Futures and Interest Rate

Lecture 1(07/05 am) Interest Rate
Review and Introduction to Futures
• Types of rates and measuring interest rates
• Zero rates and zero coupon bond prices
• Forward rates
• Futures exchanges & types of contracts
• Specifications of a futures contract
• Margining of futures contracts
• Examples of futures contracts
• Convergence of futures price to spot price
• Forward vs. futures
Futures and Options FINC-GB.3335 Summer 2014 1-1
Lecture 1 (07/05 pm): Futures Hedging
Strategies


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•General hedging principles
•Basis risk
•Cross hedging
•Stock index futures and use in hedging
•Tracking error
•Hedging mishaps
Types of Rates
• Treasury rates: yield on Treasury securities
(no counterparty credit risk)
• Libor rates: London Interbank Offered Rate:
this has been the basis for Eurodollar futures
and interest rate swaps. Recent problems
have appeared since the banks behind this
market have experienced credit issues.
• Fed fund rate, Fed fund futures and Overnight
Indexed Swaps: Recently, this has become
the new benchmark for swaps with zero
counterparty risk.
Futures and Options FINC-GB.3335 Summer 2014 1-3
Measuring Interest Rates
• Ra is an annual rate compounded annually
• Rq is an annual rate compounded quarterly
• Z(t) is the value today of $1 received at t
• t is expressed in years


( )
( )
) (
1
4
1 1
) (
4
1 1
4
4
t z
Rq
Ra
t z
Rq
Ra
t
t
t
t
=
|
.
|

\
|
+ = +
=
|
.
|

\
|
+ = +
÷
÷
Futures and Options FINC-GB.3335 Summer 2014 1-4
<= Future valuation
<= Present valuation
Measuring Interest Rates:
Continuously Compounded Rate
) ( 1
) (
1
1
t z e
n
r
t z
e
n
r
t r
nt
c
t r
nt
c
c
c
= ~
|
.
|

\
|
+
= ~
|
.
|

\
|
+
÷
÷
Futures and Options FINC-GB.3335 Summer 2014 1-5
• is an annual rate compounded continuously.
• As n becomes large:

c
r
Measuring Interest Rates:
Continuously Compounded Rate
( ) ( )
( )
( )
t
t z
r
t z t r
t z e t z e
c
c
t r t r
c c
) ( ln
) ( ln
) ( ln ln ) (
= ÷ ÷
= ÷ ÷
= ÷ =
÷ ÷
Futures and Options FINC-GB.3335 Summer 2014 1-6
Annual rate continuously compounded that you will receive today, for the period
of period „t‟ years.
Measuring Interest Rates:
Continuously Compounded Rate
( )
( )
( )
t
t z
r
t z t r
t z
e
t z
e
ly equivalent
c
c
t r t r
c c
) ( ln
) ( ln
) (
1
ln ln
) (
1
÷
= ÷
÷ = ÷
|
|
.
|


\
|
= ÷ =
Futures and Options FINC-GB.3335 Summer 2014 1-7
Measuring Interest Rates: Example
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Annual rate 5.00%
PV of $1 in 1 year 0.95238
Semiannual rate 4.9390% 2
Quarterly rate 4.9089% 4
Monthly rate 4.8889% 12
Daily rate 4.8793% 365
Continuous rate 4.8790%
a
R
a
R
Z
+
=
1
1
) 1 (
n
)) 1 ( ln( Z ÷
( ) 1 ) 1 (
) / 1 (
÷
÷ n
Z n
( ) 1 ) 1 (
) / 1 (
÷
÷ n
Z n
( ) 1 ) 1 (
) / 1 (
÷
÷ n
Z n
( ) 1 ) 1 (
) / 1 (
÷
÷ n
Z n
The rate changes while the zero coupon bond price
stays the same.
Measuring Interest Rates
• What are unique are the Z(t)‟s, the zero
coupon bond prices for different maturities t.
• Rates will be a function of the compounding
assumptions(annual, semiannual, continuous).
• Rates will also be a function of market
conventions related to how they are paid
(Treasury daycount, Annual/360 daycount,
30/360 daycount, Annual/365 daycount).
Futures and Options FINC-GB.3335 Summer 2014 1-9
Measuring Interest Rates: Forward
Rates


Futures and Options FINC-GB.3335 Summer 2014 1-10

• The forward rates are such that if you
compound your money first at the t
i
zero rate
from time 0 to time t
i
and then at the forward
rate from t
i
to t
i+1
, you will get the same as if
you compounded your money at the t
i+1
zero
rate from time 0 to t
i+1
.

Measuring Interest Rates: Forward
Rates


Futures and Options FINC-GB.3335 Summer 2014 1-11

Time Zero rate PVF Forward rate
0.5 0.0404 0.9800 0.0404 0.0404
1 0.0408 0.9600 0.0412 0.0412
1.5 0.0466 0.9325 0.0582 0.0582
2 0.0502 0.9045 0.061 0.061
1
1
) (
) (
ln
÷
÷
÷
(
¸
(

¸

÷
i i
i
i
t t
t PVF
t PVF
1
1 1
) ( * ) ( *
÷
÷ ÷
÷
÷
i i
i i i i
t t
t rate zero t t rate zero t
Measuring Interest Rates: Example
• The annual 1-year, 2-year, 3-year and 3.5-
year zero rates are 4%, 4.3% 4.7% and 5%
on a semiannual compounding basis.
a) What are the corresponding rates assuming
continuous compounding?
b) What is the forward rate for the 6 month
period starting in 3 years (assume continuous
compounding)?


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Measuring Interest Rates: Example
Futures and Options FINC-GB.3335 Summer 2014 1-13
t rate rate/2 FVF=1/Z(t) LN(FVF)/t
1 4.00% 2.00% 1.0404 3.9605%
2 4.30% 2.15% 1.0888135 4.2544%
3 4.70% 2.35% 1.1495479 4.6456%
3.5 5.00% 2.50% 1.1886858 4.9385%
a) see column e
b) exp(r(3yr)*3+forward*.5)=exp(r(3.5yr)*3.5)
r(3yr)*3+forward*.5=r(3.5yr)*3.5
forward=(r(3.5yr)*3.5-r(3yr)*3)/.5
forward 6.696% 6.696%
Measuring Interest Rates: Bootstrap
• Calculate zero rates for maturities of 6
months, 12 months, 18 months and 24
months. (Assume continuously
compounded rates).
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Bond Principal
($)
Time to Maturity
(years)
Annual Coupon
Paid Semiannually
($)
Bond Price
($)
100 .5 0 98
100 1.0 0 96
100 1.5 4 99
100 2.0 4 98
Measuring Interest Rates: Bootstrap
Futures and Options FINC-GB.3335 Summer 2014 1-15
Bond time coupon anPrice
100 0.5 0 98
100 1 0 96
100 1.5 4 99
100 2 4 98
.5 y rate 4.0405% -LN(D2/A2)/B2
1 y rate 4.0822% -LN(D3/A3)/B3
zero rate for 1.5 year will be obtained by bootstrapping. Taking the 1.5 y
bond, you compute the PV of all the flows you know how to compute.
These are the coupons at .5 and 1.
PV cpn .5 1.96 (C4/2)*EXP(-B6*B2)
PV cpn 1 1.92 (C4/2)*EXP(-B7*B3)
The remainder of the payment to be received at time 1.5 is 4/2 plus 100
so 99-PV cpn .5 - PV cpn 1=PVF(1.5)*(102)
PVF(1.5) 0.93255
1.5 y rate 4.6556%
same approach can be followed to get the 2 y rate
PV cpn .5 1.96
PV cpn 1 1.92
PV cpn 1.5 1.865098
PVF(2) 0.90446
2 y rate 5.0209%
Measuring Interest Rates: Par Yield
• In the last example the value of the two year
bond was given by:


• The par yield is equal to the coupon, c, of a
bond with a par of 1 which is worth par:


Futures and Options FINC-GB.3335 Summer 2014 1-16
2 * 0502 . 5 . 1 * 0466 . 1 * 0408 . 5 . * 0404 .
2
4
100
2
4
2
4
2
4
98
÷ ÷ ÷ ÷
|
.
|

\
|
+ + + + = e e e e
2 * 0502 . 5 . 1 * 0466 . 1 * 0408 . 5 . * 0404 .
2
1
2 2 2
1
÷ ÷ ÷ ÷
|
.
|

\
|
+ + + + = e
c
e
c
e
c
e
c
Measuring Interest Rates: Par Yield







Futures and Options FINC-GB.3335 Summer 2014 1-17
( ) | |
( )
| |
| |
0506 . 0253 .
9045 . 9325 . 9600 . 9800 .
) 9045 . 1 ( 1
2
1 1
2
2
1 1
2
1
2 2 2
1
2 * 0502 . 5 . 1 * 0466 . 1 * 0408 . 5 . * 0404 .
2 * 0502 .
2 * 0502 . 5 . 1 * 0466 . 1 * 0408 . 5 . * 0404 . 2 * 0502 .
2 * 0502 . 5 . 1 * 0466 . 1 * 0408 . 5 . * 0404 .
= ¬ =
+ + +
÷
=
+ + +
÷
=
+ + + = ÷
|
.
|

\
|
+ + + + =
÷ ÷ ÷ ÷
÷
÷ ÷ ÷ ÷ ÷
÷ ÷ ÷ ÷
c
c
e e e e
e c
e e e e
c
e
e
c
e
c
e
c
e
c
Measuring Interest Rates: Par Yield
• It is also true that :





• By construction, the annual par yield is
semiannually compounded in this example.


Futures and Options FINC-GB.3335 Summer 2014 1-18
4 3
2
2
0506 .
1
1
*
2
0506 .
1
2
0506 .
1
1
*
2
0506 .
2
0506 .
1
1
*
2
0506 .
2
0506 .
1
1
*
2
0506 .
1
|
.
|

\
|
+
|
.
|

\
|
+ +
|
.
|

\
|
+
+
|
.
|

\
|
+
+
|
.
|

\
|
+
=
Major Futures Exchanges (2004)
Futures and Options FINC-GB.3335 Summer 2014 1-19
Source: CME, An Introduction to Futures and Options
Major Exchanges (March 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-20
Source: CME, An Introduction to Futures and Options
Source: http://www.futuresindustry.org/downloads/FI-2012_Volume_Survey.pdf
Major Exchanges (March 2014)
Futures and Options FINC-GB.3335 Summer 2014 1-21
Source: http://www.futuresindustry.org/downloads/FIA_Annual_Volume_Survey_2013.pdf
* Open interest for these exchanges does not include options traded in the U.S. and cleared by OCC # Includes NYSE Euronext
Futures Exchanges
• Futures US/America:
– CME Group (CME, CBOT, NYMEX) www.cmegroup.com
– ICE www.theice.com
– BM&FBOVESPA www.bmf.com.br
– Mexican Derivatives Exchange www.mexder.com.mx
– Rosario Futures Exchange (Argentina) www.rofex.com.ar
• Futures Europe:
– EUREX www.eurexchange.com
– NYSE EURONEXT www.euronext.com
– London Metal Exchange www.lme.com
– Moscow Exchange (includes MICEX) www.moex.com
• Futures Asia:
– Dalian Commodity Exchange (DCE) www.dce.com.cn
– Shangai Futures Exchange (SHFE) www.shfe.com.cn
– Zengzhou Commodity Exchange (ZCE) www.czce.com.cn
– National Stock Exchange of India www.nse-india.com
– SGX (SIMEX) www.sgx.com
– TSE www.tse.or. jp
Futures and Options FINC-GB.3335 Summer 2014 1-22
Role of the Exchange
– The exchanges provide an environment for
fair price discovery and for liquidity.
– The exchanges create a level playing field
and provide pricing transparency.
– To create an orderly market, the exchange
may institute price limits and position limits.
– To insure solvability, the exchange will control
counterparty risk by imposing performance
bonds/margins (initial and maintenance)
Futures and Options FINC-GB.3335 Summer 2014 1-23
Type of Contracts
Financials – 3
rd

– Interest rate
– Equity indices
– Currencies

Agricultural – Started 1st

Metals – 2nd

Energy -4
th
last


Futures and Options FINC-GB.3335 Summer 2014 1-24
Market Statistics - BIS (June 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-25
Source: http://www.bis.org/statistics/r_qa1306_hanx23a.pdf
Market Trend - BIS (June 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-26
http://www.bis.org/statistics/dt1920a.csv
0
5000
10000
15000
20000
25000
30000
35000
Financial Futures Outstanding (in $ Billion)
North America Europe Asia and Pacific
Other Markets Total
Market Statistics - FIA (March 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-27
Source: http://www.futuresindustry.org/downloads/FI-2012_Volume_Survey.pdf
These two
are mostly
options
„Interest rates‟ account for largest volume for futures
Major U.S. Futures Contracts (2004)
Futures and Options FINC-GB.3335 Summer 2014 1-28
Source: CME, An Introduction to Futures and Options
Major Futures Contracts (2013)
Futures and Options FINC-GB.3335 Summer 2014 1-29
Source: http://www.futuresindustry.org/downloads/FI-2012_Volume_Survey.pdf
Type of Contracts
– Interest rate
• Eurodollar
• Euribor
• T-Note
• Bund futures
• Fed funds
• Eonia
– Equity indices
• S&P 500
• Nikkei
– Currencies
• Euro


Futures and Options FINC-GB.3335 Summer 2014 1-30
– Agricultural
• Corn
• Soybeans
– Metals
• Precious metals
• Industrial metals
– Energy
• Light Sweet Crude
Oil (WTI)
• Crude Oil (Brent)
• Natural Gas
• Heating Oil
CME Leading Products (Q2 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-31
Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf
CME Leading Products (Q2 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-32
Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf
E-mini: eg. Only 50 times the SnP 500, instead of 250 times
CME Leading Products (Q2 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-33
Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf
CME Leading Products (Q2 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-34
Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf
CME Products (Q2 2013)
Futures and Options FINC-GB.3335 Summer 2014 1-35
Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf
Specifications of a Futures Contract
– Legal and binding contract to buy or sell, at a
specified expiration date, a standardized
quantity of a product of a given quality at an
agreed upon price.
– The contract is marked to market at the end of
every trading day. The end of day gain (or
loss) is credited (or debited) from the account.
– Margins (initial and maintenance) are required.
– Maximum daily price changes may be imposed.

Futures and Options FINC-GB.3335 Summer 2014 1-36
Specifications of a Futures Contract(Cont.)
– At maturity, the contract can be cash settled,
closed by physical delivery or EFP.
– In case of physical delivery, the seller delivers
the commodity to the buyer.
– The seller has often different potential options
at delivery (quality, quantity, timing).
Futures and Options FINC-GB.3335 Summer 2014 1-37
Marked to Market: Long Futures Contract
Gain
Loss
F
t+1
0
F
t
*


Futures and Options FINC-GB.3335 Summer 2014 1-38
Suppose the contract settled at F
t
*
=100 at time t . Table shows the gain
(credit) or loss (debit) on the next day (t+1).
F
t+1
F
t+1
– F
t
*

90 -10 Debit
95 -5 Debit
100 0 -
105 5 Credit
110 10 Credit
Marked to Market: Short Futures Contract
Gain
Loss
F
t+1
0
F
t
*


Futures and Options FINC-GB.3335 Summer 2014 1-39
Suppose the contract settled at F
t
*
=100 at time t. Table shows the gain
(credit) or loss (debit) on the next day (t+1).

F
t+1
F
t+1
– F
t
*

90 10 Credit
95 5 Credit
100 0 -
105 -5

Debit
110 -10 Debit
Margin Calls: S&P 500 Example
– On 02/03 a hedger with a diversified stock portfolio of
$13.25 Million decides to sell 40 Mar 2012 futures
contracts on the S&P 500 index at the open:

• Contract size 250 times the S&P futures price (the index)
• Futures price at open at $1323 per contract
• Settlement price on 02/03 $1339.10 per contract
• Initial margin requirement $25,000 per contract
• Maintenance margin $20,000 per contract
Futures and Options FINC-GB.3335 Summer 2014 1-40
Margin Calls: Hypothetical Outcome
Futures and Options FINC-GB.3335 Summer 2014 1-41
Date Price change Gain/Loss Beg Balance Cum Gain Margin Call End Balance Available
2/03 am 1,323.00 $ 1,000,000 $ - $ - $ 1,000,000 $ - $
2/03 settl 1,339.10 $ 16.10 $ (161,000) $ 1,000,000 $ (161,000) $ - $ 839,000 $ - $
2/06 settl 1,350.00 $ 10.90 $ (109,000) $ 839,000 $ (270,000) $ 270,000 $ 1,000,000 $ - $
2/07 settl 1,360.00 $ 10.00 $ (100,000) $ 1,000,000 $ (370,000) $ - $ 900,000 $ - $
2/08 settl 1,320.00 $ (40.00) $ 400,000 $ 900,000 $ 30,000 $ - $ 1,300,000 $ 300,000 $
contract -40
multiplier 250
Init. Mg. 25,000.00 $
Maint Mg. 20,000.00 $
Example: Corn Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-42
Source: www.cmegroup.com
Delivery Option
If the value of #1 Yellow is
less 1.5c higher than #2.
Then seller should buy and
sell #1 corn.
Example: Corn Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-43
Source: www.cmegroup.com
It is 8
th
of a cent. 642.4 => $64.25
Example: Natural Gas (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-44
Source: www.cmegroup.com
Example: Natural Gas (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-45
Source: www.cmegroup.com
Example: Light Sweet Crude Oil (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-46
Source: www.cmegroup.com
Light Sweet is essentially WTI.
Example: Light Sweet Crude Oil (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-47
Source: www.cmegroup.com
Example: Brent Crude Futures (ICE)
Futures and Options FINC-GB.3335 Summer 2014 1-48
Source: www.theice.com
Example: Brent Crude Futures (ICE)
Futures and Options FINC-GB.3335 Summer 2014 1-49
Note that the contract is cash settled at maturity. Physical delivery is
ensured through EFP (exchange for physical). In an exchange for physical,
you can close a long futures position by selling the futures and buying the
physical in one transaction.
EFS is exchange for swap.
Source: www.theice.com
Example: Eurodollar Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-50
Source: www.cmegroup.com
We should know everything
in this slide.
Example: Eurodollar Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-51
Source: www.cmegroup.com
Example: S&P 500 Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-52
Example: S&P 500 Futures (CME)
Futures and Options FINC-GB.3335 Summer 2014 1-53
Source: www.cmegroup.com
Futures-Spot Convergence: Hypothetical
360 Day Futures

Futures and Options FINC-GB.3335 Summer 2014 1-54
Futures-Spot Convergence:
Hypothetical 360 Day Futures

Futures and Options FINC-GB.3335 Summer 2014 1-55
Futures-Spot Convergence:
Hypothetical 360 Day Futures
– If convergence does not happen, an arbitrageur can
benefit by selling the more expensive of the two and
buying the other as a hedge.
– Convergence may not exactly happen because of the
inability to measure the value of the underlying at the
expiry of the contract. See discussion on SOQ for SP
500 contract.
– Convergence may not exactly happen because
• of delivery costs for physical delivery.
• of a delivery option.
• of timing of delivery by the seller.


Futures and Options FINC-GB.3335 Summer 2014 1-56
Forward vs. Futures
– Forward contracts are OTC contracts. OTC contracts
are tailored to the customer‟s needs and are not
standardized. OTC contracts are exposed to the risk
of the counterparty and are governed by a contract
like ISDA (www.isda.org)
– Like a futures contract, a forward contract is a firm
agreement to buy or sell an underlying asset or
commodity at a future time at a price set at inception.
– Unlike futures, forward contracts are not cash settled
daily at the settlement price. In addition, there may
be poor price visibility.
– Forwards are not subject to margin calls but may be
subject to a CSA.
Futures and Options FINC-GB.3335 Summer 2014 1-57
Forward vs. Futures
Forward Futures
Contract between two parties Contract with an exchange
Non standard contract
(but ISDA, FXC)
www.newyorkfed.org/fxc/
Standard contract
Settled at end of contract Daily settled
Counterparty risk Exchange risk
CSA (Collateral Service Agreement) Margins
Self Regulation (ISDA,FXC) Regulated by Commodity
Futures Trading Commission,
National Futures Association and
Exchanges
Futures and Options FINC-GB.3335 Summer 2014 1-58
Regulations
• Commodity Exchange Act (title 7 USC)
• CFTC defines Designated Contract Markets (DCM) and Exempt Commercial
Markets (ECM). CME is an example of a DCM.
• Institutional and retail customers can trade on DCM. ECM can only deal with
institutional customers.
• As a rule customers will access DCM or ECM through FCM and CTA. FCM
can be clearing and non clearing members of an exchange.
• Most exchanges have separate clearing houses. CME and ICE are
somewhat of an exception to the extent that the clearing house is part of the
exchange.
• CME is a DSRO and in this regard will self regulate and run due diligence on
its members who must respect the rules of the exchange.
• Recent events (MF Global) have shown a potential weakness of the system.
Customer funds must be held by law in segregated accounts. An FCM could
be in good standing with an exchange (on a net basis) but may not have kept
customers‟ funds in segregated accounts in violation of the Commodity
Exchange Act (17 CFR 1.20). This was solved by CFTC and NFA.
http://www.cftc.gov/PressRoom/PressReleases/pr6303-12
• http://uscode.house.gov/ go to title 7, chapter 1
• http://www.ecfr.gov/cgi-bin/ECFR?page=browse go to title 17
• http://www.gpo.gov/fdsys/granule/CFR-2012-title17-vol1/CFR-2012-title17-
vol1-sec1-20
Futures and Options FINC-GB.3335 Summer 2014 1-59
Regulations
• Designated Self-Regulatory Organization (DSRO): Self-regulatory
organizations (i.e., the commodity exchanges and registered futures
associations) must enforce minimum financial and reporting requirements
for their members, among other responsibilities outlined in the CFTC's
regulations.
• Contract Market: A board of trade or exchange designated by the
Commodity Futures Trading Commission to trade futures or options under
the Commodity Exchange Act. A contract market can allow both
institutional and retail participants and can list for trading futures contracts
on any commodity, provided that each contract is not readily susceptible
to manipulation. Also called designated contract market.
• Exempt Commercial Market: An electronic trading facility that trades
exempt commodities on a principal-to-principal basis solely between
persons that are eligible commercial entities.
Futures and Options FINC-GB.3335 Summer 2014 1-60
Source: http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_c
Regulations
• Exempt Commodity: The Commodity Exchange Act defines an exempt
commodity as any commodity other than an excluded commodity or an
agricultural commodity. Examples include energy commodities and
metals.
• Excluded Commodity: In general, the Commodity Exchange Act defines
an excluded commodity as: any financial instrument such as a security,
currency, interest rate, debt instrument, or credit rating; any economic or
commercial index other than a narrow-based commodity index; or any
other value that is out of the control of participants and is associated with
an economic consequence.
• Futures Commission Merchant (FCM): Individuals, associations,
partnerships, corporations, and trusts that solicit or accept orders for the
purchase or sale of any commodity for future delivery on or subject to the
rules of any exchange and that accept payment from or extend credit to
those whose orders are accepted.
• Commodity Trading Advisor (CTA): A person who, for pay, regularly
engages in the business of advising others as to the value of commodity
futures or options or the advisability of trading in commodity futures or
options, or issues analyses or reports concerning commodity futures or
options
Futures and Options FINC-GB.3335 Summer 2014 1-61
Source: http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_c
General Hedging Principles
• A long futures hedge is appropriate when you
are short an asset (long a liability) today and you
want to protect against its appreciation. A long
futures position is also appropriate when you
know you will purchase an asset in the future
and want to lock in its price today.
• A short futures hedge is appropriate when you
own an asset today and want to protect against
its depreciation. A short futures hedge is also
appropriate when you know you will issue a
liability in the future and want to lock its
price/rate today .
Futures and Options FINC-GB.3335 Summer 2014 1-62
General Hedging Principles
Futures and Options FINC-GB.3335 Summer 2014 1-63
Existing Asset


SELL FUTURES
Existing Liability


BUY FUTURES
Anticipated Asset
(to be bought later at then price)


BUY FUTURES
Anticipated Liability
(to be sold later at then price)
SELL FUTURES
Eg. Issuing a Mortgage. You expect
that the rate is going to go up. But I
want to lock in the low rate now. So
sell the futures.
Example: Basic Futures Hedging
Futures and Options FINC-GB.3335 Summer 2014 1-64
EXISTING ASSET
SELL FUTURES
Protect against price going down
• S&P500 futures to hedge a stock
portfolio
• ED futures to hedge fixed rate
payments on a bond investment
• Corn futures to hedge the value of a
crop
EXISTING LIABILITY
BUY FUTURES
Protect against price going up
• ED futures to hedge fixed rate
payments on a bond issue
• T-bond futures to hedge an existing
bond issue
• NG futures to hedge a delivery of
natural gas at a given price
ANTICIPATED ASSET
(to be bought later at then price)
BUY FUTURES
Protect against price going up
• T-Bond futures to hedge a future
bond purchase
• ED futures to hedge future floating
interest rate receipts
• Heating Oil futures to hedge future
purchase of jet fuel (Example 3.3)
ANTICIPATED LIABILITY
(to be sold later at then price)
SELL FUTURES
Protect against price going down
• T-Bond to hedge a future corporate
bond issue
• ED futures to hedge future floating
interest rate payments
• Heating Oil futures to hedge future
sales
Simple S&P Hedging Example
• On Feb 10, a portfolio manager wants to hedge
a $400 Million portfolio exactly duplicating the
S&P 500 index.
– The S&P 500 index is at 1343
– The March futures maturing on 3/16/2012 is at 1340
• The portfolio manager will be able to hedge its
value by selling:
400 Million/ (250*1340) = 1194 S&P 500 futures
(see formula 3.4 page 64)

Futures and Options FINC-GB.3335 Summer 2014 1-65
Basis Risk
• Hedging may not be as straightforward as in the
previous example. In particular:
– The asset underlying the futures contract may not be
exactly the same as the asset being hedged.
– The hedging horizon may not coincide exactly with
the maturity of the futures contract.
– The hedge may require rolling futures contracts to
another maturity at a future date.
• These issues will expose the hedging process to
basis risk.
Futures and Options FINC-GB.3335 Summer 2014 1-66
Basis Risk
• The basis is the difference between the spot
price of the asset being hedged and the futures
price of the contract which is used.
• The basis will converge to zero if the futures
price converges to the spot price.
• The effectiveness of the hedge will be directly
related to the volatility of the basis and to its
value at the time where the hedge matures or is
closed.
Futures and Options FINC-GB.3335 Summer 2014 1-67
Long Hedge
• We define
F
1
: Initial Futures price at hedge inception
F
2
: Final Futures price at a future date
S
2
: Final asset price at a future date
• If you hedge the future purchase price of an asset
by entering into a long futures contract today, then
Asset Cost = S
2
– (F
2
– F
1
) = F
1
+ (S
2
– F
2
) = F
1
+
Basis


Futures and Options FINC-GB.3335 Summer 2014 1-68
If Basis: S2-F2 is zero. You achieved your goal. But if it is going all over the place
than you run the risk.
Short Hedge
• Again we define
F
1
: Initial Futures price at hedge inception
F
2
: Final Futures price at a future date
S
2
: Final asset price at a future date
• If you hedge the future sale price of an asset by
entering into a short futures contract then
Price Realized=S
2
+ (F
1
– F
2
) = F
1
+ Basis


Futures and Options FINC-GB.3335 Summer 2014 1-69
Sell the future today to lock in a price at a future date.
Minimizing Basis Risk
• Choose if possible a delivery month that is as
close as possible to, but later than, the horizon of
the hedge.
• If you need to roll the hedge, monitor the value of
the roll (price difference between consecutive
futures) and its fair value.
• 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
and adjust the hedge to reflect this correlation.
Futures and Options FINC-GB.3335 Summer 2014 1-70
Cross Hedging: Example 3.3 (Page 61)
• An airline expects to buy in the near future 2 million
gallons of jet fuel and hedges the anticipated
purchase buying heating oil futures. A futures
contract is written on 42,000 gallons.
• If the futures was also written on jet fuel, the airline
would have to buy
– futures contracts

– where Q
A
is the size of the position being hedged in units
– and Q
B
is the size of 1 futures contract in units
• The airline should buy 2,000,000/42,000 = 47.62
contracts



Futures and Options FINC-GB.3335 Summer 2014 1-71
B
A
Q
Q
Cross Hedging: Example 3.3 (Page 61)
• However since the contract is written on heating oil,
there will be basis risk.
• The 1-1 hedge ratio must be adjusted to minimize
basis risk.
• This is equivalent to creating a minimum variance
(risk) portfolio with both the hedge and the future
asset.
• To find this minimum variance portfolio, we regress
the change in jet fuel price on the change in the
heating oil futures (invoice)price and adjust the
hedge ratio computed on the previous slide by the
beta of the regression.



Futures and Options FINC-GB.3335 Summer 2014 1-72
Cross Hedging: Example 3.3 (Page 61)
Futures and Options FINC-GB.3335 Summer 2014 1-73
Month Spot Pr. Ch. Fut. Pr. Ch.
1 0.029 0.021 LINEST(B2:B16,C2:C16,TRUE,TRUE)
2 0.02 0.035 beta alpha
3 -0.044 -0.046 value 0.777651 0.000874
4 0.008 0.001 Std Dev 0.086343 0.002616
5 0.026 0.044 R Square 0.861875 0.010126 Std Dev residuals
6 -0.019 -0.029 F statistics 81.11779 13
7 -0.01 -0.026 Expl. Sum of Sq 0.008317 0.001333 Residual Sum of Sq
8 -0.007 -0.029
9 0.043 0.048 Corr Coeff 0.92837
10 0.011 -0.006 Std Dev futures 0.03134
11 -0.036 -0.036 Std Dev spot 0.02625
12 -0.018 -0.011 beta 0.777651
13 0.009 0.019 Total Sum of Sq 0.00965
14 -0.032 -0.027 Rsquare 0.861875
15 0.023 0.029 confidence lev. 0.025 0.975
beta 0.591118 0.964183
Cross Hedging: Example 3.3 (Page 61)
• The beta of the regression is: (Hedge Ratio)


• where
 o
S
is the standard deviation of AS, the change in the
spot price during the hedging period,
 o
F
is the standard deviation of AF, the change in the
futures price during the hedging period,
 µ is the coefficient of correlation between AS and AF.
Futures and Options FINC-GB.3335 Summer 2014 1-74
F
S
h
o
o
µ =
*
Cross Hedging: Example 3.3 (Page 61)
• o
S
the std dev. of the change in jet fuel spot = .0263
• o
F
the std dev. of the change in HO futures = .0313
• µ the correlation coeff. between the two changes =
.928


• The number of heating oil futures to buy is:
Futures and Options FINC-GB.3335 Summer 2014 1-75
78 .
0313 .
0263 .
* 928 .
*
=
|
.
|

\
|
= =
F
S
h
o
o
µ
( )
37 14 . 37
000 , 42
000 , 000 , 2 * 78 .
1
*
*
*
~ = = =
futures of size
hedge to quantity h
N
Cross Hedging: Tailing the Hedge
• Previous analysis assumes that we are hedging
gallon of spot for gallon of futures.
• To generalize the hedge, we need to compare
the dollar value of the spot position being
hedged with the dollar value of the underlying
futures position (futures invoice price times the
multiplier/size of the futures contract)

Futures and Options FINC-GB.3335 Summer 2014 1-76
( )
( )
( )
36 21 . 36
99 . 1 * 000 , 42
94 . 1 * 000 , 000 , 2 * 78 .
* 1
* *
1 $
$ *
*
*
*
~ = =
=
=
price f utures f utures of size
price spot hedge to quantity h
f utures underlying value
hedge to value h
N
Hedging : a Bad Idea
Futures and Options FINC-GB.3335 Summer 2014 1-77
Month Spot Pr. Ch. Fut. Pr. Ch.
1 -0.029 -0.005 LINEST(B2:B16,C2:C16,TRUE,TRUE)
2 0.04 0.01 beta alpha
3 -0.044 -0.046 value 0.504542 0.0100718
4 0.016 0.01 Std Dev 0.2867813 0.0072562
5 0.026 0.044 R Square 0.1923074 0.026483 Std Dev residuals
6 0.019 -0.029 F statistics 3.0952318 13
7 -0.01 -0.026 Expl. Sum of Sq 0.0021708 0.0091176 Residual Sum of Sq
8 -0.007 0.005
9 0.06 -0.02 Corr Coeff 0.4385287
10 0.011 -0.006 Std Dev futures 0.0246804
11 -0.02 -0.036 Std Dev spot 0.0283957
12 -0.018 -0.011
13 0.03 -0.019 confidence level 0.025 0.975
14 -0.01 -0.027 beta -0.1150112 1.1240953
15 0.023 0.029
Equity Portfolio & Index Futures
• Index futures will be useful in many ways to
modify or hedge the return of an equity portfolio:
– A futures hedge will help an investor eliminate the
market risk of a portfolio and keep the stock specific
risk of this portfolio.
– A futures hedge eliminating the market risk of a
portfolio can be implemented and reversed quickly.
– Futures can help the investor bring the market risk or
beta of a portfolio of stocks closer to the market risk of
a benchmark.
– More generally, futures can be used to modify the
market risk or beta of a portfolio of stocks.
Futures and Options FINC-GB.3335 Summer 2014 1-78
Hedging an Equity Portfolio (Page 64)
• Following the approach developed above, a
stock portfolio with a value P can be hedged by
selling a number of futures contracts equal to:


• where | is the beta of the portfolio derived by
regressing the rate of return on the portfolio on
the rate of return on the futures,
• and F is the futures price times the contract
size.

Futures and Options FINC-GB.3335 Summer 2014 1-79
F
P
|
Hedging an Equity Portfolio: Example p 65
• S&P 500 index is 1000
• S&P futures price is 1,010
• Value of Portfolio is $5.05 million
• Beta of portfolio is 1.5
• What short position in futures contracts (4 month) on
the S&P 500 is necessary to hedge the portfolio?


• How do you simulate potential values of your
hedged portfolio in three months?
Futures and Options FINC-GB.3335 Summer 2014 1-80
30
2525 . 0 $
05 . 5 $
5 . 1
250 * 010 , 1 $
05 . 5 $
5 . 1 = = = =
million
million million
F
P
N |
Hedging an Equity Portfolio: Example p 65
• Assume that:
– S&P index is 900 and futures price is 902 in three months,
– the risk free rate is 4% per annum
– the dividend yield is 1% per annum
• Need to compute the return on the market/index.


• Use CAPM to compute expected return on portfolio
Futures and Options FINC-GB.3335 Summer 2014 1-81
% 75 . 9 0025 . 1 .
4
01 .
1
1000
900
÷ = + ÷ = + ÷ = return Market
15125 . )
4
04 .
0975 . ( * 5 . 1
4
04 .
÷ = ÷ ÷ + = ret Port Expected
Hedging an Equity Portfolio: Example p 65
• Compute expected portfolio value in three months:


– This would be the value of your portfolio with no hedge.
• Compute the P&L on your short futures position:


• The total expected value in 3 months of the portfolio
and its hedge:
Futures and Options FINC-GB.3335 Summer 2014 1-82
5 . 187 , 096 , 5 $ 000 , 810 $ 50 . 187 , 286 , 4 $ = +
50 . 187 , 286 , 4 $ 05 . 5 $ * ) 15125 . 1 ( = ÷
=
million
val Port Expected
000 , 810 $ ) 000 , 27 $ ( * 30
) 1010 902 ( * 250 * 30 &
= ÷ ÷ =
÷ ÷ = L P Futures
Hedging an Equity Portfolio: Table 3.4
Futures and Options FINC-GB.3335 Summer 2014 1-83
Port. Val. $5,050,000
S0 1000
F0 1010
multiplier 250
r$ 4.000%
d 1.000%
T 0.25
beta 1.5
# contracts 30
Index in 3 months 900 950 1000 1050 1100
futures price 902 952 1003 1053 1103
Gain on futures 810,000 435,000 52,500 (322,500) (697,500)
cap gain return -10.00% -5.00% 0.00% 5.00% 10.00%
dividend return 0.250% 0.250% 0.250% 0.250% 0.250%
return on market -9.7500% -4.7500% 0.2500% 5.2500% 10.2500%
expected port return -15.125% -7.625% -0.125% 7.375% 14.875%
expected port value 4,286,188 $ 4,664,938 $ 5,043,688 $ 5,422,438 $ 5,801,188 $
Total Value Position 5,096,188 $ 5,099,938 $ 5,096,188 $ 5,099,938 $ 5,103,688 $
Changing Portfolio Risk Using Index Futures
• In the previous example, we hedged the portfolio
fully by selling 30 futures contract. This means that
the beta of the portfolio including the hedge is zero.
• We may not want to fully hedge the portfolio. What
position is necessary to reduce the beta of the
portfolio from 1.5 to 0.5 instead of zero?
• To do that we will only sell

Futures and Options FINC-GB.3335 Summer 2014 1-84
( ). 5 .
) 5 . 1 (
20
500 , 252 $
05 . 5 $
) 5 . 5 . 1 (
*
level
beta desired the is and portfolio the of beta
the is where contracts
million
|
| = ÷
( ) = ÷
F
P
*
| |
Changing Portfolio Risk Using Index
Futures
• In the opposite way, you may desire to increase the
sensitivity of a portfolio with respect to the S&P 500
by buying futures. Using the previous example, you
may want to increase the beta of this portfolio relative
to the S&P 500 from 1.5 to 2.
• You will do that by buying

Futures and Options FINC-GB.3335 Summer 2014 1-85
( ). 0 . 2
) 5 . 1 (
10
500 , 252 $
05 . 5 $
) 5 . 1 2 (
*
level
beta desired the is and portfolio the of beta
the is where contracts
million
|
| = ÷
( ) = ÷
F
P
| |
*
Hedge Tracking Error
• The tracking error is defined as the standard error
of the residuals of the regression of the rate of
return of the asset being hedged on the rate of
return of the hedging instrument.
• There are two measures of the tracking error: the
predicted (ex ante) tracking error and the realized
(ex post) tracking error.
• The predicted tracking error is the standard
deviation of the residuals obtained when estimating
the beta.
• The realized tracking error is the standard deviation
of the realized residuals after the beta has been set.
Futures and Options FINC-GB.3335 Summer 2014 1-86
Hedge Tracking Error: Ex Ante
• We are on July 1, 2009. In the next slide, we derive
the beta using rate of return data from June 1 to
June 30, 2009 on the Russell 3000 index (RUA)
and SPU9 (September 2009 S&P 500 futures).
• The beta is equal to 1.0651. Thus the dollar value
of the hedge will be 1.0651 times the dollar value of
the asset.
• The hedge is good since the R^2 is 97%.
• This is confirmed by comparing the actual rates of
return on RUA with the explained rates of return
and by looking at the standard deviation of the
residuals, .2% daily and 3.45% annually.
Futures and Options FINC-GB.3335 Summer 2014 1-87
Hedge Tracking Error: Ex Ante
Futures and Options FINC-GB.3335 Summer 2014 1-88
Date RUA SPU9 RUA SPU9 EX ANTE Expl Ret. Res
6/1/2009 549.68 934.6 Ret. Ret. beta alpha
6/2/2009 551.14 938.1 0.27% 0.37% value 1.0651 -0.0002 0.38% -0.11%
6/3/2009 543.41 927.2 -1.41% -1.17% StDev 0.0433 0.0005 -1.26% -0.15%
6/4/2009 550.01 936 1.21% 0.94% R Square 0.9696 0.0022 StDev Res 0.99% 0.22%
6/5/2009 548.73 936.1 -0.23% 0.01% F statistics 605.5351 19.0000 -0.01% -0.23%
6/8/2009 547.38 934.6 -0.25% -0.16% Expl. SOS 0.0029 0.0001 Res SOS -0.19% -0.06%
6/9/2009 549.91 935.5 0.46% 0.10% 0.08% 0.38%
6/10/2009 547.94 936.4 -0.36% 0.10% 0.08% -0.44%
6/11/2009 551.18 938.2 0.59% 0.19% 0.19% 0.40%
6/12/2009 551.57 940.7 0.07% 0.27% annual 0.26% -0.19%
6/15/2009 538.16 919.4 -2.46% -2.29% Stdev RUA 0.0122 19.29% -2.46% 0.00%
6/16/2009 530.95 907.8 -1.35% -1.27% Stdev SPU9 0.0112 17.83% -1.37% 0.02%
6/17/2009 530.42 905.3 -0.10% -0.28% correlation 0.9847 -0.31% 0.21%
6/18/2009 534.47 913.2 0.76% 0.87% beta 1.0651 0.91% -0.15%
6/19/2009 536.38 915.7 0.36% 0.27% R Square 0.9696 0.27% 0.08%
6/22/2009 519.38 888.6 -3.22% -3.00% -3.22% 0.00%
6/23/2009 520.04 890.2 0.13% 0.18% StDev Res (n-2) 0.0022 3.45% 0.17% -0.05%
6/24/2009 523.98 898 0.75% 0.87% 0.91% -0.16%
6/25/2009 535.56 916.6 2.19% 2.05% 2.16% 0.02%
6/26/2009 535.56 913.9 0.00% -0.30% -0.33% 0.33%
6/29/2009 539.73 921.2 0.78% 0.80% 0.83% -0.05%
6/30/2009 535.62 915.5 -0.76% -0.62% -0.68% -0.08%
Hedge Tracking Error: Ex Post
• Given the beta computed on July 1, we check the
result of the regression ex post on July data.
• The hedge remains good since the ex post R^2
computed as the ratio of the ex post explained
SSR to the total SSR is equal to 96.2%
comparable to the 97% ex ante R^2.
• This is confirmed by comparing the actual rates of
return on RUA with the explained rates of return
and by looking at the standard deviation of the
residuals, .3% daily and 4.4% annually which
remain close to their ex ante values.
Futures and Options FINC-GB.3335 Summer 2014 1-89
Hedge Tracking Error: Ex Post
Futures and Options FINC-GB.3335 Summer 2014 1-90
Date RUA SPU9 RUA SPU9 EX ANTE BETA 1.0651 Expl Ret. Res
7/1/2009 538.65 919.2 Ret. Ret. EX POST
7/2/2009 522.57 893.3 -3.03% -2.86% -3.04% 0.01%
7/6/2009 522.95 895.5 0.07% 0.25% SOSR 0.00016 0.26% -0.19%
7/7/2009 512.51 879.3 -2.02% -1.83% EXPL SOS 0.00402 -1.94% -0.07%
7/8/2009 511.18 873.7 -0.26% -0.64% R^2 96.2% -0.68% 0.42%
7/9/2009 513.05 878.9 0.37% 0.59% 0.63% -0.27%
7/10/2009 511.36 874.3 -0.33% -0.52% Daily Annual -0.56% 0.23%
7/13/2009 523.87 895.6 2.42% 2.41% Stdev RUA 1.4% 21.5% 2.56% -0.15%
7/14/2009 526.93 901.4 0.58% 0.65% StDev SPU9 1.3% 20.5% 0.69% -0.11%
7/15/2009 542.91 927.2 2.99% 2.82% StDev Res 0.3% 4.4% 3.01% -0.02%
7/16/2009 548.08 935.7 0.95% 0.91% 0.97% -0.02%
7/17/2009 547.61 936.9 -0.09% 0.13% 0.14% -0.22%
7/20/2009 554.38 949 1.23% 1.28% 1.37% -0.14%
7/21/2009 556.07 953.4 0.30% 0.46% 0.49% -0.19%
7/22/2009 556.23 949.4 0.03% -0.42% -0.45% 0.48%
7/23/2009 569.71 968.9 2.39% 2.03% 2.17% 0.23%
7/24/2009 571.82 977.8 0.37% 0.91% 0.97% -0.60%
7/27/2009 573.58 979.9 0.31% 0.21% 0.23% 0.08%
7/28/2009 572.23 975.9 -0.24% -0.41% -0.44% 0.20%
7/29/2009 569.23 974.9 -0.53% -0.10% -0.11% -0.42%
7/30/2009 576.31 982.2 1.24% 0.75% 0.79% 0.44%
7/31/2009 576.56 984.4 0.04% 0.22% 0.24% -0.19%
Hedging : Where it Could Break Down-MG
• Assume you have agreed to sell an amount of
commodity at a fixed price every month for a long
period of time and you decide to hedge the risk. To
hedge, you need to buy a strip of futures contracts of
different maturities.
• Assume that the liquidity of the back maturity months
is poor and that most of the liquidity is in the front
contracts. You will stack the futures in the front
month and roll them to execute the hedge.
• You have the perfect set up for potentially large
losses: Metallgesellschaft AG. in 1993
• Metallgesellschaft (MG) had similar contracts on oil.


Futures and Options FINC-GB.3335 Summer 2014 1-91
Hedging : Where it Could Break Down - MG
• MG was not hurt by increasing prices of oil. On the
contrary, they started losing money when oil prices fell.
• MG lost money on the basis risk they ran between their
liability (selling a strip of forward contracts) and their
hedge (buying front contract).
• These losses came from transaction costs on the roll
given the size of their positions and from the fact that
the futures curve went from backwardation (decreasing)
to contango (increasing).
• Finally the size of the margin calls overwhelmed MG.
MG was not under hedge accounting and was at risk of
not finding the liquidity to post margins.

Futures and Options FINC-GB.3335 Summer 2014 1-92
Hedging : Where it Could Break Down -
Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-93
• Amaranth Advisors, LLC was created in 2000 as a multi-
strategy hedge fund. Amaranth started operations with
approximately $600 Million of capital and focused on
arbitrage strategies around convertible bonds and mergers.
• Amaranth started energy commodity trading in 2002, hiring
former employees of Enron.
• At the beginning of 2006, Amaranth started to position in
natural gas because its management believed that market
forces should justify higher spreads between natural gas
prices in the winter and natural gas prices in the
Summer/Fall.
• The strategy was successful until the Summer of 2006 but
led to extremely large positions on NYMEX and ICE.

Hedging : Where it Could Break Down -
Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-94
Source: Excessive speculation in the Natural Gas Market, Staff Report, US
Senate, June 25 & July 9, 2007 Hearings
Hedging : Where it Could Break Down -
Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-95
• In August 2006, NYMEX took more forceful action to limit
Amaranth‟s trading. Amaranth increased its trading on ICE
which was not subject to the same regulation.

Source: Same, page 98
Hedging : Where it Could Break Down -
Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-96
In late August the market turns against Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-97
Hedging : Where it Could Break Down -
Amaranth
Hedging : Where it Could Break Down -
Amaranth
Futures and Options FINC-GB.3335 Summer 2014 1-98
• Margin calls accumulated because of the collapse of
the spread positions.
• On September 20, Amaranth formally sold its energy
book to JP Morgan Chase and Citadel and liquidated
the remainder of its $8 billion portfolio to meet its
margin calls.
• One of the consequences of this collapse was the
extension of the CFTC regulatory powers on markets
like ICE that offer futures contract on exempt
commodities. This was done through the Farm Bill,
passed by Congress in 2008.
Hedging : Where it Could Break Down-Barrick
• A similar situation arose with gold producers selling
their gold production forward.
• Several of them sold many years of production
forward when gold prices appeared to be on a
secular down trend in the 1990‟s.
• As gold rose in more recent years prior to 2009, they
needed to post margin on several years of hedged
production. Several firms faced a tight funding
situation because they “could not get their gold out of
the ground fast enough”. Barrick issued $3 billion of
equity and additional debt to close the hedge book.
http://www.barrick.com/investors/news/news-
details/2009/BarrickAnnouncesPlantoEliminateGoldHedges1121019/default.aspx
Futures and Options FINC-GB.3335 Summer 2014 1-99