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1、ThorstenHensandMarcOliverRiegerSolutionsFinancialEconomicsAConciseIntroductiontoClassicalandBehavioralFinanceSPINSpringer’sinternalprojectnumber,ifknownJuly21,2010Springer2DecisionTheory2.1Considerthegambleoftossingafairdie.Ifa6occursthenyouwin6,000,0
2、00monetaryunits.Inanyoftheotheroutcomesyoudon’twinanything.Thislotteryisgivenas1/6oooo6,000,000oooogoOOOOOOO5/6OO0Ifyourinitialwealthisw0=10,000monetaryunitsandyourexpectedutilityfunctionisu(x)=lg(x)(xisyourfinalwealthafterplayingthegame,lgisthelogfunc
3、tioninbase10)thentheutilityoftheabovelotterywillbenX=61u(g)=lg(ai)·,6i=1whereaiisthefinalwealthwhenoutcomeioccurred,forexampleoutcomea1=10,000asyoudon’twinorloseanything.Thevalueoftheexpectedutilityisu(g)=4.46.Ifwewin1,000,000forsure,thisisequivalentto
4、adegenerategamble.Theexpectedutilityofthisgambleisu(g0)=lg(1,010,000)=6.0043.Inthethirdcaseyouhavetopay10monetaryunitstobeabletopartic-ipateinthegambleoffering61monetaryunitswithprobability1/6.Inthiscasethen42DecisionTheory1/6oooo10,051oog00oOooOOOOOO5
5、/6OO9,990Tocomputethecertaintyequivalentofthisgamblewerecallitsdefinitionu(CE)=u(g00)nX=61lg(CE)=lg(ai)·≈4.00.6i=1ThusCE≈104=10,000.Soforsmallgamblesascomparedtoyouroverallwealthyouarepracticallyindifferentbetweenthegambleanditscertaintyequivalent.2.2Le
6、ta,b,cbetheprobabilitymeasurescorrespondingtothelotteriesA,B,C.ThenwehavebyasssumptionZZZuda≥udb≥udc.LetusdenotethethreeintegralsbyU(A),U(B),U(C).Thenthereisobviouslyap∈[0,1]suchthatU(B)=pU(A)+(1−p)U(C).(Ifyoudonotbelievethat,tryp=(U(B)−U(C))/(U(A)−U(
7、C)).)Theutilityofpa+(1−p)cisnowZZZU(pa+(1−p)c)=ud(pa+(1−p)c)=puda+(1−p)udc=pU(A)+(1−p)U(C)=U(B).2.3Tosolvethisexercisewewillmakeuseofthefactthattheslopeoftheutilityfunctionatanypoint(−fforexample)willbehigherthantheslopeofthehypothenuseofthetriangleco
8、nstructedfrom−fandu(−f)thus0u(−f)u(−f)>.−fBytheFundamentalTheoremofCalculuswehavethefollowingrelation:Z−fu0(s)ds=u(−f)−u(−2f).−2f2DecisionTheory5R−f0Solvingforu(−f)weobtainu(−f)=u(−2f)+u(s)ds.Usingthepre-−2fviousinequalitywehavethefollowingrel
