Lecture 9: time and assets market
Lecture 9: time and assets market
Contents Inter-temporal preferences TWo periods Several periods Asset market CAPM APT Complete market Pure arbitrage
Contents • Inter-temporal preferences – Two periods – Several periods • Asset market – CAPM – APT – Complete market – Pure arbitrage
Inter-temporal preferences Utility function of inter-temporal U(c(…c)=∑。h(c) Every period consumption c, depend on how much he consumed and invested in period t-1
Inter-temporal preferences • Utility function of inter-temporal • Every period consumption ct depend on how much he consumed and invested in period t-1. 1 1 1 ( ) ( ) T t T t t U c c u c − = =
Inter-temporal preferences Two periods model In the case with out any uncertainty max(co, CD=u(co)+du(c) St.(v-c0)(1 0 +r)=c1 First order condition u(c n)=6(+r) u(c If Co=c, means 8 1+r
Inter-temporal preferences • Two periods model : • In the case with out any uncertainty • First order condition: • If means 0 1 0 1 0 1 max ( , ) ( ) ( ) . . ( )(1 ) U c c u c u c s t w c r c = + − + = 0 1 ( ) (1 ) ( ) u c r u c = + 1 1 r = + 0 1 c c =
Inter-temporal preferences TWO periods model with uncertainty investment Endowment wealth w Period 1: consume c, invest the rest wealth in two assets,(1-x) percentage has a certain return of Ro and x pays a random return of R Period2: C2=W2=(W-C[RX+Ro(1-x)]=(w-CR Utility function: U(, C2)=u(C)+DEu(C2)
Inter-temporal preferences • Two periods model with uncertainty investment. – Endowment wealth w. – Period1: consume c1 , invest the rest wealth in two assets, (1-x) percentage has a certain return of R0 and x pays a random return of – Period2: – Utility function: R1 2 2 1 1 0 1 c w w c R x R x w c R = = − + − = − ( )[ (1 )] ( ) 1 2 1 2 U c c u c Eu c ( , ) ( ) ( ) = +
Inter-temporal preferences Two periods model Indirect utility function of period 1 with w y(w)=maxu(C)+SEu(w-CR First order condition (c1)=6El(C2)R El(2)(R-R)=0
Inter-temporal preferences • Two periods model: – Indirect utility function of period 1 with w. – First order condition: 1 1 1 , ( ) max ( ) ( ) c x V w u c Eu w c R = + − 1 2 2 1 0 ( ) ( ) ( )( ) 0 u c Eu c R Eu c R R = − =
Inter-temporal preferences several periods model Period t: consume c. invest the rest wealth in two assets,(1-x) percentage has a certain return of Ro and x, pays a random return of R Periodt+ 1: C=W=(W-CR Utility function (1…n)=∑oEl(c) t=0
Inter-temporal preferences • several periods model – Period t: consume ct , invest the rest wealth in two assets, (1-xt ) percentage has a certain return of R0 and xt pays a random return of – Periodt+1: – Utility function: R1 1 1 ( ) t t t t c w w c R + + = = − 1 0 ( , ) ( ) T t T t t U c c Eu c = =
Inter-temporal preferences Several periods model Indirect utility function of period T-1 V_(Wr-1)=max u(c-1+SEu(w DR T-12x7-1 First order condition U(CTD=SEu(CR E(xn)(R1-R0)=0
Inter-temporal preferences • Several periods model: – Indirect utility function of period T-1. – First order condition: 1 1 1 1 1 1 1 , ( ) max ( ) ( ) T T T T T T T c x V w u c Eu w c R − − − − − − − = + − 1 1 0 ( ) ( ) ( )( ) 0 T T T u c Eu c R Eu c R R − = − =
Inter-temporal preferences Several periods model For period T-2, when we got (C-2, x-2)then T-2T-2 DR So VI-2(Wr-2)=max u(C-2)+SEVT-WT-2-CT-R CT-2 T-2 The first order condition u(C -2)+SEV(W-R=0 E(w1)(R1-R0)=0
Inter-temporal preferences • Several periods model: – For period T-2, when we got then – So – The first order condition: 1 2 2 ( ) w w c R T T T − − − = − 2 2 ( , ) T T c x − − 2 2 2 2 2 1 2 2 , ( ) max ( ) ( ) T T T T T T T T c x V w u c EV w c R − − − − − − − − = + − 2 1 1 1 0 ( ) ( ) 0 ( )( ) 0 T T T u c EV w R EV w R R − − − + = − =
Asset market CAPM: Capital Asset Pricing Model Consumption of the next period depend on how to invest the wealth in different assets c=(-c∑xR=(m-cxR+∑xR a= R is the return of asset a and x, is the percentage of it. Asset 0 is the no risky
Asset market • CAPM: Capital Asset Pricing Model – Consumption of the next period depend on how to invest the wealth in different assets. – is the return of asset a and is the percentage of it. Asset 0 is the no risky. 0 0 0 1 ( ) ( )[ ] A A a a a a a a c w c x R w c x R x R = = = − = − + R a a x
