Asset pricing

463 Pages·2001·3.019 MB·English

by  John H. Cochrane

Asset Pricing
John H. Cochrane
June 12, 2000
1
Acknowledgments
ThisbookowesanenormousintellectualdebttoLarsHansenandGeneFama. Mostofthe
ideas in the book developed from long discussions with each of them, and trying to make
senseofwhateachwassayinginthelanguageoftheother. Iamalsogratefultoallmycol-
leaguesinFinanceandEconomicsattheUniversityofChicago,andtoGeorgeConstantinides
especially,formanydiscussionsabouttheideasinthisbook.IthankGeorgeConstantinides,
Andrea Eisfeldt, Gene Fama, Wayne Ferson, Owen Lamont, Anthony Lynch, Dan Nelson,
Alberto Pozzolo, Michael Roberts, Juha Seppala, Mike Stutzer, PietroVeronesi, an anony-
mous reviewer, and several generations of Ph.D. students at the University of Chicago for
manyusefulcomments. I thanktheNSFandtheGraduateSchoolofBusinessfor research
support.
Additionalmaterialandbothsubstantiveandtypographicalcorrectionswillbemaintained
at
http://www-gsb.uchicago.edu/fac/john.cochrane/research/papers
Comments and suggestions are most welcome This book draft is copyright c John H.
°
Cochrane1997,1998,1999,2000
JohnH.Cochrane
GraduateSchoolofBusiness
UniversityofChicago
1101E.58thSt.
ChicagoIL60637
7737023059
[email protected]
June12,2000
2
Contents
Acknowledgments 2
Preface 8
PartI. Asset pricingtheory 12
1 Consumption-basedmodelandoverview 13
1.1 Basicpricingequation 14
1.2 Marginalrateofsubstitution/stochasticdiscountfactor 16
1.3 Prices,payoffsandnotation 17
1.4 Classicissuesinfinance 20
1.5 Discountfactorsincontinuoustime 33
1.6 Problems 38
2 Applyingthebasicmodel 41
2.1 Assumptionsandapplicability 41
2.2 GeneralEquilibrium 43
2.3 Consumption-basedmodelinpractice 47
2.4 Alternativeassetpricingmodels: Overview 49
2.5 Problems 51
3 ContingentClaimsMarkets 54
3.1 Contingentclaims 54
3.2 Riskneutralprobabilities 55
3.3 Investorsagain 57
3.4 Risksharing 59
3.5 Statediagramandpricefunction 60
4 Thediscountfactor 64
4.1 Lawofonepriceandexistenceofadiscountfactor 64
4.2 No-Arbitrageandpositivediscountfactors 69
3
4.3 Analternativeformula,andx incontinuoustime 74
∗
4.4 Problems 76
5 Mean-variancefrontierandbetarepresentations 77
5.1 Expectedreturn-Betarepresentations 77
5.2 Mean-variancefrontier: IntuitionandLagrangiancharacterization 80
5.3 Anorthogonalcharacterizationofthemean-variancefrontier 83
5.4 Spanningthemean-variancefrontier 88
5.5 AcompilationofpropertiesofR ,Re andx 89
∗ ∗ ∗
5.6 Mean-variancefrontiersform: theHansen-Jagannathanbounds 92
5.7 Problems 97
6 Relationbetweendiscountfactors,betas,andmean-variancefrontiers 98
6.1 Fromdiscountfactorstobetarepresentations 98
6.2 Frommean-variancefrontiertoadiscountfactorandbetarepresentation 101
6.3 Factormodelsanddiscountfactors 104
6.4 Discountfactorsandbetamodelstomean-variancefrontier 108
6.5 Threeriskfreerateanalogues 109
6.6 Mean-variancespecialcaseswithnoriskfreerate 115
6.7 Problems 118
7 Implicationsofexistenceandequivalencetheorems 120
8 Conditioninginformation 128
8.1 Scaledpayoffs 129
8.2 Sufficiencyofaddingscaledreturns 131
8.3 Conditionalandunconditionalmodels 133
8.4 Scaledfactors:apartialsolution 140
8.5 Summary 141
8.6 Problems 142
9 Factorpricingmodels 143
9.1 CapitalAssetPricingModel(CAPM) 145
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9.2 IntertemporalCapitalAssetPricingModel(ICAPM) 156
9.3 CommentsontheCAPMandICAPM 158
9.4 ArbitragePricingTheory(APT) 162
9.5 APTvs.ICAPM 171
9.6 Problems 172
PartII. Estimatingandevaluatingassetpricingmodels 174
10 GMMinexplicitdiscountfactormodels 177
10.1 TheRecipe 177
10.2 InterpretingtheGMMprocedure 180
10.3 ApplyingGMM 184
11 GMM:generalformulasandapplications 188
11.1 GeneralGMMformulas 188
11.2 Testingmoments 192
11.3 Standarderrorsofanythingbydeltamethod 193
11.4 UsingGMMforregressions 194
11.5 Prespecifiedweightingmatricesandmomentconditions 196
11.6 Estimatingononegroupofmoments,testingonanother. 205
11.7 Estimatingthespectraldensitymatrix 205
11.8 Problems 212
12 Regression-basedtestsoflinearfactormodels 214
12.1 Time-seriesregressions 214
12.2 Cross-sectionalregressions 219
12.3 Fama-MacBethProcedure 228
12.4 Problems 234
13 GMMforlinearfactormodelsindiscountfactorform 235
13.1 GMMonthepricingerrorsgivesacross-sectionalregression 235
13.2 Thecaseofexcessreturns 237
13.3 HorseRaces 239
5
13.4 Testingforcharacteristics 240
13.5 Testingforpricedfactors: lambdasorb’s? 241
13.6 Problems 245
14 Maximumlikelihood 247
14.1 Maximumlikelihood 247
14.2 MLisGMMonthescores 249
14.3 Whenfactorsarereturns,MLprescribesatime-seriesregression 251
14.4 Whenfactorsarenotexcessreturns,MLprescribesacross-sectional
regression 255
14.5 Problems 256
15 Timeseries,cross-section,andGMM/DFtestsoflinearfactormodels 258
15.1 ThreeapproachestotheCAPMinsizeportfolios 259
15.2 MonteCarloandBootstrap 265
16 Whichmethod? 271
PartIII. Bondsandoptions 284
17 Optionpricing 286
17.1 Background 286
17.2 Black-Scholesformula 293
17.3 Problems 299
18 Optionpricingwithoutperfectreplication 300
18.1 Ontheedgesofarbitrage 300
18.2 One-periodgooddealbounds 301
18.3 Multipleperiodsandcontinuoustime 309
18.4 Extensions,otherapproaches,andbibliography 317
18.5 Problems 319
19 Termstructureofinterestrates 320
19.1 Definitionsandnotation 320
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19.2 Yieldcurveandexpectationshypothesis 325
19.3 Termstructuremodels–adiscrete-timeintroduction 327
19.4 Continuoustimetermstructuremodels 332
19.5 Threelineartermstructuremodels 337
19.6 Bibliographyandcomments 348
19.7 Problems 351
PartIV. Empirical survey 352
20 Expectedreturnsinthetime-seriesandcross-section 354
20.1 Time-seriespredictability 356
20.2 TheCross-section:CAPMandMultifactorModels 396
20.3 Summaryandinterpretation 409
20.4 Problems 413
21 Equitypremiumpuzzleandconsumption-basedmodels 414
21.1 Equitypremiumpuzzles 414
21.2 Newmodels 423
21.3 Bibliography 437
21.4 Problems 440
22 References 442
PartV. Appendix 455
23 Continuoustime 456
23.1 BrownianMotion 456
23.2 Diffusionmodel 457
23.3 Ito’slemma 460
23.4 Problems 462
7
Preface
Assetpricingtheorytriestounderstandthepricesorvaluesofclaimstouncertainpayments.
Alow priceimplies a highrateof return, soonecanalsothinkof thetheoryas explaining
whysomeassetspayhigheraveragereturnsthanothers.
Tovalueanasset,wehavetoaccountforthedelayandfortheriskofitspayments. The
effects of time are not too difficult to work out. However, corrections for risk are much
moreimportantdeterminantsofanmanyassets’values. Forexample,overthelast50years
U.S.stockshavegivenarealreturnofabout9%onaverage. Ofthis,onlyabout1%isdue
to interest rates; the remaining 8% is a premium earned for holding risk. Uncertainty, or
correctionsforriskmakeassetpricinginterestingandchallenging.
Assetpricingtheorysharesthepositivevs. normativetensionpresentintherestofeco-
nomics. Does it describe the waytheworlddoes workor thewaythe worldshould work?
Weobservethepricesorreturnsofmanyassets. Wecanusethetheorypositively,totryto
understandwhypricesorreturnsarewhattheyare.Iftheworlddoesnotobeyamodel’spre-
dictions,wecandecidethatthemodelneedsimprovement.However,wecanalsodecidethat
theworld iswrong, thatsomeassets are“mis-priced”andpresent tradingopportunitiesfor
theshrewdinvestor. Thislatteruseofassetpricingtheoryaccountsformuchofitspopular-
ityandpracticalapplication. Also,andperhapsmostimportantly,thepricesofmanyassets
orclaimstouncertaincashflowsarenotobserved,suchaspotentialpublicorprivateinvest-
mentprojects, newfinancialsecurities, buyout prospects, andcomplexderivatives. Wecan
applythetheorytoestablishwhatthepricesoftheseclaimsshould beaswell; theanswers
areimportantguidestopublicandprivatedecisions.
Asset pricing theory all stems from one simple concept, derived in the first page of the
firstChapterofthisbook: priceequalsexpecteddiscountedpayoff. Therestiselaboration,
special cases, and a closet full of tricks that make the central equation useful for one or
anotherapplication.
Therearetwopolarapproachestothiselaboration. Iwillcallthemabsolutepricingand
relativepricing. Inabsolutepricing,wepriceeachassetbyreferencetoitsexposuretofun-
damental sources of macroeconomic risk. Theconsumption-basedandgeneral equilibrium
modelsdescribedbelowarethepurestexamplesofthisapproach. Theabsoluteapproachis
mostcommoninacademicsettings, inwhichweuseassetpricingtheorypositivelytogive
aneconomicexplanationforwhypricesarewhattheyare,orinordertopredicthowprices
mightchangeifpolicyoreconomicstructurechanged.
Inrelativepricing,weaskalessambitiousquestion. Weaskwhatwecanlearnaboutan
asset’svaluegiventhepricesofsomeotherassets.Wedonotaskwherethepriceoftheother
set of assets came from, and we use as little informationabout fundamental risk factors as
possible.Black-Scholesoptionpricingistheclassicexampleofthisapproach.Whilelimited
inscope,thisapproachoffersprecisioninmanyapplications.
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Assetpricingproblemsaresolvedbyjudiciouslychoosinghowmuchabsoluteandhow
muchrelativepricingonewilldo,dependingontheassetsinquestionandthepurposeofthe
calculation. Almostnoproblemsaresolvedbythepureextremes. Forexample,theCAPM
anditssuccessorfactormodelsareparadigmsoftheabsoluteapproach. Yetinapplications,
theypriceassets“relative”tothemarketorotherriskfactors,withoutansweringwhatdeter-
mines the marketor factor riskpremiaandbetas. The latter are treated as freeparameters.
Ontheotherendofthespectrum,mostpracticalfinancialengineeringquestionsinvolveas-
sumptions beyond pure lack of arbitrage, assumptions about equilibrium “market prices of
risk.”
Thecentralandunfinishedtaskofabsoluteassetpricingistounderstandandmeasurethe
sourcesofaggregateormacroeconomicriskthatdriveassetprices.Ofcourse,thisisalsothe
centralquestionof macroeconomics, andthis is a particularlyexcitingtimefor researchers
whowant toanswer thesefundamentalquestions inmacroeconomicsandfinance. Alotof
empiricalworkhasdocumentedtantalizingstylizedfactsandlinksbetweenmacroeconomics
andfinance. For example, expectedreturns varyacrosstimeandacross assets inways that
arelinkedtomacroeconomicvariables,orvariablesthatalsoforecastmacroeconomicevents;
a wide class of models suggests that a“recession” or “financial distress”factor lies behind
manyassetprices. Yettheorylagsbehind; we donot yet haveawell-describedmodel that
explainstheseinterestingcorrelations.
In turn, I think that what we are learning about finance must feed back on macroeco-
nomics. To take a simple example, we have learned that the risk premium on stocks – the
expectedstock return less interest rates –is muchlarger than the interest rate, andvaries a
gooddealmorethaninterestrates.Thismeansthatattemptstolineinvestmentupwithinter-
estratesareprettyhopeless–mostvariationinthecostofcapitalcomesfromthevaryingrisk
premium. Similarly,wehavelearnedthatsomemeasureofriskaversionmustbequitehigh,
or people would all borrow like crazy to buy stocks. Most macroeconomics pursues small
deviationsaboutperfectforesightequilibria,butthelargeequitypremiummeansthatvolatil-
ityisafirst-ordereffect,notasecond-ordereffect. Standardmacroeconomicmodelspredict
thatpeoplereallydon’tcaremuchaboutbusinesscycles(Lucas1987). Assetpricesarebe-
ginningtorevealthattheydo–thattheyforegosubstantialreturnpremiatoavoidassetsthat
fallinrecessions.Thisfactoughttotellussomethingaboutrecessions!
This book advocates a discount factor / generalized method of moments view of asset
pricingtheoryandassociatedempiricalprocedures. Isummarizeassetpricingbytwoequa-
tions:
p =E(m x )
t t+1 t+1
m =f(data,parameters).
t+1
wherep =assetprice,x =assetpayoff,m =stochasticdiscountfactor.
t t+1 t+1
9