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5 Profits to a Naked (Unhedged) Long Spot Position

# 5 Profits to a Naked (Unhedged) Long Spot Position

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46

TABLE 2.3

FORWARD CONTRACTS AND FUTURES CONTRACTS

Profits to a Naked (Unhedged)
Long Spot Position with the
Current Spot Price=\$5.9875/bu

Wheat Spot Price
@ Expiration

Profits from Wheat
Position Unhedged

5.8
5.85
5.9
5.95
6
6.05
6.1
6.15
6.2
6.25
6.3
6.35
6.4
6.45
6.5
6.55
6.6
6.65
6.7
6.75

If wheat prices rise above the current level, the wheat farmer proﬁts, because
he can later sell his wheat for more than the going price today. We are ignoring
carrying charges at the moment by considering gross proﬁts. Later, we will
consider net proﬁts, which is deﬁned as gross proﬁts net (minus) carrying
charges.
From this exercise, we also see that the proﬁts for an unhedged position
are just the proﬁts from holding the commodity long. This amounts to (the
number of units held long)*(the price change over the period). Or in our
notation, (500,000)*(+PT(␻)–Pt ).

HEDGING WITH FORWARD CONTRACTS

47

2.6 PROFITS TO A FULLY HEDGED CURRENT LONG SPOT
POSITION
We want to recalculate the proﬁts from ownership of the spot commodity
combined with the forward sale of the farmer’s (grain merchant’s) wheat. Then,
we will compare them to those from the unhedged position.
First, we need to discuss what goes into this proﬁt calculation. As in the
case of the unhedged position, the farmer calculates the proﬁt from his
forward sale relative to the current spot wheat price of \$5.9875 per bu. This
time he is not selling his wheat at the unknown going spot wheat price, PT (␻).
Instead, he is selling it forward for the ﬁxed price Ft,T .
Per bushel proﬁts on this strategy are+Ft,T–Pt where Pt is the current (today)
spot wheat price. Note the use of Pt because it represents the per unit
price of the spot commodity today. For our case, Ft,T=\$7.315/bu. and
Pt=\$5.9875/bu. The per bushel proﬁt on the hedged position should be
\$7.315–\$5.9875=\$1.3275. Multiplying by the scale factor (number of bushels)
we obtain, 500,000*\$1.3275=\$663,750, just as before.
In this case, the farmer (grain merchant) is guaranteed a gross proﬁt of
\$663,750, no matter what happens in the spot market for wheat over the
4-month period. The forward sale is a far safer way of selling his wheat. In
fact, it is riskless. That is its major beneﬁt.
In order to obtain this beneﬁt of risk reduction, the farmer has to give up
some upside potential. That is, the forward sale gives him downside protection
in return for giving up upside potential. Since the farmer probably doesn’t
want to gamble with his wheat crop, the forward sale is a useful alternative.
We have already seen the payoff graphs for a long and a short forward
position, and for a naked (unhedged) long position. What does the picture
look like if we combine a long spot position with a short forward position?
This is a fairly easy exercise, but it is worth delving into it more thoroughly.
A fully hedged position will have a constant proﬁt equal to (the number
of units)*(+Ft,T –Pt ), which in our wheat example is 500,000*(\$7.315–
\$5.9875) or \$663,750. Let’s take this apart a bit.
There are two components to a fully hedged position. First, there is the
underlying commodity which we wish to hedge, the 1000 shares of IBM for
the stock example or the 500,000 bushels of wheat.
Next comes the position one has in the underlying commodity. In both
examples, the investor was long 1000 shares of IBM and the wheat farmer
(grain merchant) was long 500,000 bushels of wheat. The proﬁts to a naked

48

FORWARD CONTRACTS AND FUTURES CONTRACTS

long commodity position are equal to (the number of units)*(+PT(␻)–Pt ). The
reason they are long is that both individuals anticipate selling their underlying
commodity in the future. Even though the farmer may not have his crop
available, he is still implicitly long.
The primary worry is that IBM’s stock price will fall and the spot wheat
price will fall by the time of the sale. To protect themselves, market participants
take the opposite position in a derivative security; in this case, a short position
in a forward contract. That is, they sell forward at time t their respective spot
commodity, at the going forward price Ft,T .
This makes economic sense. If you are worried about price volatility of a
commodity, sell it in a market for a price that has no volatility, a ﬁxed price
like the forward price, Ft,T .
The payoff to a short forward position depends on the spot price at
expiration of the forward contract and is equal to (the number of units)*
(+Ft,T–PT(␻)). To get the profit from a short forward position, we would have
to subtract the current cost of the forward contract. That cost is always zero,
as we will shortly see, when the forward contract is initiated. So there is nothing
to subtract.
Now we can combine both proﬁt ﬁgures to get the proﬁt ﬁgure for a fully
hedged long spot position,
(the number of units)*(+PT(␻)–Pt)+(number of units)*(+Ft,T–PT(␻))
=(the number of units)*(+PT(␻)–Pt+Ft,T–PT(␻))
=(the number of units)*(–Pt+Ft,T)
=(the number of units)*(+Ft,T–Pt)
Miraculously, the unknown spot price at expiration, PT(␻), cancels out and
we get what we derived before which is a constant proﬁt that has nothing to
do with what happens in the spot market between now and expiration of the
forward contract. This means that we have successfully hedged our long spot
position, if success means we have removed all of its risk.
n

CONCEPT CHECK 8

a. Fill in the third column of Table 2.4 below. Proﬁt means: proﬁt for the
sale of all 500,000 bu. of wheat subsequent to a short forward position
taken today.

HEDGING WITH FORWARD CONTRACTS

TABLE 2.4

49

Profits to a Long Spot Position Sold Forward at Ft,T

Ft,T

Wheat Spot Price
@ Expiration

\$7.315/bu

5.8

Profit to a Spot Position in
500,000 bu. of Wheat Sold Forward
by a Short Forward Contract

5.85
5.9
5.95
6
6.05
6.1
6.15
6.2
6.25
6.3
6.35
6.4
6.45
6.5
6.55
6.6
6.65
6.7
6.75

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CONCEPT CHECK 9

a. Draw a Microsoft Excel chart of the total proﬁts to the fully hedged position
in Concept Check 8.
The horizontal axis is the spot wheat price at expiration. The vertical
axis is the proﬁt for 5,000 bushels of the fully hedged wheat position.

50

FORWARD CONTRACTS AND FUTURES CONTRACTS

2.7 ADDING PROFIT TABLES TO DETERMINE PROFITS FROM
A FULLY HEDGED POSITION
There is another very useful method for determining proﬁts on relatively
complex positions like hedged positions. The key is to recognize that such
positions involve combinations of positions. For our wheat farmer (grain
merchant) the components are,
1. A long position in 500,000 bu. of spot wheat. The current spot price is
Pt=\$5.9875/bu.
2. A short forward position to sell 500,000 bushels of wheat in 4 months’
time at the current forward price of \$7.315/bu.
You have already derived the proﬁt Table 2.3 for 1. in Concept Check 6,
but we will reproduce it here in Table 2.5. The formula from which the
numbers in Table 2.5 are derived is,
(the number of units)*(+PT(␻)–Pt)=500,000*(+PT(␻)–Pt).
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CONCEPT CHECK 10

a. Fill in the third column of Table 2.6. Proﬁt means: proﬁt to a naked short
forward position in 500,000 bushels of wheat. Note that this is a naked
short forward position.

Table 2.6 is derived from the formula,
(the number of units)*(+Ft,T–PT(␻))
=500,000*(+Ft,T–PT(␻))
=500,000*(+\$7.315–PT(␻)),
since the forward price today for delivery of spot wheat 4 months from today
is Ft,T =\$7.315/bu.
Now, we can combine the two tables, Table 2.5 and Table 2.6, and see
what hedging does for us in this simple scenario. The combined Table 2.7
represents the proﬁt from the fully hedged spot position. Table 2.7 is just the
sum (combination of Table 2.5 and Table 2.6).

HEDGING WITH FORWARD CONTRACTS

TABLE 2.5

Profits from a Fully Unhedged
Spot Position in Wheat

Wheat Spot Price
@ Expiration

Profits from Wheat
Position unhedged

5.8

–93750

5.85

–68750

5.9

–43750

5.95

–18750

6

51

6250

6.05

31250

6.1

56250

6.15

81250

6.2

106250

6.25

131250

6.3

156250

6.35

181250

6.4

206250

6.45

231250

6.5

256250

6.55

281250

6.6

306250

6.65

331250

6.7

356250

6.75

381250

Table 2.7 is exactly what we obtained before, but now we can get some
further insight into what hedging does. Let’s ﬁrst summarize our price data
in Table 2.8.

52

FORWARD CONTRACTS AND FUTURES CONTRACTS

TABLE 2.6

Profits from a Short Forward Position in Wheat

Ft,T

Wheat Spot Price
@ Expiration

\$7.315 /bu

5.8

Profit to a Naked Short Forward Position
in 500,000 bu. of wheat

5.85
5.9
5.95
6
6.05
6.1
6.15
6.2
6.25
6.3
6.35
6.4
6.45
6.5
6.55
6.6
6.65
6.7
6.75

A little scenario analysis will serve to illustrate what is going on.
Scenario 1

First, suppose that the spot wheat price drops to PT(␻)=\$5.85/bu at the end
of 4 months. Compared to the current wheat spot price level, that is a loss
of 500,000*(+PT(␻)–Pt )=500,000*(+\$5.85–\$5.9875)=–\$68,750.
Fortunately, the farmer was hedged in the forward market where he made
a proﬁt of,
500,000*(+Ft,T–PT(␻))=500,000*(+\$7.315–\$5.85)
=+\$732,500.
Overall, his position locked in a proﬁt of \$732,500–\$68,750=\$663,750.

HEDGING WITH FORWARD CONTRACTS

TABLE 2.7

53

Profit from the Fully Hedged Spot Position

Wheat Spot Price
@ Expiration

Profits from a Naked
(Unhedged) Long Spot
Wheat Position

Profit To a Naked
Profits to the
Short Forward Position Combined (Fully
in 500,000 bu. of wheat Hedged) Position:
Long Spot,
Short Forward

5.8

–93750

757500

663750

5.85

–68750

732500

663750

5.9

–43750

707500

663750

5.95

–18750

682500

663750

6250

657500

663750

6.05

31250

632500

663750

6.1

56250

607500

663750

6.15

81250

582500

663750

6.2

106250

557500

663750

6.25

131250

532500

663750

6.3

156250

507500

663750

6.35

181250

482500

663750

6.4

206250

457500

663750

6.45

231250

432500

663750

6.5

256250

407500

663750

6.55

281250

382500

663750

6.6

306250

357500

663750

6.65

331250

332500

663750

6.7

356250

307500

663750

6.75

381250

282500

663750

6

TABLE 2.8

Price Data Summary

Current Spot Price

Pt

\$5.9875 bu

Current Forward Price

Ft,T

\$7.315 /bu

Ultimate Spot Price

PT(␻)

?

54

FORWARD CONTRACTS AND FUTURES CONTRACTS

Scenario 2

Next, suppose that the spot price of wheat goes up in the next 4 months to
\$7.50/bu. Then the proﬁt on the naked spot position would be,
500,000*(+\$7.50–\$5.9875)
=500,000*\$1.5125
=\$756,250.
However, we have to remember that we are hedged, so that we have to settle
up on our obligatory short forward position on which there was a negative
proﬁt (loss) of 92,500=500,000*(+\$7.315–\$7.50).
The reason for the loss is that we pre-committed to sell the wheat forward
at \$7.315/bu and the spot market turned out to be higher by \$7.50–\$7.315
per bu. So we lost \$.185 per bushel which is a grand total loss of
500,000*\$.185=\$92,500.
Overall, our total proﬁt is \$756,250–\$92,500=\$663,750, which is the same
as what it was under the ﬁrst scenario.
Scenario 2 seems to imply that we could be ‘worse off’ by hedging. If we
knew in advance that the spot price of wheat would be \$7.50/bu with complete
certainty, then clearly we would be better off in ﬁnal wealth by not hedging.
However, we do not know. So, we have to be prudent and protect our
spot wheat position against downside risk. In return for the protection afforded
by forward contracting, we have to give up some upside proﬁt potential in
the event that the spot price of wheat goes up above the forward price.
These two scenarios for the ultimate spot price, indicate that the short hedge
locks in the proﬁt ﬁgure equal to (the number of units)*(+Ft,T –Pt).
2.8 COMBINING CHARTS TO SEE PROFITS FROM THE
HEDGED POSITION
Rather than just adding up the proﬁts to the two components of the hedged
position entry by entry, long spot wheat and short a forward contract, we can
add up the charts for each of these components. This reduces the amount of
work we have to do and will prove to be a useful technique when we get to
options.
Figure 2.1 indicates the results of this process for our example.

HEDGING WITH FORWARD CONTRACTS

55

FIGURE 2.1 Graphical Method to Get Hedged Position Profits
900,000
800,000
700,000
600,000
Profit to Naked Spot

500,000
400,000

Profits from Hedged
Position

300,000
200,000

Profit to a Naked Short
Forward Position

100,000
0
–100,000
–200,000

5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7
Spot Wheat Price in 4 Months

The orange line plus the blue line equals the purple line. In a sense, the
orange line cancels out the blue line, the downward slope of the orange line
(representing losses on the short forward position) are being exactly offset by
the upward slope of the blue line representing proﬁts on the long spot
position.
That is, losses (gains) on the long spot position are being exactly offset by
gains (losses) on the short forward position. The net effect is the lock-in of
proﬁts on the hedge at the ﬁxed amount of \$663,750. This amount is called
the time t basis and will be discussed in detail in Chapter 6.

n

n

n

56

FORWARD CONTRACTS AND FUTURES CONTRACTS

n

KEY CONCEPTS

1.
2.
3.
4.
5.
6.
7.
8.
9.

Motivation for Hedging.
Proﬁts vs. Payoffs.
Payoff to a Long Forward Position.
Payoff to a Short Forward Position.
Hedging with Forward Contracts.
Proﬁts to a Naked (Unhedged) Long Spot Position.
Proﬁts to a Fully Hedged Current Long Spot Position.
Adding Tables to Determine Proﬁts from a Fully Hedged Position.
Combining Charts to Determine Proﬁts from the Hedged Position.

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END OF CHAPTER EXERCISES FOR CHAPTER 2

1. You hold 1000 shares of IBM stock currently trading for Pt=\$185. You
intend to hold the position for the next 3 months.
a. Ignoring any dividends or payouts over the next 3 months, map out
the proﬁt per share diagram for your position.
Note that PT(␻) is on the horizontal axis and proﬁts per share is on the
vertical axis.
2. Referring back to Kansas City red winter wheat on the quote date, suppose
that \$7.2650/bu. was the current spot wheat price.
a. Fill in the second column of Table 2.9. The position is fully unhedged
and anticipates selling 500,000 bu. of wheat in 4 months in the spot
market.
b. Draw a Microsoft excel chart of the proﬁt per bushel to the fully
unhedged position. The horizontal axis is the spot wheat price in 4
months. The vertical axis is the proﬁt per bushel to the unhedged wheat
position.