44
0.79 0.6
0.04 -0.06
0.15 0.01
0.7 0.8
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
-0.51 -0.56
-0.41 -0.46
4.40
Hedging Using Index Futures (Page 87) • To hedge the risk in a portfolio the number of contracts that should be shorted is
m* = β • where
P A
– P is the value of the portfolio − β is its CAPM beta, and - A is the value of the assets underlying one futures contract Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.41
Reasons for Hedging an Equity Portfolio • Desire to be out of the market for a short period of time. – Hedging may be cheaper than selling the portfolio and buying it back.
• Desire to hedge systematic risk – Appropriate when you feel that you have picked stocks that will outperform the market. Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.42
Example • • •
Value of S&P 500 is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio? Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.43
Answer • Note that the futures contract value is the index times $250 per unit of index – This gives a futures contract value of $250 x 1k=$250k
• Then we have P A = (1. 5)( 5m) /( 250)(1k )
m* = β
= 30
futures contracts need to be sold. Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.44
Changing Beta • What position is necessary to reduce the beta of the portfolio to 0.75? • What position is necessary to increase the beta of the portfolio to 2.0?
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.45
Answers
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.46
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
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.47
Example
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.48
Key points • A long futures hedge is appropriate when – you know you will purchase an asset in the future and – You want to lock in the price
• A short futures hedge is appropriate when – you know you will sell an asset in the future & – you want to lock in the price
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.49
Key points cont’d • The optimal hedge ratio is typically less than one and given by the formula: h* =
σS ρ σF
• Hedging reduces the variance of the basis (profits) to zero only if the futures and spot prices are perfectly correlated Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.50
Key points cont’d • The efficiency of a hedge is defined as the proportion of the basis risk it eliminates • The efficiency of the hedging strategy can be derived from the parameters of an OLS regression – See assignment Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.51
Assignment 1:
Hedge efficiency • Suppose that we wish to calculate the efficiency of the hedge in Exercise 5: – Show that the proportion of the basis risk that the hedge eliminates, or the Hedge Efficiency, HE, is given by
HE = R2
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.52
I.e. the goodness-of-fit statistic from the OLS regression of ∆S on ∆F (unadjusted for degrees of freedom!): ∆S = α + β∆F + ε
Copyright Robert Cressy, 2003Fundamentals of Futures and Options Markets, 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.53
Assignment 2:
Optimal hedge regression • I estimate the following OLS regression on cotton price data: ∆S t = α + β ∆Ft + ε t
• I get the following results: ^
α = 0. 231 ^
β = 0. 45 R 2 = 0. 88 Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.54
• • •
What is the optimal hedge ratio? What is the reduction in risk afforded by this hedge? Suppose that – a futures contract is 5,000 bushels – my inventory is 1m bushels
•
What would you advise me to do? Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
4.55
Assignment 3:
Perfect hedge • A perfect hedge is defined as a situation where basis risk can be optimally reduced to zero • Show that this requires that the correlation coefficient between S and F be unity.
Fundamentals of Futures and Options Markets , 4th edition © 2001 by John C. Hull with additional notes by Robert Cressy, copyright 2003
