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. futuresFutures and Options FINC-GB.3335 Summer 2014 1-1

Lecture 1 (07/05 pm): Futures Hedging Strategies

Futures and Options FINC-GB.3335 Summer 2014 1-2

• 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.


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

411

)(4

11

4

4

tz

RqRa

tzRq

Ra

tt

tt


<= Future valuation

<= Present valuation

Measuring Interest Rates: Continuously Compounded Rate

)(1

)(

11

tzen

r

tze

n

r

trnt

c

trnt

c

c

c


• is an annual rate compounded continuously.• As n becomes large:

cr


t

tzr

tztr

tzetze

c

c

trtr cc

)(ln

)(ln

)(lnln)(


Annual rate continuously compounded that you will receive today, for the periodof period ‘t’ years.


t

tzr

tztr

tze

tze

lyequivalent

c

c

trtr cc

)(ln

)(ln

)(

1lnln

)(

1


Measuring Interest Rates: Example


Annual rate 5.00%

PV of $1 in 1 year 0.95238

Semiannual rate 4.9390% 2Quarterly rate 4.9089% 4Monthly rate 4.8889% 12Daily rate 4.8793% 365Continuous rate 4.8790%

aR

aRZ

1

1)1(

n

))1(ln( Z

1)1( )/1( nZn

1)1( )/1( nZn

1)1( )/1( nZn

1)1( )/1( nZn

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).


Measuring Interest Rates: Forward Rates


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

Measuring Interest Rates: Forward Rates


Time Zero rate PVF Forward rate0.5 0.0404 0.9800 0.0404 0.0404

1 0.0408 0.9600 0.0412 0.04121.5 0.0466 0.9325 0.0582 0.0582

2 0.0502 0.9045 0.061 0.061

1

1)()(

ln

ii

i

i

tt

tPVFtPVF

1

11 )(*)(*

ii

iiii

tt

tratezerottratezerot


• 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)?




t rate rate/2 FVF=1/Z(t) LN(FVF)/t1 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 eb) 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)/.5forward 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).


Bond Principal($)

Time to Maturity(years)

Annual CouponPaid 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


Bond time coupon anPrice100 0.5 0 98100 1 0 96100 1.5 4 99100 2 4 98

.5 y rate 4.0405% -LN(D2/A2)/B21 y rate 4.0822% -LN(D3/A3)/B3zero rate for 1.5 year will be obtained by bootstrapping. Taking the 1.5 ybond, 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 100so 99-PV cpn .5 - PV cpn 1=PVF(1.5)*(102)PVF(1.5) 0.932551.5 y rate 4.6556%

same approach can be followed to get the 2 y ratePV cpn .5 1.96PV cpn 1 1.92PV cpn 1.5 1.865098PVF(2) 0.904462 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:


2*0502.5.1*0466.1*0408.5.*0404.

2

4100

2

4

2

4

2

498

eeee

2*0502.5.1*0466.1*0408.5.*0404.

21

2221

e

ce

ce

ce

c

Measuring Interest Rates: Par Yield


0506.0253.9045.9325.9600.9800.

)9045.1(1

2

11

2

211

21

2221

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.

cc

eeee

ec

eeeec

e

ec

ec

ec

ec

Measuring Interest Rates: Par Yield• It is also true that :

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


43

2

20506.

1

1*

2

0506.1

20506.

1

1*

2

0506.

20506.

1

1*

2

0506.

20506.

1

1*

2

0506.1

Major Futures Exchanges (2004)


Source: CME, An Introduction to Futures and Options

Major Exchanges (March 2013)



Source: http://www.futuresindustry.org/downloads/FI-2012_Volume_Survey.pdf

http://www.bis.org/statistics/r_qa1306_hanx23a.pdf

http://www.futuresindustry.org/downloads/FI-2012_Volume_Survey.pdf

Major Exchanges (March 2014)


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


http://www.futuresindustry.org/downloads/FIA_Annual_Volume_Survey_2013.pdf

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


http://www.cmegroup.com/

http://www.theice.com/

http://www.bmf.com.br/

http://www.mexder.com.mx/

http://www.rofex.com.ar/

http://www.eurexchange.com/

http://www.euronext.com/

http://www.lme.com/

http://www.moex.com/

http://www.dce.com.cn/

http://www.shfe.com.cn/

http://www.czce.com.cn/

http://www.nse-india.com/

http://www.sgx.com/

http://www.tse.or.jp/

http://www.tse.or.jp/

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)


Type of Contracts

Financials – 3rd – Interest rate– Equity indices– Currencies

Agricultural – Started 1st

Metals – 2nd

Energy -4th last


Market Statistics - BIS (June 2013)


Source: http://www.bis.org/statistics/r_qa1306_hanx23a.pdf



Market Trend - BIS (June 2013)


http://www.bis.org/statistics/dt1920a.csv

Mar.93

Dec.93

Sep.94

Jun.95

Mar.96

Dec.96

Sep.97

Jun.98

Mar.99

Dec.99

Sep.00

Jun.01

Mar.02

Dec.02

Sep.03

Jun.04

Mar.05

Dec.05

Sep.06

Jun.07

Mar.08

Dec.08

Sep.09

Jun.10

Mar.11

Dec.11

Sep.12

0

5000

10000

15000

20000

25000

30000

35000

Financial Futures Outstanding (in $ Billion)

North America Europe Asia and Pacific Other Markets Total

http://www.bis.org/statistics/dt1920a.csv

Market Statistics - FIA (March 2013)



These two are mostly options

‘Interest rates’ account for largest volume for futures



Major U.S. Futures Contracts (2004)



Major Futures Contracts (2013)





Type of Contracts

– Interest rate• Eurodollar• Euribor• T-Note• Bund futures• Fed funds• Eonia

– Equity indices• S&P 500• Nikkei

– Currencies• Euro


– 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)


Source: http://www.cmegroup.com/education/files/cme-group-leading-products-2013-q2.pdf


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 Products (Q2 2013)





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.


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).


Marked to Market: Long Futures Contract

Gain

Loss

Ft+1

0Ft*


Suppose the contract settled at Ft* =100 at time t . Table shows the gain

(credit) or loss (debit) on the next day (t+1).

Ft+1 Ft+1 – Ft*

90 -10 Debit

95 -5 Debit

100 0 -

105 5 Credit

110 10 Credit

Marked to Market: Short Futures Contract

Gain

Loss

Ft+1

0Ft*


Suppose the contract settled at Ft* =100 at time t. Table shows the gain

(credit) or loss (debit) on the next day (t+1).

Ft+1 Ft+1 – Ft*

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


Margin Calls: Hypothetical Outcome


Date Price change Gain/Loss Beg Balance Cum Gain Margin Call End Balance Available2/03 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 -40multiplier 250Init. Mg. 25,000.00$ Maint Mg. 20,000.00$

Example: Corn Futures (CME)

Futures and Options FINC-GB.3335 Summer 2014 1-42Source: www.cmegroup.com

Delivery OptionIf the value of #1 Yellow is less 1.5c higher than #2.Then seller should buy and sell #1 corn.

Example: Corn Futures (CME)


Source: www.cmegroup.com

It is 8th of a cent. 642.4 => $64.25

Example: Natural Gas (CME)


Example: Light Sweet Crude Oil (CME)


Source: www.cmegroup.comLight Sweet is essentially WTI.

Example: Light Sweet Crude Oil (CME)


Example: Brent Crude Futures (ICE)

Futures and Options FINC-GB.3335 Summer 2014 1-48Source: www.theice.com

Example: Brent Crude Futures (ICE)


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)


We should know everything in this slide.

Example: Eurodollar Futures (CME)



Example: S&P 500 Futures (CME)


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.


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.


Forward vs. FuturesForward 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


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-vol

1-sec1-20


http://www.cftc.gov/PressRoom/PressReleases/pr6303-12

http://www.cftc.gov/PressRoom/PressReleases/pr6303-12

http://uscode.house.gov/

http://uscode.house.gov/

http://www.ecfr.gov/cgi-bin/ECFR?page=browse

http://www.ecfr.gov/cgi-bin/ECFR?page=browse

http://www.gpo.gov/fdsys/granule/CFR-2012-title17-vol1/CFR-2012-title17-vol1-sec1-20

http://www.gpo.gov/fdsys/granule/CFR-2012-title17-vol1/CFR-2012-title17-vol1-sec1-20

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.


Source: http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_c

http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_s

http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_co


http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_m

http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_e



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


Source: http://www.cftc.gov/ConsumerProtection/EducationCenter/CFTCGlossary/glossary_c




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 .


General Hedging Principles


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


EXISTING ASSETSELL 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 LIABILITYBUY 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 FUTURESProtect 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 FUTURESProtect 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)


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.


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.


Long Hedge • We define

F1 : Initial Futures price at hedge inception

F2 : Final Futures price at a future date

S2 : 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 = S2 – (F2 – F1) = F1 + (S2 – F2) = F1 + Basis


If Basis: S2-F2 is zero. You achieved your goal. But if it is going all over the placethan you run the risk.

Short Hedge• Again we define

F1 : Initial Futures price at hedge inception

F2 : Final Futures price at a future date

S2 : 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=S2+ (F1 – F2) = F1 + Basis


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.


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 QA is the size of the position being hedged in units

– and QB is the size of 1 futures contract in units

• The airline should buy 2,000,000/42,000 = 47.62 contracts


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.


Cross Hedging: Example 3.3 (Page 61)


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 alpha3 -0.044 -0.046 value 0.777651 0.0008744 0.008 0.001 Std Dev 0.086343 0.0026165 0.026 0.044 R Square 0.861875 0.010126 Std Dev residuals6 -0.019 -0.029 F statistics 81.11779 137 -0.01 -0.026 Expl. Sum of Sq 0.008317 0.001333 Residual Sum of Sq8 -0.007 -0.0299 0.043 0.048 Corr Coeff 0.92837

10 0.011 -0.006 Std Dev futures 0.0313411 -0.036 -0.036 Std Dev spot 0.0262512 -0.018 -0.011 beta 0.77765113 0.009 0.019 Total Sum of Sq 0.0096514 -0.032 -0.027 Rsquare 0.86187515 0.023 0.029 confidence lev. 0.025 0.975

beta 0.591118 0.964183


• The beta of the regression is: (Hedge Ratio)

• where sS is the standard deviation of DS, the change in the

spot price during the hedging period, sF is the standard deviation of DF, the change in the

futures price during the hedging period, r is the coefficient of correlation between DS and DF.


F

Sh*


• sS the std dev. of the change in jet fuel spot = .0263

• sF the std dev. of the change in HO futures = .0313

• r the correlation coeff. between the two changes = .928

• The number of heating oil futures to buy is:


78.0313.

0263.*928.*

F

Sh

3714.37

000,42

000,000,2*78.

1

***

futuresofsize

hedgetoquantityhN

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)


3621.36

99.1*000,42

94.1*000,000,2*78.

*1

**

1$

$*

*

**

pricefuturesfuturesofsize

pricespothedgetoquantityh

futuresunderlyingvalue

hedgetovaluehN

Hedging : a Bad Idea


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 alpha3 -0.044 -0.046 value 0.504542 0.01007184 0.016 0.01 Std Dev 0.2867813 0.00725625 0.026 0.044 R Square 0.1923074 0.026483 Std Dev residuals6 0.019 -0.029 F statistics 3.0952318 137 -0.01 -0.026 Expl. Sum of Sq 0.0021708 0.0091176 Residual Sum of Sq8 -0.007 0.0059 0.06 -0.02 Corr Coeff 0.4385287

10 0.011 -0.006 Std Dev futures 0.024680411 -0.02 -0.036 Std Dev spot 0.028395712 -0.018 -0.01113 0.03 -0.019 confidence level 0.025 0.97514 -0.01 -0.027 beta -0.1150112 1.124095315 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.


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 b 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.


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?


302525.0$

05.5$5.1

250*010,1$

05.5$5.1

million

millionmillion

F

PN

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


%75.90025.1.4

01.1

1000

900returnMarket

15125.)4

04.0975.(*5.1

4

04.retPortExpected

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:


5.187,096,5$000,810$50.187,286,4$

50.187,286,4$05.5$*)15125.1(

million

valPortExpected

000,810$)000,27$(*30

)1010902(*250*30&

LPFutures

Hedging an Equity Portfolio: Table 3.4


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 1100futures price 902 952 1003 1053 1103Gain 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


.5.

)5.1(

20500,252$

05.5$)5.5.1(

*

level

betadesiredtheisandportfoliotheofbeta

theiswherecontractsmillion

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


.0.2

)5.1(

10500,252$

05.5$)5.12(

*

level

betadesiredtheisandportfoliotheofbeta

theiswherecontractsmillion

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.


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.


Hedge Tracking Error: Ex Ante


Date RUA SPU9 RUA SPU9 EX ANTE Expl Ret. Res6/1/2009 549.68 934.6 Ret. Ret. beta alpha6/2/2009 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.


Hedge Tracking Error: Ex Post


Date RUA SPU9 RUA SPU9 EX ANTE BETA 1.0651 Expl Ret. Res7/1/2009 538.65 919.2 Ret. Ret. EX POST7/2/2009 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.


Hedging : Where it Could Break Down - Amaranth


• 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.



Source: Excessive speculation in the Natural Gas Market, Staff Report, US Senate, June 25 & July 9, 2007 Hearings



• 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



In late August the market turns against Amaranth



• 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


http://www.barrick.com/investors/news/news-details/2009/BarrickAnnouncesPlantoEliminateGoldHedges1121019/default.aspx

http://www.barrick.com/investors/news/news-details/2009/BarrickAnnouncesPlantoEliminateGoldHedges1121019/default.aspx

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