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RiskandUncertaintyII
BenjaminChiao
March31,2011(afterclassversion)LectureNotesforUndergraduate
LevelInformationEconomics,ShanghaiUniversityofFinanceand
Economics
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IDe…nitionsofRiskAttitudes
IMeasurementofRiskAttitudes
IComparisonofDistributions
2/19
IDe…nitionsofRiskAttitudes
!
IMeasurementofRiskAttitudes
IComparisonofDistributions
3/19
IDe…nitionsofRiskAttitudes
!
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RiskAversev.RiskLoving
ISetting
IMonetaryoutcomes.Denotemoneyasacontinuousrandom
variablexdistributedaccordingtothecumulativedistribution
functionF(x).
IThelotteryspaceisallpossiblec.d.fovernonnegative
amountofmoneyL
ISothecontinuousversionoftheEUtheoremgives:
U(F)=u(x)dF(x)(1)
Z
ITheBernoulliutilityfunctionu(x)isassumedtobeincreasing
andcontinuous.
IAndalsoboundedness(referstotheSt.Petersburg-Menger
paradox)
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RiskAttitudeDe…nitions
IDe…nitionArisklovingpersonweaklyprefersalotteryF()
overthelotterythatgivesasureoutcomehavingthevalue
thesameastheexpectedvaluexdF(x).
IIfthepersonisindi¤erent,he’scalledRriskneutral
IIfthepersonpreferstheotherdirection,he’scalledriskaverse
IIntheexpectedutilitycontext,itfollowsfromtheabovethata
riskaversepersonsatis…estheJensen’sinequality(orconcavity
ofu):
F:u(x)dF(x)u(xdF(x))(2)
8ZZ
ISeeFigure.
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IDe…nition
ICertaintyequivalentc(F,u)istheamountofmoneythat
satis…es
u(x)dF(x)=u(c(F,u))(3)
Z
IThatis,youareindi¤erenttotakingcorthegambleF.
ISeeFig.
IExercise1:Provethatapersonisriskaversei¤
c(F,u)xdF(x)
R
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IDe…nition
IProbabilitypremiumπ(x,e,u)istheprobabilitythatsatis…es,
foralle>0,
11
u(x)=(+π(x,e,u))u(x+e)+(π(x,e,u))u(xe)
22
IGivenasureoutcomex,theprobabilitypremiumπ(x,e,u)is
theprobabilityyouwouldsubstractfrom1/2ofprobabilityof
gettingaslightlyworseo¤outcome(xe)andaddtoa1/2
probabilityofgettingaslightlybettero¤ outcome(x+e)so
thatyouareindi¤erentbetweenthisandx.
ISeeFig.
IExercise2:Provethatapersonisriskaversei¤π(x,e,u)0
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IDe…nitionsofRiskAttitudes
IMeasurementofRiskAttitudes
!
IComparisonofDistributions
9/1
内容提供者
2024-09-11
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160KB
19页
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