gupea_2077_19299_1

AnempiricalevaluationofValueatRisk

MasterThesisIndustrialandfinancialmanagementUniversityofGothenburg,SchoolofBusiness,Economics&LawJanuary2009Instructor: ZiaMansouriAuthors: MartinGustafsson1982 CarolineLundberg1984

AnempiricalevaluationofValueatRiskGustafsson&Lundberg

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ABSTRACT

In lightoftherecentfinancialcrisis,riskmanagementhasbecomeaverycurrent issue.Oneofthe

most intuitiveandcomprehendibleriskmeasures isValueatRisk(VaR).VaRputsamonetaryvalue

on the risk that arises from holding an asset and is defined as the theworst loss over a target

horizonwithagiven levelofconfidence.Therearecurrentlynumerous techniques forcalculating

VaRonthemarketandnostandardissetonhowitshouldbedone.ThiscanmakethefieldofVaR

hardtooverlookforsomeonenotinitiatedintheworldofeconometrics.

InthispaperweexaminethreebasicwidelyusedapproachesusedtocalculateVaR.Theapproaches

examinedarethenonHistoricalSimulationapproach,theGARCHapproachandtheMovingAverage

approach. This paper has twomain purposes, the first is to test the different approaches and

comparethemtoeachotherintermsofaccuracy.Thesecondistoanalyzetheresultsandseeifany

conclusions can be drawn from the accuracy of the approach with respect to the return

characteristicsoftheunderlyingassets.TheaccuracyofaVaRapproach istestedbythenumberof

VaRbreaks that itproduces i.e. thenumberof times that theobserved asset returnexceeds the

predictedVaR.ThenumberofVaRbreaksisevaluatedwiththeKupiectestwhichdefinesaninterval

of VaR breaks in which the approachmust perform to be accepted. By the term asset return

characteristicswemean the statistical properties of the returns of the assets such as volatility,

kurtosisandskewness.

The study is conducted on three fundamentally different assets, Brent crude oil, OMXs30 and

Swedish threemonths treasurybills.Daily returndatahasbeencollectedstarting from January1st

1987 toSeptember30th2008 forall theassets.Observations spanning from1987 to1995willbe

usedashistoricalinputfortheapproachesandobservationsspanningfrom1996toSeptember30th

2008willbethecomparativeperiodinwhichtheapproachesperformancewillbemeasured.

Theresultsofthestudyshowthatnoneofthethreeapproachesaresuperiortotheothersandthat

more complex approaches do not guarantee more accurate results. Instead it seems that the

characteristicsoftheassetreturnsincombinationwiththedesiredconfidenceleveldeterminehow

well a certain approach performs on a certain asset.We show that the choice of VaR approach

shouldbeevaluatedindividuallydependingontheassetstowhichitistobeapplied.

Keywords: Value at Risk, return characteristics, historical simulation, moving average, GARCH,

normaldistribution,Brentoil,OMXs30,Swedishtreasurybills.


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GLOSSARYOFMAINTERMS

Autocorrelation:Isthecorrelationbetweenthereturnsatdifferentpointsintime.

Backtesting:Theprocessoftestingatradingstrategyonpriortimeperiods.

Fattails:Tailsofprobabilitydistributionsthatarelargerthanthoseofnormaldistribution.

GARCHapproach toVaR:Anapproach for forecastingvolatilities thatassumes thatvolatilitynext

perioddependsonlaggedvolatilitiesandlaggedsquaredreturns.

Historical simulation approach to VaR: An approach that estimates VaR from a profit and loss

distributionsimulatedusinghistoricalreturnsdata.

Kupiectest:IsavaluationmethodtoevaluateVaRresults.

Kurtosis:Measuresthepeakednessofadatasample.Ahighvalueofkurtosismeansthatmoreofthe

datasvariancecomesfromextremedeviations.

MovingaverageapproachtoVaR:Aparametricapproachthatassumesnormaldistributedreturns

anddisregardsthefactofvolatilityclustering.

Normaldistribution:TheGaussianorbellcurveprobabilitydistribution.

Outlier:Anextremerarereturnobservation.

Risk:Theprospectofgainandloss.Riskisusuallyregardedasquantifiable.

Skewness:Tellsuswhetherthedataissymmetricornot.

ValueatRisk(VaR):Themaximumlikelylossoversomeparticularholdingperiodataparticularlevel

ofconfidence.

Volatility:Thevariabilityofaprice,usuallyinterpretedasitsstandarddeviation.

Volatility clustering:High volatility observations seems to be clusteredwith other highvolatility

observationsandthesameistrueforlowvolatilityobservationsthatclusterwithotherlowvolatility

observations.


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LISTOFFIGURES

Figure1.5 Dispositionofthispaper.

Table2.3.2 Relatingstandarddeviationtochosenconfidencelevel.

Figure3.2.3a MLEestimationgraphforBrentOil.

Figure3.2.3b MLEestimationgraphforOMXs30.

Figure3.2.3c MLEestimationgraphforSTB3M.

Figure3.3 Graphshowinganegativerespectivelyapositiveskew.

Figure3.4 ProbabilitydensityfunctionforthePearsontypeVIIdistributionwithkurtosisofinfinity(red);2(blue);and0(black).

Table3.5 Statisticalcharacteristicsoftheassetreturns.Figure3.6a DailyreturnsforBrentoil.

Figure3.6b Histogramshowingthedailyreturnscombinedwithanormaldistributioncurve.

Figure3.7a DailyreturnsforOMXs30.


Figure3.8a DailyreturnsforSTB3M.


Table3.9 Resultsfromautocorrelationtest.

Table3.10 TargetnumberofVaRbreaksandKupiectestintervals.

Table4.1.1a Backtestingresultswithhistoricalsimulationapproach.

Figure4.1.1b VaRforBrentOil,HistoricalSimulation99.9%confidence19962008.

Table4.1.2 Backtestingresultswithmovingaverageapproach.

Table4.1.3 BacktestingresultswithGARCHapproach.

Table5.1 Summaryofbacktestingresults.


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TABLEOFCONTENTS

1.INTRODUCTION...................................................................................................................................7

1.1WHATISVALUEATRISK................................................................................................................7

1.2BACKGROUNDOFVaR...................................................................................................................8

1.3PROBLEMDISCUSSION..................................................................................................................8

1.4PURPOSE......................................................................................................................................11

1.5DISPOSITION................................................................................................................................11

2.THEORY..............................................................................................................................................12

2.1VALUEATRISKVaR....................................................................................................................12

2.2THENEEDFORVaR......................................................................................................................13

2.3VaRAPPROACHES........................................................................................................................14

2.3.1HISTORICALSIMULATIONAPPROACH..................................................................................14

2.3.2MOVINGAVERAGEAPPROACH............................................................................................15

2.3.3GARCHAPPROACH...............................................................................................................17

2.4THEUNDERLYINGASSETS............................................................................................................18

2.4.1BRENTOIL.............................................................................................................................18

2.4.2OMXs30................................................................................................................................18

2.4.3THREEMONTHSWEDISHTREASURYBILLS..........................................................................18

2.5BACKTESTINGWITHKUPIEC........................................................................................................19

3.METHOD............................................................................................................................................20

3.1ANALYTICALAPPROACH..............................................................................................................20

3.1.1QUANTITATIVEAPPROACH..................................................................................................20

3.1.2DEDUCTIVEAPPROACH........................................................................................................20

3.1.3RELIABILITY...........................................................................................................................21

3.1.4VALIDITY...............................................................................................................................21

3.2CALCULATIONOFVaR.................................................................................................................21



3.2.3GARCHAPPROACH...............................................................................................................23

3.3SKEWNESS...................................................................................................................................25


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3.4KURTOSIS.....................................................................................................................................26

3.5THESOURCEOFDATA.................................................................................................................26

3.6BRENTOIL....................................................................................................................................27

3.7OMXs30.......................................................................................................................................29

3.8THREEMONTHSWEDISHTREASURYBILLS.................................................................................30

3.9AUTOCORRELATION....................................................................................................................31

3.10BACKTESTINGWITHKUPIEC......................................................................................................32

4.RESULTS&ANALYSIS.........................................................................................................................33

4.1BACKTESTINGRESULTS................................................................................................................33



4.1.3GARCHAPPROACH...............................................................................................................37

4.2EXACTNESS&SIMPLICITY............................................................................................................38

4.3VALIDITY......................................................................................................................................39

5.CONCLUSIONS...................................................................................................................................41

5.1CONCLUSIONS.............................................................................................................................41

5.2FURTHERRESEARCH....................................................................................................................43

REFERENCES..........................................................................................................................................44

LITERARYSOURCES............................................................................................................................44

INTERNETSOURCES...........................................................................................................................45

APPENDIX..............................................................................................................................................46


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1.INTRODUCTION

Thischaptergivesabackgroundtothesubjecttreated inthisstudyValueatRisk. Itfurtherstates

theproblems associatedwith the calculationofValue atRisk and thepurposeof thispaper.The

chapterendswiththedispositionofthepaper.

1.1WHATISVALUEATRISK

Toanswerthisquestionwecanstartoffbyaskingourselvesthemorefundamentalquestionofwhat

risk is. Ineconomicstherearemanydifferenttypesofrisksuchas legalrisk,marketrisk,company

specificriskandsoon.Buthowdowedefinewhatrisk isandhowdowenumericallymeasure its

size? Infinance,risk isoftenthoughtofasthevolatilityorstandarddeviationofanassetsreturns.

Butallvolatilityreallytellsusishowmuchthereturnsofanassetvariesarounditsmean.Thislabels

bothpositive andnegativepricemovements as risk, althoughmost investorswould consider risk

somethingnegativeandupwardpricemovementsassomethingpositive(Jorion2001).

It isnotavery intuitivemeasureand canbehard tograsp for thosenot initiated in theworldof

finance.Whatthevalueatriskmeasurementdoesisthatitputsanumber,eitheramoneyvalueora

percent value of the worst loss over a target horizon with a given level of confidence (Jorion

2001:22).ValueatriskorVaRaswewilladdressitfromhereonthusconsistsofthreecomponents,a

confidencelevel,atimehorizonandavalue.Thetimecomponenttellsushowfarintothefuturewe

are looking,thefurtherwe lookthe largerthepotential loss.Theconfidence leveldetermineswith

whichcertainty themeasurement ismade,higherconfidence levelsmeanshigherpotential losses.

Finally,thevaluecomponentsimply is themonetaryvalue thatwerisk losingduringtheproposed

timehorizongiventheproposedconfidencelevel.

Forexamplea$1000,oneday,95percentconfidencelevelVaRvalueforastockmeansthatduring

thenextdaywecanbe95percentcertainthatthevalueofourholdingsinthisparticularstockwill

notdecreasebymorethan$1000.

AneventwherethereturnofanassetexceedstheestimatedVaRmeasureiscalledaVaRbreak.The

chosenconfidenceleveloftheVaRapproachdetermineshowmanyVaRbreaksanapproachshould

produceifperformingwell.TheaimofaVaRapproachisthereforenottoproduceasfewVaRbreaks

aspossible.AnaccurateVaRapproachproducesanumberofVaRbreaksascloseaspossibletothe

number of VaR breaks specified by the confidence level. Therefore if a VaR approachwith 95%

confidenceiscalculateditshouldproduceVaRmeasuresthatareexceededbythereturnoftheasset

fivepercentofthetimesitisapplied.ThusifVaRpredictionsaremadefor1000individualdaysthe

estimatedVaRshouldbeexceeded50times.AnydeviationfromtheexpectednumberofVaRbreaks

regardlessifitisnegativeorpositiveisasignofinaccuracyormissspecification.


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1.2BACKGROUNDOFVaR

Riskmanagement isan importantpart forall financial institutionsandalso forcompanies thatare

exposedtorisk.Duringthe lastdecadetherehasbeenarevolution inriskmanagementandVaR is

oneofthemeasuresthathavegottena lotofattention.Holton(2003)writesthattherootsofVaR

howeverdatebacktoasearlyas1922,atwhichtimeTheNewYorkStockExchangeimposedcapital

requirementsonmemberfirms.

ResearchonVaRdidhowevernot takeoffuntil1952when two researchers,MarkowitzandRoy,

almostsimultaneouslybutindependentofeachotherpublishedquitesimilarmethodsofmeasuring

VaR.According toHolton (2003), theywereworkingondevelopingameansofselectingportfolios

thatwouldbeabletooptimizetherewardforagiven levelofrisk.Hefurtherpointsoutthatafter

that ittookanother40yearsuntiltheVaRmeasurementbegantobewidelyusedamong financial

institutionsandcompanies.

AccordingtoFernandez (2003),past financialdisasterssuchastheone in1987andthecrisesthat

followedledtoadecisionbytheBaselCommitteethatallbanksshouldkeepenoughcashtobeable

tocoverpotential losses in their tradingportfoliosovera tendayhorizon,99percentof the time.

TheamountofcashtobekeptwastobecalculatedusingVaR.Pastcriseshasshownthatenormous

amounts of money can be lost over a day as a result of poor supervision and financial risk

management.TheVaRmeasurethereforegotitsbreakthroughbecauseofhistoricalmistakesinrisk

management.

Today,theutilizationofVaRiswidelyspreadinfinancialinstitutions,howeveritisnotaswidespread

in non financial firms. This can be explained by the fact that non financial firms do not usually

forecastprofitsand lossesonadailybasis.NeverthelessMauro(1999)pointsoutthatVaRcanbe,

andisused,innonfinancialfirmsthatareaffectedbyvolatilityinprices,particularlyinashorttime

horizon.

AnadvantageofVaRisthatitisameasurethatcanbeappliedtoalmostanyasset.Adisadvantage

however,isthatthereisalmostaninfinitenumberofwaystocalculateVaR,eachoftheapproaches

with its own advantages and disadvantages. This makes different VaR measurements hard to

comparewitheachotheriftheyhavenotbeencalculatedusingthesameapproach.

1.3PROBLEMDISCUSSION

ValueatRisk isameasurethat iswidespreadwithin financial institutionswherethe importanceof

strictriskmanagementhasbecomevital.AccordingtoJorion (2002)VaRhasbecomethestandard

benchmarkformeasuringfinancialrisk.Bankswithlargetradingportfoliosandinstitutionsthatdeal

withnumeroussourcesoffinancialriskhavebeenheadingtheuseofriskmanagement.Jorion(2001)

writesthattoputanumberonthepossiblemaximumlossunderaspecifictimehorizonhasformany

banksbecomeanobligation.Anexampleof this isLesleyDanielsWebster, formerheadofmarket

riskatChaseManhattanBank,whosawitasanecessity.Everymorninghereceiveda30pagereport


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that summarized theVaRof thebank. Theneat little reportquantified the riskof all the trading

positionsofthebank.VaRisanimportantmeasurewithinriskmanagementbecauseitisanintuitive

riskmeasure as it puts amonetary number on the risk exposure a company faces.According to

Linsmeier&Pearson (1996) thestrengthsofVaR is that itprovidesasimplerandaccurateoverall

measure of themarket risk the company is taking. This is donewithout going into specific and

complexdetailsaboutthecompanyspositions.Thisisagreatadvantagesinceriskassessmentsare

mostoftenproducedforseniormanagementwithnoorlittleexperiencewitheconometrics.

Theredoesnot,however,appeartobeanyconsensusonhowtocalculateVaR.Todaythereseems

to be asmanyways to calculateVaR as there are practitioners using it. There are some general

approaches tochoose from,suchasparametric,semiparametricornonparametricapproaches.A

parametric approach means that you use a parameter or a model to describe reality, often

generalizing andmaking assumptions.Anonparametricapproachhowevermay forexampleonly

lookathistoricaldatatomakepredictionsaboutthefuture.Undereachofthesecategoriesthereare

manydifferentapproachesthatcanbeusedtocalculateVaRandeachandeveryoneofthemcanin

theirturnbeappliedindifferentwaysundervariousassumptionsorgeneralizations.Inthisstudythe

historicalsimulationapproachwillberepresentingthegroupofnonparametricapproaches.Moving

averageandGARCHwillberepresentingtheparametricapproaches.TheGARCHapproachconsiders

volatility clustering i.e. the fact that days with high volatility are often clustered together. The

approacheshavebeenchosenbecausetheyarethemostwidelyusedapproachesofcalculatingVaR.

Thereareseveralearlierstudiesonthesubject,howevermostofthemaimtodevelopnewandmore

advancedwaysofcalculatingVaR.LinsmeierandPearson (1996)concludes that there isnosimple

answertowhichVaRapproachisthebest.Theyalldifferintheirabilityofcapturingriskandfurther

theydifferintheireaseofimplementationandtheeaseofexplanationtoseniormanagement.They

state that the historical simulation approach is easy to implement but it does require a lot of

historical data. It is also an approach that is easy to explain to seniormanagementwhich is an

essential part of the purpose of theVaRmeasure. In another studymade byGoorbergh&Vlaar

(1999)themostimportantcharacteristicofstockreturnswhenusingVaRisvolatilityclustering.This

isaphenomenomthatcanbeeffectivelymodeledbymeansofGARCH.

Thestudiesmade in thepastarenotunanimous, theyproducecontradictory results.Forexample

Cabedo&Moya (2003) finds thatVaRs fromanautoregressivemovingaverageapproach (ARMA)

outperforms theGARCH.Costelloetal. (2008) finds that theopposite is true,GARCHoutperforms

the ARMA and they suggest that the conclusion drawn by Cabedo & Moya is based on the

assumptionofnormaldistribution.Thedifferentapproachesareappliedondifferentassetsandthe

approachesareoftenadaptedtotheassetstowhichtheyareapplied.Thisindicatethattheresults

fromdifferentstudiescanbehard to interpretandcompare.We therefore think that itwouldbe

interesting to conduct a study by using the threemostwidely used approaches. Theywill all be

appliedonthesamethreeassetstomaketheresultsmorecomparable.

TheVaRwillbe estimatedon adailybasisusing threedifferent confidence levels,95%,99% and

99.9%toseehowtheapproachesperformatdifferentconfidence levelsondifferentassets.Jorion


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(2001)andmanyotherswithhimusestheseconfidence levelswhencalculatingVaR.Thedifferent

levels fitdifferentpurposesanddependson themanagements relation to risk.Choosingahigher

levelofconfidencewill result inahigherVaR.Thedifferencebetweenaconfidence levelof99or

99.9percentmightnotseembigbuthasalargeeffectontheVaRassessment.

Differentcharacteristicsoftheassetsreturns,i.e.thestatisticalpropertiesoftheassetsreturnssuch

asvolatility,kurtosisandskewness,mightaffectthecalculatedVaRmakingsomeapproachesmore

preferablewithspecificassets.This isespecially truewith theparametricapproaches thatassume

thereturnstobenormallydistributed.This leadstotheproblemthatthisstudy isfocusedon,the

connection between an assets return characteristics and its effect on the calculated VaR. Three

different assets have been chosen; Brent crude oil, Swedish stockmarket index (OMXs30) and

Swedishthreemonthtreasurybills(STB3M).Themotivesforchoosingtheseassetsarethattheyare

fundamentallydifferentassetswithfundamentallydifferentcharacteristics.Thereasonwhythisisso

important in this study is that it provides a better prerequisite to examine how the return

characteristicsaffectVaRapproaches.

VaRcanbeappliedonanyassetthathasameasurablereturn.Theassetsusedinthisstudyallhave

incommonthattheirreturnscaneasilybemeasured,theircharacteristicsdohoweverdiffer.Brent

oilhasmanyusesonebeingpetrolproductionandanotherisheatingpurposes.Oilcompaniessitting

on largereservesareverysensitivetochanges inoilpricesmakingVaRasuitableriskmeasurefor

assessingpotentialfuturelosses.Thesamegoesforanyonetradingwithoilregardlessofthembeing

buyersorsellers.Theoilmarketdiffersfromthestockmarketinthesenseofbuyers.Oilistradedin

hugeamountsby largeoilcompanies incontrasttostocksthatcanbetraded insmallquantitiesby

small investors. Stocks are perhaps themost common asset used in VaR calculations. The risks

connected to stock returns are quite easy to understand. Financial institutions invest in huge

portfoliosconsistingofvariousstockswithvaryingrisks.Themarketrisktheyfacemustbeevaluated

and according to Jorion (2002) this isbestdonebyusingVaR.When it comes to treasurybills it

should represent a stable and secure investment instrument. Fluctuating interest rates however

affects themarketvalueof thebills.Thismeans that themarketvalueof thebillscanvaryduring

theirholdingtimewhichmakesVaRausefulmeasureinordertoestimatepotentiallosses.

HowdothedifferentcharacteristicsofthereturnsoftheunderlyingassetsaffectVaR?AnotherimportantaspectisthecomplexityoftheVaRapproachesthemselves.Wewilltakeacloser

lookon theperformancedifferencebetweenmorecomplexandsimplerapproaches tocalculating

VaR. Thiswill result in an evaluation ofwhether the extra effort put in to the use of themore

complexapproachesisworththewhileintermsofaccuracy.Theaccuracyoftheapproacheswillbe

measuredintermsofthenumberofVaRbreakstheyproduce

WhichoftheapproachesgivesthemostaccurateVaRmeasure?


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1.4PURPOSE

Thepurposeof this study is tocompare threedifferentapproaches tocalculatingVaRnamely the

Historical Simulation approach, the Moving Average approach and the GARCH approach. The

approacheswillbeappliedon the threedifferentassetswith theconfidence levels95%,99%and

99.9%.Theperformanceandaccuracyoftheapproacheswillthenbeanalyzedwithrespecttoasset

returncharacteristicsandcomplexityoftheapproach.

1.5DISPOSITION

Introduction. The first chapter presents the subject examined in this

studyandgivesanintroductiontoVaRasameasureinriskmanagement

andhowitcanbeused.

Theory. In the upcoming chapter, the development and functions of

ValueatRiskaredescribedalongwiththechosenmethodsofcalculating

ValueatRisk. It isplacedbefore themethodchapter inorder togivea

better understanding for the approaches usedmaking the calculations

moreunderstandable.

Method.Thethirdchapterdealswiththemethodusedinthestudy.First,

the analytical approach is described and then we move on to the

methodsusedtocalculatetheValueatRisk.Thedataoftheassets,which

Value at Risk are calculated on, is also presented and their return

characteristicsareillustratedanddescribed.

Results&Analysis. In the fourth chapter, the results are presented in

formof theproducedValueatRiskestimations.Back testing ismade in

order to see how the methods have performed. It also contains the

analysiswheretheresultsareanalyzedtoseehowthecharacteristicsof

theassetreturnsaffectthecalculatedValueatRisk

Conclusions. In the fifth chapter, we present the conclusions that we

havedrawnfromthecalculationsanddiscussfurtherresearch.

Figure 1.5 Disposition of thispaper.


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2.THEORY

InthischapterwewillpresenttheriskmeasureValueatriskandthedifferentVaRapproachesthat

wewill be using in this thesis alongwith themathematicalmodels that they are based on. The

underlyingassetswillalsobepresented.

2.1VALUEATRISKVaR

Asmentionedearlier thedefinitionofVaRby Jorion (2001:22) is:VaR summarizes theworst loss

overatargethorizonwithagivenlevelofconfidence.Themostcommonapproachistheparametric

underwhichthemathematicsofVaRcanbedescribedbythefunctionbelow.

C $

The daily standard deviation times the number of standard deviations that corresponds to the

selectedconfidencelevel,C,timesthesquarerootofthetimehorizontimesthemonetarysizeofthe

investmentresultsintheVaR.InthisthesiswewillbecalculatingonedayVaRestimatessothetime

component can be excluded. Also, the monetary size of the investment is just there to put a

monetaryfigureontheriskandcanalsobeexcluded.Whatwehaveleftisthestandarddeviationof

historicalreturnsandtheconfidencelevel.

Ifchoosinga lowerconfidence level,as95percentor90percent, theVaRwilldecrease.Different

levelswillfitdifferentfirmsandpurposesandwillbechosenaccordingtothemanagementsrelation

torisk.Themoreriskaversethefirmisthehigherconfidencelevelwillbeselected.HoweverDowd

(1998) claims that VaR users and than in first hand banks will inmost cases prefer the lower

confidencevalueinordertodecreasethecapitalneededtocoverthepotentialfuturelosses.

InordertoprovideanoverviewofthemeasureVaR,asimpleillustrationisgiven.Supposethatthe

oilpricetodayis100USDperbarrelandthatthedailystandarddeviation()is20USD.Companies

thatbuylargeamountsofoilmightwanttoknowhowmuch,givenacertainconfidencelevel,they

canpossiblylosewhenbuyingtheoiltodaycomparedtotomorrow.Ifthechosenconfidenceinterval

is 99 percent, thatwouldmean that one day out of hundred the losswill be greater than the

calculatedVaR.This is truewhen thepricechange isnormaldistributedaround theaverageprice

change.

Valueduetoanincreaseintheoilprice:100 2.33 146.6USD

Valueduetoandecreaseintheoilprice:100 2.33 53.4USD

Thismeansthatwith99percentchancethelosswillnotbegreaterthan10053.4=46.6USDwhich

istheVaRforaconfidencelevelof99percent.

Often some assumption aremade inorder to calculate theVaR and ifwe suppose that thedaily

changesinoilpriceshasadensityfunctionthisfunctionismostoftenassumedtoberepresentedby


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the normal distribution. The assumption has the benefit of making the VaR estimations much

simpler. However it has some disadvantages. The price changes do not always fit the normal

distribution curve and when more observations are found in the tails, normalbased VaR will

understate the losses that canoccur.A solution couldbe touseanotherdistribution that regards

fatter tails.The fatter tails that forexamplecomeswith the tdistributionmakeshigh lossesmore

commonaccordingtoDowd(1998)andthereforeitalsogiveshigherVaR.

When usingVaR, the assets have to be valued at theirmarket value. Thiswill not be a problem

accordingtoPenza&Bansal(2001)withassetstradedinaliquidmarket,wherepriceseasilycanbe

derived.ThisiscalledmarkingtomarketandDowd(1998)definesthetermasthepracticeofvaluing

andfrequentlyrevaluingpositionsinmarketablesecuritiesbymeansoftheircurrentmarketprices.

Financialassetsareoften tradedonadailybasisand therefore, it iseasier toobtain theircurrent

value.Forsomenonfinancialassets,thepricemayhavetobeestimated,whichmakethecalculations

moreuncertain.Theassetsusedinthisstudycausesnoprobleminthisarea,howeveronecanclaim

that the pricing does differ among them. Some claim to be able to predict the future price of a

specificstockbecauseofavailable informationconcerningthestock,whilefuturepricesonoilmay

beharder topredict.Reasons for thiscanbe the strong influenceOPEChason thepriceofoilby

changingtherelationofdemandandsupply.Sadeghi&Shavvalpour(2006)callattentiontotheneed

ofVaRasa tool forquantifyingmarketriskwithinoilmarkets.Futureassetpricesandreasons for

themareacomplexareahoweverasstatedbeforealltheassetsusedinthisstudyaretradeddaily,

makingitnoproblemtoderivetheneededmarketprices.

2.2THENEEDFORVaR

Holton(2002)putsthebreakthroughforVaRinthe1970sand1980s.Duringthistimemanychanges

tookplaceanddiversefinancialcrisesbecameafact.TheBrettonWoodssystemcollapsed in1971

makingtheexchangeratesflow.BecauseoftheoilcrisiscausedbyOPEC,oilpriceswentthroughthe

roofbygoingfromUSD2toUSD35.Alongwiththecrisisalsocamefinancialinnovationssuchasthe

proliferationofleverage.Beforethistime,theavenuesforcompoundingriskwerelimitedaccording

toHolton(2002).Withnewinstrumentsandnewformsoftransactions,newleveragewaysopened

up. Holton also brings up that when this happened, trading organizations sought new ways to

managerisktaking.Duringthistime,banksstartedtocreateameasureasaresultofmorecomplex

riskmanagement.Theriskshadtobeaggregatedsowhenfacingthisproblem,thewantedsolution

wasameasurethatcouldprovidethecompanieswithabetterunderstandingfortheriskexposure.

Itwas JPMorganwhobecame themost successful.Theycreated theRiskMetrics systemwhich is

mentionedby Jorion (2001). It revealed riskmeasures for300 financial instruments and thedata

characterizeavariancecovariancematrixofriskandcorrelationmeasuresthatprogress through

time.Thedevelopmentofthemeasuretookalongtimeandwasfinishedin1990.Fouryearslaterit

wasmadefreetothepublicsomethingthatcreatedbigattention.TheuseofVaRsystemincreased

enormously and apositive effecton the companies riskmanagementwere found. The important

contributionofRiskMetricswasthatitpublicizedVaRtoawideaudience.


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AnimportantlandmarkinriskmanagementthatmadeVaRknowntothemasswastheBaselAccord.

Itwasconcludedin1988andfullyimplementedintheG10countries1in1992.Thisismentionedby

Saunders (1999) and itwas a regulation on commercial banks to provide amore secure system

throughminimumcapitalrequirementsforbanksmarketsrisk.

2.3VaRAPPROACHES

Thethreedifferentapproachesused inthisstudywillbepresentedbelow.Theyallhavetheirown

advantages and disadvantageswhich affects the VaR values that they produce. The assumptions

madeby theapproachesdealswith the returncharacteristics indifferentways.Theeffectsof this

willbefurtherdiscussedintheanalysis.

2.3.1HISTORICALSIMULATIONAPPROACH

Anonparametricmethoddoesnotassumethereturnsoftheassetstobedistributedaccordingtoa

specificprobabilitydistribution. It isasimplewaytocalculatetheValueatRiskthat iswidelyused

justbecausethefactthatitignoresmanyproblemsandstillproducesrelativelygoodresults(Dowd

1998). The non parametric historical simulation approach used in the study is the historical

simulationapproach.

Theideaofthehistoricalsimulationapproachistousethehistoricaldistributionsofpricechangesin

order to calculate theVaR.Historicaldata is collected fora chosenperiodand then thehistorical

price changes are assumed to be a good assessment of future price changes. The function for

historicalsimulationcanbedescribedasbelow(Goorbergh&Vlaar1999).

|

The|istheVaRvalueattimet+1where istheinitialvalueoftheassetandisthep:th

percentile of each subsample, which means taking the asset return between time t and

preceding returns and calculating thepercentileof these values that corresponds toour selected

confidencelevel.

Themeritsofthisapproach,accordingtoDowd(1998),areitssimplicitywhichmakesitagreathelp

forriskmanagersandthedataneededshouldbeavailablefrompublicsources.Another important

benefitisthatitdoesnotdependontheassumptionaboutthedistributionofreturns.Whetheritis

thenormaldistributionornotthatgivesthebestestimatesisnotthequestionhere.Eventhoughthe

approach is indifferent towhichdistribution the returnshave is it important to remember that it

doesassumethedistributionofreturnstoremainthesameinthefutureasithasbeeninthepast.

Dowd(1998)statesthatthismakestheapproachlessrestrictivebecauseweneitherhavetoassume

1 Belgium, Canada, France, Germany, Italy, Japan, theNetherlands, Sweden, the United Kingdom and theUnitedStates.


15

thatthechangesare independentgivingtheapproachnoproblems inaccommodatingthefattails

thattormentnormalapproachestoVaR.

A possible disadvantage, argued by Jorion (2001), is that the approach requires a lot of data to

performwellathigherconfidencelevels.WhenestimatingVaRata99%confidencelevel,intuitively

at least100historicalvalueshave tobe inputted.Buteven then theapproachonlyproducesone

observation in the tail. Perhaps not enough historical data is available to produce a good VaR

estimate,aproblemthatontheotherhandcanoccurformostoftheVaRapproaches.Jorion(2001)

furtherargues that there isa tradeoff in theapproachwhen includingmoreandmorehistorical

values.Ononehand, theapproachbecomesmoreaccurateathigherconfidence levels,butat the

sametime,theriskofincludingoldvaluesthatarenotrelevantforfuturereturnsishigher.

The historical simulation approach also assumes that the past is identical to the future,making

historicalrisksthesameasfuturerisks,whichmightnotbethecaseiftherehavebeenextraordinary

eventsintherecentpast.Perhapsthisismostoftennotthecase,buttheimportanceofgettingdata

thattrulyreflectsthepastbecomescritical.Ifthechosendataperiodistooshort,itmightreflecta

moreor less volatileperiod thatdoesnot give a goodpictureof thehistorical volatility. Itmight

includeperiodsofunusualkindthatisnotrepresentative.

Goorbergh&Vlaar (1999)arguesthat it isnotgenerallypossibletomakehistoricalsimulationVaR

predictionsonwindowsizessmallerthanthereciprocaloftheselectedconfidencelevel.Thismeans

that forevaluating a99.9% confidence youwouldneed awindow sizeof at least

. 1000

observations.

The conclusions that can be drawn is that the approach does have both advantages and

disadvantages which might make it important to complement it with other tests that pick up

plausiblerisknotrepresentedinthehistoricaldata.

2.3.2MOVINGAVERAGEAPPROACH

Aparametricapproachassumes that theassets return followaprobabilitydistribution.Themore

naveapproach,herecalled themovingaverageapproach, isaparametricapproach thatassumes

thatthereturnsofassetsarenormallydistributed.Theapproachmeasuresthestandarddeviationof

pastreturnsandusesthistogetherwiththestandardnormaldistributiontodescribetheprobability

ofdifferentoutcomesinthefuture.Howeverthisapproachasmentionedabovedisregardsthewell

establishedphenomenonof volatility clustering.Dowd (1998) sees this as abigproblem because

there is a tendency for high volatility observations to be clustered with other highvolatility

observationsandthesameistrueforlowvolatilityobservationsthatclusterwithotherlowvolatility

observations.Thisphenomenahasbeenconfirmedbymanystudiessincethefirstobservationmade

byMandelbrot(1963).

Under thisapproach, the sample standarddeviation is firstcalculated fromanumberofhistorical

observationsusingthisfunction.


16

1

1

Thenumberofobservationsisrepresentedbyn,standsforthedailylogpricefordayiand isthe

averagedailylogprice.

Calculating theactualVaRvalue isthendonebyretrievingthenumberofstandarddeviationsthat

corresponds to the selected confidence level from the normal cumulative distribution function

presentedbelow,andmultiplyingitwiththestandarddeviationofthehistoricalobservations(Jorion

2001).

Thefunctionbelowisthecumulativedistributionfunction(cdf)ofthestandardnormaldistribution.

This function calculates the probability that a random number drawn from the standard normal

distribution is less than x.The inverseof this functiongives the valueof x that is the limitunder

which a random value drawn from the standard normal distribution will fall with the inputted

probability(Lee,Lee&Lee2000).Erfstandsfortheerrorfunction.

12 1

2

Basedon this the confidence levels thatwehave chosen corresponds to the followingnumberof

standarddeviations(Penza&Banzal2001).

Confidencelevel Standarddeviations95,00% 1,64599,00% 2,32699,90% 3,090

Table2.3.2Relatingstandarddeviationtochosenconfidencelevel.

Animportantpartofthisapproachischoosingthenumberofdaystobasethevolatilityestimateon.

Ifthevolatilityoftheunderlyingassetisestimatedfromtoomanyhistoricalobservations,thereisa

riskof includingobservations thatare toooldand irrelevant.On theotherhand,ahistoricalvalue

based on too few valesmeans a risk of getting values that are too heavily influenced by recent

events. As a rule of thumb,Hull (2008)mentions that you should base your historical volatility

estimateonequallymanydaysasyouaretryingtoforecast intothefuture,butno lessthan30.In

thisstudy,thenumberofdayswouldthereforebe30.

Thedistributionofthereturnscanbedifferentforvaryingassets.Themostwidelyuseddistribution

isthenormaldistributionbecauseofitssimplicity,however,itdoesseldomexactlyfitthereturnsof

an asset. Dowd (1998) states that normality gives us a simple and tractable expression for the

confidenceintervalfortheVaRestimate,andwithanormalreturn,theVaRisjustthemultipleofthe

standarddeviation.Itisonlyabouttwoparameters,theholdingtimeandtheconfidencelevel.These

parameterscanvarydependingontheuser.Also,goingfromaVaRestimateforaspecificconfidence


17

leveltoanotheronecaneasilybedone justbychangingthenumbersfromthenormaldistribution

associatedwiththeconfidencelevel.

2.3.3GARCHAPPROACH

GARCHisnotreallyaVaRmethodinitselfbutratheramoreadvancedwaytocomputethestandard

deviationofpast returns that in turn is intended togivemorepreciseVaRestimates.TheGARCH

approach is considered to be semi parametric, meaning that it has both parametric and none

parametricproperties. It isparametric in thesense that theestimatedvarianceof theapproach is

usedwiththenormaldistribution,but it isatthesamenonparametricsincethe inputtedvariables

arerealreturnsandnotestimatedvolatilities.

TheGARCHapproachforvariance,h,lookslikethis(Jorion2001):

Where , and areweights. is theweight forhowmuch of todays variance influencesour

forecast, weights howmuch of yesterdays estimationweights into the forecast. Jorion (2001)

pointsoutthatthesumofandmustbelessthanoneandisusuallycalledthepersistenceofthe

approach. It iscalledpersistence, sinceduringamultiperiod forecast into the future, theweights

woulddecayastheyareboth lessthanoneandthecalculatedvalueofvariancewouldrevertback

towardstheestimatedlongtermvariance.

Thoughtheuseof itsnonparametric input,theGARCHapproachcantakesvolatilityclustering into

consideration.Thismeansthatittakesintoaccountthatlargechangesinpricesataspecifictimeare

usually followed by more large changes and vice versa. This is done by weighting historical

observationsandrecentobservationsunequally.

There isnowayofsolvingforthecorrectvaluesof,andmathematicallyso ithastobedone

numericallyusingmaximumlikelihoodestimation(MLE).Toestimatetheparameters,andthe

logarithmofthelikelihoodfunctionofthenormaldistributionisused(Jorion2001).

max 1

2

2

,andgives thevaluesofht,somaximizing this functionbyadjusting,andgivesus the

optimal values of , and . This optimization is done for each and every one of the historical

observationonwhichtheVaRmeasureistobebased.Theparameters,andarethenadjusted

togiveamaximumvalueofallofthoseobservationscombinedforthechosenwindowintime.

ItshouldbenotedthatastheMLEfunctionisderivedfromthenormaldistributionfunction,italso

assumesthatthereturnsoftheassetsarenormallydistributed.TheMLEfunctionmaytherefore, if

appliedtononenormallydistributedreturns,producequestionableresults(Jorion2001).


18

2.4THEUNDERLYINGASSETS

The underlying assets, on which our VaR calculations are based, are chosen because of their

fundamentallydifferenceswhichwouldindicatefundamentallyvaryingreturndistributions.Allthree

assetsshowdifferentpropertiesmakingtheminterestingforcomparisonaccordingtothepurposeof

thestudy.

Wheneverholdinganassetyou face the riskof theassetgainingor losing value in relation to its

purchaseprice.Thefieldofriskmanagement isallaboutassessingandmitigatingrisk.Aneffective

risk management is according to Saunders & Cornett (2007) central to a financial institutions

performance. This is also true for any company exposed to risk. As discussed in the problem

discussionVaRisausefulmeasureforallofourassets.

2.4.1BRENTOIL

Crudeoil isthemosttradedcommodityontheworldmarketandthemost importanthubsforthis

trading are New York, London and Singapore. The prices of the Brent oil, which is the world

benchmark for crudeoiland ispumped from theNorth Seaoilwells is theoilused in the study.

AccordingtoEydeland&Wolyniec(2003),abouttwothirdsoftheworldsupplyofoil ispricedwith

referencestothisbenchmark.

Theoilprice isveryvolatile,andareasonforthis isthe influenceofOrganizationofthePetroleum

ExportingCountries(OPEC).Thisisanoilcartelthatwascreatedin1960andnowconsistsoftwelve

membercountries.Theiroilproductionstandsfornearlyhalfoftheworldproduction,andbyraising

or lowering theirproduction, they can affect theprice. There aredifferentopinions aboutOPECs

influenceonthebalanceofdemandandsupply.Somepeopleclaimthattheypreventtheforcesof

themarkettosettherealpricewhileotherssaythattheycanonlyaffectthesupplyofoil,notthe

demand,makingtheforcesofthemarketthroughthedemandcreateanequilibriumprice.

2.4.2OMXs30

OMXs30 isanabbreviationofOMXStockholm30and is thedenominationof the30most traded

stocks on the Stockholm stock exchange. It is a capitalweighted index thatmeasures the price

developmentof thestocks.Thevalueof thestock for theseparatecompanies is thebasisof their

partofthe index. Theassemblyofthe index isconductedtwiceperyearandtherevenuesofthe

companiesthenworkasagroundforthecompanieselected.(www.omxnordicexchange.com)

2.4.3THREEMONTHSWEDISHTREASURYBILLS

TheSwedishtreasurybillsareissuedthroughtheSwedishNationalDebtOfficeandthebidsarethen

submittedtodealersauthorizedbythesame.Thebillsaretradedonthesecondarymarket,whichis

consideredtobeahealthymarketwhen lookingatthe liquidity.Thematurityvarieswithinayear,

however,thetreasurybillsusedinthisstudyarethethreemonthbills(STB3M).(www.riksbank.se)


19

2.5BACKTESTINGWITHKUPIEC

WhencalculatingtheVaR,one isonly interested inthe lefttailthatrepresentsthecaseswhenthe

returnsareworse thanexpected.Toevaluate themeasuresused,aback test canbe conducted.

Goorbergh&Vlaar(1999)explainsthetestascountingthenumberofdaysintheevaluationsample

thathadaresultworsethanthecalculatedVaR.Anobservationwheretheactualreturnexceedsthe

VaRiscalledaVaRbreak.Thebacktestisimportanttouse,andbyperformingitandcomparingthe

differentapproaches,onecanensurethattheapproachesareproperlyformedaccordingtoCostello

&Asem&Gardner(2008).

Jorion(2001)statethatthenumberofVaRbreaksisexpectedtobethesameasoneminusthelevel

ofconfidence.So forasampleof100observationswherea95%confidenceVaR iscalculated,we

would expect five 100% 95% 100 5VaRbreaks tooccur. In the study this is called the

targetnumberofVaRbreaks. If therearemoreor lessVaRbreaks thanexpected, it isbecauseof

deficiencies intheVaRapproachortheuseofan inappropriateVaRapproach.Awidelyusedback

test is theKupiec test.This testuses thebinominaldistribution tocalculate theprobability thata

certainnumberofVaRbreakswilloccurgivenacertainconfidencelevelandsamplesize.TheKupiec

testfunctionis(Veiga&McAleer,2008):

|,

1

Thevariable is thenumberofVaRbreaks,N the sample sizeand corresponds to the levelof

confidencechosenfortheVaRapproach(makinga95%confidence levela5%probability inputfor

theapproach). Ifthesamplesize is inputtedand issettooneminusthe levelofconfidence,the

binomialfunctionproducesthelikelihoodthataspecificnumberofVaRbreaksistooccur.Byusing

thecumulativebinomialdistribution, it ispossibletocalculatean intervalwithinwhichthenumber

ofVaRbreaksmust fall, inorder for theapproach tobeaccepted.This isdoneby calculating for

whichvaluesof that thecumulativebinomialdistributionproducesprobabilities thatare in the

interval 2.5% 97.5% (which corresponds to a 95% test confidence). VaR approaches that

producevaluesofthatlieswithinthisrangecanthereforebeaccepted.Iftheapproachproduces

valuesofoutsidethisspan,theapproachisrejected.Arejectionmeansthattheconfidencelevel

thatweused intheVaRapproachdidnotmatchtheactualprobabilityofaVaRbreak.This inturn

indicatesthattheapproachisnotperformingwellandthatitshouldberejected.


20

3.METHOD

In this chapter, themethods used in the study are presented. First, the nature of this study is

describedandthenthecalculationsofVaRareexplainedforthedifferentapproaches.

3.1ANALYTICALAPPROACH

This study takes an analytical approach and assumes an independent reality. This is a method

approach that iswidelyusedand thehistoryof itgoeswayback in time.Themostdistinguishing

characteristic concerning theprocedureof this approach is, in accordancewithArbnor&Bjerkne

(1994),itscyclicnature.Itstartswithfacts,endswithfactsandthesefactscanbethestartofanew

cycle.Themeaning istoextractgoodmodels inordertodescribetheobjectivereality.Themodels

withinthisapproacharemostoftenofquantitativecharacterandthisisalsotrueforthisstudy.Itis

naturalthatlogicandmathematicshasaparttoplayinthesearchforthebestmodeltoapply.

3.1.1QUANTITATIVEAPPROACH

In this study, the testsarebasedupona lotofhistoricaldata.Dailypricechangesare studied for

three different assets that are widely traded, this form of studymakes us apply a quantitative

approach.ThisapproachcomparedtoaqualitativeapproachisaccordingtoArbnor&Bjerkne(1994)

oftenmore about clear variables and can cover a larger number of observations,which fits our

compositionbetter.Itfurtherassumesthatwecanmakethetheoreticalconceptsmeasurable.Inthe

past, the quantitative approach has inmany caseswrongly been seen as the only real scientific

method, or as Holsti said in 1969, If you cant count it, it doesnt count. There are however

limitations connected to theapproachandHolme&Solvang (1991)pointsout that ifyouarenot

awareof them, the resultcaneasilybemisjudged.Numbersareoften seenas the truth,and this

trustcancreateproblemsmakingtheanalysisofthenumbersveryimportant.

Historicaldatahavebeencollectedinthisstudy,andbasedonthese,anumberofcalculationshave

beenmade in order to extract the VaR. The purpose is to be able to seewhich approach that

performsbest,andtheresultisbasedonthefiguresfromthecalculations.Howevertheresultisnot

justanumberitisalsoputintocontexttobettergetanunderstandingofitsmeaning.

3.1.2DEDUCTIVEAPPROACH

When taking a deductive approach one starts from an already existing theory and then either

strengthenorweakentheconfidenceofthetheory,meaningthatonestartsfromagenerallawand

thenmovestoaseparatecase.Inourstudy,weusethreewellknownapproachesforcalculatingVaR

and test their performance on assetswith different return characteristics. The purpose is to test

models,nottocreatenewones.


21

Asaresultofthisstudy,thefunctionoftheapproacheswillperhapsbestrengthenedbecauseofa

specific character, however, the same approach applied on the same asset can turn out to be

unreliable for a certain confidence level. The approachs can from the start be rather simple,

however, by adding a dimension like the GARCH approach, one can make the approach more

complex andmaybebetter fitted to a certain asset.The testof the approacheswillbebasedon

backtesting, we will count the numbers of VaR breaks,meaning the number of times the loss

exceededthecalculatedVaR.ReasonsfortheactualnumberofVaRbreakswillbeanalyzed.

3.1.3RELIABILITY

Inordertodecidethereliabilityoftheresults,oneshouldbeabletodothetestanumberoftimesto

seeifthesameresultsareachieved.Arbnor&Bjerkne(1994)callsthisatestretest,anditishelpful

intellingwhetherthestudyanditsresultsarereliable.Allthedatausedinthestudytocalculatethe

VaRispublicinformation,makingthetestreproducible.

3.1.4VALIDITY

Themostimportantfactorwhenjudgingwhethertheresultsarejustifiableintheanalyticalapproach

isthroughthevalidity.Ifyoucannotanswerthequestionwhethertheresultsgivesatruepictureof

thereality,theresultscaneasilybecomemeaningless.Thecloserwegettothetruesituation,givena

specifieddefinition,thehigherthevalidity.Therelationbetweentheoryanddataisvital,andHolme

&Solvang(1991)pointsoutthatthevaliditycanbeenhancedbyacontinuouslyadaptionbetween

thetheoriesandthemethodsused intheexamination.This isdonebychoosingmethodsthathas

the ability to handle or show the impacts that different assets characteristics can have on the

measuredVaR.Arbnor&Bjerkne (1994)stresses the importanceof threekindsofvalidityand the

needforcombiningthem.Thesearethesurfacevaliditythatdealswiththereasonabilityassessment

in relation toearlier results, the internalvaliditydealingwith thequestion ifwehadexpected the

resultwehaveconcludedandfinallytheexternalvalidityandtherebytheusefulnessoftheresultin

otherareas.Thesequestionswillbefurtherdealtwithintheanalysis.

Itisfurtherimportanttohaveinmindthatanapproachofthekindusedinthestudycangiveyoua

numberoftheVaRbutitisalwaysanestimationofapossiblefutureloss.Givenacertainconfidence

leveltheVaRcanbecalculated,however,thenumberreceivedisnottheabsolutetruth,itcannotbe

guaranteedthatthefuturelosseswontbelargerthanexpected.

3.2CALCULATIONOFVaR

Withmanydifferentapproachesandmodels,thechoicethatVaRusersfaceisthechoiceofpicking

the rightone thatmatches theirpurposebest.Theapproachesshouldmakeestimates that fit the

futuredistributionofreturns. IfanoverestimationofVaR ismade,thenoperatorsendsupwithan

overestimateoftherisk.Thiscouldresultintheholdingofexcessiveamountsofcashtocoverlosses,

asinthecasewithbanksundertheBaselIIaccord.Thesamegoesfortheoppositeevent,whenVaR

hasbeenunderestimatedresultinginfailuretocoverincurredlosses.


22

Inthisessay,wewillbecalculatingVaRforthreedifferentunderlyingassetsandcomparetheresults.

TheassetsthatwillbeusedforourcalculationsareBrentOil,OMXs30stockindex,andthreemonth

Swedish treasurybills.TheVaRapproaches thatwewillcalculateare thehistoricalsimulation, the

movingaverageapproach,assumingnormaldistribution,andtheGARCHapproach,assumingnormal

distribution.Whenusing theparametricapproaches,wedosuspect that the returns thatwehave

basedthecalculationsonaremost likelynotperfectlynormallydistributed. Infact,economictime

seriesrarelyare(Jorion2001).Theperformanceoftheseparametricapproacheswillthereforepartly

bedeterminedbyhowwell thenormaldistribution assumption fits the actualdistributionof the

returns.

The tool thatwehaveused toperformourcalculations isMicrosoftExcel.Excel isaveryversatile

toolthatcanbehelpfulifoneknowhowtouseitright.However,thedownsideofExcelisthatitis

muchslowerthansoftwarethat isspecificallydesignedforfinancialcalculations.Therearealsono

automated functions forcalculatinga lotof themoreadvancedoperations,oftenused in financial

timeseriesanalysis,suchasautocorrelationsetc.Thesemoreadvancedcalculationshavetobedone

manually, which can be very time consuming and also limits us somewhat when it comes to

computingmoreadvancedapproaches.


CalculatingaVaRmeasureusing thehistorical simulationapproach isnotmathematicallycomplex

but can be trying since the calculations require a lotofhistoricaldata. The first task is topick a

historical time frame on which to base the simulation. Themore historical values we base our

calculationson themoreobservationswewillget in the tailsand thus themoreaccurate theVaR

measurewillbeathigherconfidencelevels.

Thedownsideisthataddingmorehistoricaldatameansaddingolderhistoricaldatawhichcouldbe

irrelevant to the future development of the underlying asset.Our simulationwill be based on a

slidingwindowoftheprevious2000observationswhichcorrespondsto8calendaryears.Wehave

selectedthisratherlargewindowforthehistoricalsimulationapproachaswewanttheapproachto

performwellatthe99%and99.9%confidence levels.Asargued inthetheorychapter itmakesno

sensetosetthewindowsizetoanylessthan1000observationsforthe99.9%confidencelevel,but

evenatthislevelwetheoreticallyonlyhaveoneobservationinthetail.Wethereforedecidedtoset

the window size to 2000 observations to enhance the performance of the approach at higher

confidenceintervals.

Extracting the VaR measure from the historical data simply requires us to choose the desired

confidence level andpickout then:thobservation in thehistoricaldata that corresponds to that

confidencelevel.Forexample,a95%confidencelevelmeansthatweareinterestedintheworst5%

oftheobservations. Ifweforexampleareusing1000observations,thiswouldmeanthatthe95%

VaRwouldbethe50thworstobservation(1000 5% 50.


23

ThePERCENTILEfunctionweused inExcelcalculatesthen:thpercentileofthevalues inachosen

dataset.Ifthedesiredpercentileisnotamultipleofthenumberofvaluesinthedataset,Excelwill

doalinearinterpolationbetweenthetwoclosestvaluestofindthedesiredvalue.


Underthisapproach,wecontinuouslymeasuredthestandarddeviationofreturnsoverawindowof

thelast30days.Thisstandarddeviationwasthenmultipliedwiththenumberofstandarddeviations

extractedfromthestandardnormaldistributionthatcorrespondstotheselectedconfidence level.

Inotherwords,thestandarddeviationchangesasthewindowmovesalongbutthemeanisassumed

tobezero.

3.2.3GARCHAPPROACH

ThechallengewiththeGARCHapproach istoestimatetheparameters,and.Asdescribed in

the theory chapter, theseparametershas tobe solved fornumerically,usingmaximum likelihood

estimation. The literature that we have studied describes the MLE function but provides little

practicalinformationonhowtoimplementit.Themathbehindmaximumlikelihoodestimationcan

becomplicatedtounderstandforthosenotfamiliarwithbusinessstatistics.Manyanalyststhatdo

these types of calculations use preconfigured software and seldom have to engage in the

mathematicsthatliesbehindthecalculations.

TheMLEformula itself isnotcomplicatedandwas implementedforeachandeveryobservation in

ourstudy.MaximizingthefunctionwasdoneusingtheSOLVERfunction inExcel.Thefirstproblem

thatweencounteredwashowtodecidehowmanyhistoricalobservationsthatweshouldbaseour

MLEon.At first,wehadplanned tobase theMLEonamovingwindowof250values,matchinga

year in tradingdays,but thisapproachproducedvery inconsistent resultsoftensettingoneof the

parameters,or,closetozeroandtheotherclosetoone.Theconclusionwedrewwasthatthe

MLEwasbasedon too fewvalues tobeconsistent.Asmentioned in the theorychapter, theMLE

functionalsoassumes that the returnsarenormallydistributed.Thus, thesmaller thesample that

theMLEestimation isbasedon, the larger the risk that thosevaluesmightbe significantlyparted

fromthenormaldistribution.

Welookedthroughagreatdealofarticlesandbooksinhopeoffindingsomekindofdocumentation

onhowmanyobservationstoincludeinourMLEbutwewereunabletofindanyrecommendations

togoby.Wethereforemadeourownestimationofanappropriatetimeframeonatrialanderror

basis.Thiswasdonebyconstructingamacro inExcel thatwouldmaximize theMLE function, first

basedonthe250firstobservationsandthenadd100observationsatatimeeachtimereestimating

theparameters,anduntilwehad testedallof theobservations.Themacro isshown in the

appendix1.Theresultingvaluesof,andareshowninthegraphs3.2.3a,bandc.


24

Figure3.2.3aMLEestimationgraphforBrentOil

Figure3.2.3bMLEestimationgraphforOMXs30.

Figure3.2.3cMLEestimationgraphforSTB3M.

Thegraphsshowthevaluesofandinblueandredontheleftverticalaxisandthevaluesofin

greenontherightverticalaxis.Thehorizontalaxisshowsthenumberofobservationsincludedinthe

estimate.Sincethesumofandmustbelessthanoneandthereforemustgodownfortogo

up.Thisiswhythevaluesdivergeduringthefirstestimationstolaterleveloutorbecomestable.


25

Onecanclearlyseehowvaluesof,andstabilizeasweaddmoreandmorevaluestotheMLE.

We also interpret these results as an indication of to what degree the returns are normally

distributedornot.

By lookingatthegraphs,weseethatataround2000observationstheMLEhasstabilizedforallof

our three assets, which lead us to conclude that 2000 observations is a suitable number of

observationstobaseourMLEon.

Afterdeciding theappropriatenumberofobservations tobaseourMLEon,we then recalculated

theMLEfunction forourdatasets,nowwitha lagof2000observationsfrom19960101onwards.

Newgraphshavebeenplottedforthevaluesof,and,andtheycanbefoundinappendix3.

After thevaluesof,andhadbeenestimated, theywere inputted into theapproachand the

resultswillbeshownintheresultsandanalysischapter.

3.3SKEWNESS

Theskewnesstellsuswhetherthedataissymmetricornot.Normallydistributeddataisassumedto

besymmetricallydistributedarounditsmean,i.e.ithasaskewof0.Adatasetwitheitherapositive

or negative skew therefore deviates from the normal distribution assumptions. This can cause

parametricapproaches, inourcase themovingaverageapproachand theGARCHapproach, tobe

lesseffectiveifassetreturnsareheavilyskewed,sincetheseapproachesassumethatthereturnsare

normally distributed. The result can be an overestimation or underestimation of the VaR value

dependingontheskewoftheunderlyingassetsreturn(Lee,Lee&Lee2000).

Figure3.3Graphshowinganegativerespectivelyapositiveskew.


26

3.4KURTOSIS

Thekurtosismeasuresthepeakednessofadatasampleanddescribeshowconcentratedthereturns

arearoundtheirmean.Ahighvalueofkurtosismeansthatmoreofthedatasvariancecomesfrom

extremedeviations.ThekurtosisofthenormaldistributionsalsomentionedbyLee,Lee&Lee(2000)

asmesokurtic is three soaswith the skewnessanydeviations from thenormaldistributioncause

problems for theparametricapproaches.Ahighkurtosismeans that theassets returns consistof

more extreme values than modeled by the normal distribution. This positive excess kurtosis is

accordingtoLee&Lee&Lee(2000)calledleptokurticandlowerkurtosismeaninganegativeexcess

iscalledplatykurtic.LowkurtosisproducesVaRvaluesthataretoosmall.

3.5THESOURCEOFDATA

Thedataused in thestudyare timeseries thatreflects thehistoricalpricechanges for thechosen

assets.Theperiod,whichthecalculationsarebasedon,stretchesfrom19870101to20080930for

allthethreeassets.However,thedatafrom1987to19951231hasbeencollectedtoenableusto

baseourcalculationsthatrequirehistoricaldataon.

Different sources have been used to collect the data.Daily oil prices are taken from the Energy

InformationAdministration (www.eia.doe.gov),whoprovidesofficialenergy statistics from theUS

government.TheOMXs30datacomesfromOMXNordicExchange(www.omxnordicexchange.com),

and the Treasury bill data can be found on the homepage of the Swedish central bank

(www.riksbank.se).Thecharacteristicsofthedifferentassetsaredescribedbelow.

Wehavecalculatedsomecharacteristickeyfiguresaboutthedatasetsinordertobetterbeableus

to compare them. The key figures thatwe have calculated are skewness, kurtosis, volatility, and

Figure3.4ProbabilitydensityfunctionforthePearsontypeVIIdistributionwithkurtosisofinfinity(red);2(blue);and0(black).


27

average price change. Table 3.5 shows some of the statistical characteristics of each dataset,

measuredduringtheentireperiodof19872008.

BrentOil OMXs30 3MSTB

Skewness: 0.83 0.02 3.77

Kurtosis: 19.14 7.28 182.51

Dailyvolatility: 2.32% 1.47% 1.49%

Annualvolatility: 36.78% 23.37% 23.66%

Averagelogpricechange: 1.63% 1.03% 0.58%

Table3.5Statisticalcharacteristicsoftheassetreturns.

3.6BRENTOIL

Figure3.6aDailyreturnsforBrentoil.

The oil prices are themost volatile prices in this study,with a 2.32% daily volatility and 36.78%

annualvolatility.Thekurtosis is19.14,whichmake it farnarrowerwithmuch fatter tails than the

normaldistribution.

TheskewnessoftheBrentoilreturnshaveanegativeskewof0.83,whichindicatesthatthereturns

areskewedtotheright.Thismeansyetanotherdeparturefromtheassumptionsofnormalityinthe

parametricapproachescausingthesetoperformlesseffective.Thedailyaveragelogpricechangeis

1.63%,whichbasicallymeansthatoverthemeasuredtimeperiodtheaveragereturnwas1.63%.


28

Ingraph3.6b,wehavefittedanormaldistributionprobabilitydistributioncurveagainstahistogram

ofthereturns.Themostvisuallyprominentdeviationfromthenormaldistributionassumptionisthe

kurtosis, which can be seen from the middle bars of the histogram rising above the normal

distribution.Therearealsoseveraloutlierstorightand leftofthehistogram(whichcanbehardto

seebecauseofthescaleofthepicture)which layoutsidetheboundsoftheboundsofthenormal

distributioncurve.

Figure3.6bHistogramshowingthedailyreturnscombinedwithanormaldistributioncurve.


29

3.7OMXs30

Figure3.7aDailyreturnsforOMXs30.

Sincethereturnsofthe indexcorrespondstoawelldiversifiedportfolio,theyarenotveryvolatile.

Theaveragevolatilityis1.47%dailyand23.37%annually.Thekurtosisis7.28,whichsuggestthatthe

distributionofthevaluesisnarrowerthanthenormaldistributionandwithsomewhatfattertails.It

ishowevernotashighasitwasfortheBrentoil,whichmightindicateabettercompatibilitywiththe

parametricapproaches,atleastatlowerlevelsofconfidence.

Theskewness isonly 0.02,whichandtheaveragedaily logpricechange is1.03%.Apartfromthe

kurtosisthenormaldistributioncurveisaprettynicefitonthesereturns.

Therearehowever largeoutliers in thatarenot clearlyvisible in thegraphwhich layoutside the

confinementsofthenormaldistributioncurve.Thereisalsonosignificantsignofautocorrelationas

shownbythetableinthepreviouschapter.



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3.8THREEMONTHSWEDISHTREASURYBILLS

Figure3.8aDailyreturnsforSTB3M.

The returnsof the treasurybillshave themostextremecharacteristicsofall thedatasets. Ithasa

strongpositive skewnessof3.77, and a very large kurtosisof182.51 suggesting apoor fit to the

normaldistributionwhichalsocanbeobservedfromthehistogram infigure3.8b.Theaverage log

pricechangeis0.58%butfromfigure3.8awecanseethatvolatilityvariesfromlowtoextreme.The

mostextremereturnsarefromtheearly1990swhenthebankingcrisisoccurredinSweden.



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3.9AUTOCORRELATION

Intimeseriesanalysis,autocorrelation,orserialcorrelationas it isalsocalled, isameasurementof

whether the data correlateswith itself over time.Autocorrelation in financial time series data is

measuredbyregressionoftheobservedreturnswithalaggedversionofthemselves.Regressingthe

returnswith themselves like thiscanbedescribedas testing if it ispossible todescribe returnsof

todayasalinearfunctionofthereturnsfromyesterday.Theresultingresidualsfromtheregression

are then tested for autocorrelation. The presence of autocorrelation means that the applied

approachwillhaveapoorfittotheactualdatawhichleadstheanalysttoconcludethatthereturns

oftodaycannotbeaccuratelydescribedasa linearfunctionofthereturnsofyesterday.Thusthere

must be other factors besides the historical returns that effect todays returns and thereby

approachesthattrytopredictfuturereturnsbylookingatthepastwillbelesseffective(Tsay,2002).

Wehavechosentoperformaone laggedDurbinWatson(DW)testonourdatatosee ifthereare

anyfirstorderautocorrelation.TheDWtestusesthefollowingformulatocalculateitsstatistic:

Theparameterdenotes the residuals from the regression.TheDW statistic canassumeavalue

between0and4.ValuesoftheDWstatisticlargerthan2indicatenegativeautocorrelationwhereas

valueslessthantwoindicatepositiveautocorrelation.Avaluecloseto2indicatesthattheresiduals

areprobablynotautocorrelated(Tsay2002).

Whentestingforautocorrelation,aconfidence interval iscalculatedaround2withinwhichtheDW

statisticmustfallinorderfortheresidualstobeconsiderednottobenotcorrelatedwiththegiven

degree of significance. Harvey (1990) has shown that for large sample sizes, this interval is

approximatelynormallydistributedwithameanoftwoandavarianceof

.

We have used the statistical softwareprogramR to test our time series for autocorrelation. The

autocorrelation tests results have been compiled in table 3.9 and some additional regression

statisticscanbefoundinappendix2.

RESULTSFROMAUTOCORRELATIONTESTAsset No. Lag Autocorrelation dL DW dU PBRENTOIL 5454 1 0,000660287 1,955455 1,998650 2,044545 96,40%OMXs30 5475 1 0,000758127 1,955540 1,997565 2,044460 94,80%STB3M 5468 1 0,009438294 1,955512 1,980684 2,044488 36,80%

Table3.9Resultsfromautocorrelationtest.

ThepvalueisaprobabilityratiothatRcalculateswhichdonatestheprobabilitythattheDWstatistic

wouldoccurrandomly.AlowpvaluethusindicatesthattheDWstatisticinquestionisveryunlikelyif

theapproachiscorrect.Thenullhypothesisis:


32

Thenullhypothesiswillbe rejected if thep value is less thanoneminus the selected confidence

intervalwhichinourcaseis95%.Thusifpwouldhavebeenlessthan5%,thenullhypothesiswould

havebeenrejected,whichwouldhavemeantthatthereturnswereautocorrelated.Asstatedbythe

pvaluesandtheDWstatisticsinthetable,thereisnosignofautocorrelationintheresiduals.

3.10BACKTESTINGWITHKUPIEC

InordertomeasurewhichoftheVaRapproachesthathasbeenmostsuccessful,wewillbacktest

theVaRvaluesagainsttheactualreturnsofeachday.Intheeventthattheactuallossonaparticular

dayislessthantheprojectedVaRvalue,wesaythataVaRbreakhasoccurred.

TheexpectednumberofVaRbreaksisoneminustheselectedlevelofconfidencetimesthenumber

ofobservations.Thenumbershavebeen roundedoff to thenearest integer.Wehavecompileda

tableofthedesirednumberofVaRbreaksintable3.10.

DESIREDNO.OFVaRBREAKSANDKUPIECTESTINTERVALS

95% 99% 99.9%

Asset No. MINTARGET MAX MIN TARGET MAX MIN TARGET MAX

BrentOil 3259 139 163 187 22 33 43 0 3 6

OMXs30 3215 137 161 184 22 32 43 0 3 6

STB3M 3225 137 161 185 22 32 43 0 3 6

Table3.10TargetnumberofVaRbreaksandKupiectestintervals.

ThenearerthedesirednoofVaRbreaksanapproachproducesthebettertheapproachisconsidered

toperform.WhethertheproducednumberofVaRbreaksisgreaterorsmallerthatthetargetvalue

makesnodifference, it istheabsolutenumberofVaRbreaksthattheapproachdeviateswiththat

determineshoweffectiveitis.


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4.RESULTS&ANALYSIS

Inthischapter,wewillconcludeourresultsandthereafteranalyzetheminrelationtothepurposeof

thestudy.Theanalysiswillbedivided,treatingtheapproachesseparately, inordertogetabetter

overviewoftheapproachesandtheirresults.

4.1BACKTESTINGRESULTS

Thebacktestingresultsarepresentedforthethreedifferentassetsunderthethreeapproaches.The

outcomes depend on the characteristics of the asset, and by examining them, the result can be

explained.


Thebacktestingresultsforthethreeassets,calculatedusingthehistoricalsimulationapproach,are

shown in table4.1.1a.The table shows theminimumandmaximumvaluesallowedby theKupiec

test,thetargetnumberofVaRbreaksandtheresultingnumberofVaRbreaksfromthesimulation.

VaRbreaksthatarewithintherangeallowedbytheKupiectestaremarkedgreenandthosethatare

notaremarkedred.

KUPIECTESTHISTORICALSIMULATION,2000DAYLAG 95% 99% 99.9%

Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 209 187 22 33 37 43 0 3 7 6OMXs30 3215 137 161 216 184 22 32 54 43 0 3 5 6STB3M 3225 137 161 109 185 22 32 24 43 0 3 7 6

Table4.1.1aBacktestingresultswithhistoricalsimulationapproach.

Theresultsshowthattheapproachperformspoorlyonthe95%confidence levelwhile itproduces

betterresultsonthehigherconfidence levels.Onthe95%confidencelevel,theapproachproduces

toomanyVaRbreaks,whichindicatethattheapproachoverestimatestheVaROnthe99%levelthis

approachperformsequallycomparedtotheGARCHapproachandoutperformsthemovingaverage

approach.Whenitcomestothehighestlevelthisapproachistheonlyonethatcomesevencloseto

beingacceptedbytheKupiectest.

This approach looks at the latest2000daily returnswhen calculating the volatility,which in turn

makes theapproach react slowly tonew informationand changes in thedaily returns.There isa

tradeoff between new information and old historical information that we presented in earlier

chapters.Ifashorterintervalwouldhavebeenchosen,thiswouldhaveincreasedtheimpactofnew

observations,puttinghigherfocusonrecentmarketconditions.


34

Table4.1.1ashowstheVaRestimatedbyhistoricalsimulationatthe99.9%confidencelevelforBrent

Oil.Outofall theassets,thehistoricalsimulationapproachperformsbestresultswithBrentoil. It

doeshoweverseem that theapproachgenerallyunderestimates theValueatRisk, thusproducing

toomany VaR breaks. The characteristics of Brent oil showed that the normal distribution, and

thereforetheparametricapproaches,wouldmakeapoorfitandthatthehistoricalsimulationwould

suittheassetbetter.Brentoilisthemostvolatileoftheassetsbutthevolatilityisratherpredictable,

ormore correctly, highbut stabile. The reason towhyhistorical simulation performsbetterwith

Brentoil isbecauseof itsstability.Asdiscussedearlierthehistoricalsimulationapproachdoesnot

assume the returns to followaspecificprobabilitydistribution,but itdoesassume that the return

distributionisthesameinthepast,inourcase2000days,asitwillbetomorrow.Thestabilityofthe

Brentoilreturnsisthereforethereasonforthegoodresults.

Whentestingtheaccuracyofanapproachitisequallyimportantwithunderestimationsasitiswith

overestimations.Theresultsforthetwootherassetsareaboutthesamewiththedifferencelyingin

overestimationsoftheValueatRiskfortheSTB3MandunderestimationsofthevalueforOMXs30.

Thevolatility for theseassets isnotashighas forBrentoilbut itchangesmoredramaticallyover

time. If a shorter time period had been chosen, the results for STB3Mwould have looked very

different ignoringthegreatvolatilitychangesthatappearedaround1993.Thehistoricalsimulation

would in thatcasehaveproducedbetterresultssince the fluctuations inreturnswouldhavebeen

less.

Figure4.1.1bVaRforBrentOil,HistoricalSimulation99.9%confidence19962008.

The look of the graph 4.1.1b is characteristic for historical simulationVaR estimates. The sudden

jumpsupanddownandtheplateausinbetweensignaltheentranceandexitofanextremehistoric

return that affects theVaR as long as it remains in thewindow. The longwindow thatwe have

chosen seems to be the cause to why the approach overestimates the VaR for STB3M and

underestimatesitforBrentOilandOMXs30.Atthe95%,confidencelevelthelargewindowchosen

seemstoberesultingintoofewVaRbreaksforSTB3M.Itwouldseemthatthelargewindowistoo

large for the95%confidence level,adding toomanyextremeobservations in the tails.Thiscauses

theapproachtooverestimatetheVaR,andproducetoofewVaRbreaks.ForBrentoilandOMXs30,


35

the opposite seams to apply. The large window puts toomuch weight on themore frequently

occurringsmallerreturnsanddrawsthecenterofgravityintheapproachawayfromtheinteresting

outliers,causingtheapproachtoproducetolowVaRandthustomanyVaRbreaks.Thesameseems

toapplyatthe99%confidencelevel,buttheeffectsseemmilder,perhapsbecausethewindowsize

ismoreappropriatehere.

As canbe seen in table4.4.1a theapproach is rejectedby theKupiec testat the95% confidence

level.Webelievethisisduetothewindowsizeof2000observationsbeingtoolargeforthehistorical

simulation approach at the 95% confidence level. The approach could probably have performed

betterhadwechosenamoreappropriatewindowsize.

Theapproachdoes,however,performbetteratthetwohigherconfidencelevels,wheretheselected

windowsizeisbettersuited.Sincethehistoricalsimulationapproachisnonparametric,ittakesallof

theoutliersof the returns intoaccount thatare ignoredby theparametricapproaches,using the

normaldistribution.This isalsooneof the reasons towhy theapproachperformsbetterat these

higherconfidencelevelscomparedtotheothertwoapproachesaswewilldiscoverlater.

Byweighing all the returns equally, it takes time for the approach to react to fluctuations in the

returns.ThismeansthatoldextremeoutliersinthereturnsaffectthecalculatedVaRforalongtime.

WhendealingwithreturnsthatsufferfromextremeleptokurtosisasisthecasewiththeSTB3M,one

endsupwithanaverageVaRthatisconsiderablyhigherthantheaveragereturn,inotherwordsthe

fewextremeoutliershaveatoobigimpactontheVaRvaluegivenacertainwindowsizeandlevelof

confidence.Thehistoric simulationapproach isgenerally considered toperformwellwith returns

thatareleptokurtic,i.e.hasfattails,butthereseemstobealimitwhentheleptokurtosisbecomes

toomuchalsoforthehistoricalsimulationapproachtohandle.Wethereforethinkthatthesizeof

thehistoricalwindowhastobechosenwithrespectnotonlytotheconfidencelevelbutalsotothe

kurtosis.

Thehistorical simulation approach assumes that thedistributionof returnsdoesnot changeover

time.Withoutthisassumption,therewouldbenoreasonatalltolookatthepastreturnsinhopeof

predictingthefuture.Inotherwords,onecontributingcauseoferrorintheapproachcouldbethat

thedistributionofreturnsisnotstationary.

Over all this approach performs best on high confidence levels, and this is because the chosen

historicalwindow size is better suited for these confidence levels. The reason that the approach

performs relativelybetterathighconfidence intervalscompared to theother threeapproaches is

thatitconsiderstheextremevaluesthatfallsoutofthenormaldistribution.Thiscanclearlybeseen

inthehistogramsinchapterthreeshovingthedailyreturnsandthenormaldistribution.ForBrentoil

andtheOMXs30,asignificantnumberofobservationscanbefoundinthetails.FortheSTB3M,most

observationsfallinthemiddleofthenormaldistribution,causingtheapproachtooverestimatethe

VaR.HistoricalsimulationcanberecommendedwhenestimatingVaRathighlevelsofconfidencefor

returnsthatarestationaryandnottooleptokurtic.


36


ThemovingaverageapproachisthemostbasicapproachforcalculatingVaRandithasbeenaround

thelongest.Eventhoughitiselementary,itperformsquitewellatlowerconfidencelevelsandunder

therightconditions.Thebacktestresultsareshownintable4.1.2.

KUPIECTESTMOVINGAVERAGEAPPROACH,30DAYLAG 95,00% 99,00% 99,90%Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 178 187 22 33 48 43 0 3 13 6OMXs30 3215 137 161 181 184 22 32 65 43 0 3 21 6STB3M 3225 137 161 170 185 22 32 96 43 0 3 47 6

Table4.1.2Backtestingresultswithmovingaverageapproach.

ThemovingaverageapproachunderestimatestheVaRathigherconfidencelevelsduetothefatter

tailofthereturnsthanassumedbythenormaldistribution.Theapproachseamstoperformwellon

lowerconfidencelevelswherethenormalityassumptionismorevalid.Consequently,theKupiectest

rejectsthemovingaverageapproachatthehigherconfidencelevelsandacceptstheapproachatthe

95%level.

Asdiscussedinthemethodchapter,thecharacteristicsoftheOMXs30returnsgiveanindicationthat

theassumptionofnormalityisnotthatunreasonable.TheOMXs30returnsarenotthatskewedand

thekurtosis isnot thatgreat.Despite this, theapproachperformsbetteron theBrentoil returns

eventhoughthecharacteristicsofthosereturns looktobefurtherfromthenormalityassumption.

This canperhapsbeexplainedby lookingat theplotof the returnsof theassets.The returns for

BrentoilismoreskewedandleptokurticthanthoseoftheOMXs30,butthereturnsoftheBrentoil

aremorestablewhereastheOMXs30returnsshowclearsignsofvolatilityclustering.

The extreme volatilitypeaksof the STB3M returnsmakes forpoor compatibilitywith themoving

average approach at higher confidence levels, which can be seen in table 4.1.2. The approach

producesapproximately thedoubleamountofVaRbreaks, compared to theother twodata sets.

Thisisaresultoftheoutliersoftenappearalonewithoutapriorriseinvolatilitythedaysbeforethe

VaRbreak,as isusualwhenvolatility is clustered.This isalso the reason for the returnsbeing so

leptokurtic.Thismeansthattheapproachhasnowayofforecastingthesesuddenrises involatility

and thus a VaR break occurs. The day after the VaR break when the outlier passes in to the

measurement window the VaR measure rises significantly. Since the VaR break was a single

occurrenceitisnotfollowedbyadditionalextremeoutcomes.


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4.1.3GARCHAPPROACH

Table4.1.3showsthebacktestingresultsoftheGARCHapproach.

KUPIECTESTGARCH 95,00% 99,00% 99,90%Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 156 187 22 33 43 43 0 3 16 6OMXs30 3215 137 161 161 184 22 32 47 43 0 3 17 6STB3M 3225 137 161 95 185 22 32 38 43 0 3 19 6

Table4.1.3BacktestingresultswithGARCHapproach.

TheGARCHapproachproducesgood results for the95percentconfidence levelwithexceptionof

theSTB3M.Theresultsforthe99%confidence levelarealsoreasonable.Theapproachproducesa

bittoomanyVaRbreaksasexpectedfromaapproachbasedonthenormaldistribution.Thebetter

performanceoftheGARCHapproachonthe99%confidencelevel,comparedtothemovingaverage

approach, is as we interpret it solely due to the GARCH approaches more advanced way of

estimating the daily variance. The GARCH approach does however also suffer from the same

deficiency as themoving average approachwhich is that it assumes the returns to be normally

distributed. This becomes apparent at the 99.9% confidence level, where the GARCH approach

constantlyproducedtomanyVaRbreakstobeacceptedbytheKupiectest.

TheKupiectestrejectstheapproachforalloftheassetsatthe99.9%level.Atthe99%level,theonly

asset forwhich theapproachwas rejectedwas theOMXs30.Thiswassurprisingat first,since the

GARCHapproachperformedsowellontheOMXs30atthe95%confidencelevel.Thereasontowhy

the approach produced too many VaR breaks is probably that the assumption of normality is

acceptable at lower confidence levels but not at higher ones. The further out in the tail of the

distributionof returnswecome, the lessacceptable theassumptionofnormalitybecomes.This is

howeverconflictedby the fact that theKupiec testaccepted theapproachat the99%confidence

level for the Brent Oil and STB3M returns. This can be explained by the observation that the

approach produces fewerVaR breaks than the target for these returns at the 95% level, i.e. the

approach produces to high VaR values. At the 99% level itwould seem that the assumption of

normality that generally causes the approach to produce toomany VaR breaks is countered by

another error in the approach, which cancels both errors out, resulting in the approach being

acceptedbytheKupiectest.

Thenextquestion is to figureoutwhat the error in the approach is that results in the approach

producingtoofewVaRbreaksatthe95%confidencelevelforBrentOilandSTB3M.Wereasonthatit

is yet again the assumption of normality that is throwing our calculations off. The maximum

likelihoodestimationformulathatweusedtoestimatethevaluesof,andalsoassumedthat

the returns that itwasevaluatingwerenormal.As argued in themethod chapter, the returnsof

BrentoilandSTB3Mexhibitedthecharacteristicsleastcompatiblewiththenormaldistribution.We


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thereforesuspectthattheassumptionofnormalityhascausedtheMLEfunctiontoproducevalues

of,and thatare slightlyoff theiroptimum,which in turncouldbe thecauseof theGARCH

functionproducingtoofewVaRbreaks.Thisargumentissupportedbylookingatthegraphsofthe

valuesof,andinappendix3forallofourthreeassets.Thegraphsshowhowthevaluesof,

and changeasweestimate themwitha rollingwindowof theprevious2000observations.The

values for theOMXs30 returns are stablewhereas the values for the two other assets fluctuate

somewhat,indicatingthattheMLEfunctionmightnotfunctionproperlywiththereturnsasaresult

oftheassumptionsofnormality.

Thesetwoerrorsbothspringfromthenormalityassumption,butseemtobychanceaffecttheVaR

inoppositedirection,causingtheapproachtoperformbetterthanexpectedatthe99%confidence

level.SoeventhoughtheapproachisacceptedbytheKupiectestforBrentoilandSTB30atthe99%

confidencelevelwehaveourreservationsofwhethertheoutcomereallyisaresultoftheapproach

performingwellorifitisjustarandomoccurrence.

Altogether,we feel that theGARCHapproachdoesagood jobofhandlingvolatilityclusteringand

estimatingavarianceforecasts.Wedo,however,feelthattheMLEfunctionthatwehaveuseddoes

notworkproficientlyandthatitwouldhavetobeadjustedtobetterfittheassetsreturnsinorderto

furtherincreasetheperformanceoftheGARCHapproach.

According to Goorbergh & Vlaar the most important return characteristic to account for when

calculatingVaRisvolatilityclustering.Wefindthatvolatilityclusteringindeedisimportantbutdonot

agree that it is themost important characteristic.Aswe have shown in this chapter themoving

averageapproachperformswellatthe95%confidence leveleventhough itdoesnottakevolatility

clustering into consideration. In our opinion the most important return characteristic is the

distribution of the returns and howwell theymatch the distribution assumed by the approach.

Volatilityclusteringisaccordingtoourresultsthesecondmostimportantcharacteristic.

4.2EXACTNESS&SIMPLICITYThe tradeoff between exactness and simplicity is an interesting but hard dilemma to solve or to

decide about. People will argue that if amodel do not provide exact and accurate answers or

estimations,theyarenotuseful.Somemodelshavetoprovideresultsornumbersthat istoa100

percentcorrect,otherwise itcan result in severeoutcomes.A straight forwardexample ismodels

usedintheconstructionofairplanes

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