1、Lecture 9: time and assets market,Contents,Inter-temporal preferences Two periods Several periods Asset market CAPM APT Complete market Pure arbitrage,Inter-temporal preferences,Utility function of inter-temporal Every period consumption ct depend on how much he consumed and invested in period t-1.,

2、Inter-temporal preferences,Two periods model : In the case with out any uncertainty First order condition: If means,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

3、return of R0 and x pays a random return of Period2: Utility function:,Inter-temporal preferences,Two periods model: Indirect utility function of period 1 with w. First order condition:,Inter-temporal preferences,several periods model Period t: consume ct, invest the rest wealth in two assets, (1-xt)

4、 percentage has a certain return of R0 and xt pays a random return of Periodt+1: Utility function:,Inter-temporal preferences,Several periods model: Indirect utility function of period T-1. First order condition:,Inter-temporal preferences,Several periods model: For period T-2, when we got then So T

5、he first order condition:,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.,Asset market,CAPM: Mean-variance efficient: minimize the Vari

6、ance when the Means are same.,Asset market,CAPM: The first order condition: If a portfolio is mean-variance efficient, means invest 100% in asset e and 0 in others is M-V efficient too. Then we got:,Asset market,CAPM: For a=0 and a=e we got: Then we have: That means if we have a M-V efficient portfo

7、lio asset an a risky-free asset, we can achieve efficient portfolio set by taking convex combinations of them. (see the fig.),Asset market,CAPM: Let e=m, where is the market portfolio of risky assets. And,Asset market,APT: Arbitrage pricing theory One factor: Construct a portfolio of two assets a an

8、d b with x and (1-x) The return will be,Asset market,APT: Arbitrage pricing theory One factor: If this portfolio is risky-free, means And , We have or So Finally we got:,Asset market,APT: Arbitrage pricing theory Two factor: Construct a risky-free portfolio The matrix must be singular So,Asset marke

9、t,Expect utility: We have got: Rubinstein(1976) For individual i ,Asset market,Expect utility: When a=c, We got,Asset market,Complete market: An asset a valued Vas in state s, how much is this asset worth in period 0? Construct a portfolio: holding Vas unit Arrow-Debreu Security s, means a A-D secur

10、ity valued $1 while the portfolio valued Vas in state s. so:,Asset market,Complete market: Let be the probability of s its the value of a portfolio that pays off $1 for certain, and R0 be the risk-free return of it. Then:,Asset market,Complete market:,Asset market,Pure arbitrage: No opportunity for pure arbitrage. No arbitrage principle: if then There exist price such that the value of any asset a is give