FNCE 30007
Derivative Securities
Lecture – Hedging with Futures
Outline
 Short/Long Hedges
 Should companies hedge?
 Basis Risk
 Cross Hedging
 Use of Stock Index Futures
 Hedge Ratio
Reading: Chapter 3
Short and Long Hedges
 Futures can be used by speculators to gain highly
levered, risky positions (lecture 1)
 This lecture focuses on the use of futures by hedgers
 Hedger’s objective: take a futures position to
minimize risk.
 Short futures hedge when
 sell an asset in the future & want to lock in price
 asset is owned or will be owned
 Long futures hedge when
 purchase an asset in the future & want to lock in
price
Short Hedge
Profit
Spot price:
hedge completion
date
Spot position: long
(hedger owns the asset)
S0 (spot today)
Profit
Futures position: short
Futures price:
hedge completion
date
F0 (futures today)
Spot and futures are rarely equal, but they move together.
The hedge commences at time t=0
If the hedger gains in the spot they lose in the futures and vice versa
Long Hedge
Profit
Spot price:
hedge completion
date
Spot position: short
(hedger buys asset in the future)
S0 (spot today)
Profit
Futures position: long
Futures price:
hedge completion
date
F0 (futures today)
Spot and futures rarely equal, but they move together.
The hedge commences at time t=0
If the hedger gains in the spot they lose in the futures and vice versa
Short and long hedges
 We are going to use three approaches
 All provide the same answer
 Each provides different insights
 Approach 1: What transactions actually occur
 Approach 2: Emphasizes offsetting payoffs and
assists in the development of the hedge ratio
 Approach 3: Emphasizes basis risk (the key risk
when hedging)
Short hedge
 April 20: Farmer negotiates to sell 50,000 bu of corn
at the spot price on June 20
 June 20 is futures maturity date – no basis risk (
 Quoted:
 Spot price of corn: $3.50/bu
 June corn futures price: $3.35/bu (each contract 5,000 bu)
 After futures gains & losses, price received is
$3.35/bu.
Apr 20 June 20
Short 10 June futures
Sell corn at spot price
Close out futures
= )
Short hedge
Scenario 1: Spot price June 20 $3.10/bu
Approach 1: Actual transactions
Spot: Farmer sells corn at ($3.10)(50,000) = 155,000
Futures: Gain ($3.35 – $3.10)(50,000) = $12,500
Total received: $155,000 + $12,500 = $167,500.
price/bu = $167,500 / 50,000 = $3.35
Approach 2: Offsetting payoffs
$3.50 $3.35
Profit Profit
Spot at T Futures at T
Position at t=0: $175,000
(50,000bu x $3.50)
Adjust for hedge profit/loss
Spot loss: ($20,000)
(50,000bu x $0.40)
Futures gain: $12,500
(50,000bu x $0.25)
Net $167,500
The hedge loss is $-20,000+$12,500 = $7500
Short hedge
Scenario 2: Spot price June 20 $3.70/bu
Approach 1: Actual transactions
Spot: Sell corn at ($3.70)(50,000) = $185,000
Futures: Loss ($3.70 – $3.35)(50,000) = $17,500
Total received: $185,000 – $17,500 = $167,500.
Price/bu = $167,500 / 50,000 = $3.35
Approach 2: Offsetting payoffs
$3.50 $3.35
Profit Profit
Spot at T Futures at T
Position at t=0: $175,000
(50,000bu x $3.50)
Adjust for hedge profit/loss
Spot gain: $10,000
(50,000bu x $0.20)
Futures loss: ($17,500)
(50,000bu x $0.35)
Net $167,500
The hedge loss of $7500 is locked in at time t=0
Long hedge
 April 20: Food company to buy 50,000 bu of corn at
the spot price on June 20
 June 20 is futures maturity date – no basis risk (
 Quotes:
 Spot price of corn: $3.50/bu
 June corn futures price: $3.35/bu (each contract 5,000 bu)
 After futures gains & losses the price paid should be
$3.35/bu.
Apr 20 June 20
Long 10 June futures
Buy corn at spot price
Close out futures
= )
Long hedge
Scenario 1: Spot price June 20 $3.10/bu
Approach 1: Actual transactions
Spot: Company buys corn ($3.10)(50,000) = 155,000
Futures: Loss ($3.35 – $3.10)(50,000) = $12,500
Total paid: $155,000 + $12,500 = $167,500.
price/bu = $167,500 / 50,000 = $3.35
Approach 2: Offsetting payoffs
Profit
Spot at T$3.50
Profit
$3.35 Fut at T
Position at t=0: $175,000
(50,000bu x $3.50)
Adjust for hedge profit/loss
Spot gain: $20,000
(50,000bu x $0.40)
Futures loss: ($12,500)
(50,000bu x $0.25)
Net $167,500
Profit from hedge of $7,500 offsets cost.
Recall short hedge loss of $7500.
Long hedge
Scenario 2: Spot price June 20 $3.70/bu
Approach 1: Actual transactions
Spot: Buy corn at ($3.70)(50,000) = $185,000
Futures: Gain ($3.70 – $3.35)(50,000) = $17,500
Total paid: $185,000 – $17,500 = $167,500.
Price/bu = $167,500 / 50,000 = $3.35
Approach 2: Offsetting payoffs
Profit
Spot at T$3.50
Profit
$3.35 Fut at T
Position at t=0: $175,000
(50,000bu x $3.50)
Adjust for hedge profit/loss
Spot loss: $10,000
(50,000bu x $0.20)
Futures gain: ($17,500)
(50,000bu x $0.35)
Net $167,500
Profit from hedge of $7,500 offsets cost - this is locked in.
Recall short hedge loss of $7500 – this is also locked in.
Question
 Should the food company buy the corn it needs today in
the spot market?
 If company doesn’t need corn until June 20, better not to
buy today.
 A lower price will be paid on June 20 ($3.35 c.f $3.50).
 Storage costs avoided.
 But what if F(t)>S(t)? If buy in spot now incur
 Time value of $ (+)
 Storage costs (+)
 Convenience yield (-)
 Most of the time long hedgers do not take delivery
 close out before the delivery date and buy in the spot.
 futures delivery can be very expensive.
Should companies hedge?
 Arguments in favor of hedging
 Companies should focus on the main business they are in & minimize
risks from interest rates, exchange rates, & other market variables
 Arguments against hedging
 Shareholders are usually well diversified and can make their own
hedging decisions
 Explaining a situation where there is a loss on the hedge and a gain on
the underlying can be difficult
 It may increase risk to hedge when competitors don’t (e.g gold
jewellery manufacturer in a perfectly competitive market).
Profit S0 F0Profit
Spot increase: costs passed on to consumers
Spot decrease: gains passed on to consumers
Futures position is speculative
Gold spot Gold futures
Basis risk
 Perfect hedge: eliminates all risk.
 Previous examples were perfect
 Most hedges are imperfect because:
 hedge requires futures contract to be closed out
before expiration date
 hedger may not be sure about the exact date the
asset will be bought or sold.
 cross hedge: asset to be hedged not the same as
the asset underlying the futures contract (e.g
hedge jet fuel with oil futures)
Basis risk
 These issues create basis risk.
 Basis(t) = spot price of asset to be hedged(t) – futures price of
contract used(t)
 Basis risk arises because of uncertainty about the basis when the
hedge is closed out
Futures
Price
Spot
Price
Time
Close out
Convergence explained by
the cost of carry model
(next week)
Enter hedge
Basis risk: Long hedge
 Approach 3: Emphasizes basis risk
 Suppose
F0: Initial Futures Price
F1: Final Futures Price
S1: Final Asset Price
 Hedge future asset purchase by taking a long futures
position
 Cost of Asset = S1 – (F1 – F0) = F0 + (S1 -F1)= F0 +Basis1
Approach 3
Up to now we closed out at time T
and therefore Basis1 =0. This
meant a perfect hedge was achieved.
Unknown at time t=0
Approach 1
Profit:
F1-F0 F0
F1
Basis risk: Short hedge
 Suppose
F0 : Initial Futures Price
F1 : Final Futures Price
S1 : Final Asset Price
 Hedge the future asset sale via a short futures contract
 Price Realized=S1+ (F0 – F1) = F0 + (S1 - F1)= F0 + Basis1
Approach 1 Approach 3
Up to now we closed out at time T
and therefore Basis1 =0. This
meant a perfect hedge was achieved.
Profit:
F0-F1 F0
F1
Basis risk: Example 1
(close out before expiration date)
t=0: Spot = $3.50, futures = $3.35.
t=1: Spot = $3.30, futures = $3.05.
Basis:
b0 = 3.50 – 3.35 = 0.15
b1 = 3.30 – 3.05 = 0.25
 Approach 3
 Short hedger: price rec’d: F0 + b1 = 3.35 + .25 = $3.60
 Long hedger: price paid: F0 + b1 = 3.35 + .25 = $3.60
 b1 is unknown basis risk (F0 is certain)
 Seek to remove spot price risk but introduce basis risk
 Basis risk can improve or worsen hedger’s position.
 Approach 2
 Short hedger: Position at t=0 $3.50 plus hedge profit $0.10 = $3.60
S0=$3.50
$3.30
F0=$3.35$3.05
Hedge profit:
Fut gain $0.30-spot loss $0.20 = $0.10
Basis risk: Example 2
(unsure about close out date)
 It is July 25.
 Company knows it will purchase 40,000 bu of corn
some time in September or October.
 Current December corn futures price is $2.55/bu.
 Each corn futures contract is for 5,000 bushels.
 Hedging strategy:
 Long position in 40,000/5,000 = 8 December corn
futures contracts on July 25 at a futures price of
$2.55.
 Close out contract when ready to purchase the corn.
Basis risk: Example 2
(unsure about close out date)
 Assume company ready to purchase corn on October 15.
 Spot price on October 15 = $2.90
 December futures price on October 15 = $2.80
 Basis on October 15: $2.90 - $2.80 = $0.10
 Net cost of corn:
 Approach 1:
Spot price on October 15 - gain on futures
= 2.90 – (2.80 – 2.55) = $2.65
 Approach 3:
Futures price on July 25 + basis on October 15
= 2.55 + (2.90 – 2.80) = $2.65.
 Approach 2:
 S0 unavailable
Basis risk: cross hedge
 Hedge by taking position in related futures contract.
 no derivatives contract for asset being hedged OR
 futures contract exists but market highly illiquid/expensive.
 Success depends on relationship between
 i) asset being hedged and
 ii) asset which underlies the derivatives contract.
 Should cointegrate with fast rate of equilibrium adjustment.
 Basis risk likely to increase.
F0 + (S*1 - F1) + (S1 - S*1)
(S*1 - F1) = basis that exists if the asset being hedged were the same as
the asset underlying the futures
(S1 - S*1) = basis from difference between two assets.
Additional source of uncertainty
Basis risk: cross hedge
 Illustrate with a jet fuel hedge using oil futures
 F0+B1 = F0+(S1-F1) = F0+(S*1-F1)+ (S1-S*1)
Basis:
close
out
Oil Jet
fuel
Jet
fuel
Oil Basis:
S*1: oil
Oil Oil
Cost/revenue for
regular oil hedge
Additional source of
uncertainty.
Want diff to be stable
Time
Price
Oil: spot/fut
Jet fuel spot
Hedge commences
Short hedge (sell fuel): loss spot, loss futures
Long hedge (buy fuel): gain spot, gain futures
If jet fuel increases and oil (spot/fut) decreases
fortunes reversed. This is not acting like a hedge
anymore
Index futures hedging
 Desire to be out of the market for a
short period of time.
 Hedging may be cheaper than selling
the portfolio & buying it back.
 Desire to hedge systematic risk
 Feel you have picked stocks that will
outperform the market.
 Seek exposure to idiosyncratic risk only
Index futures hedging
 To hedge the risk in a portfolio the number of
contracts shorted is
 beta = 1, portfolio return ≈ market return
 beta = 2, portfolio return ≈ twice market return
 beta = 0.5, portfolio return ≈ half market return
 Note when beta≠1, cross hedge because futures contract not
written against the spot exposure
# contracts for index fund
* 0
0
0
0
current portfolio value
current valueof onefutures contract
A
F
A
F
PVN
V F
V P
V F
β β= =
= =
= =
Example: Index futures hedge
Value of ASX 200 = 5,500
Futures price = 5,600
Size of portfolio = $10 million
Beta of portfolio = 1.2
Risk-free rate = 8% per annum
Dividend yield on the index = 1% p.a
Dividend yield on the portfolio = 4% p.a
One contract is $25 times the futures index points
What position in ASX 200 futures to hedge the portfolio for 3
months?
(1.2)(10million/(25*5,600)) = 85.71~ 86 short contracts
If index fund short 71, scale up by 20% to 86 given the higher spot risk.
Example: Index futures hedge
 Suppose the index is 5300 in three months & the futures price is 5350.
 Approach 2:
 Adjust current value of the position for the hedged gain/loss
 Short futures (gain)
(86)(5,600 – 5,350 )(25) = 537,500
 Index (loss)
(5,500 – 5,300) / 5,500 = 3.64%.
 Index pays dividend of 1% p.a (0.25% per 3 months).
 Taking dividends into account loss reduces to 3.64 – 0.25 = 3.39%
S0=5500
ST=5300
Spot Futures
F0=5600
FT=5350
Example: Index futures hedge
 CAPM
ri = rf + Beta(rm –rf)
 Expected (%) portfolio loss
rp = 0.02 + 1.2(-0.0339 – 0.02) = -4.46%
 Expected portfolio value (incl dividends):
$10,000,000(1 – 0.0446) = $9,554,000
 Expected value of the hedgers position:
$ 9,554,000 + $537,500 = $10,091,500
Scale up the market loss via CAPM
Index futures : changing beta
 Position to reduce beta of the portfolio to 0.90. (Assume
initial beta of 1.5)
 Short 45 contracts
 Position to increase beta of the portfolio to 2.5 (Assume
initial beta of 1.5)
 Buy 75 contracts
45~86.44
)25)(350,5(
000,000,10)9.05.1()( * =−=−
F
Pββ
75~76.74
)25)(350,5(
000,000,10)5.15.2()( * =−=−
F
Pββ
Minimum variance hedge ratio (MVHR)
 Typically measured in quantities (commodity
contracts) and $ values (financial contracts)
 So far the hedge ratio = 1.0 (naïve hedge)
 A naïve hedge may be sub-optimal with basis risk i.e
 cross hedging and/or
 hedge completion date not same as futures expiration date
sizeof futures positionHedge ratio (HR)
sizeof spot exposure
=
Minimum variance hedge ratio (MVHR)
 We can derive a minimum variance hedge ratio
 It recognizes basis risk and seeks to minimize the
variation in the hedged profit/loss.
*
2
*
2
optimal hedge ratio
covariance between thespot exposureand futures used
futures variance
spot std deviation
correlation between spot and futures
sf s
sf
f f
sf
f
s
sf
h
h
σ σ
ρ
σ σ
σ
σ
σ
ρ
= =
=
=
=
=
=
Minimum variance hedge ratio (MVHR)
 Example 1: Hedge S&P500 (S&P500.xlsx) for index fund
 Calculate variances, covariance and correlations using
continuously compounded returns
 Calculate the cont compounded hedged returns each day via
 Determining the time period for estimation
 Comparison of naïve v MVHR
 Can calculate the optimal number of contracts via
[ ], 1r 100 ln( ) ln( )s t t tS S −= × −
*
, , ,h t s t f tr r h r= − ×
* * A
F
VN h
V
= VA=$10M, VF=$1M
Naïve hedge N*=10
If h*=0.9, N*=9
Minimum variance hedge ratio (MVHR)
We consider a min variance hedger as at 31/12/2019 seeking to
hedge a $200M investment in the S&P500 index over 2020.
Minimum variance hedge ratio (MVHR)
 Example 2: Fuel hedge (Fuel hedge.xls)
F
A
Q
QhN ** =
QA = size of position being hedged (units)
QF = size of one futures contract
N* = optimal number of futures contracts
Minimum variance hedge ratio
 We have just seen
 Hull argues that this implies
 Equation 1 however assumes hedge close out is
close to the maturity of the futures contract.
 If this is not the case, because also
accounts for the risk associated with early close out.
( )* * 2A
F
VN h
V
=( )* 0
0
1A
F
PVN
V F
β β= =
*h β=
*h β≠ *h