Hedging Strategies
Using Futures
Chapter 3
0
HEDGERS
OPEN POSITIONS IN THE FUTURES
MARKET IN ORDER TO
ELIMINATE THE RISK
ASSOCIATED WITH THE
SPOT PRICE
OF THE UNDERLYING ASSET
1
Spot price risk
Pr
Sj
St
t
j
time
2
HEDGERS
PROBLEM: TO OPEN
A LONG HEDGE
OR
A SHORT HEDGE?
There are two ways to determine
whether to open a short or a long
hedge:
3
1.
A LONG HEDGE
OPEN A LONG FUTURES POSITION
IN ORDER TO HEDGE THE PURCHASE OF
THE PRODUCT AT A LATER DATE.
THE HEDGER LOCKS IN THE PURCHASE
PRICE.
A SHORT HEDGE
OPEN A SHORT FUTURES POSITION
IN ORDER TO HEDGE THE SALE OF THE
PRODUCT AT A LATER DATE.
THE HEDGER LOCKS IN THE SALE PRICE
4
2.
A LONG HEDGE
OPEN A LONG FUTURES POSITION
WHEN THE FIRM HAS A
SHORT SPOT POSITION.
A SHORT HEDGE
OPEN A SHORT FUTURES POSITION
WHEN THE FIRM HAS A
LONG SPOT POSITION.
5
Example:
A LONG HEDGE
Date
Spot market
Futures market
t
St = $800/unit
Contract to buy
Gold on k.
Ft,T = $825/unit
-25
long one gold
futures for delivery
at T
k
Buy the gold
Sk = $816/unit
Short one gold
futures for delivery
at T.
Fk,T = $842/unit
-26
Basis
1
T
Amount paid:
or
816 + 825 – 842 = $799/unit
825 + (816 – 842)
= $799/unit
6
Example:
A SHORT HEDGE
Date
Spot market
Futures market
t
St = $800/unit
Contract to sell
Gold on k,
Ft,T = $825/unit
-25
short one gold
futures for delivery
at T
k
Sell the gold
Sk = $784/unit
Long one gold
futures for delivery
at T.
Fk,T = $812/unit
-28
Basis
3
T
Amount received: 784 + 825 – 812 = $797/unit
or
825 + (784 – 812)
= $797/unit
7
NOTATIONS:
t<T
t = current time; T = delivery time
F t,T =
THE FUTURES PRICE AT TIME t FOR
DELIVERY AT TIME T.
St =
THE SPOT PRICE AT TIME t.
k=
THE DATE UPON WHICH THE FIRM
TRADES THE ASSET IN THE SPOT
MARKET.
k≤T
Sometimes t = 0 denotes the date the
hedge is opened.
8
THE HEDGE TIMING
k = is the date on which the hedger
conducts the firm spot business and
simultaneously closes the futures
position. This date is almost always
before the delivery month; k ≤ T.
Today
Open the hedge:
open a futures
position
Trade spot and
Close the futures
position
Delivery
t
k
T
Time
9
THE HEDGE TIMIMG
Date k is (almost) always before the
delivery month.
WHY?
1. Often k is not in any of the delivery
months available.
2. From the first trading day of the delivery
month, the SHORT can decide to send a
delivery note. Any LONG with an open
position may be served with this
10
delivery note.
Spot and Futures prices over time
Commodities and assets are traded in the
spot and futures markets simultaneously.
Thus, the relationship between the sport
and futures prices:
At any point in time
And
Over time
Is of great importance for traders.
11
The Basis
The basis at any time point, j, is the
difference between the asset’s spot price
and the futures price on j.
BASISj = SPOT PRICEj - FUTURES PRICEj
Notationally: Bj = Sj - Fj,T
j < T.
When discussing a basis, one must specify
the futures in question, i.e., a specific
delivery month. Usually, however, it is
understood that the futures is for the
nearest month to delivery.
12
A LONG HEDGE
TIME
t
SPOT
Contract to buy
FUTURES
LONG Ft,T
B
Bt
SHORT Fk,T
Bk
Do nothing
k
BUY Sk
T delivery
Actual purchase price = Sk + Ft,T - Fk,T
= Ft,T + [Sk - Fk,T]
= Ft,T + BASISk
13
A SHORT HEDGE
TIME
t
SPOT
Contract to sell
FUTURES
SHORT Ft,T
B
Bt
LONG Fk,T
Bk
Do nothing
k
SELL Sk
T delivery
Actual selling price
= Sk + Ft,T - Fk,T
= Ft,T + [Sk - Fk,T]
= Ft,T + BASISk
14
In both cases,
Long hedge and short hedge
the hedger’s purchase/sale price, when
the hedge is closed on date k, is:
Ft,T + BASISk
This price consists of two portions:
a known portion:
and a random portion: the
We return to this point later.
Ft,T
BASISk
15
ALSO NOTICE:
t
k
T
The purchase/sale price when the hedge
is closed on date k is: Ft,T + BASISk
Which may be rewritten:
=
Ft,T + BASISk + St – St
=
St – [St – Ft,T - Bk]
=
St + [Bk – Bt]
16
Spot prices and futures prices over time
The key to the success of a hedge is the
relationship between the cash and the
futures price over time:
Statistically, Futures prices and Spot
prices of any underlying asset, co vary
over time. They tend to co move
“together” ; not in perfect tandem and
not by the same amount, nevertheless,
these prices move up and down
together most of the time, during the
17
life of the futures.
Open
close Long hedge Short hedge
the hedge
Fk,T
Ft,T
Sk
a success
a failure
Loss on
the hedge
St
Fk,T
Sk
a failure
a success
Loss on
the hedge
18
Example: A LONG HEDGE
TIME
t
SPOT
St= $3.40
Do nothing
k
BUY Sk=$3.80
FUTURES
LONG
BASIS
Ft,T=$3.50
-$.10
SHORT
F k,T=3.85
-$.05
T delivery
Actual purchase price:
NO hedge:
$3.80
With hedge:
$3.45  (Successful hedge)
19
Example: A LONG HEDGE
TIME
t
SPOT
St= $3.40
Do nothing
k
BUY Sk=$3.00
FUTURES
LONG
BASIS
Ft,T=$3.50
-$.10
SHORT
F k,T=3.05
-$.05
T delivery
Actual purchase price:
NO hedge:
$3.00
With hedge:
$3.45  (Unsuccessful hedge)20
The basis upon delivery: BT = 0
On date k, the basis is
Bk = Sk - Fk,T
k < T.
If k coincides with the delivery date,
however, k = T. The basis is:
BT = ST - FT, T
at T.
BUT, FT,T is the futures price on date T for
delivery on date T, which implies that:
FT,T = ST

BT = 0.
21
Convergence of Futures to Spot
over the life of the futures
Futures
Price
Spot Price
Spot Price
Futures
Price
Time
(a)
Time
(b)
22
Basis Risk
The Basis is the difference between the
spot and the futures prices. I.e., the Basis
is a RANDOM VARIABLE. Thus,
Basis risk
arises because of the uncertainty about the
Basis when the hedge is closed out on k.
The basis, however, is the difference of two
random variables and thus, the Basis is
LESS RISKY than each price by itself.
Moreover, we do know that BT = 0
upon delivery.
23
Generally, the basis fluctuates less than
both, the cash and the futures prices.
Hence, hedging with futures reduces
risk. Basis risk exists in any hedge,
nonetheless.
Sk
Pr
Bk
Ft,T
St
t
BT = 0
Bt
k
T
time
24
We showed that for both types of hedge
A SHORT HEDGE or A LONG HEDGE,
The price received/paid by the hedger:
Ft,T + BASISk
This price consists of two parts:
Part one:
Ft,T is KNOWN when the
hedge is opened.
Part two:
BASISk is risky.
25
Conclusion:
At time t, WITHOUT HEDGING
cash-price risk.
WITH HEDGING,
basis risk.
Hedging with futures is nothing more than
changing the firm’s spot price risk
Into a smaller risk, namely,
The basis risk.
26
A CROSS HEDGE:
When there is no futures contract on the
asset being hedged,
choose the contract whose futures price is
most highly correlated with the spot asset
price.
NOTE, in this case, the hedger creates a
two components basis:
one component associated with the asset
underlying the futures
and one component associated with the
spread between the two spot prices.
27
A CROSS HEDGE:
Let S1t
be the spot asset price at time t.
Remember! - This is the asset that the
hedger is trying to hedge; e.g. jet fuel.
Let S2t
be the spot price at time t of the asset
underlying the futures. E.g., natural gas.
This, of course, is a different asset and
that is why this hedge is called a
CROSS HEDGE
28
A CROSS HEDGE
TIME
CASH
FUTURES
t
Contract to trade S1
Do nothing
Ft,T(2)
k
Trade for S1K
Fk,T(2)
T delivery
PAY/RECEIVE= S1K + Ft,T(2) - Fk,T(2)
= Ft,T(2) +[S2k - Fk,T(2)] +[S1k - S2k]
= Ft,T(2) + BASIS(2)k + SPREADK
29
Arguments in Favor of Hedging
Companies should focus on the main
business they are in and take steps to
minimize risks arising from interest
rates, exchange rates, and other
market variables
30
Arguments against Hedging
• Explaining a situation where there
is a loss on the hedge and a gain on
the underlying can be difficult.
• Shareholders are usually well
diversified and can make their own
hedging decisions.
31
Delivery month? MOSTLY, the hedge is
opened with a futures for the delivery
month closest to the firm’s spot trading of
the asset, or the nearest month beyond
that date. The key factor in choosing the
futures’ delivery month is the correlation
between the spot and futures prices or
price changes.
Statistically, in most cases, the spot price
highest correlation is with the nearest
delivery month futures price, which is
closest to the firm’s cash activity.
32
The number of Futures to use in the
hedge
Open a hedge.
Questions:
Long or Short?
Delivery month?
Commodity to use?
How many futures to use in the hedge?
33
HEDGE RATIOS, NOTATION:
NS
=
The number of units of the
commodity to be traded in the
SPOT market.
NF
=
The number of units of the
commodity in ONE FUTURES
CONTRACT.
n
=
The number of futures contracts
to be used in the hedge.
h
=
The hedge ratio.
34
HEDGE RATIOS:
Open a hedge.
Question:
Given that the firm has a contract to trade
NS units of the underlying commodity on
date k in the spot market and given that
one futures covers NF units of the
underlying commodity:
How many futures to use in the hedge?
i.e., what is n?
35
HEDGE RATIOS, DEFINITION:
The number of units in the futures position
h
The number of units in the spot position
nN F
h
NS

NS
nh
.
NF
The hedge ratio, h, determines the number
of futures to hold, n.
36
THE NAÏVE HEDGE RATIO:
h = 1.
The total number of units covered
by the futures position = nNF , exactly
covers the number of units to be traded
in the spot market = NS.
nN F
h
1
NS
NS
 n
NF
37
Examples:
NAÏVE HEDGE RATIO:
h = 1.
1. A firm will sell NS = 75,000
barrels of crude oil.
NYMEX WTI: NF = 1,000 barrels.
SHORT:
n
= 75,000/1,000
= 75 NYMEX futures.
38
2. A firm will buy NS = 200,000
bushels of wheat.
CBT wheat futures:
NF = 5,000.
LONG:
n = 200,000/5,000
= 40 CBT futures.
39
3. A firm will sell NS = 3,600
ounces of gold.
NYMEX gold futures: NF = 100 ounces.
SHORT:
n
= 3,600/100
= 36 CBT futures.
40
How to open a long hedge with multiple
future spot trading? A Strip.
DATE
SPOT MARKET
Sep1,07
Contract to buy 75,000bbls
of WTI crude oil.
on: Oct 1,07;
Nov 1,07;
Dec 1,07;
Jan 2,08.
41
A STRIP.
A STRIP is a hedge in which there are
several long (or several short) positions
opened simultaneously with equal time
span between the delivery months of the
positions.
Each one of these futures exactly hedges
a specific future trade in the spot market
42
Open a long STRIP with h = 1
DATE
SPOT MARKET
S
FUTURES MARKET
F FUTURES POSITIONS
Sep1,07
contract to
92.00
buy 75,000bbls on
Oct 1,07;
Nov 1,07;
Dec 1,07;
Jan 2, 08.
Long 75 NOV 07
93.00
long 75 NOV 07
Long 75 DEC 08
93.50
long 75 DEC 08
Long 75 JAN 08
93.85
long 75 JAN 08
Long 75 FEB 08
94.60
long 75 FEB 08
43
Date
SPOT MARKET
S
FUTURES MARKET
Sep1,07 contract to
92.00
buy 75,000bbls
Long
Long
Long
Long
Oct1,07 buy 75,000bbls 93.00
Nov1,07 buy 75,000bbls
75
75
75
75
NOV 2007
DEC 2007
JAN 2008
FEB 2008
F
FUTURES POSITIONS
93.00
93.50
93.85
94.60
long 75 NOV 2007
long 75 DEC 2007
long 75 JAN 2008
long 75 FEB 2008
short 75 NOV 07
93.10
long 75 DEC 2007
long 75 JAN 2008
long 75 FEB 2008
92.90
short 75 DEC 07
93.05
long 75 JAN 2008
long 75 FEB 2008
Dec1,07 buy 75,000bbls
94.00
short 75 JAN 08
94.15
long 75 FEB 2008
Jan2,08 buy 75,000bbls
94.75
short 75 FEB 08
94.95
NO POSITION
The average price for the un hedged strategy : (93+92.90+94+94.75)/4 = 93.660
The average price for the hedged strategy:
93.00 + (93.00 - 93.10) =
92.90
93.50 + (92.90 – 93.05) =
93.35
93.85 + (94.00 – 94.15) =
93.609
94.60 + (94.75 - 94.95) =
94.40
44
93.5625
ROLLING THE HEDGE FORWARD
Lack of sufficient liquidity in contracts for
later delivery months may cause firms to
hedge a long-term business trade
employing shorter term hedges. In this
case, the shorter term hedges must be
rolled over until the firm trade in the
cash market.
45
Roll over hedge with h = 1
DATE
SPOT MARKET
DEC, 07 contract to sell
S
89.00
FUTURES MARKET
F
FUTURES POSITIONS
Short 100 NYMEX WTI; 88.20
100,000bbls on
Futures for delivery on
JAN, 09.
MAY 08
SHORT 100 MAY 08 Fs.
And Roll over the hedge on
APR 2008
And
AUG 2008
46
Date
SPOT MARKET
DEC, 07 contract to
S
FUTURES MARKET
89.00
sell 100,000 bbls
F
short 100 MAY WTI
FUTURES POSITIONS
88.20
Oct1,07 buy 75,000bbls Short 100 MAY 2008
APR 08
long 100 MAY 2008
Short 100 SEP 2008
87.40
87.00
Short 100 SEP 2008
AUG 08
Long 100 SEP 2008
Short 100FEB 2009
86.50
86.30
Short 100 FEB 2009
Long 100 FEB 2009
85.90
NO POSITION
JAN, 09 sell 100,000bbls
86.00
The selling price without the rolling hedge:
$86.00/barrel
The selling price with the rolling hedge:
$87.70/barrel
$86.00 + (88.20 – 87.40) + (87.00 – 86.50) + (86.30 – 85.90) = 87.70.
47
Other hedge ratios.
Suppose that the relationship between
the spot and futures prices over time is:
Spot
Futures
case one:
 $1
 $2
Case two:
 $1
 $0.5
Clearly, the Naïve hedge ratio is not
appropriate in these cases.
48
THE MINIMUM VARIANCE HEDGE RATIO
OBJECTIVE: To minimize the risk
associated with the hedge
RISK = VOLATILITY.
THE VOLATILITY MEASURE:
THE VARIANCE
49
THE MINIMUM VARIANCE HEDGE RATIO
Restating the hedge goal,
OBJECTIVE: Given that the firm will trade
NS units in the spot market,
find the number of futures,
n*
THAT MINIMIZES THE VARIANCE OF THE
CHANGE OF THE HEDGED POSITION’S
VALUE.
50
NOTATIONS
t
=
St =
k
=
T
=
Fj,T=
The hedge opening date.
Spot market price.
The hedge closing date.
The futures delivery date.
The futures price on date j for
delivery at T. t ≤ j ≤ T.
51
NOTATIONS
n =
h =
NF =
NS =
The number of futures contracts
used in the hedge.
The hedge ratio.
The number of units of the asset in
one contract.
The number of units of the asset
to be traded spot on k.
52
FROM THE GENERAL RELATIONSHIP BETWEEN
n and h (SLIDE 36) the optimal number of
futures, n* is determined by h*:
NS
n h
.
NF
*
*
Thus, we find h* and thereby determine
the optimal number of futures to be
held in the hedge, n*.
53
Derivation of the result:
The initial and terminal hedged position
values:
VPt = StNS +nNFFt,T
VPk = SkNS +nNFFk,T
The position value change:
(Vp) = VPk - VPt
= (SkNS +nNFFk,T) - (StNS +nNFFt,T)
= NS(Sk- St) +nNF(FK,T - Ft,T).
54
Rewriting the last result:
(VP) = NS(Sk- St) +nNF(Fk,T - Ft,T).
[
]
(VP) = NS (Sk- St) +[nNF/NS](Fk,T - t,T)
[
]
(VP) = NS (Sk- St) +h(Fk,T - Fy,T)
PROBLEM: Find h* so as to minimize
the Variance of (VP).
55
VAR(VP) = NS2 VAR[(Sk- St) +h(Fk,T - Ft,T)]
= NS2[VAR(S)+VAR(hF)+2COV(S;hF)]
= NS2 [VAR(S)+h2VAR(F)+2hCOV(S;F)].
Set:
d[VAR(VP)]/dh = 0:
2h*VAR (F) + 2COV(S; F) = 0.
h* = - COV(S;F)/VAR(F)
56
THE MINIMUM RISK HEDGE RATIO IS:
cov( S; F)
h*  .
var( F)
σ ΔS
n * NF
h*  - ρΔS,ΔF
.
σ ΔF
NS
NS
n  h
.
NF


57
This result can be rewritten as:
cov(S; F)
h*  .
var(F)
σS n * N F
h*  - ρS,F

.
σF
NS
NS
n  h
.
NF


58
The negative sign in the formula for h*,
only indicates that in the hedge position
the SPOT and the FUTURES positions
are in opposite directions.
If the hedger is short spot,
the hedge is long.
If the hedger is long spot,
the hedge is short.
59
EXAMPLE 1: A company will buy 800,000
gallons of diesel oil in 2 months. It opens
a long cross hedge using NYMEX heating
oil futures. An analysis of price changes
over a 2 month interval yields:
(ΔS) = 0.025; (ΔF)=0.033;and
ρ(ΔS;ΔF) = 0.693.
The risk minimizing hedge ratio:
h* = -(.693)(0.025)/0.033 = -0.525.
One NYMEX heating oil contract is for
NS = 42,000 gallons, so
Long
n* = (0.525)[800,000/42,000]
60
= 10futures.
Notice that in this case, a NAÏVE HEDGE
ratio would have resulted in taking a long
position in:
n* = 800,000/42,000 = 19 futures.
Taking into account the correlation
between the spot price changes and the
futures price changes, allows the use of
The minimum variance hedge ratio and
thus, n* = 10 futures.
Of course, if the correlation and the
standard deviations take on other values
the risk-minimizing hedge ratio may
61
require more futures than the naïve ratio.
EXAMPLE 2: A firm will buy 1 million
gallons of jet fuel in 3 months. The firm
chooses to long cross hedge with NYMEX
heating oil futures. σ(S)=0.04,
σ(F)=0.02; ρ(S;F) = 0.42.
The optimal hedge ratio:
h* = - (0.42)(0.04)/(0.02) = - 0.84.
Thus, to minimize the risk long 20 futures:
n* = (0.84)[1,000,000/42,000] = 20.62
h* , using Regression:
DATA:
n+1 weeks.
S1
F1,t
S1
F1
S2
F2,t
S2
F2
S3
F3,t
S3
F3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Sn
Fn
Sn+1
Fn+1,t
ΔS  α  βΔF  e
i

β  h*
i
i
i1,2, ..., n.
63
EXAMPLE 3. Hedging for copper: A STRIP.
On SEP 4, 2005 A U.S. firm has a contract to
purchase NS = 1,000,000 pounds of copper
on the first trading day of each of the
following months:
FEB 06, AUG06, FEB07 and AUG07.
The firm decides to hedge these purchases
with NYMEX copper futures.
One NYMEX copper futures is for:
NF = 25,000 pounds of copper.
Following a regression analysis, the firm
decides to use: h* = - 0.7.
64
Date:
SEP 04 2005
Spot price:
USD2.72/pound
Futures prices, USD/pound were:
For Delivery: MAR 2006
2.723
SEP 2006
2.728
MAR 2007
2.716
SEP 2007
2.695
65
How to open the long Strip:
The number of futures to LONG is:
n* = (0.7)[1,000,000/25,000] = 28.
All prices are USD/pound.
Date
SPOT
SEP 05 contract
Do nothing
FUTURES MARKET
F
FUTURES POSITIONS
Long 28 MAR 2006 2.723
Long 28 MAR 2006
Long 28 SEP 2006 2.728
Long 28 SEP 2006
Long 28 MAR 2007 2.716
Long 28 MAR 2007
Long 28 SEP 2007 2.695
Long 28 SEP 2007
66
The following prices have materialized on
the first trading days of the given months:
All prices are USD/pound
DATE
SEP05
FEB06 AUG06 FEB07 AUG07
SPOT
2.72
2.69
2.65
PRICE
Futures prices for delivery
2.77
MAR06
2.723
2.691
SEP06
2.728
2.702
2.648
MAR07
2.716
2.707
2.643
2.767
SEP07
2.695
2.689
2.642
2.765
2.88
2.882
67
Date
SPOT MARKET
FUTURES MARKET
F
FUTURES POSITIONS
SEP 05
NOTHING
Long 28 MAR 2006 2.723
long 28 MAR 2006
Long 28 SEP 2006 2.728
long 28 SEP 2006
Long 28 MAR 2007 2.716
long 28 MAR 2007
Long 28 SEP 2007 2.695
long 28 SEP 2007
Feb 06
buy 1M units 2.69
short 28 MAR 06
2.691
long 28 SEP 2006
long 28 MAR 2007
long 28 SEP 2006
Aug 06
buy 1M units 2.65
short 28 SEP 06
2.648
long 28 MAR 2007
long 28 SEP 2007
Feb 07
buy 1M units 2.77
short 28 MAR 07
2.767
long 28 SEP 2007
Aug 07
buy 1M units 2.88
short 28 SEP 07
2.882
NO POSITION
The average price for the un hedged strategy : (2.69+2.65+2.77+2.88)/4 = $2.7475/pound
The average price for the hedged strategy:
(.3)2.69 + (.7)(2.69 + 2.723 – 2.691) = 2.7124
(.3)2.65 + (.7)(2.65 + 2.728 – 2.648) = 2.7060
(.3)2.77 + (.7)(2.77 + 2.716 – 2.767) = 2.7343
(.3)2.88 + (.7)(2.88 + 2.695 – 2.882) = 2.7498
$2.725625/pound
68
Cost saving: 4M[2.7457 – 2.7256625] = $127,500.
Stock index futures.
Foreign currency futures.
In each case, we first describe the
SPOT MARKET
And then analyze the
FUTURES MARKET.
69
STOCK INDEX FUTURES
The first stock index futures began trading in 1982
on the KCBT. The underlying was the
VALUE LINE INDEX.
Soon afterwards, the CBT, tried to launch a DJIA
futures. It lost its court battle with the Dow Jones
Co. and could not establish that futures. Instead, it
started trading futures on the
MAJOR MARKET INDEX, the MMI.
Today, Stock Index Futures are traded on dozens of
different indexes.
70
STOCK INDEXES (INDICES)
A STOCK INDEX IS A SINGLE NUMBER
BASED ON INFORMATION ASSOCIATED
WITH A
PORTFOILO OF STOCKS.
A STOCK INDEX IS SOME KIND OF AN
AVERAGE OF THE PRICES AND THE
QUANTITIES OF THE SHARES OF THE
STOCKS THAT ARE INCLUDED IN THE
PORTFOLIO THAT UNDERLYING THE
INDEX.
71
STOCK INDEXES (INDICES)
THE MOST USED INDEXES ARE
A SIMPLE PRICE AVERAGE
AND
A VALUE WEIGHTED AVERAGE.
72
STOCK INDEXES - THE CASH MARKET
A. AVERAGE PRICE INDEXES: DJIA, MMI:
N = The number of stocks in the index
Sj = Stock j market price; j = 1,…,N.
D = Divisor
S

I=
;
j
D
j = 1,..., N.
Initially, D = N and the Index is set at an
agreed upon level. To assure continuity, the
Divisor is adjusted over time.
73
EXAMPLES OF INDEX ADJUSMENTS
STOCK SPLITS: 2 FOR 1:
(S1S2 ,..., SN ) / D1I1
1.
2.
1
(S1 S2 ,...,SN ) / D 2 I1
2
Before the split:
(30 + 40 + 50 + 60 + 20) /5 = 40
I = 40 and D = 5.
An instant later:
(30 + 20 + 50 + 60 + 20)/D = 40
The new divisor is D = 4.5
74
CHANGE OF STOCKS IN THE INDEX
1.
(S1S2 (ABC) ,..., SN ) / D1I1
2.
(S1S2 (XYZ) ...SN ) / D2 I1
Before the change:
(31 + 19 + 53 + 59 + 18)/4.5 = 40
I = 40
and D =4.5.
An instant later:
(30 + 150 + 50 + 60 + 20)/D = 40
The new divisor is D = 7.75
75
A STOCK DIVIDEND DISTRIBUTION
Firm 4 distributes 40% stock dividend.
Before the distribution:
(32 + 113 + 52 + 58 + 25)/7.75 = 36.129
D = 7.75.
An instant later:
(32 + 113 + 52 + 34.8 + 25)/D = 36.129
76
The new divisor is D = 7.107857587.
STOCK # 2 SPLIT 3 FOR 1.
Before the split:
(31 + 111 + 54 + 35 + 23)/7.107857587
= 35.7351
An instant later:
(31 + 37 + 54 + 35 + 23)/D = 35.73507
The new Divisor is D = 5.0370644.
77
ADDITIONAL STOCKS
1.
(S1S2 ,...,SN ) / D1I1
2.
(S1S2 ,...,SN SN+1 ) / D2 I1
Before the stock addition:
(30 + 39 + 55 + 33 + 21)/5.0370644
= 35.338
An instant later:
(30 + 39 + 55 + 33 + 21 + 35)/D = 35.338
D = 6.0275.
78
A price adjustment of Altria Group Inc.
(MO), (due to a distribution of Kraft
Foods Inc. (KFT) shares,) was effective
for the open of trade on trade date April
2, 2007.As a result, the new divisor for
the DJIA became:
D = 0.123051408.
The last revision of the DJIA’s Divisor
was on AUG 2007 and the Divisor was set
at:
D = 0.123017848
79
VALUE WEIGHTED INDEXES
S & P500, NIKKEI 225, VALUE LINE
NS

I
N S
tj tj
t
j  1,2,......, n
Bj Bj
B = SOME BASE TIME PERIOD
Initially:
t=B
The initial value of the Index is set at an
80
arbitrarily chosen value: M.
** The S&P500 index base period was
1941-1943 with initial value: M = 10.
** The NYSE index base period was
Dec. 31, 1965 with initial value: M = 50.
** The NASDAQ composite index base
period was FEB 5 1971 With initail value:
M = 100.
81
The rate of return on ANY PORTFOLIO:
The return on a PORTFOLIO in any period
t, is:
the weighted average of the individual
stocks returns. The weights are the
percentages of the stocks value in the
portfolio.
R Pt   w tjR tj.
w tj 
N tjStj
N S
tj tj

Vtj
VtP
.
82
The Rate of Return on a portfolio
VPt +1VPt  N t +1jSt +1j   N tjStj
R Pt 

VPt
 N tjStj
R Pt
N


S  N tjStj
t +1j t +1j
N S
;
tj tj
but, N t +1j  N tj. Thus,
R Pt
N (S S )


N S
tj
t +1j
tj tj
tj
83
 St 1j  Stj 
 N tjStj  S 
tj

,
R Pt 
 N tjStj
N S R 


. Rewrite
N S
tj tj
tj
this as :
tj tj
 N tjStj 
 
 R tj , or
  N tjStj 
 Vtj 
 
 R tj . Finally,
 VtP 
N tjStj Vtj
w tj

.
R Pt   w tjR tj.
 
 
 N tjStj VtP
84
THE BETA OF A PORTFOLIO
THEOREM: Consider a portfolio consisting
of shares of N stocks.
The portfolio’s BETA is the weighted
average of the stock’s betas. The weights
are the dollar value weights of the stocks
in the portfolio.
R
85
THE BETA OF A PORTFOLIO
Proof:
We use a well diversified index as a proxy
portfolio for the
market portfolio.
R
Let:
P denote the portfolio underlying
the Index, I.
Let:
j denote the individual stock in the
portfolio.
j = 1, 2, …,N.
86
By the definition of BETA:
COV(R P ; R M )
βP 
.
VAR(R M )
The Index is a proxy for the Market :
COV(R P ; R I )
βP 
.
VAR(R I )
From the previous theorem : R P   w jR j,
βP 
COV([ w jR j ]; R I )
VAR(R I )
.
87
Recall that th e covariance is
a linear operator, thus :
βP
w COV(R ; R )


, or :
j
j
I
VAR(R I )
 COV(R j ; R I ) 
βP   w j 
 w jβ j.

 VAR(R I ) 
88
STOCK PORTFOLIO BETA
STOCK NAME
PRICE
FEDERAL MOUGUL
MARTIN ARIETTA
IBM
US WEST
BAUSCH & LOMB
FIRST UNION
WALT DISNEY
DELTA AIRLINES
P =
18.875
73.500
50.875
43.625
54.250
47.750
44.500
52.875
SHARES
9,000
8,000
3,500
5,400
10,500
14,400
12,500
16,600
VALUE
169,875
588,000
178,063
235,575
569,625
687,600
556,250
877,725
3,862,713
WEIGHT
BETA
.044
.152
.046
.061
.147
.178
.144
.227
1.00
.80
.50
.70
1.1
1.1
1.4
1.2
(.044)(1.00) + (.152)(.8) + (.046)(.5) + (.061)(.7)
+ (.147)(1.1) + (.178)(1.1) + (.144)(1.4) + (.227)(1.2)
= 1.06
89
A STOCK PORTFOLIO BETA
STOCK NAME
BENEFICIAL CORP.
CUMMINS ENGINES
GILLETTE
KMART
BOEING
W.R.GRACE
ELI LILLY
PARKER PEN
P
PRICE
40.500
64.500
62.000
33.000
49.000
42.625
87.375
20.625
SHARES
11,350
10,950
12,400
5,500
4,600
6,750
11,400
7,650
VALUE
459,675
706,275
768,800
181,500
225,400
287,719
996,075
157,781
3,783,225
WEIGHT BETA
.122
.187
.203
.048
.059
.076
.263
.042
.95
1.10
.85
1.15
1.15
1.00
.85
.75
= .122(.95) + .187(1.1) + .203(.85) + .048(1.15) + .059(1.15) + .076(1.0)
+ .263(.85) + .042(.75)
= .95
90
Sources of calculated Betas and calculation inputs
Example: ß(GE) 6/20/00
Source
ß(GE)
Value Line Investment Survey
1.25
NYSECI
Weekly Price
5 yrs (Monthly)
Bloomberg
1.21
S&P500I
Weekly Price
2 yrs (Weekly)
Bridge Information Systems
1.13
S&P500I
Daily Price
2 yrs (daily)
Nasdaq Stock Exchange
1.14
Media General Fin. Svcs. (MGFS)
Quicken.Excite.com
1.23
MSN Money Central
1.20
DailyStock.com
1.21
Standard & Poors Compustat Svcs
S&P Personal Wealth
1.2287
S&P Company Report)
1.23
Index Data
S&P500I
Horizon
Monthly P ice
3 (5) yrs
S&P500I
Monthly Price
5 yrs (Monthly)
S&P500I
Daily Price
5 yrs (Daily)
S&P500I
Monthly Price
5 yrs (Monthly)
Charles Schwab Equity Report Card 1.20
S&P Stock Report
AArgus Company Report
1.23
1.12
Market Guide
YYahoo!Finance
1.23
Motley Fool
1.23
91
STOCK INDEX FUTURES
1.
The monetary value of ONE CONTRACT
is:
(THE INDEX VALUE)($MULTIPLIER)
or
(I)($m)
2.
Accounts are settled by
CASH SETTLEMENT
92
A Stock Index Futures
• Can be viewed as an investment asset
paying a dividend yield
• The futures price and spot price
relationship is therefore
Ft.T = Ste
(r–q )(T-t)
.
q = the annual dividend yield on the
portfolio represented by the index
93
A Stock Index Futures
• For the formula to be true it is
important that the index represents
an investment asset
• In other words, changes in the index
must correspond to changes in the
value of a tradable portfolio
• The Nikkei index viewed as a dollar
number does not represent an
investment asset
94
STOCK INDEX HEDGING
Stock index hedgers may use the NAÏVE
hedge ratio, h = 1. Mostly, however,
hedgers use the minimum variance hedge
ratio. In this case, the underlying asset is a
stock index; actually the portfolio that
underlie the index. Thus, the parameter
that relates the spot asset and the index is
the Beta of the spot asset’s with the Index.
Remember: The index is the proxy for the
Market portfolio.
95
RECALL THAT THE MINIMUM VARIANCE HEDGE RATIO IS:
cov( S; F)
h* .
var( F)
σ ΔS
h *  - ρΔS,ΔF

σ ΔF
n * NF
.
NS
NS
n  h
.
NF


96
S Sk - St
=
= rS
St
St

F Fk,T - Ft,T
=
= rF 
Ft,T
Ft,T
ΔS = St rS
ΔF = Ft,T rF
COV(St rS , Ft,T rF )
COV(S; F)
h*=  VAR( F)
VAR(Ft,T rF )
COV(rS , rF ) [St Ft,T ]
h*= 2
VAR(r F )
Ft,T
97
COV(rS , rF ) [St Ft,T ]
h*= 2
VAR(r F ) Ft,T
COV(rS , rF ) St
St
h*=  -β
.
VAR(r F ) Ft,T
Ft,T
NS
St N S
Spot Value at t
n* h*
 β
 -β
NF
Ft,T N F
Futures Value at t
VS
n  β
VF
*
98
STOCK PORTFOLIO HEDGE
STOCK NAME
PRICE
FEDERAL MOUGUL
MARTIN ARIETTA
IBM
US WEST
BAUSCH & LOMB
FIRST UNION
WALT DISNEY
DELTA AIRLINES
βP
18.875
73.500
50.875
43.625
54.250
47.750
44.500
52.875
SHARES
9,000
8,000
3,500
5,400
10,500
14,400
12,500
16,600
VALUE
169,875
88,000
178,063
235,575
569,625
687,600
556,250
877,725
3,862,713
WEIGHT BETA
.044
.152
.046
.061
.147
.178
.144
.227
1.00
.80
.50
.70
1.1
1.1
1.4
1.2
= .044(1.00) + .152(.8) + .046(.5)
+ .061(.7) + .147(1.1) + .178(1.1)
+ .144(1.4)+ .227(1.2)
= 1.06
99
TIME
CASH
MAR.31 VS = $3,862,713
FUTURES
SEP SP500I FUTURES. F = 1,052.60.
VF = 1,052.60($250) = $263,300
3,862,713
n = - 1.06
= - 16.
263,300
*
SHORT 16 SEP SP500I Fs.
JUL.27
VS = $3,751,307
LONG 16 SEP SP500I Fs
F = 1,026.99
GAIN = (1,052.60 - 1,026.99)($250)(16)
= $102,440.00
TOTAL VALUE
$3,853,747.00
100
ANTICIPATORY HEDGE OF A TAKEOVER
A firm intends to purchase 100,000 shares of XYZ
ON DEC.17.
DATE
SPOT
FUTURES
NOV.17
S = $54/SHARE
MAR SP500I FUTURES IS F = 1,465.45
β = 1.35
VF = 1,465.45($250)
VS = (54)100,000
= $5,400,000
= $366,362.50
5,400,000
n = - 1.35
= - 20
366,362.50
*
LONG 20 MAR SP500I Futures.
DEC.17
S = $58/SHARE
SHORT 20 MAR SP500I Futures
PURCHASE 100,000 SHARES.
F = 1, 567.45
COST = $5,800,000
Gain:
20(1,567.45 - 1,465.45)$250
= $510,000
$5,800,000 - $510,000
= $52.9/SHAR E
Actual purchasing price:
100,000
101
HEDGING A ONE STOCK PORTFOLIO
SPECIFIC STOCK INFORMATION INDICATES THAT THE STOCK SHOULD
INCREASE IN VALUE BY ABOUT 9%. THE MARKET IS EXPECTED TO DECREASE
BY 10%, HOWEVER. THUS, WITH BETA = 1.1 THE STOCK PRICE IS EXPECTED TO
REMAIN AT ITS CURRENT VALUE. SPECULATING ON THE UNSYSTEMATIC RISK,
WE OPEN THE FOLLOWING STRATEGY:
TIME
SPOT
FUTURES
JULY 1 OWN 150,000 SHARES
S = $17.375
VS = $2,606,250
β = 1.1
DEC. IF PRICE F = 1,090
VF = 1,090($250) = $272,500
2,606,250
n = - 1.1
= - 11
272,500
*
SHORT 11 DEC. SP500I Futures
SEP.30
S = $17.125
V = $2,568,750
LONG 11 DEC SP500I Futures
F = 1,002.
Gain: $250(11)(1,090 - 1,002) = $242,000
102
ACTUAL
V = $2,810,750. An increase of about 8%
MARKET TIMING USING BETA
When we believe (speculate) that the
market trend is changing, we can change
the beta of our portfolio. We may purchase
high beta stocks and sell low beta stocks,
when we believe that the market is turning
upward; or purchase low beta stocks and
sell high beta stocks, when we believe that
the market is moving down. Instead we
may try to change the beta of our spot
position by using the INDEX FUTURES
103
The Minimum Variance Hedge Ratio in our
case is: h* = -(VS/VF). Assume that the
current position is a portfolio with current
spot market value of VS and n stock index
futures. Then:
The BETA of the spot position may be
altered from its current value, , to a
Target Beta = T, buying or selling n
futures:
VS
n*  [β T  β] .
VF
104
Proof:
VP  VS  nVF
Δ(VP )  Δ(VS )  n * (VF )
Δ(VP ) Δ(VS )
Δ(VF ) Δ(VS )
VF Δ(VF )

 n*

 n*
.
VS
VS
VS
VS
VS VF
DEFINE
Δ(VP )
rP 
;
VS
Δ(VS )
rS 
;
VS
VF
ErP  ErS  n * ErF  ErT
VS
and
Δ(VF )
rF 
.
VF
105
Again :
VF
E(rT )  E(rS )  n * E(rF ).
VS
Following the CAPM, we can write :
E(rT )  rf  β T [E(rM )  rf ] and
E(rS )  rf  β[E(r M )  rf ] and
E(rF )  E(rM )  rf .
Notice that in the last equation F is a futures
on the index and thus, it' s β  1. Hence, it requires
no initial outlay and no opportunit y cost is needed
in the furmula.
106
VF
E(rT )  E(rS )  n
E(rF )
VS
*
VF
 rf  β[E(r M )  rf ]  n
[E(rM )  rf ]
VS
*
 rf  β T [E(rM )  rf ]
*
and solve for n :
VS
n*  [β T  β]
VF
107
MARKET TIMING HEDGE RATIO (page 66)
The rule: In order to change the BETA of
the spot position from  to T, the stock
index futures may be used as follows:
If β T  β; we wish to reduce β to β T
VS
SHORT n*  [β  β T ]
contracts .
VF
If β T  β; we wish to increase β to β T
LONG
VS
n*  [β T  β]
contracts .
VF
108
MARKET TIMING HEDGE; EN EXAMPLE
STOCK NAME
PRICE
BENEFICIAL CORP.
CUMMINS ENGINES
GILLETTE
KMART
BOEING
W.R.GRACE
ELI LILLY
PARKER PEN
40.500
64.500
62.000
33.000
49.000
42.625
87.375
20.625
SHARES
11,350
10,950
12,400
5,500
4,600
6,750
11,400
7,650
VALUE
459,675
706,275
768,800
181,500
225,400
287,719
996,075
157,781
3,783,225
WEIGHT
BETA
.122
.187
.203
.048
.059
.076
.263
.042
.95
1.10
.85
1.15
1.15
1.00
.85
.75
β(portfolio) = .122(.95) + .187(1.1) + .203(.85)
+ .048(1.15) + .059(1.15) + .076(1.0)
+ .263(.85) + .042(.75)
= .95
109
The portfolio manager speculates that the
market has reached a turning point and
is on its way up.
The idea is that in this case it is possible to
increase the portfolio’s Beta employing
Stock Index futures.
Suppose that the portfolio manager wishes
to increase the current Beta from
β = .95 to
βT = 1.25.
110
TIME
SPOT
FUTURES
AUG.29
V = $3,783,225.
DEC SP500I Fs
 = 0.95.
= 1,079.8($250) = $269,950
3,783,225
n * = (1.25 - .95)
=4
269,950
LONG 4 DEC SP500I Futures
NOV.29
V = $4,161,500
F = 1,154.53
SHORT 4 DEC SP500I Futures
GAIN (1,154.53 - 1,079.8)(250)(4)
= $74,730
TOTAL PORTFOLIO VALUE $4,236,230
THE MARKET INCREASED ABOUT 7% AND
THE PORTFOLIO VALUE INCREASED ABOUT 12%
111
FOREIGN CURRENCY:
THE SPOT MARKET
EXCHANGE RATES:
THE PRICE OF ONE CURRENCY IN
TERMS OF ANOTHER CURRENCY IS
THE EXCHANGE RATE BETWEEN THE
TWO CURRENCIES.
112
SPOT EXCHANGE RATES:
THERE ARE TWO QUOTE FORMATS:
1. S(USD/FC) =
THE NUMBER OF USD IN
ONE UNIT OF THE
FOREIGN CURRENCY.
2. S(FC/USD) =
THE NUMBER OF THE
FOREIGN CURRENCY
UNITS IN ONE USD.
113
1
S(FC /FC ) =
1 2 S(FC /FC )
2 1
Example :
S(USD/EUR) = 1.4821
S(EUR/USD)  .67476
1
1

= 1.4821 S(USD/EUR)
S(EUR/USD) .67476
114
WHEN WE HAVE BID AND ASK QUOTES :
1
S(FC/USD)

ASK S(USD/FC)
1
S(FC/USD)

BID S(USD/FC)
S(GBP/USD)
S(USD/GBP)
S(USD/GBP)
S(GBP/USD)
ASK
BID
ASK
BID
BID
ASK
 GBP.5USD, buy ONE USD pay .50GBP.
 USD2.00/G BP, sell ONE GBP get USD2.
 USD2.083/ GBP, buy ONE GBP pay USD2.0 83.
 GBP.480/USD, sell ONE USD get .48GBP.
115
BUY
USD
PAY
GBP
S(GBP/USD)ASK S(USD/GBP)BID
= GBP 0.50
= USD 2.083
RECEIVE S(GBP/USD)BID S(USD/GBP)BID
= GBP 0.48
= GBP 2.000
USD
GBP
SELL
116
CURRENCY CROSS RATES
LET FC1, FC2 AND FC3 DENOTE THREE
DIFFERENT CURRENCIES.
IN THE ABSENCE OF ARBITRAGE :
S(FC1/FC3)
S(FC1/FC2) =
S(FC2/FC3)
S(FC3/FC2)
=
S(FC3/FC1)
117
CURRENCY CROSS RATES – DEC 17.07
(www.x-rates.com)
USD
GBP
CAD
EUR
MXN
USD
1
2.01400
0.989609
1.439200
0.0920801
GBP
0.496524
1
0.491364
0.714597
0.045720
CAD
1.010500
2.035151 1
1.454310
0.093047
EUR
0.694830
1.399380 0.687611
1
0.063980
MXN
10.860109 21.87230 10.747300 15.629900 1
118
CURRENCY CROSS RATES
EXAMPLE:
FC1 = USD; FC2 = MXN;
FC3 = GBP.
USD
GBP
USA 1
2.01400
UK 0.496524 1
MEX 10.860109 21.87230
MXN
0.0920801
0.045720
1
119
CURRENCY CROSS RATES
EXAMPLE
Let FC1  USD; FC2  MXN; FC3  GBP.
S(GBP/MXN)
S(USD/MXN) =
S(GBP/USD)
S(USD/GBP)
=
.
S(MXN/GBP)
120
CURRENCY CROSS RATES
EXAMPLE
S(GBP/MXN)
S(GBP/USD)
S(USD/GBP)
S(MXN/GBP)
 0.045720.
 0.496524.
 2.014000.
 21.872300.
S(GBP/MXN)
0.045720

 0.092080.
S(GBP/USD)
0.496524
S(USD/GBP)
2.014000

 0.092080.
S(MXN/GBP)
21.872300
121
AN EXAMPLE OF CROSS SPOT RATES ARBITRAGE
COUNTRY
USD
GBP
CHF
U.S.A
1.0000
1.5640
0.5580
U.K
0.6394
1.0000
0.3546
SWITZERLAND
1.7920
2.8200
1.0000
THEORY :
S(GBP/USD)
= S(CHF/USD)
S(GBP/CHF)
BUT : 0.6394 = 1.8031  1.7920
0.3546
SIMILARLY :
BUT :
S(USD/GBP)
= S(CHF/GBP)
S(USD/CHF)
1.5640
= 2.8029 < 2.8200
0.5580
122
THE CASH ARBITRAGE ACTIVITIES:
Start:
End.
USD1,000,000
USD1,006,134
0.6394
0.5580
GBP639,400
CHF1,803,108
2.8200
123
Forward rates, An example:
GBP
DEC 17, 2007
SPOT
USD1.997200/GBP
1 Month forward
USD1.995300/GBP
2 Months forward
USD1.993760/GBP
3 Months forward
USD1.992010/GBP
6 Months forward
USD1.986500/GBP
12 Months forward
USD1.972630/GBP
2 Years
USD1.947750/GBP
124
forward
FOREIGN CURRENCY CONTRACT SPECIFICATIONS
CURRENCY
SIZE
MINIMUM
FUTURES
CHANGE USD/FC
CHANGE F
JAPAN YEN
12.5M
.000001
USD12.50
CANADIAN DOLLAR
100,000
.0001
USD10.00
62,500
.0002
USD12.50
SWISS FRANC
125,000
.0001
USD12.50
AUSTRALIAN DOLLAR
100,000
.0001
USD10.00
MEXIAN PESO
500,000
.000025
USD12.50
BRAZILIAN REAL
100,000
.0001
USD10.00
EURO FX
125,000
.0001
USD12.50
BRITISH POUND
* MUST CHECK FOR DAILY PRICE LIMITS
* CONTRACT MONTHS FOR ALL CURRENCIES:
MARCH, JUNE, SEPTEMBER, DECEMBER
* LAST TRADING DAY: FUTURES TRADING TERMINATES AT 9:16 AM ON THE SECOND
BUSINESS DAY IMMEDIATELY PRECEEDING THE THIRD WEDNESDAY OF THE CONTRACT
MONTH.
* DELIVERY BY WIRED TRASFER. 3RD WEDNESDAY OF CONTRACT MONTH
125
SPECULATION: TAKE RISK FOR EXPECTED PROFIT
AN OUTRIGHT NAKED POSITION WITH CANADIAN DOLLAS:
t - MARCH 1.
S(USD/CD) = .6345 <=> S(CD/USD) = 1.5760
T- SEPTEMBER
F(USD/CD) = .6270 <=> F(CD/USD) = 1.5949
SPECULATOR:
“THE CD WILL NOT DEPRECIATE TO THE
EXTENT IMPLIED BY THE SEP. FUTURES.
INSTEAD, IT WILL DEPRECIATE TO A PRICE
HIGHER THAN USD.6270/CD.”
TIME
MAR 1
CASH
DO NOTHING
FUTURES
LONG n, CD SEP FUTURES
AT USD.6270/CD
AUG 20
DO NOTHING
SHORT n, CD SEP FUTURES
AT USD.6300/CD
PROFIT = (USD.6300/CD - USD.6270/CD)(CD100,000)(n) = USD300(n).
126
HEDGING
IN THE FOLLOWING EXAMPLES WE USE
THE NAÏVE HEDGE RATIO:
h = 1.
Two ways:
1. n = NS/NF
2. n = VS/VF
127
BORROWING U.S. DOLLARS SYNTHETICALLY ABROAD
OR
HOW TO BEAT THE DOMESTIC BORROWING RATE
A U.S. FIRM NEEDS TO BORROW USD200M FROM MAY 25, 2003 TO DECEMBER 20,
2003, FACES THE FOLLOWING DATA:
BID
ASK
SPOT:
USD1.25000/EUR
USD1.25100/EUR
DEC FUTURES:
USD1.25850/EUR
USD1.26000/EUR
ITALY:
6.7512%
6.9545%
(365-day year)
USA:
8.6100%
8.75154%
(360-day year)
Interest rates:
128
TIME
SPOT
FUTURES
MAY 25
(1) BORROW EUR160,000,000
LONG 1,332 DEC EUR FUTURES FOR
FOR 6.9545% FOR 209 DAYS
(2) EXCHANGE THE EUR INTO
INTO USD200,000,000 AND USE
F = 1.26000
166,500,000
n
= 1,332
125,000
THIS SUM TO FINANCE THE PROJECT
DEC 20 LOAN VALUE ON DEC. 20
160,000,000e(0.069545)(209/365)
= EUR166,500,000
TAKE DELIVERY OF EUR166,500,000
PAYING USD209,790,000
REPAY THE LOAN.
THE IMPLIED REVERSE REPO RATE FOR 209 DAYS =
1
209,790,000
ln[
] = .0823, or 8.23%.
209/360 200,000,000
129
EXAMPLES OF HEDGING FOREIGN CURRENCY
EXAMPLE 1: A LONG HEDGE.
ON JULY 1, AN AMERICAN AUTOMOBILE DEALER ENTERS INTO A CONTRACT TO IMPORT
100 BRITISH SPORTS CARS FOR GBP28,000 EACH. PAYMENT WILL BE MADE IN BRITISH
POUNDS ON NOVEMBER 1. RISK EXPOSURE: IF THE GBP APPRECIATES RELATIVE TO
THE USD THE IMPORTER’S COST WILL RISE.
TIME
SPOT
FUTURES
JUL. 1
S(USD/GBP) = 1.3060
LONG 46 DEC BP FUTURES
CURRENT COST = USD3,656,800
FOR F = USD1.2780/GBP
DO NOTHING
NOV. 1
h  1;
3,656,800
n=
= 46
62,500(1.2780)
S(USD/GBP) = 1.4420
SHORT 46 DEC BP FUTURES
COST = 28,000(1.4420)(100)
FOR F = USD1.4375/GBP
= USD4,037,600
PROFIT: (1.4375 - 1.2780)62,500(46)
= USD458,562.50
ACTUAL COST = USD3,579,037.50
130
EXAMPLE 2: A LONG HEDGE
ON MARCH 1, AN AMERICAN WATCH RETAILER AGREES TO PURCHASE 10,000 SWISS
WATCHES FOR CHF375 EACH.
THE SHIPMENT AND THE PURCHASE WILL TAKE PLACE ON AUGUST 26.
TIME
SPOT
FUTURES
MAR. 1
S(USD/CHF) = .6369
LONG 30 SEP CHF FUTURES
CURRENT COST 10,000 (375)(.6369) F(SEP) = USD.6514/CHF
AUG. 25
= USD2,388,375
CONTRACT = (.6514)125,000
DO NOTHING
= USD81,425.
S=USD.6600/CHF
SHORT 30 SEP CHF FUTURES
BUY 10,00 WATCHES FOR
F(SEP) = USD.6750/CHF
(375)(.6600)(10,000)
PROFIT(.6750 - .6514)125,000(30)
TOTAL $2,475,000.
= USD88,500.
2,388,375
n=
= 30
81,425
ACTUAL COST USD2,386,500
131
EXAMPLE 3: A LONG HEDGE
ON MAY 1, AN ITALIAN EXPORTER AGREES TO SELL 1,000 SPORTS CARS TO AN
AMERICAN DEALER FOR USD50,000 EACH.
THE SHIPMENT AND THE PAYMENT WILL TAKE PLACE ON OCT 26.
TIME
SPOT
FUTURES
MAY. 1
S(EUR/USD) = .87000
LONG 298 DEC EUR FUTURES
CURRENT VALUE:
F(DEC) = USD1.17EUR
= EUR43,500,000
43,500,000
n=
= 348
125,000
S=EUR.81300/USD
SHORT 348 DEC EUR FUTURES
DELIVER THE CARS FOR
F(DEC) = USD1.29000/EUR
PAYMENT: EUR40,650,000.
PROFIT(1.29 – 1.17)(125,000)(348)
OCT. 26
=USD5,220,000
ACTUAL PAYMENT IN EUR:
40,650,000 + 5,220,000(.813) = EUR44,893,860.
132
EXAMPLE 4: A LONG HEDGE: PROTECT AGAINST DEPRECIATING DOLLAR
ON MAY. 23, AN AMERICAN FIRM AGREES TO BUY 100,000 MOTORCYCLES FROM A JAPANESE
FIRM FOR JY202,350 . Payment and delivery will take place on DEC 20.
CURRENT PRICE DATA: ASK
BID
SPOT:
USD.007020/JY
USD.007027/JY
(142.4501245)
142.3082396)
USD.007190/JY
USD.007185/JY
DEC FUTURES:
ON DECEMBER 20 THE FIRM WILL NEED THE SUM OF JY20,235,000,000.
TODAY, THIS SUM IS VALUED AT 20,235,000,000(.007027) = USD142,191,345
N = USD142,191,345/(JY12,500,000)(USD.007190/JY) = 1,582.
133
TIME
CASH
FUTURES
MAY 23
DO NOTHING
LONG 1,582 JY FUTURES FOR
V = USD142,191,345
F(ask) = USD.007190/JY
S = USD.0080/JY
SHORT 1,582JY Fs.
BUY MOTORCYCLES
FOR USD.0080/JY
FOR USD161,880,000
PROFIT: (.0080-.00719)12,500,000(1,582)
CASE I:
DEC 20
= USD16,017,750
NET COST: USD161,880,000 - USD16,017,750 = USD145,862,250.
CASE II:
DEC 20
S = USD.0065/JY
SHORT 1,582 JY Fs.
BUY MOTORCYCLES
FOR USD.0065/JY
USD131,527,500
LOSS: (.00719-.0065)12,500,000(1,582)
= USD13,644,750
NET COST: USD145,172,250.
134
EXAMPLE 5: A SHORT HEDGE
A US MULTINATIONAL COMPANY’S ITALIAN SUBSIDIARY WILL GENERATE EARNINGS
OF EUR2,516,583.75 AT THE END OF THE QUARTER - MARCH 31. THE MONEY WILL BE
DEPOSITED IN THE NEW YORK BANK ACCOUNT OF THE FIRM IN U.S. DOLLARS.
RISK EXPOSURE: IF THE DOLLAR APRECIATES RELATIVE TO THE EURO THERE WILL
BE LESS DOLLARS TO DEPOSIT.
TIME
CASH
FEB. 21 S(USD/EUR) = 1.18455
FUTURES
F(JUN) = USD1.17675/EUR
CURRENT SPOT VALUE
F = 125,000(1.17675) = USD147,093.75
= USD2,981,019.28
n = 2,981,019.28/147,093.75 = 20.
DO NOTHING
SHORT 20 JUN EUR FUTURES
MAR 31 S(EUR/USD) = 1.1000
DEPOSIT 2,768,242.125
LONG 20 JUN EUR FUTURES
F(JUN) = USD1.10500
PROFIT: (1.17675 -1.10500)125,000(20) =
USD179,375
TOTAL AMOUNT TO DEPOSIT USD2,947,617.125
135