4. Hedging strategies using futures
104
Hedging Strategies Using Futures Chapter 4
Transcript of 4. Hedging strategies using futures
Hedging Strategies Using Futures
Chapter 4
HEDGERS
OPEN POSITIONS IN THE FUTURES MARKET
IN ORDER TO ELIMINATE THE RISK ASSOCIATED WITH THE
PRICE OF THE UNDERLYING ASSET IN THE SPOT MARKET.
There are two ways to determine whether to open a short or a long
hedge:
1. A LONG HEDGE
OPEN A LONG FUTURES POSITION IN ORDER TO HEDGE THE PRODUCT PURCHASE TO BE
MADE AT A LATER DATE.
In this case, it is known that a purchase will be made at a future date.
THW HEDGWR LOCKS IN THE PURCHASE PRICE.A SHORT HEDGE
OPEN A SHORT FUTURES POSITION
IN ORDER TO HEDGE THE SALE OF THE PRODUCT TO BE MADE AT A LATER DATE.
In this case, it is known that a sale will be made at a future date.
THE HEDGER LOCKS IN THE SALE PRICE
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.
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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
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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.
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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.
Some times we use t = 0 for the initial futures purchase or sale. .
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DELIVERYk = 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. WHY?
1. 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.
2. Often k is not in a delivery month.
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DEFINITION
The basis is the difference between the asset’s spot and the futures price.
BASISk = SPOT PRICEk - FUTURES PRICEk
Notationally:
Bk = Sk - Fk,T k < T.
In general, one must specify a specific futures, i.e., a specific delivery month.
Usually, however, it is understood that the futures is for the nearest
month to delivery.
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A LONG HEDGE
TIME SPOT FUTURESt Contract to buy LONG F t,T
Do nothing
k BUY Sk SHORT F k,T
T delivery
ACTUAL PAYMENT = Sk + Ft,T - Fk,T
= Ft,T + [Sk - Fk,T]
= Ft,T + BASISk
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A SHORT HEDGE
TIME CASH FUTURES
t Contract to sell SHORT Ft,T
Do nothing
k SELL Sk LONG Fk,T
T delivery
ACTUAL SELLING PRICE = Sk + Ft,T - Fk,T
= Ft,T + [Sk - Fk,T]
= Ft,T + BASISk
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Observe that in both cases,
Long hedge and short hedge
The cash flow for the hedger when the hedge is closed on date k, is:
Ft,T + BASISk
This cash flow consists of a known price Ft,T
and a random value BASISk
We return to this point later.
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THE KEY FOR THE SUCCESS OF HEDGING WITH FUTURES IS:
the relationship between the cash and the futures price over time:
Futures and spot prices of any underlying asset, co vary over
time.
Although not in perfect tandem and not by the same amount, these
prices move up and down together most of the time, during the life of
the futures.
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Example: A LONG HEDGE
Date Spot market Futures market
t St = $400/unit Ft,T = $425/unitContract to buy long one
gold Gold on k. futures for delivery at T
K Buy the gold Short one gold Sk = $416/unit futures
for delivery at T.
Fk,T = $442/unit
T Amount paid: 416 + 425 – 442= $399/unit
or 425 + (416 – 442) = $399/unit
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Example: A SHORT HEDGE
Date Spot market Futures market
t St = $400/unit Ft,T = $425/unitContract to sell short one
gold Gold on k, futures for delivery at T
K Sell the gold Long one gold Sk = $384/unit futures
for delivery at T.
Fk,T = $412/unit
T
Amount received: 384 + 425 – 412 = $397/unit
or 425 + (384 – 412) = $397/unit
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BASISk = SPOT PRICEk - FUTURES PRICEk
Notationally:
Bk = Sk - Fk,T k < T.
But on the delivery date k = T:
BT = ST - FT, T = 0 k = T.
T is the nearest month of delivery which is at or following k.
The latter equation indicates that the basis converges to zero on the delivery date. FT,T is the price of the commodity on date T for delivery and payment on date T. Hence, FT,T
= ST .
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Convergence of Futures to Spot
Time Time
(a) (b)
FuturesPrice
FuturesPrice
Spot Price
Spot Price
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The relationship between the cash and the futures price over time:
1.The basis is the difference between two random variables. Thus, it varies in an unpredictable way. Over time, it narrows, widens and may change its sign.
2.The basis converges to zero at the futures maturity.
3.The basis is less volatile than either price.
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Basis Risk
• Basis is the difference between spot & futures prices.
• Basis risk arises because of the uncertainty about the basis when the hedge is closed out at time k.
• We do know, however, that BT = 0 at delivery.
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Generally, the basis fluctuates less than both, the cash and the futures prices. Hence,
hedging with futures reduces risk. Nonetheless,
Basis risk exists in any hedge.
Bt
t
Pr
St
Ft,T
k T time
BT = 0
Bk
Sk
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We showed above that for both types of hedge
A SHORT HEDGE or A LONG HEDGE,
The final payment received or paid to the hedger is:
Ft,T + BASISk
This cash flow consists of two parts: the first, Ft,T is KNOWN when the hedge is opened. The second part - BASISk – is a random element. Conclusion: At time t, the firm faces the cash-price risk. Upon opening a hedging position, the firm locks in the futures price, but it still remains exposed to the
basis risk, because the basis at time k is random.
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Bt
t
Pr
St
Ft,T
k T time
BT = 0
Bk
Sk
We thus, proved that: hedging amounts to the reducing the
firm’s risk exposure because the basis is less risky than the
spot price risk.
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Choice of contract for hedging• Choose a delivery month that is as close
as possible to, but later than, the end of the life of the hedge
• When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. Then, there are two components to the basis: The basis associated with the asset underlying the futures and the spread between the two spot prices. Let S1 be the spot asset price. This is the asset that the hedger is trying to hedge. Let S2 be the spot price of the asset underlying the futures. This is a different asset.
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A CROSS HEDGE
TIME CASH FUTURES
t Contract to trade Ft,T
Do nothing
k Trade for S1K Fk,T
T delivery
PAY OR RECEIVE = S1K + Ft,T - Fk,T
= Ft,T + [S2k - Fk,T] +[S1k - S
2k]
= Ft,T + BASISk + SPREADK
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Delivery month? Normally, the hedge is opened with futures for the delivery month closest to the firm operation date in the cash market or the nearest month beyond that date.
The key factor here is the correlation between the cash and futures prices or price changes.
Statistically, it is known that in most cases, the highest correlation is with the futures prices of the delivery month nearest to the cash activity.
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HEDGE RATIOS
Open a hedge.
Questions: Long or Short?
Delivery month? Commodity to use? How many futures to use?
The number of futures in the position is determined by the
HEDGE RATIO
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HEDGE RATIOS; DEFINITION:
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.
.N
Nhn
N
nNh
F
S
S
F
positionspot in the units ofnumber
position futures in the units ofnumber h
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NAÏVE HEDGE RATIO:
ONE - FOR - ONE
F
S
S
F
N
N n 1
N
nNh
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Examples: NAÏVE HEDGE RATIO: ONE - FOR – ONE
1. Intends to sell NS = 50,000 barrels of crude oil. NF = 1,000 barrels
Short n = 50,000/1,000 = 50 NYMEX futures.
2. Intend to borrow $10M for ten years. Hedge with CBT T-bond futures. NS = 10,000,000; NF = 100,000.
Short n = 10,000,000/100,000 = 100 CBT T-bond
futures.
3. Intend to buy NS = 200,000 bushels of wheat.CBT wheat futures: NF = 5,000.
Long n = 200,000/5,000 = 40 CBT futures.
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Other hedge ratios.
Suppose that the relationship between the spot and futures prices over time is:
Spot Futures
$1 $2
$1 $0.5
Clearly, the Naïve hedge ratio is not appropriate in these cases.
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OPTIMAL HEDGE RATIOS
THE MINIMUM VARIANCE HEDGE RATIO
OBJECTIVE: TO MINIMIZE THE RISK ASSOCIATED WITH THE HEDGE.
RISK IS MEASURED BY: VOLATILITY.
THE VOLATILITY MEASURE IS THE VARIANCE
OBJECTIVE:
FIND THE NUMBER OF FUTURES, n, THAT MINIMIZES THE VARIANCE OF THE CHANGE OF
THE HEDGED POSITION VALUE.
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THE MATHEMATICS
St = Spot market price.
t = 0 The hedge opening date.t = 1 The hedge closing date.T = The delivery date.F0,T = The futures price on date 0 for delivery at T.n = The number of futures 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 t = 1.
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The initial and terminal hedged position values:
VP0 = S0NS +nNFF0,T
VP1 = S1NS +nNFF1,T
The position value change:
(Vp) = VP1 - VP0
= (S1NS +nNFF1,T)
- (S0NS +nNFF0,T)
= NS(S1- S0) +nNF(F1,T - F0,T).
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AGAIN:
(VP) = NS(S1- S0) +nNF(F1,T - F0,T).
(VP) = NS[(S1- S0) +nNF/NS(F1,T - 0,T)]
(VP) = NS[(S1- S0) +h(F1,T - F0,T)]
PROBLEM: (VP) is a random variable because the prices at t=1 are unknown. Find h* so as to minimize the risk associated with (VP).
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VAR(VP) = = NS
2 VAR [(S1- S0) +h(F1,T - F0,T)]
= NS2[VAR(S)+VAR(hF)
+2COV(S;hF)]
= NS2 [VAR(S) + h2VAR(F)
+2hCOV(S;F)].
TO MINIMIZE VAR(VP) take it’s derivative
with respect to h and equate it to zero:
2h*VAR (F) + 2COV(S; F) = 0.
h* = - COV(S;F)/VAR(F)
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THE MINIMUM RISK HEDGE RATIO IS:
.N
Nh - n
:is hedge in the futures ofnumber optimal then*,for Solving
.N
N*n -
σ
σρ- h*
thus,,σσ
y)cov(x;ρ
:y and x , variablesrandom any twofor But
.F)var(
F)S;cov(- h*
F
S
S
F
ΔF
ΔSΔFΔS,
yxyx,
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The negative sign only indicates that 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 short,
the hedge is short.
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S1 F1,t S1 F1
S2 F2,t S2 F2
S3 F3,t S3 F3
. . . .
. . . .
. . . .
. . Sn Fn
Sn+1 Fn+1,t
DATA (SAY DAILY) n+1 DAYS.
*hβ
n. ..., 1,2,i α eβΔFΔS iii
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EXAMPLE 1: A company needs to buy 800,000 gallons of diesel
oil in 2 months. It opens a long hedge using heating oil futures. An analysis of price changes ΔS and ΔF over a 2 month interval yield:
(ΔS) = 0.025; (ΔF) = 0.033; 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 42,000 gallons, so
Long n* = (0.525)(800,000)/42,000 = 10 futures.
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EXAMPLE 1, continued: 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, the optimal number of futures: 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.
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EXAMPLE 2: A company knows that it will buy 1 million gallons of jet fuel in 3 months. The company chooses to long hedge with heating oil futures. The standard deviation of the change in the price per gallon of jet fuel over a 3-month period is calculated as 0.04. The standard deviation of the change in the futures price over a 3-month period is 0.02 and the coefficient of correlation between the 3-month change in the price of jet fuel and the 3-month change in the futures price is 0.42. The optimal hedge ratio:
h* = (0.42)(0.04)/(0.02) = 0.84,
And the risk-minimizing number of futures
n* = (0.84)(1,000,000)/42,000 = 20.
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EXAMPLE 3. A Hedging example for copper:
Date: OCT03 FEB04AUG04FEB05AUG05
Spot: 72.00 69.00 65.00 77.00 88.00
Futures
For Delivery: MAR 2004 72.30 69.10
SEP 2004 72.80 70.20 64.80
MAR 2005 71.60 70.70 64.30 76.70
SEP 2005 69.50 68.90 64.20 76.50 88.20
Today is OCT 2003. A US firm has a contract to purchase 1,000,000 pounds of copper in FEB 04, AUG04, FEB05 and AUG05.
The firm decides to hedge these purchases with NYMEX copper futures. One NYMEX copper futures is for 25,000 pounds of copper.
The firm decides to use h* = .7.
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We now analyze three potential Hedging strategies.
HEDGING POLICY I:
The hedge ratio is h* = .7.
No other restrictions.
The firm uses a STRIP . That is, a sequence of futures that are equally spaced, each one hedging one spot future trade.
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Date SPOT MARKET FUTURES MARKETF FUTURES POSITIONS
Oct 03 NOTHING Long 28 MAR 2004 72.30
long 28 MAR 2004 Long 28 SEP 2004 72.80 long 28 SEP 2004
Long 28 MAR 2005 71.60 long 28 MAR 2005
Long 28 SEP 2005 69.50 long 28 SEP 2005
Feb 04 buy 1M units 69.00 short 28 MAR 04 69.10 long 28 SEP 2004
long 28 MAR 2005long 28 SEP 2005
Aug 04 buy 1M units 65.00 short 28 SEP 04 64.80 long 28 MAR 2005
long 28 SEP 2005
Feb 05 buy 1M units 77.00 short 28 MAR 05 76.70 long 28 SEP 2005
Aug 05 buy 1M units 88.00 short 28 SEP 05 88.20 NO POSITION
The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75The average price for the hedged strategy:(.3)69 + (.7)(69 + 72.30 – 69.10) = 71.24(.3)65 + (.7)(65 + 72.8 – 64.8) = 70.60(.3)77 + (.7)(77 + 71.6 – 76.7) = 73.43(.3)88 + (.7)(88 + 69.5 – 88.2) = 74.98
72.5625
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Hedging strategy II:
The hedge ratio is h* = .7
The firm will not use futures with delivery months which are more than 13 months hence.
Thus, the firm uses a STACK.
All the futures needed to be stacked will be stacked to the latest delivery months futures used.
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Hedging strategy II:
Date SPOT MARKET FUTURES MARKET F FUTURES POSITIONSOct 03 NOTHING Long 28 MAR 2004 72.30 long 28 MAR 2004
Long 84 SEP 2004 72.80 long 84 SEP 2004
Feb 04 buy 1M units 69.00 short 28 MAR 04 69.10short 56 SEP 04 70.20 long 28 MAR 2005long 56 MAR 05 70.70 long 28 SEP 2005
Aug 04 buy 1M units 65.00 short 28 SEP 04 64.80 long 28 SEP 2004short 28 MAR 05 64.30 long 56 MAR 2005long 28 SEP 05 64.20
Feb 05 buy 1M units 77.00 short 28 MAR 05 76.70 long 28 SEP 2005
Aug 05 buy 1M units 88.00 short 28 SEP 05 88.20 NO POSITION
The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75The average price for the hedged strategy:(.3)69 + (.7)(69 + 72.30 – 69.10) +(1.35)(72.80 – 70.20) = 74.75(.3)65 + (.7)(65 + 72.8 – 64.80) + (.7)( 70.70 – 64.30) = 75.08(.3)77 + (.7)(77 + 70.7 – 76.7) = 72.80(.3)88 + (.7)(88 + 64.2 – 88.2) = 71.20
73.4575
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Hedging strategy III. The hedge ratio is h* = .7 The firm uses the following stacking policy:
On OCT 2003 Stack all the futures needed to be stacked onto the SEP 2004
position.
On FEB 2004 Do not change your SEP2004 position. That is, keep the entire stack on the SEP2004 position.
On AUG 2004 Enter into whatever futures positions the firm needs, which are not restricted and continue with them to the hedge termination date.
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Hedging policy III.Date SPOT MARKET FUTURES MARKET F FUTURES POSITIONSOct 03 NOTHING Long 28 MAR 2004 72.30 long 28 MAR 2004
Long 84 SEP 2004 72.80 long 84 SEP 2004
Feb 04 buy 1M units short 28 MAR 04 69.10 long 84 SEP 2004
69.00
Aug 04 buy 1M units short 84 SEP 04 64.80
65.00 long 28 MAR 05 64.30 long 28 MAR 2005
long 28 SEP 05 64.20 long 28 SEP 2005
Feb 05 buy 1M units short 28 MAR 05 6.70 long 28 SEP 2005
77.00Aug 05 buy 1M units short 28 SEP 05 88.20 NO POSITION
88.00
The average price for the un hedged strategy : (69+65+77+88)/4 = 74.75The average price for the hedged strategy:(.3)69 + (.7)(69 + 72.30 – 69.10) = 71.2465 + 2.1)( 72.8– 64.80) = 81.80(.3)77 + (.7)(77 + 64.3 – 76.7) = 68.32(.3)88 + (.7)(88 + 64.2 – 88.2) = 71.20
73.14
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HEDGE RATIOS
As we move from one type of underlying asset to another, we will use these hedge ratios as well as new ones to be developed later.
The two underlying assets that we analyze next are:
1. Stock index futures.
2. Foreign currency futures.
In each case, we first describe the SPOT MARKET and
then analyze the FUTURES MARKET.
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STOCK INDEX FUTURES
The first stock index futures began trading in 1982 on the KCBT. The underlying was the
VALUE LINE INDEX.
Soon after, the CBT, after losing its battle with the Dow Jones Co., started trading futures on the
MAJOR MARKET INDEX, the MMI.
Today, Stock Index Futures are traded one dozens of indexes.
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STOCK INDEXES (INDICES)
A STOCK INDEX IS A SINGLE NUMBER BASED ON INFORMATION ASSOCIATED WITH A SET OF
STOCK PRICES AND QUANTITIES.
A STOCK INDEX IS SOME KIND OF AN AVERAGE OF THE PRICES AND THE QUANTITIES OF THE STOCKS THAT ARE
INCLUDED IN THE PORTFOLIO THAT REPRESENT THE INDEX.
THE MOST USED INDEXES ARE
A SIMPLE PRICE AVERAGE
AND
A VALUE WEIGHTED AVERAGE.
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STOCK INDEXES - THE CASH MARKETA. AVERAGE PRICE INDEXES: DJIA, MMI:
N = The number of stocks in the index
P = Stock market price
D = Divisor
INITIALLY, D = N AND THE INDEX IS SET AT A GIVEN LEVEL. TO ASSURE INDEX CONTINUITY, THE DIVISOR IS ADJUSTED OVER TIME.
N.1,..., = i ;D
P = I i
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EXAMPLES OF INDEX ADJUSMENTS
STOCK SPLITS: 2 TO 1.
1.
2.
1. (30 + 40 + 50 + 60 + 20) /5 = 40
I = 40 and D = 5.
2. (30 + 20 + 50 + 60 + 20)/D = 40
The new divisor is D = 4.5
(P P P D I1 2 N 1 1 ... ) /
(P P P D I1 2 N 2 1 1
2... ) /
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CHANGE OF STOCKS IN THE INDEX
1.
2.
1. (30 + 20 + 50 + 60 + 20)/4.5 = 40
I = 40and D =4.5.
2. (30 + 120 + 50 + 60 + 20)/D = 40
The new divisor is D = 7.75
(P P ABC) P D I1 2 N 1 1 ( ... ) /
(P P XYZ) P D I1 2 N 2 1 ( ... ) /
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STOCK #4 DISTRIBUTED 40% STOCK DIVIDEND
(32 + 113 + 52 + 58 + 25)/7.75 = 36.1203
D = 7.75.
Next, (32 + 113 + 52 + 34.8 + 25)/D = 36.12903
The new divisor is D = 7.107857587.
STOCK # 2 SPLIT 3 TO 1.
(31 + 111 + 54 + 35 + 23)/7.107857587 = 35.73507
(31 + 37 + 54 + 35 + 23)/D = 35.73507
The new Divisor is D = 5.0370686.
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ADDITIONAL STOCKS
1.
2.
1. (30 + 39 + 55 + 33 + 21)/5.0370686 = 35.338013
2. (30 + 39 + 55 + 33 + 21 + 35)/D = 35.338013
D = 6.0275035.
(P P P D I1 2 N 1 1 ... ) /
121+NN21 ID/)PP,...,P(P
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VALUE WEIGHTED INDEXES
S & P500, NIKKEI 250, VALUE LINE
B = SOME BASIS TIME PERIOD
INITIALLY t = B THUS, THE INITIAL INDEX VALUE IS SOME
ARBITRARILY CHOSEN VALUE: M. Examples:
The S&P500 index base period was 1941-1943 and its initial value was set at M = 10.
The NYSE index base period was Dec. 31, 1965 and its initial value was set at M = 50.
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58
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The return on a value weighted index in any period t, is the weighted average of the individual stock returns; the weights
are the dollar value of the stock as a proportion of the entire index value.
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59
THE RATE OF RETURN ON THE INDEX
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THE BETA OF A PORTFOLIO
THEOREM:
A PORTFOLIO’S BETA IS THE WEIGHTED AVERAGE OF THE BETAS OF THE STOCKS THAT COMPRISE THE PORTFOLIO. THE WEIGHTS ARE THE DOLLAR VALUE
WEIGHTS OF THE STOCKS IN THE PORTFOLIO.
Proof: Assume that the index is a well diversified portfolio, I.e., the index represents the market portfolio.
Let P denote the portfolio representing the index; let I denote the index and let i denote the individual stock; i = 1, 2, …,N.
R
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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
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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
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
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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
66
STOCK INDEX FUTURES
ONE CONTRACT VALUE =
(INDEX VALUE)($MULTIPLIER)
One contract = (I)($m)
ACCOUNTS ARE SETTLED BY CASH SETTLEMENT
67
Stock Index (Page 52)
• Can be viewed as an investment asset paying a dividend yield
• The futures price and spot price relationship is therefore
F0,T = S0e(r–q )T .
q = the annual dividend yield on the portfolio represented by the index
68
Stock Index (continued)
• For the formula to be true it is important that the index represent 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
69
Index Arbitrage
• When F0>S0e(r-q)T an arbitrageur buys
the stocks underlying the index and sells futures
• When F0<S0e(r-q)T an arbitrageur buys
futures and shorts or sells the stocks underlying the index
70
Index Arbitrage (continued)
• Index arbitrage involves simultaneous trades in futures and many different stocks
• Very often a computer is used to generate the trades
• Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold
71
STOCK INDEX HEDGINGAs before, Stock index hedgers may usethe NAÏVE hedge ratio. Mostly, howeverhedgers use the minimum risk (variance)hedge ratio. In the current case, the underlying asset is a stock index, or theportfolio that represents the index. Thus, the parameter that connects the spotasset and the index is the spot asset’sBETA, where the portfolio representing the index is the proxy for the marketportfolio.
72
RECALL THAT THE MINIMUM RISK HEDGE RATIO IS:
.N
Nh- n
:is hedge in the futures ofnumber optimal theAnd
.N
N*n -
σ
σρ- h*
OR
.F)var(
F)S;cov(- h*
F
S
S
F
ΔF
ΔSΔFΔS,
We now apply this formula to the case of stock indexes
73
Value Futures
ValueSpot β -
NF
NSβ
N
N*h *n
.F
Sβ-
F
S
)VAR(r
)r,COV(r - = *h
F
FS
)VAR(r
)r,COV(r - = *h
)rVAR(F
)rF,rCOV(S -
F)VAR(
F)S;COV( - = *h
rF = ΔF F
ΔF =
F
F - F = r
rS = ΔS S
ΔS =
S
S - S = r
F0
S0
F
S
0
0
0
0
F
FS
20
00
F
FS
F0
F0S0
F000
01F
S000
01S
74
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 AT 1,465.45
β = 1.35 F = 1,465.45($250)
V = (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
75
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 1,090
S = $17.375 F = 1,090($250) = $272,500
V = $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) = $242,000ACTUAL
V = $2,810,750. An increase of about 8%
11- = 272,500
2,606,2501.1- =n
76
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
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
77
TIME CASH FUTURES
MAR.31 V = $3,862,713 SEP SP500I FUTURES
F = 1,052.60($250) = $263,300
SHORT 16 SEP SP500I Fs.
JUL.27 V = $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
78
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 RATIO
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
β(portfolio) = .122(.95) + .187(1.1) + .203(.85)
+ .048(1.15) + .059(1.15) + .076(1.0)
+ .263(.85) + .042(.75)
= .95
79
MARKET TIMING HEDGE RATIO
When we believe that the market trend is changing, we need to 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 position by using the INDEX FUTURES without changing the portfolio’s composition.
80
The Minimum Variance Hedge Ratio in our case is:
h* = -(S/F)
Assume that the current position is a portfolio with current spot market value of VS and n stock index futures.
81
Next, we prove:
The BETA of the spot position may be altered from its current value, , to a Target Beta = T, buying or selling the following the number of futures:
.F
Sβ][βn T
82
).E(rS
Fn)E(r)E(r
.F
Δ(VF)r and ;
S
Δ(VP)r ;
S
(VP)r
DEFINE
.F
Δ(VF)
S
Fn
S
Δ(VS)
S
Δ(VF)n
S
Δ(VS)
S
Δ(VP)
(VF)n Δ(VS)Δ(VP)
nVFVS VP
FSP
FSP
83
F
S]β[βn
r)E(rS
Fn]r)β[E(rr
]r)[E(rβr
:Substitue ].r)[E(rβr)E(r
.r)E(r)E(r ; ]r)β[E(rr)E(r
:can write weCAPM, theFollowing
).E(rS
Fn)E(r)E(r :Again
T
fMfMf
fMTf
fMTfP
fMFfMfS
FSP
84
MARKET TIMING HEDGE RATIO
The rule: In order to change the BETA of the spot position from to T, the stock index futures may be used as follows:
.ββ if contracts F
Sβ][βnLONG
.ββ if contracts F
S]β[βn SHORT
TT
TT
85
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
86
FOREIGN CURRENCY: THE SPOT MARKET
EXCHANGE RATES:
THE VALUE (PRICE) OF ONE CURRENCY IN TERMS OF ANOTHER CURRENCY IS THE EXCHANGE RATE
BETWEEN THE TWO CURRENCIES.
THERE ARE TWO QUOTE FORMATS FOR QUOTATIONS:
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.
87 NZD2.get USD1sell SD,NZD2.000/U BIDS(NZD/USD)
cents. 48get NZD1 sell ZD, USD.480/N BIDS(USD/NZD)
NZD2.083.pay buy USD1 SD,NZD2.083/U ASKS(NZD/USD)
cents. 50pay NZD1buy USD.5NZD, ASKS(USD/NZD)
ASKS(FC/EUR)
1 BIDS(EUR/FC)
BIDS(FC/USD)
1 ASKS(EUR/FC)
:QUOTESASK AND BID HAVE WEWHEN
.5945 S(GBP/USD)S(GBP/USD)
1 =
.5945
1 = 1.6821 = S(USD/GBP)
S(FC/EUR)
1 = S(EUR/FC)
88
CURRENCY CROSS RATES
LET FC1, FC2 AND FC3 DENOTE 3 DIFFERENT CURRENCIES. THEN, IN THE ABSENCE OF
ARBITRAGE OPPORTUNITIES, THE FOLLOWING EQUALITY MUST HOLD:
S(FC3/FC1)
S(FC3/FC2) =
S(FC2/FC3)
S(FC1/FC3) = S(FC1/FC2)
89
CURRENCY CROSS RATES
EXAMPLE: FC1 = USD; FC2 = MXP;FC3 = GBP.
USD MXP GBP
USA 1.0000 0.0997 1.6603
MEXICO 10.0301 1.00016.6530
UK 0.6023 0.06005 1.000
90
CURRENCY CROSS RATESEXAMPLE
0.0997. 16.653
1.6603
S(MXP/GBP)
S(USD/GBP)
0.0997. 0.6023
0.06005
S(GBP/USD)
S(GBP/MXP)
16.653. S(MXP/GBP)1.6603 S(USD/GBP)
0.0997; S(USD/MXP)
.S(MXP/GBP)
S(USD/GBP) =
S(GBP/USD)
S(GBP/MXP) = S(USD/MXP)
GBP. FC3 MXP; FC2 USD; FC1Let
91
AN EXAMPLE OF CROSS SPOT RATES ARBITRAGE
COUNTRY USD GBP CHF
SWITZERLAND 1.7920 2.8200 1.0000
U.K 0.6394 1.0000 0.3546
U.S.A 1.0000 1.5640 0.5580
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
92
THE CASH ARBITRAGE ACTIVITIES:
USD1,000,000 USD1,006,134.26
0.6394 0.5580
GBP639,400 CHF1,803,108
2.8200
93
FOREIGN CURRENCY CONTRACT SPECIFICATIONS
CURRENCY SIZE MINIMUM MINIMUM FUTURES
CHANGE CHANGE
JAPAN YEN 12.5M .000001 $12.50
CANADIAN DOLLAR 100,000 .0001 $10.00
BRITISH POUND 62,500 .0002 $12.50
SWISS FRANC 125,000 .0001 $12.50
AUSTRALIAN DOLLAR 100,000 .0001 $10.00
MEXIAN PESO 500,000 .000025 $12.50
BRAZILIAN REAL 100,000 .0001 $10.00
EURO FX 125,000 .0001 $12.50
* THERE ARE NO 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
94
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).
95
INTERCURRENCY FUTURES SPREAD
A FUTURES CROSS-CURRENCY SPREAD IS THE PURCHASE OF ONE CURRENCY FUTURES AND THE SIMULTANEOUS SALE OF ANOTHER CURRENCY FUTURES; BOTH
FUTURES ARE FOR THE SAME DELIVERY MONTH.
A POSITION TRADER OBSERVES THE FOLLOWING RATES AND CROSS RATES:
MARCH 1: USD1.7225/GBP USD.6369/CHF GBP.3698/CHF
JUNE Fs USD1.7076/GBP USD.6448/CHF G BP.3776/CHF
(Currently: 1GBP = 2.7042CHF. JUN FUTURES: 1BBP = 2.6483CHF)
SPECULATOR: “THE BRITISH POUND WILL DEPRECIATE RELATIVE TO THE SWISS FRANK BY LESS THAN WHAT IS EXPECTED ACCORDING TO THE JUNEFUTURES CROSS RATE. IN FACT, I BELIEVE THAT THE BRITISH POUND WILL APPRECIATE AGAINST THE SWISS FRANC BETWEEN NOW AND THE END OF MAY TO AROUND GBP.3600/CHF OR, GBP2,7778/CHF.”
IN OTHER WORDS:THE SPREAD USD1.7076/GBP - USD.6448/CHF = USD1.0628 WILL INCREASE.
BUY THIS SPREAD!
LONG THE GBP JUNE FUTURES AND SIMULTANEOUSLY, SHORT THE CHF JUNE FUTURES
96
TIME CASH FUTURES
MAR 1 DO NOTHING SHORT 1 JUNE CHF FUTURES FOR $.6448/CHF (Fs = 125,000CHF)
LONG 2 JUNE BP FUTURESFOR $1.7076/BP BP (Fs = 62,5000BP)
SPREAD BOUGHT = USD1.7076 - USD.6448 = USD1.0628
MAY 20 DO NOTHING CLOSE YOUR SPREAD:
LONG 1 JUNE CHF FUTURES FOR USD.630/CHF
SHORT 2 JUNE BP FUTURES FOR usd1.730/GBP
SPREAD SOLD = $1.730 - $.6300 = $1.1000
PROFIT = (USD1.1000 - $USD1.0628)(125,000) = USD4,650/CONTRACT
NOTICE THAT THE GBP HAS APPRECIATED
FROM GBP.3698/CHF ( 1GBP = 2.7042CHF) IN MARCH
TO
USD.6300/CHF/USD1.730/GBP = GBP.3642/CHF (1GBP = 2.7457CHF) IN JUNE
97
BORROWING U.S. DOLLARS SYNTHETICALLY ABROAD OR
HOW TO BEAT THE DOMESTIC BORROWING RATE
A CASE OF QUASI-ARBITRAGE,
A US 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)
98
IN THE SPIRIT OF
REVERSE CASH-AND-CARRYTIME 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.
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
99
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
100
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
101
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 298 DEC EUR FUTURES
DELIVER THE CARS FOR F(DEC) = USD1.26000/CHF
PAYMENT: EUR40,650,000. PROFIT1.29 – 1.17)(125,000)(298)
=USD4,470,000
ACTUAL PAYMENT IN EUR:
40,650,000 + 4,470,000(.813) = EUR44,284,110.
298 = 17)125,000(1.
43,500,000 =n
102
EXAMPLE 4: A LONG HEDGE: PROTECT AGAINST DEPRECIATING DOLLAR
AN AMERICAN FIRM AGREES TO BUY 100,000 MOTORCYCLES FROM A JAPANESE FIRM FOR JY202,350 .
CURRENT PRICE DATA: ASK BID
SPOT: USD.007020/JY USD.007027/JY
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.
103
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.
104
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 DEUTCHE MARK THERE WILL BE LESS DOLLARS TO DEPOSIT.
TIME CASH FUTURES
FEB. 21 S(EUR/USD) = .8442 F(JUN) = EUR.8498/USD
CURRENT SPOT VALUE F = 125,000(.8498) = USD106,225
= USD2,124,500 n = 2,124,500/106,225 = 20.
DO NOTHING SHORT 20 JUN EUR FUTURES
MAR 31 S(EUR/USD) = .8400 LONG 20 JUN EUR FUTURES
DEPOSIT 2,516,583.75(.84) F(JUN) = $.5449/DM
= USD2,113,930.35 PROFIT: (.8498 - .8449)125,000(20) = USD12,250.
TOTAL AMOUNT TO DEPOSIT USD2,126,180.35
