Hedging Strategies Using Futures

1
Hedging Strategies Using Futures

• Chapter 3

2
HEDGERS OPEN POSITIONS IN THE FUTURES MARKET IN
ORDER TO ELIMINATE THE RISK ASSOCIATED WITH
THE SPOT PRICE OF THE UNDERLYING ASSET
3
Spot price risk
Pr
Sj
St
time
j
t
4
HEDGERSPROBLEM TO OPEN A LONG HEDGEOR A
SHORT HEDGE? There are two ways to determine
whether to open a short or a long hedge
5
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
6
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.
7
Example A LONG HEDGE Date Spot market Futures
market Basis t St 800/unit Ft,T
825/unit -25 Contract to buy long one gold
Gold on k. futures for delivery at
T k Buy the gold Short one gold Sk
816/unit futures for delivery at T.
Fk,T 842/unit -26
1 T Amount paid 816 825 842
799/unit or 825 (816 842) 799/unit
8
Example A SHORT HEDGE Date Spot market Futures
market Basis t St 800/unit Ft,T
825/unit -25 Contract to sell short one
gold Gold on k, futures for delivery
at T k Sell the gold Long one gold
Sk 784/unit futures for delivery at
T. Fk,T 812/unit -28
3 T Amount received 784 825 812
797/unit or 825 (784 812) 797/unit
9

• NOTATIONS
• t lt 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.

10

• 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 Trade spot and Delivery
• Open the hedge Close the futures
• open a futures position
• position
• t k T Time

11

• THE HEDGE TIMIMG
• Date k is (almost) always before the delivery
  month.
• WHY?
• 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 delivery note.

12

• 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.

13
The Basis The basis at any time point, j, is the
difference between the assets spot price and the
futures price on j. BASISj SPOT PRICEj -
FUTURES PRICEj Notationally Bj Sj - Fj,T j lt
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.
14

• A LONG HEDGE
• TIME SPOT FUTURES B
• t Contract to buy LONG Ft,T Bt
• Do nothing
• k BUY Sk SHORT Fk,T Bk
• T delivery
• Actual purchase price Sk Ft,T - Fk,T
• Ft,T Sk - Fk,T
• Ft,T BASISk

15

• A SHORT HEDGE
• TIME SPOT FUTURES B
• t Contract to sell SHORT Ft,T Bt
• Do nothing
• k SELL Sk LONG Fk,T Bk
• T delivery
• Actual selling price Sk Ft,T - Fk,T
• Ft,T Sk - Fk,T
• Ft,T BASISk

16
In both cases, Long hedge and short hedge the
hedgers purchase/sale price, when the hedge is
closed on date k, is Ft,T BASISk This price
consists of two portions a known
portion Ft,T and a random portion
the BASISk We return to this point later.
17
ALSO NOTICE 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
t k T
18
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 life of the
futures.
19
Open close the hedge
Long hedge Short hedge a success a
failure Loss on the hedge a failure a
success Loss on the hedge
Fk,T Sk Fk,T Sk
Ft,T St
20

• Example A LONG HEDGE
• TIME SPOT FUTURES BASIS
• t St 3.40 LONG
• Do nothing Ft,T3.50 -.10
• k BUY Sk3.80 SHORT
• F k,T3.85 -.05
• T delivery
• Actual purchase price
• NO hedge 3.80
• With hedge 3.45 ? (Successful hedge)

21

• Example A LONG HEDGE
• TIME SPOT FUTURES BASIS
• t St 3.40 LONG
• Do nothing Ft,T3.50 -.10
• k BUY Sk3.00 SHORT
• F k,T3.05 -.05
• T delivery
• Actual purchase price
• NO hedge 3.00
• With hedge 3.45 ? (Unsuccessful hedge)

22
The basis upon delivery BT 0 On date k, the
basis is Bk Sk - Fk,T k lt 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.
23
Convergence of Futures to Spot over the life of
the futures

Futures Price
Spot Price
Futures Price
Spot Price
Time
Time
(a)
(b)
24

• 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.

25
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
BT 0
Bt
time
k
T
t
26
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.
27
Conclusion At time t, WITHOUT
HEDGING cash-price risk. WITH HEDGING, basis
risk. Hedging with futures is nothing more than
changing the firms spot price risk Into a
smaller risk, namely, The basis risk.
28

• 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.

29

• 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

30
A CROSS HEDGE TIME CASH FUTURES t Contract
to trade S1 Ft,T(2) Do nothing 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
31
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

32
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.

33
Delivery month? MOSTLY, the hedge is opened with
a futures for the delivery month closest to the
firms 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 firms cash activity.
34
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?
35
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.
36
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?
37
HEDGE RATIOS, DEFINITION
The hedge ratio, h, determines the number of
futures to hold, n.
38
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.
39

• Examples NAÏVE HEDGE RATIO h 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.

40

• 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.

41

• A firm will sell NS 3,600
• ounces of gold.
• NYMEX gold futures NF 100 ounces.
• SHORT
• n 3,600/100
• 36 CBT futures.

42
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.
43
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
44
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
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Date SPOT MARKET S FUTURES MARKET
F FUTURES POSITIONS Sep1,07 contract
to 92.00 Long 75 NOV 2007 93.00
long 75 NOV 2007 buy 75,000bbls Long 75 DEC
2007 93.50 long 75 DEC 2007 Long 75
JAN 2008 93.85 long 75 JAN 2008 Long
75 FEB 2008 94.60 long 75 FEB
2008 Oct1,07 buy 75,000bbls 93.00 short 75 NOV
07 93.10 long 75 DEC 2007
long 75 JAN 2008 long 75 FEB
2008 Nov1,07 buy 75,000bbls 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 (9392.909494.75)/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 93.5625
46
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.
47
Roll over hedge with h 1 DATE SPOT MARKET
S FUTURES MARKET F FUTURES
POSITIONS DEC, 07 contract to sell 89.00 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
48
Date SPOT MARKET S FUTURES MARKET F FUTURES
POSITIONS DEC, 07 contract to 89.00 short
100 MAY WTI 88.20 sell 100,000
bbls Oct1,07 buy 75,000bbls Short 100 MAY
2008 APR 08 long 100 MAY
2008 87.40 Short 100 SEP 2008 87.00 Sh
ort 100 SEP 2008 AUG 08 Long 100 SEP 2008
86.50 Short 100FEB 2009 86.30 Short
100 FEB 2009 JAN, 09 sell 100,000bbls
86.00 Long 100 FEB 2009 85.90 NO
POSITION 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.
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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.
50
THE MINIMUM VARIANCE HEDGE RATIO OBJECTIVE To
minimize the risk associated with the
hedge RISK VOLATILITY. THE VOLATILITY
MEASURE THE VARIANCE
51
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 POSITIONS
VALUE.
52

• NOTATIONS
• t The hedge opening date.
• St Spot market price.
• k The hedge closing date.
• T The futures delivery date.
• Fj,T The futures price on date j for
  delivery at T. t j T.

53

• NOTATIONS
• n The number of futures contracts used in
  the hedge.
• h The hedge ratio.
• NF The number of units of the asset in one
  contract.
• NS The number of units of the asset to be
  traded spot on k.

54

• FROM THE GENERAL RELATIONSHIP BETWEEN n and h
  (SLIDE 36) the optimal number of futures, n is
  determined by h

Thus, we find h and thereby determine the
optimal number of futures to be held in the
hedge, n.
55

• 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).

56

• 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).

57

• VAR(?VP) NS2 VAR(Sk- St) h(Fk,T - Ft,T)
• NS2VAR(?S)VAR(h?F)2COV(?Sh?F)
• NS2 VAR(?S)h2VAR(?F)2hCOV(?S?F).
• Set dVAR(?VP)/dh 0
• 2hVAR (?F) 2COV(?S ?F) 0.
• h - COV(?S?F)/VAR(?F)

58
THE MINIMUM RISK HEDGE RATIO IS
59
This result can be rewritten as
60
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.
61

• 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
• ?(?S) 0.025 ?(?F)0.033and ?(?S?F)
  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
• 10futures.

62

• 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 require more
  futures than the naïve ratio.

63
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.
s(?S)0.04, s(?F)0.02 ?(?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.
64
h , using Regression DATA n1 weeks.
65
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.
66
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
67
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 FUTU
RES MARKET F FUTURES POSITIONS SEP
05 contract Long 28 MAR 2006 2.723 Long 28 MAR
2006 Do nothing 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
68
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 PRICE 2.72 2.69 2.65 2.77 2.88
Futures prices for delivery Futures prices for delivery Futures prices for delivery Futures prices for delivery Futures prices for delivery Futures prices for delivery
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 2.882
69
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
(2.692.652.772.88)/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 Cost saving 4M2.7457
2.7256625 127,500.
70
Stock index futures. Foreign currency
futures. In each case, we first describe
the SPOT MARKET And then analyze the FUTURES
MARKET.
71
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.
72
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.
73
STOCK INDEXES (INDICES) THE MOST USED INDEXES
ARE A SIMPLE PRICE AVERAGE AND A VALUE
WEIGHTED AVERAGE.
74
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 Initially, D N and the Index is set
at an agreed upon level. To assure continuity,
the Divisor is adjusted over time.
75
EXAMPLES OF INDEX ADJUSMENTS STOCK SPLITS 2 FOR
1 1. 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
76
CHANGE OF STOCKS IN THE INDEX 1. 2. 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
77
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 The new divisor is D 7.107857587.
78
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.
79
ADDITIONAL STOCKS 1. 2. 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.
80
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 DJIAs
Divisor was on AUG 2007 and the Divisor was set
at D 0.123017848  
81
VALUE WEIGHTED INDEXES S P500, NIKKEI 225,
VALUE LINE B SOME BASE TIME
PERIOD Initially t B The initial value of the
Index is set at an arbitrarily chosen value M.
82
The SP500 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.
83
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.
84
The Rate of Return on a portfolio
85
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86
THE BETA OF A PORTFOLIO THEOREM Consider a
portfolio consisting of shares of N stocks. The
portfolios BETA is the weighted average of the
stocks betas. The weights are the dollar value
weights of the stocks in the portfolio.
87
THE BETA OF A PORTFOLIO Proof We use a well
diversified index as a proxy portfolio for the
market portfolio. Let P denote the
portfolio underlying the Index, I. Let j
denote the individual stock in the portfolio. j
1, 2, ,N.
88
By the definition of BETA
89
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90
STOCK PORTFOLIO BETA
STOCK NAME PRICE SHARES
VALUE WEIGHT BETA
?P (.044)(1.00) (.152)(.8) (.046)(.5)
(.061)(.7) (.147)(1.1) (.178)(1.1)
(.144)(1.4) (.227)(1.2) 1.06
91
A STOCK PORTFOLIO BETA STOCK NAME PRICE
SHARES VALUE WEIGHT
BETA
?P .122(.95) .187(1.1) .203(.85)
.048(1.15) .059(1.15) .076(1.0) .263(.85)
.042(.75) .95
92
Sources of calculated Betas and calculation
inputs Example ß(GE) 6/20/00 Source ß(GE)
Index Data Horizon Value Line Investment
Survey 1.25 NYSECI Weekly
Price 5 yrs (Monthly) Bloomberg
1.21 SP500I Weekly Price
2 yrs (Weekly) Bridge Information Systems
1.13 SP500I Daily Price 2
yrs (daily) Nasdaq Stock Exchange
1.14 Media General Fin. Svcs. (MGFS)
SP500I Monthly P ice 3 (5) yrs
Quicken.Excite.com 1.23 MSN
Money Central
1.20 DailyStock.com
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93
STOCK INDEX FUTURES 1. The monetary value of
ONE CONTRACT is (THE INDEX VALUE)(MULTIPLIER) o
r (I)(m) 2. Accounts are settled by CASH
SETTLEMENT
94
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(rq )(T-t) .
• q the annual dividend yield on the
  portfolio represented by the index

95
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

96
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 assets with the Index.
• Remember The index is the proxy for the
• Market portfolio.

97
RECALL THAT THE MINIMUM VARIANCE HEDGE RATIO IS
98
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99
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100
STOCK PORTFOLIO HEDGE
STOCK NAME PRICE
SHARES VALUE WEIGHT BETA
ßP .044(1.00) .152(.8) .046(.5)
.061(.7) .147(1.1) .178(1.1)
.144(1.4) .227(1.2) 1.06
101
TIME CASH FUTURES MAR.31 VS 3,862,713
SEP SP500I FUTURES. F 1,052.60. VF
1,052.60(250) 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
102
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
ß 1.35 VF 1,465.45(250) VS
(54)100,000 366,362.50
5,400,000 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 Actual
purchasing price
103
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 DEC. IF PRICE F 1,090 S 17.375
VF 1,090(250) 272,500 VS 2,606,250
ß 1.1 SHORT 11 DEC. SP500I
Futures SEP.30 S 17.125 LONG 11 DEC
SP500I Futures V 2,568,750 F 1,002.
Gain 250(11)(1,090 - 1,002)
242,000 ACTUAL V 2,810,750. An increase
of about 8
104
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
105
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
106
Proof
107
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109
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
110
MARKET TIMING HEDGE EN EXAMPLE STOCK NAME
PRICE SHARES
VALUE WEIGHT BETA
ß(portfolio) .122(.95) .187(1.1)
.203(.85) .048(1.15) .059(1.15)
.076(1.0) .263(.85) .042(.75)
.95
111
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 portfolios Beta employing Stock
Index futures. Suppose that the portfolio manager
wishes to increase the current Beta from ß .95
to ßT 1.25.
112
TIME SPOT FUTURES AUG.29 V 3,783,225.
DEC SP500I Fs ? 0.95. 1,079.8(250)
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
113
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.
114

• SPOT EXCHANGE RATES
• THERE ARE TWO QUOTE FORMATS
• 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.

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117
PAY BUY BUY
PAY USD GBP
PAY S(GBP/USD)ASK GBP 0.50 S(USD/GBP)BID USD 2.083
RECEIVE S(GBP/USD)BID GBP 0.48 S(USD/GBP)BID GBP 2.000
RECEIVE USD GBP
RECEIVE SELL SELL
118
CURRENCY CROSS RATES LET FC1, FC2 AND FC3 DENOTE
THREE DIFFERENT CURRENCIES. IN THE ABSENCE OF
ARBITRAGE
119
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
120
CURRENCY CROSS RATES EXAMPLE FC1 USD FC2
MXN FC3 GBP. USD GBP MXN USA
1 2.01400 0.0920801 UK 0.496524 1 0.045720
MEX 10.860109 21.87230 1
121
CURRENCY CROSS RATES EXAMPLE
122
CURRENCY CROSS RATES EXAMPLE
123
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
124
THE CASH ARBITRAGE ACTIVITIES Start End. US
D1,000,000 USD1,006,134 0.6394 0.558
0 GBP639,400 CHF1,803,108 2.8200
125
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
forward USD1.947750/GBP
126
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 BRITISH POUND 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 MUST CHECK FOR DAILY PRICE
LIMITS CONTRACT MONTHS FOR ALL CURRENCIES
MARCH, JUNE, SEPTEMBER, DECEMBER LAST TRADING
DAY FUTURES TRADING TERMINATES AT 916 AM ON
THE SECOND BUSINESS DAY IMMEDIATELY PRECEEDING
THE THIRD WEDNESDAY OF THE CONTRACT MONTH.
DELIVERY BY WIRED TRASFER. 3RD WEDNESDAY OF
CONTRACT MONTH
127
SPECULATION TAKE RISK FOR EXPECTED PROFIT AN
OUTRIGHT NAKED POSITION WITH CANADIAN DOLLAS t
- MARCH 1. S(USD/CD) .6345 ltgt S(CD/USD)
1.5760 T- SEPTEMBER F(USD/CD) .6270 ltgt
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 CASH FU
TURES MAR 1 DO NOTHING 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).
128

• HEDGING
• IN THE FOLLOWING EXAMPLES WE USE THE NAÏVE HEDGE
  RATIO
• h 1.
• Two ways
• n NS/NF
• n VS/VF

129
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 Interest rates ITALY 6.7512
6.9545 (365-day year) USA 8.6100 8.75154
(360-day year)
130
TIME SPOT FUTURES MAY 25 (1) BORROW
EUR160,000,000 LONG 1,332 DEC EUR FUTURES
FOR FOR 6.9545 FOR 209 DAYS F
1.26000 (2) EXCHANGE THE EUR INTO INTO
USD200,000,000 AND USE THIS SUM TO FINANCE THE
PROJECT DEC 20 LOAN VALUE ON DEC. 20 TAKE
DELIVERY OF EUR166,500,000 160,000,000e(0.069545
)(209/365) PAYING USD209,790,000
EUR166,500,000 REPAY THE LOAN.
THE IMPLIED REVERSE REPO RATE FOR 209 DAYS
131
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 IMPORTERS 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 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
132
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
USD2,388,375 CONTRACT (.6514)125,000 DO
NOTHING USD81,425. AUG. 25
SUSD.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. ACTUAL COST USD2,386,500
133
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 OCT. 26 SEUR.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)
USD5,220,000 ACTUAL PAYMENT IN EUR
40,650,000 5,220,000(.813) EUR44,893,860.
134
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)
DEC FUTURES USD.007190/JY USD.007185/JY
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.
135
TIME CASH FUTURES MAY 23 DO NOTHING LONG
1,582 JY FUTURES FOR V USD142,191,345 F(ask)
USD.007190/JY CASE I DEC 20 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) 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(1,5
82) USD13,644,750 NET COST
USD145,172,250.
136
EXAMPLE 5 A SHORT HEDGE A US MULTINATIONAL
COMPANYS 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 FUTURES FEB.
21 S(USD/EUR) 1.18455 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 LONG 20 JUN EUR
FUTURES DEPOSIT 2,768,242.125 F(JUN)
USD1.10500 PROFIT (1.17675
-1.10500)125,000(20) USD179,375 TOTAL
AMOUNT TO DEPOSIT USD2,947,617.125