Chapter4

IntroductiontoRiskManagement

Question4.1.

Thefollowingtablesummarizestheunhedgedandhedgedpro?tcalculations:

CopperpriceinTotalcostUnhedgedpro?tPro?tonshortNetincomeon

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

forward

$0.30

$0.20

$0.10

0

hedgedpro?t

$0.10

$0.10

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.20

?$0.10

0

$0.10

$0.10

$0.10

$0.20

$0.30

?$0.10

?$0.20

$0.10

$0.10

Weobtainthefollowingpro?tdiagram:

Question4.2.

Iftheforwardpricewere$0.80insteadof$1,wewouldgetthefollowingtable:

43

Part1Insurance,Hedging,andSimpleStrategies

CopperpriceinTotalcostUnhedgedpro?tPro?tonshortNetincomeon

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

forward

$0.10

$0

hedgedpro?t

?$0.10

?$0.10

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.20

?$0.10

0

?$0.10

?$0.20

?$0.30

?$0.40

?$0.10

?$0.10

$0.10

$0.20

$0.30

?$0.10

?$0.10

Withaforwardpriceof$0.45,wehave:

CopperpriceinTotalcostUnhedgedpro?tPro?tonshortNetincomeon

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

forward

?$0.25

?$0.35

?$0.45

?$0.55

?$0.65

?$0.75

hedgedpro?t

?$0.45

?$0.45

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.20

?$0.10

0

?$0.45

?$0.45

$0.10

$0.20

$0.30

?$0.45

?$0.45

Althoughthecopperforwardpriceof$0.45isbelowourtotalcostsof$0.90,itishigherthanthe

variablecostof$0.40.Itstillmakessensetoproducecopper,becauseevenatapriceof$0.45in

oneyear,wewillbeabletopartiallycoverour?xedcosts.

Question4.3.

Pleasenotethatwehavegiventhecontinuouslycompoundedrateofinterestas6%.Therefore,

theeffectiveannualinterestrateisexp(0.06)?1=0.062.Inthisexercise,weneedto?ndthe

futurevalueoftheputpremia.Forthe$1-strikeput,itis:$0.0376×1.062=$0.04.Thefollowing

tableshowsthepro?tcalculationsforthe$1.00-strikeput.Thecalculationsforthetwootherputs

areexactlysimilar.The?gureonthenextpagecomparesthepro?tdiagramsofallthreepossible

hedgingstrategies.

CopperpriceinTotalcostUnhedgedPro?tonlong

oneyearpro?t

Put

Netincomeon

$1.00-strikeputpremiumhedgedpro?t

option

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.20

?$0.10

0

$0.30

$0.20

$0.10

0

$0.04

$0.04

$0.04

$0.04

$0.04

$0.04

$0.06

$0.06

$0.06

$0.06

$0.16

$0.26

$0.10

$0.20

$0.30

0

0

44

Chapter4IntroductiontoRiskManagement

Pro?tdiagramofthedifferentputstrategies:

Question4.4.

Wewillexplicitlycalculatethepro?tforthe$1.00-strikeandshow?guresforallthreestrikes.The

futurevalueofthe$1.00-strikecallpremiumamountsto:$0.0376×1.062=$0.04.

CopperpriceinTotalcostUnhedgedPro?tonshort

pro?t

Call

Netincomeon

$1.00-strikecallpremiumhedgedpro?t

oneyear

option

0

received

$0.04

$0.04

$0.04

$0.04

$0.04

$0.04

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.20

?$0.10

0

?$0.16

?$0.06

$0.04

0

0

$0.10

$0.20

$0.30

0

$0.14

$0.14

$0.14

?$0.10

?$0.20

45

Part1Insurance,Hedging,andSimpleStrategies

Weobtainthefollowingpayoffgraphs:

Question4.5.

XYZwillbuycollars,whichmeansthattheybuytheputlegandsellthecallleg.Wehaveto

computeforeachcasethenetoptionpremiumposition,and?nditsfuturevalue.Wehavefor

$0.0178?$0.0376×1.062=?$0.021

$0.0265?$0.0274×1.062=?$0.001

$0.0665?$0.0194×1.062=$0.050

a)

b)

c)

46

Chapter4IntroductiontoRiskManagement

a)

CopperpriceinTotalcostPro?ton.95Pro?tonshort

Net

Hedgedpro?t

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

put

$0.25

$0.15

$0.05

$0

$1.00call

0

premium

?$0.021

?$0.021

?$0.021

?$0.021

?$0.021

?$0.021

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

$0.0710

$0.0710

$0.0710

$0.1210

$0.1210

$0.1210

0

0

0

?$0.10

?$0.20

0

0

Pro?tdiagram:

b)

CopperpriceinTotalcostPro?ton.975Pro?tonshort

Net

Hedgedpro?t

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

put

$0.275

$0.175

$0.075

$0

$1.025call

0

premium

?$0.001

?$0.001

?$0.001

?$0.001

?$0.001

?$0.001

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

$0.0760

$0.0760

$0.0760

$0.1010

$0.1260

$0.1260

0

0

0

?$0.0750

?$0.1750

0

0

47

Part1Insurance,Hedging,andSimpleStrategies

Pro?tdiagram:

c)

CopperpriceinTotalcostPro?ton1.05Pro?tonshort

Net

premium

$0.05

$0.05

$0.05

$0.05

$0.05

$0.05

Hedgedpro?t

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

put

$0.35

$0.25

$0.15

$0.05

0

$1.05call

0

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

$0.1

$0.1

$0.1

$0.1

$0.1

$0.1

0

0

0

?$0.050

?$0.150

0

Weseethatwearecompletelyandperfectlyhedged.Buyingacollarwheretheputandcallleg

haveequalstrikepricesperfectlyoffsetsthecopperpricerisk.

Pro?tdiagram:

48

Chapter4IntroductiontoRiskManagement

Question4.6.

a)

CopperpriceinTotalcostPro?tonPro?tontwolong

oneyear$0.975puts

Net

premium

Hedgedpro?t

short1.025

put

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.325

?$0.225

?$0.125

?$0.025

0

$0.55

$0.35

$0.150

$0.0022

$0.0022

$0.0022

$0.0022

$0.0022

$0.0022

$0.0228

$0.0228

$0.0228

$0.0728

$0.1978

$0.2978

0

0

0

0

Wecanseefromthefollowingpro?tdiagram(andtheabovetable)thatinthecaseofafavorable

increaseincopperprices,thehedgedpro?tisalmostidenticaltotheunhedgedpro?t.

Pro?tdiagram:

49

Part1Insurance,Hedging,andSimpleStrategies

b)

CopperpriceinTotalcostPro?tontwoPro?tonthreelong

oneyear$1puts

Net

premium

Hedgedpro?t

short1.034

put

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$0.90

$0.90

$0.90

$0.90

$0.90

$0.90

?$0.6680

?$0.4680

?$0.2680

?$0.0680

0

$0.9

$0.6

$0.3

0

$0.0002

$0.0002

$0.0002

$0.0002

$0.0002

$0.0002

$0.0318

$0.0318

$0.0318

$0.0318

$0.1998

$0.2998

0

0

0

Wecanseefromthefollowingpro?tdiagram(andtheabovetable)thatinthecaseofafavorable

increaseincopperprices,thehedgedpro?tisalmostidenticaltotheunhedgedpro?t.

Pro?tdiagram:

Question4.7.

Telcoassigneda?xedrevenueof$6.20foreachunitofwire.Itcanbuyoneunitofwirefor$5plus

thepriceofcopper.Therefore,Telco’spro?tinoneyearis$6.20less$5.00lessthepriceofcopper

afteroneyear.

50

Chapter4IntroductiontoRiskManagement

CopperpriceinoneTotalcostUnhedgedpro?tPro?tononelongHedgedpro?t

year

forward

?$0.3

?$0.2

?$0.1

0

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.50

$0.40

$0.30

$0.20

$0.10

0

$0.20

$0.20

$0.20

$0.20

$0.20

$0.20

$0.10

$0.20

Weobtainthefollowingpro?tdiagrams:

Question4.8.

Inthisexercise,weneedto?rst?ndthefuturevalueofthecallpremia.Forthe$1-strikecall,itis:

$0.0376×1.062=$0.04.Thefollowingtableshowsthepro?tcalculationsofthe$1.00-strikecall.

Thecalculationsforthetwoothercallsareexactlysimilar.The?guresonthenextpagecompare

thepro?tdiagramsofallthreepossiblehedgingstrategies.

51

Part1Insurance,Hedging,andSimpleStrategies

CopperpriceinTotalcostUnhedgedPro?tonlong

Call

Netincomeon

$1.00-strikecallpremiumhedgedpro?t

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

pro?t

$0.50

$0.40

$0.30

$0.20

$0.10

0

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

0

0

$0.04

$0.04

$0.04

$0.04

$0.04

$0.04

$0.46

$0.36

$0.26

$0.16

$0.16

$0.16

0

0

$0.10

$0.20

Weobtainthefollowingpro?tdiagrams:

Question4.9.

Forthe$1-strikeput,wereceiveapremiumof:$0.0376×1.062=$0.04.Thefollowingtable

showsthepro?tcalculationsofthe$1.00-strikeput.Thecalculationsforthetwootherputsare

exactlythesame.The?guresonthenextpagecomparethepro?tdiagramsofallthreepossible

strategies.

52

Chapter4IntroductiontoRiskManagement

CopperpriceinTotalcostUnhedgedPro?tonshortReceivedNetincomeon

oneyear

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

pro?t

$0.50

$0.40

$0.30

$0.20

$0.10

0

$1.00-strikecallpremiumhedgedpro?t

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

?$0.30

?$0.20

?$0.10

$0.04

$0.04

$0.04

$0.04

$0.04

$0.04

$0.24

$0.24

$0.24

$0.24

$0.14

$0.04

0

0

0

Weobtainthefollowingpro?tdiagrams:

Question4.10.

Telcowillsellcollars,whichmeansthattheybuythecalllegandselltheputleg.Wehaveto

computeforeachcasethenetoptionpremiumposition,and?nditsfuturevalue.Wehavefor:

$0.0376?$0.0178×1.062=$0.021

$0.0274?$0.0265×1.062=$0.001

$0.0649?$0.0178×1.062=$0.050

a)

b)

c)

53

Part1Insurance,Hedging,andSimpleStrategies

a)

CopperpriceTotalcostUnhedgedPro?tonPro?tonlong

$1.00call

Net

premium

Hedgedpro?t

inoneyear

pro?t

short.95

put

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.50

$0.40

$0.30

$0.20

$0.10

0

?$0.25

?$0.15

?$0.05

$0

0

0

$0.021

$0.021

$0.021

$0.021

$0.021

$0.021

$0.2290

$0.2290

$0.2290

$0.1790

$0.1790

$0.1790

0

0

0

$0.10

$0.20

0

Pro?tdiagram:

b)

CopperpriceTotalcostUnhedgedPro?tonshortPro?tonlong

Net

$1.025callpremium

Hedgedpro?t

inoneyear

$0.70

$0.80

pro?t

$0.50

$0.40

$0.30

$0.20

$0.10

0

.95put

?$0.275

?$0.175

?$0.075

$0

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

0

$0.001

$0.001

$0.001

$0.001

$0.001

$0.001

$0.2240

$0.2240

$0.2240

$0.1990

$0.1740

$0.1740

0

0

$0.90

$1.00

0

$1.10

$1.20

0

$0.0750

$0.1750

0

54

Chapter4IntroductiontoRiskManagement

Pro?tdiagram:

c)

CopperpriceTotalcostUnhedged

Pro?ton

short.95put

?$0.25

?$0.15

?$0.05

Pro?tonlong

$.95call

0

Net

premium

$0.05

$0.05

$0.05

$0.05

$0.05

$0.05

Hedgedpro?t

inoneyear

$0.70

$0.80

pro?t

$0.50

$0.40

$0.30

$0.20

$0.10

0

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.2

$0.2

$0.2

$0.2

$0.2

$0.2

0

$0.90

$1.00

0

0

0

0

$0.050

$0.150

$0.250

$1.10

$1.20

Weseethatwearecompletelyandperfectlyhedged.Buyingacollarwheretheputandcallleg

haveequalstrikepricesperfectlyoffsetsthecopperpricerisk.

Pro?tdiagram:

55

Part1Insurance,Hedging,andSimpleStrategies

Question4.11.

a)

CopperpriceTotalcostUnhedgedPro?tonPro?tontwo

pro?t

Net

short0.95long$1.034premium

Hedgedpro?t

inoneyear

call

0

calls

0

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.50

$0.40

$0.30

$0.20

$0.10

0

?$0.0015

?$0.0015

?$0.0015

?$0.0015

?$0.0015

?$0.0015

$0.5015

$0.4015

$0.3015

$0.1765

$0.1085

$0.1085

0

0

0

0

?$0.025

?$0.125

?$0.225

0

$0.13200

$0.3320

Wecanseefromthefollowingpro?tdiagram(andtheabovetable)thatinthecaseofafavorable

decreaseincopperprices,thehedgedpro?tisalmostidenticaltotheunhedgedpro?t.

Pro?tdiagram:

56

Chapter4IntroductiontoRiskManagement

b)

CopperpriceTotalcostUnhedgedPro?ton2Pro?tonthree

inoneyearpro?t

Net

premium

Hedgedpro?t

short$1

call

long$1.034

calls

$0.70

$0.80

$0.90

$1.00

$1.10

$1.20

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.50

$0.40

$0.30

$0.20

$0.10

0

0

0

?$0.0024

?$0.0024

?$0.0024

?$0.0024

?$0.0024

?$0.0024

$0.5024

$0.4024

$0.3024

$0.2024

$0.1004

$0.1004

0

0

0

0

0

?$0.200

?$0.400

0

$0.1980

$0.4980

Wecanseefromthefollowingpro?tdiagram(andtheabovetable)thatinthecaseofafavorable

decreaseincopperprices,thehedgedpro?tisalmostidenticaltotheunhedgedpro?t.

Pro?tdiagram:

Question4.12.

Thisisaveryimportantexercisetoreallyunderstandthebene?tsandpitfallsofhedgingstrategies.

Wirconeedscopperasaninput,whichmeansthatitscostsincreasewiththepriceofcopper.We

maythereforethinkthattheyneedtohedgeagainstincreasesinthecopperprice.However,we

mustnotforgetthatthepriceofwire,thesourceofWirco’srevenues,alsodependspositivelyon

57

Part1Insurance,Hedging,andSimpleStrategies

thepriceofcopper:thepriceWircocanobtainforoneunitofwireis$50plusthepriceofcopper.

Wewillseethatthosecopperpriceriskscanceleachotherout.Mathematically,

Wirco’scostperunitofwire:

$3+$1.50+ST

Wirco’srevenueperunitofwire:$5+ST

andSTisthepriceofcopperafteroneyear.Therefore,wecandetermineWirco’spro?tsas:

Pro?t=Revenue–Cost=$5+S?$3+$1.50+S=$0.50

T

T

Weseethatthepro?tsofWircodonotdependonthepriceofcopper.Costandrevenuecopper

priceriskcanceleachotherout.Ifwebuyinthissituationalongforwardcontract,wedoinfact

introducecopperpricerisk!Tounderstandthis,addalongforwardcontracttothepro?tequation:

Pro?twithforward:=$5+S?$3+$1.50+S+S?$1=S?$0.50

T

T

T

T

Tosummarize,

CopperpriceTotalcostTotalrevenueUnhedgedPro?tonlongNetincomeon

inoneyear

$0.70

$0.80

pro?t

$0.50

$0.50

$0.50

$0.50

$0.50

$0.50

forward

?$0.30

?$0.20

?$0.10

0

‘hedged’pro?t

$0.20

$0.30

$5.20

$5.30

$5.40

$5.50

$5.60

$5.70

$5.70

$5.80

$5.90

$6.00

$6.10

$6.20

$0.90

$1.00

$0.40

$0.50

$1.10

$1.20

$0.10

$0.20

$0.60

$0.70

Question4.13.

Wedoinfactintroducecopperpricerisknomatterwhatstrategyweundertake.Therefore,nomatter

whichinstrumentweareusing,weincreasethepricevariabilityofWirco’spro?ts.Althoughthisis

asimpleexample,itisimportanttokeepinmindthatacompany’sriskmanagementshouldalways

takeplaceonanaggregatelevel,becauseotherwisecounterbalancingpositionsmaybehedged

twice.

Question4.14.

Hedgingshouldneverbethoughtofasapro?tincreasingaction.Acompanythathedgesmerely

shiftspro?tsfromgoodtobadstatesoftherelevantpriceriskthatthehedgeseekstodiminish.

Thevalueofthereducedpro?ts,shouldthegoldpricerise,subsidizesthepaymenttoGolddiggers

shouldthegoldpricefall.Therefore,acompanymayuseahedgeforoneofthereasonsstated

inthetextbook;however,itisnotcorrecttocomparehedgedandunhedgedcompaniesfroman

accountingperspective.

58

Chapter4IntroductiontoRiskManagement

Question4.15.

Iflossesaretaxdeductible(andthecompanyhasadditionalincometowhichthetaxcreditcanbe

applied),theneachdollaroflossesbearsataxcreditof$0.40.Therefore,

Price=$9Price=$11.20

(1)Pre-TaxOperatingIncome

(2)TaxableIncome

(3)Tax@40%

?$1

0

$1.20

$1.20

$0.48

0

0

(3b)TaxCredit

$0.40

?$0.60

After-TaxIncome(includingTaxcredit)

$0.72

Inparticular,thisgivesanexpectedafter-taxpro?tof:

E[Pro?t]=0.5×?$0.60+0.5×$0.72

=$0.06

andtheinef?ciencyisremoved:Weobtainthesamepayoffsasinthehedgedcase,Table4.7.

Question4.16.

a)

Expectedpre-taxpro?t

FirmA:E[Pro?t]=0.5×$1,000+0.5×?$600=$200

FirmB:E[Pro?t]=0.5×$300+0.5×

=$200

$100

Both?rmshavethesamepre-taxpro?t.

b)

Expectedaftertaxpro?t.

FirmA:

badstategoodstate

(1)Pre-TaxOperatingIncome

(2)TaxableIncome

(3)Tax@40%

?$600

$1,000

$1,000

$400

0

$0

0

(3b)TaxCredit$240

After-TaxIncome(includingTaxcredit)?$360

$600

Thisgivesanexpectedafter-taxpro?tfor?rmAof:

E[Pro?t]=0.5×?$360+0.5×

=$120

$600

59

Part1Insurance,Hedging,andSimpleStrategies

FirmB:

badstategoodstate

(1)Pre-TaxOperatingIncome

(2)TaxableIncome

(3)Tax@40%

$100

$100

$40

0

$300

$300

$120

0

(3b)TaxCredit

After-TaxIncome(includingTaxcredit)

$60

$180

Thisgivesanexpectedafter-taxpro?tfor?rmBof:

E[Pro?t]=0.5×$60+0.5×

=$120

$180

Iflossesreceivefullcreditfortaxlosses,thetaxcodedoesnothaveaneffectontheexpectedafter-tax

pro?tsof?rmsthathavethesameexpectedpre-taxpro?ts,butdifferentcash-?owvariability.

Question4.17.

a)

Thepre-taxexpectedpro?tsarethesameasinexercise4.16.a).

b)Whiletheafter-taxpro?tsofcompanyBstaythesame,thoseofcompanyAchange,because

theydonotreceivetaxcreditonthelossanymore.

c)

Wehavefor?rmA:

badstate

?$600

$0

goodstate

$1,000

$1,000

$400

(1)Pre-TaxOperatingIncome

(2)TaxableIncome

(3)Tax@40%

0

(3b)TaxCredit

notaxcredit

?$600

0

$600

After-TaxIncome(includingTaxcredit)

Andconsequently,anexpectedafter-taxreturnfor?rmAof:

E[Pro?t]=0.5×?$600+0.5×

=$0

$600

CompanyBwouldnotpayanything,becauseitmakesalwayspositivepro?ts,whichmeansthat

thelackofataxcreditdoesnotaffectthem.

CompanyAwouldbewillingtopaythediscounteddifferencebetweenitsafter-taxpro?tscalculated

in4.16.b),anditsnewafter-taxpro?ts,$0from4.17.Itisthuswillingtopay:$120÷1.1=$109.09.

Question4.18.

AuricEnterprisesisusinggoldasaninput.Therefore,itwouldliketohedgeagainstpriceincreases

ingold.

60

Chapter4IntroductiontoRiskManagement

a)Thecostofthiscollartodayisthepremiumofthepurchased440-strikecall($2.49)lessthe

premiumforthesold400-strikeput.Wecalculateacostof$2.49?$2.21=?$0.28,whichmeans

thatAuricinfactgeneratesarevenuefromenteringintothiscollar.

b)

Agoodstartingpointarethevaluesofparta).Youseethatbothputandcallareworth

approximatelythesame,thereforestartshrinkingthe440–400spansymmetricallyuntilyougeta

differenceof30,andthendosometrialanderror.Thisshouldbringyouthefollowingvalues:

Thecallstrikeis435.52,andtheputstrikeis405.52.Bothcallandputhaveapremiumof$3.425.

Question4.19.

Aswebuythecall,wewillbuyitattheask,whichis$0.25abovetheBlack-Scholesprice,and

weselltheputatthebid,whichis$0.25belowtheBlack-Scholesprice.Ournewequalpremium

conditionis:C+$0.25–(P?$0.25)=0,orC+$0.50–P=0.Sinceweknowthatthe

valueofacallisdecreasinginthestrike,andweneedaBlack-Scholescallpricethatis$.50less

valuablethantheBlack-Scholesput,weknowthatwehavetolookforapairofhigherstrikeprices.

Trialanderrorbringsustoacallstrikeof436.53,andaputstrikeof406.53.TheBlack-Scholes

callpremiumis$3.1938,andtheputhasapremiumof$3.6938.

Question4.20.

a)Sinceweknowthatthevalueofacallisdecreasinginthestrike,andweneedtoselltwocall

options,theBlack-Scholespricesofwhichequalthe440-strikecallprice,weknowthatwehave

61

Part1Insurance,Hedging,andSimpleStrategies

tolookforahigherstrikeprice.Trialanderrorresultsinastrikepriceof448.93.Thepremiumof

the440-strikecallis$2.4944,andindeedtheBlack-Scholespremiumofthe448.93strikecallis

$1.2472.

b)

Pro?tdiagram:

Question4.21.

Ifyoudonotknowhowtorunaregression,orifyouforgotwhataregressionis,youmaywant

totypethekeyword“regression”inMicrosoftExcel’shelpmenu.Itwillshowyouhowtoruna

regressioninExcel,aswellasexplaintoyouthekeyfeaturesofaregression.

Runningaregression,weobtainaconstantof2,100,000andacoef?cientonpriceof100,000.

Question4.22.

a)

Wehavethefollowingtable:

PriceQuantityRevenue

3

3

2

2

1.5

0.8

1

4.5

2.4

2

0.6

1.2

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Chapter4IntroductiontoRiskManagement

UsingExcel’sfunctionSTDEVP(4.5,2.4,2,1.2),weobtainavalueof1.2194forthestandarddevi-

ationoftotalrevenueforScenarioC.

b)

Usinganystandardsoftware’scommand(ordoingitbyhand!)todeterminethecorrelation

coef?cient,weobtainavalueof0.7586.

Question4.23.

a)

Usingequation(4.7)andthevaluesofthecorrelationcoef?cientandstandarddeviation

oftherevenuewecalculatedinquestion4.22.,weobtainthefollowingvalueforthevariance

minimizinghedgeratio:

H=?0.7586×1.2194=1.85007

0.5

Itisthusoptimaltoshort1.85millionbushelsofcorn.

b)

Ifyoudonotknowhowtorunaregression,orifyouforgotwhataregressionis,youmay

wanttotypethekeyword“regression”inMicrosoftExcel’shelpmenu.Itwillshowyouhowto

runaregressioninExcel,aswellasexplaintoyouthekeyfeaturesofaregression.

Runningsucharegression,weobtainaconstantof?2,100,000andacoef?cientonpriceof

1,850,000,thusyieldingthesameresultsasparta).

c)

PriceQuantityUnhedgedRevenueFuturesGain

Total

3

3

2

2

1.5m

0.8m

1m

4.5m

2.4m

2m

?0.5×1.85m3.575m

=?0.925m

?0.5×1.85m1.475m

=?0.925m

+0.5×1.85m2.925m

=+0.925m

0.6m

1.2m

+0.5×1.85m2.125m

=+0.925m

UsingExcel’sfunctionSTDEVP(3.575,1.475,2.925,2.125),weobtainavalueof0.7945forthe

standarddeviationoftheoptimallyhedgedrevenueforScenarioC.Weseethatwewereableto

signi?cantly