0 hedging strategies using futures chapter 3. 1 hedgers open positions in the futures market in...

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

t

Pr

St

j time

Sj

4

HEDGERS

PROBLEM: 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 POSITIONIN 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 marketBasis

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 marketBasis

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 < 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 ≤ TSometimes t = 0 denotes the date the

hedge is opened.

10

THE HEDGE TIMINGk = 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 DeliveryOpen the hedge: Close the futuresopen a futures positionposition

t k T

Time

11

THE HEDGE TIMIMGDate 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 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 timeAnd

Over timeIs of great importance for traders.

13

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.

14

A LONG HEDGE

TIME SPOT FUTURES Bt 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 Bt 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 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: Ft,T

and a random portion: theBASISk

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

Ft,T

St

Fk,T

Sk

Fk,T

Sk

Long hedgeShort hedge

a success a failure

Loss on

the hedge

a failure a success

Loss on

the hedge

Open close the hedge

20


TIME SPOT FUTURES BASISt St= $3.40 LONG

Do nothing Ft,T=$3.50 -$.10

k BUY Sk=$3.80 SHORT

F k,T=3.85 -$.05

T delivery

Actual purchase price:

NO hedge: $3.80

With hedge: $3.45 (Successful hedge)

21


TIME SPOT FUTURES BASISt St= $3.40 LONG

Do nothing Ft,T=$3.50 -$.10

k BUY Sk=$3.00 SHORT

F k,T=3.05 -$.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 < 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

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price

24

Basis RiskThe 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.

Bt

t

Pr

St

Ft,T

k T time

BT = 0

Bk

Sk

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 firm’s 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 S1Ft,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 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.

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

.N

Nhn

N

nNh

F

S

S

F

positionspot in the units ofnumber The

position futures in the units ofnumber Theh

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.

F

S

S

F

N

N n 1

N

nNh

39

Examples: NAÏVE HEDGE RATIO: h = 1.

1.A firm will sell NS = 75,000barrels of crude oil.

NYMEX WTI: NF = 1,000 barrels.

SHORT:n = 75,000/1,000

= 75 NYMEX futures.

40

2.A firm will buy NS = 200,000bushels of wheat.

CBT wheat futures: NF = 5,000.

LONG:

n = 200,000/5,000 = 40 CBT futures.

41

3.A firm will sell NS = 3,600ounces 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

45

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

Oct1,07buy 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 : (93+92.90+94+94.75)/4 = 93.660The 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, 07contract 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 MARKETF FUTURES POSITIONS

DEC, 07contract to 89.00 short 100 MAY WTI 88.20

sell 100,000 bbls Oct1,07buy 75,000bbls Short 100 MAY 2008

APR 08 long 100 MAY 2008 87.40Short 100 SEP 2008 87.00

Short 100 SEP 2008

AUG 08 Long 100 SEP 2008 86.50Short 100FEB 2009 86.30

Short 100 FEB 2009

JAN, 09 sell 100,000bbls 86.00 Long 100 FEB 2009 85.90NO 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.

49

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

POSITION’S 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*:

.N

Nh n

F

S**

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)]

= NS2[VAR(S)+VAR(hF)+2COV(S;hF)]

= 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)

58

THE MINIMUM RISK HEDGE RATIO IS:

.N

Nh n

.N

N*n -

σ

σρ- h*

.F)var(

F)S;cov(- h*

F

S

S

F

ΔF

ΔSΔS,ΔF

59

This result can be rewritten as:

.N

Nh n

.N

N*n

σ

σρ- h*

.var(F)

F)cov(S;- h*

F

S

S

F

F

SFS,

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:

(ΔS) = 0.025; (ΔF)=0.033;andρ(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]

= 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 andthus, 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. 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.

64

S1 F1,t S1 F1

S2 F2,t S2 F2

S3 F3,t S3 F3

. . . .

. . . .

. . . .

. . Sn Fn

Sn+1 Fn+1,t

h* , using Regression:

DATA: n+1 weeks.

*hβ

n. ..., 1,2,i α eβΔFΔS iii

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 FUTURES MARKET F FUTURES POSITIONS

SEP 05contract 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

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 MARKETF FUTURES POSITIONS

SEP 05 NOTHING Long 28 MAR 2006 2.723long 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 2006long 28 MAR 2007long 28 SEP 2006

Aug 06 buy 1M units 2.65 short 28 SEP 06 2.648 long 28 MAR 2007long 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/poundThe 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/poundCost saving: 4M[2.7457 – 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.

N.1,..., = j ;D

S = I j

75

EXAMPLES OF INDEX ADJUSMENTS

STOCK SPLITS: 2 FOR 1:

1.

2.

Before the split:

(30 + 40 + 50 + 60 + 20) /5 = 40I = 40 and D = 5.

An instant later:

(30 + 20 + 50 + 60 + 20)/D = 40The new divisor is D = 4.5

11N21 ID/)S,,...S(S

12N21 ID/)S,...,S2

1(S

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

11N21 ID/)S,,... ABC)(S(S

12N21 ID/)S...XYZ)(S(S

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.

11N21 ID/)S,...,S(S

121+NN21 ID/)SS,...,S(S

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 DJIA’s 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.

n,1,2,...... j SN

SNI

BjBj

tjtjt

82

** 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.

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.

.Rw R tjtjPt .V

V

SN

SN w

tP

tj

tjtj

tjtjtj

84

The Rate of Return on a portfolio

tjtj

tj1j+ttjPt

tj1j+t

tjtj

tjtj1j+t1j+tPt

tjtj

tjtj1j+t1j+t

t

t1+tPt

SN

)S(SN R

Thus, .N N but,

;SN

SNSN R

SN

SN SN

VP

VPVP R

85

.Rw R

Finally, .RV

V

or ,RSN

SN

:as thisRewrite .SN

RSN

,SN

S

SSSN

R

tjtjPt

tjtP

tj

tjtjtj

tjtj

tjtj

tjtjtj

tjtj

tj

tj1jttjtj

Pt

.V

V

SN

SNw

tP

tj

tjtj

tjtjtj

86

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

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.

R

88

.)VAR(R

)R;]RwCOV([ β

,RwR : theoremprevious theFrom

.)VAR(R

)R;COV(Rβ

:Market for theproxy a isIndex The

.)VAR(R

)R;COV(R β

I

IjjP

jjP

I

IPP

M

MPP

By the definition of BETA:

89

.β w )VAR(R

)R;COV(Rwβ

:or ,)VAR(R

)R;COV(Rwβ

: thusoperator,linear a

is covariance e that thRecall

jjI

IjjP

I

IjjP

90

STOCK PORTFOLIO BETA

FEDERAL MOUGUL 18.875 9,000 169,875 .044 1.00MARTIN ARIETTA 73.500 8,000 588,000 .152 .80IBM 50.875 3,500 178,063 .046 .50US WEST 43.625 5,400 235,575 .061 .70BAUSCH & LOMB 54.250 10,500 569,625 .147 1.1FIRST UNION 47.750 14,400 687,600 .178 1.1WALT DISNEY 44.500 12,500 556,250 .144 1.4DELTA AIRLINES 52.875 16,600 877,725 .227 1.2

3,862,713

P = (.044)(1.00) + (.152)(.8) + (.046)(.5) + (.061)(.7)

+ (.147)(1.1) + (.178)(1.1) + (.144)(1.4) + (.227)(1.2)

= 1.06

STOCK NAME PRICE SHARES VALUE WEIGHT BETA

91

BENEFICIAL CORP. 40.500 11,350 459,675 .122 .95CUMMINS ENGINES 64.500 10,950 706,275 .187 1.10GILLETTE 62.000 12,400 768,800 .203 .85KMART 33.000 5,500 181,500 .048 1.15BOEING 49.000 4,600 225,400 .059 1.15W.R.GRACE 42.625 6,750 287,719 .076 1.00ELI LILLY 87.375 11,400 996,075 .263 .85PARKER PEN 20.625 7,650 157,781 .042 .75

3,783,225

A STOCK PORTFOLIO 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 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) S&P500I Monthly P ice3 (5) yrs Quicken.Excite.com 1.23

MSN Money Central 1.20

DailyStock.com 1.21

Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly)

S&P Personal Wealth 1.2287

S&P Company Report) 1.23

Charles Schwab Equity Report Card 1.20

S&P Stock Report 1.23

AArgus Company Report 1.12 S&P500I Daily Price 5 yrs (Daily)

Market Guide S&P500I Monthly Price 5 yrs (Monthly)

YYahoo!Finance 1.23

Motley Fool 1.23

93

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

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(r–q )(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 NAÏVEhedge ratio, h = 1. Mostly, however,

hedgers use the minimum variance hedgeratio. In this case, the underlying asset is a

stock index; actually the portfolio thatunderlie the index. Thus, the parameter

that relates the spot asset and the index isthe Beta of the spot asset’s with the Index. Remember: The index is the proxy for the

Market portfolio.

97

RECALL THAT THE MINIMUM VARIANCE HEDGE RATIO IS:

.N

Nh n

.N

N*n

σ

σρ- *h

.F)var(

F)S;cov(- *h

F

S

S

F

ΔF

ΔSΔS,ΔF

98

2t,T

t,Tt

F

FS

Ft,T

Ft,TSt

Ft,TFt,T

t,Tk,T

t,T

StSt

tk

t

F

]F[S

)VAR(r

)r,COV(r - = *h

)rVAR(F

)rF,rCOV(S -

F)VAR(

F)S;COV( - = *h

rF = ΔF r = F

F - F =

F

F

rS = ΔS r = S

S - S =

S

S

99

F

S*

Ft,T

St

F

S

t,T

t

t,T

t

F

FS

2t,T

t,Tt

F

FS

V

Vβn

at t Value Futures

at t ValueSpot β -

NF

NSβ

N

N*h *n

.F

Sβ-

F

S

)VAR(r

)r,COV(r - = *h

F

]F[S

)VAR(r

)r,COV(r - = *h

100

STOCK PORTFOLIO HEDGE

FEDERAL MOUGUL 18.875 9,000 169,875 .044 1.00MARTIN ARIETTA 73.500 8,000 88,000 .152 .80IBM 50.875 3,500 178,063 .046 .50US WEST 43.625 5,400 235,575 .061 .70BAUSCH & LOMB 54.250 10,500 569,625 .147 1.1FIRST UNION 47.750 14,400 687,600 .178 1.1WALT DISNEY 44.500 12,500 556,250 .144 1.4DELTA AIRLINES 52.875 16,600 877,725 .227 1.2

3,862,713

βP = .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

16.- = 263,300

3,862,7131.06- =n*

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

β = 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:

20- = 366,362.50

5,400,0001.35- = n*

E$52.9/SHAR = 100,000

$510,000 - $5,800,000

103

HEDGING A ONE STOCK PORTFOLIOSPECIFIC 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

β = 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%

11- = 272,500

2,606,2501.1- = n*

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: .

V

Vβ][βn*

F

ST

106

TFS

FSP

F

FF

S

SS

S

PP

F

F

S

F

S

S

S

F

S

S

S

P

FSP

FSP

ErErV

V*nErEr

.V

)Δ(Vr and ;

V

)Δ(Vr ;

V

)Δ(Vr

DEFINE

.V

)Δ(V

V

V*n

V

)Δ(V

V

)Δ(V*n

V

)Δ(V

V

)Δ(V

)(V*n )Δ(V)Δ(V

nVV V

Proof:

107furmula. in the

needed iscost y opportunit no andoutlay initial no

requiresit Hence, 1.β sit' thus,andindex on the

futures a is Fequation last in the that Notice

.r)E(r )E(r

and ]r)β[E(rr)E(r

and ]r)[E(rβr)E(r

:can write weCAPM, theFollowing

).E(rV

V*n)E(r)E(r :Again

fMF

fMfS

fMTfT

FS

FST

108F

ST

*

fMTf

fMS

F*fMf

FS

F*ST

V

Vβ][βn*

:nfor solve and

]r)[E(rβr

]r)[E(rV

Vn]r)β[E(rr

)E(rV

Vn)E(r)E(r

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:

. contracts V

Vβ][βn*LONG

β toβ increase wish to weβ;β If

. contracts V

V]β[βn* SHORT

β toβ reduce wish to weβ;β If

F

ST

TT

F

ST

TT

110

BENEFICIAL CORP. 40.500 11,350 459,675 .122 .95CUMMINS ENGINES 64.500 10,950 706,275 .187 1.10GILLETTE 62.000 12,400 768,800 .203 .85KMART 33.000 5,500 181,500 .048 1.15BOEING 49.000 4,600 225,400 .059 1.15W.R.GRACE 42.625 6,750 287,719 .076 1.00ELI LILLY 87.375 11,400 996,075 .263 .85PARKER PEN 20.625 7,650 157,781 .042 .75

3,783,225

MARKET TIMING HEDGE; EN EXAMPLE


β(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 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.

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%

4 = 269,950

3,783,225.95) - (1.25 = *n

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:

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.

115

S(USD/EUR)1.4821= .67476

1S(EUR/USD)

1

.67476 S(EUR/USD) 1.4821 = S(USD/EUR)

:Example

)1

/FC2

S(FC1 = )2/FC1S(FC

116

.48GBP.get USDONE sell D,GBP.480/US BIDS(GBP/USD)

83.pay USD2.0 GBP ONEbuy GBP, USD2.083/ ASKS(USD/GBP)

get USD2. GBP ONE sell BP, USD2.00/G BIDS(USD/GBP)

.50GBP.pay USDONEbuy GBP.5USD, ASKS(GBP/USD)

ASKS(USD/FC)

1 BIDS(FC/USD)

BIDS(USD/FC)

1 ASKS(FC/USD)

:QUOTESASK AND BID HAVE WEWHEN

117

PAY

BUY USD GBP

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 2.000USD GBP

SELL

118

CURRENCY CROSS RATES

LET FC1, FC2 AND FC3 DENOTE THREE DIFFERENT CURRENCIES.

IN THE ABSENCE OF ARBITRAGE :

S(FC3/FC1)S(FC3/FC2) =

S(FC2/FC3)S(FC1/FC3) = S(FC1/FC2)

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 2.014000.0920801UK 0.496524 1 0.045720MEX 10.86010921.87230 1

121

CURRENCY CROSS RATESEXAMPLE

.S(MXN/GBP)

S(USD/GBP) =

S(GBP/USD)

S(GBP/MXN) = S(USD/MXN)

GBP. FC3 MXN; FC2 USD; FC1Let

122

CURRENCY CROSS RATESEXAMPLE

0.092080. 21.872300

2.014000 S(MXN/GBP)

S(USD/GBP)

0.092080. 0.496524

0.045720 S(GBP/USD)

S(GBP/MXN)

21.872300. S(MXN/GBP)2.014000. S(USD/GBP)

0.496524. S(GBP/USD) 0.045720. S(GBP/MXN)

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

2.8200 < 2.8029 = 0.5580

1.5640 :BUT

S(CHF/GBP) = S(USD/CHF)

S(USD/GBP) :SIMILARLY

1.7920 1.8031 = 0.35460.6394 :BUT

S(CHF/USD) = S(GBP/CHF)

S(GBP/USD) :THEORY

124

THE CASH ARBITRAGE ACTIVITIES:

Start: End.

USD1,000,000 USD1,006,134

0.6394 0.5580

GBP639,400CHF1,803,108

2.8200

125

Forward rates, An example:

GBP DEC 17, 2007

SPOT USD1.997200/GBP

1 Month forwardUSD1.995300/GBP

2 Months forwardUSD1.993760/GBP




2 Years forwardUSD1.947750/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 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

127

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 CASH FUTURES

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:

1.n = NS/NF

2.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 ASKSPOT: 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 =

1,332 = 125,000

0166,500,00n

8.23%.or .0823, = ]0200,000,00

0209,790,00ln[

209/360

1

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 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 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

46 = 780)62,500(1.2

3,656,800 =n 1;h

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 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.

ACTUAL COST USD2,386,500

30 = 81,425

2,388,375 =n

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 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)

=USD5,220,000

ACTUAL PAYMENT IN EUR:

40,650,000 + 5,220,000(.813) = EUR44,893,860.

348 = 125,000

43,500,000 =n

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(1,582)

= USD13,644,750

NET COST: USD145,172,250.

136

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 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

0 hedging strategies using futures chapter 3. 1 hedgers open positions in the futures market in...

Documents