Asset Pricing Zheng Zhenlong CHAPTER 6 Relation between Discount Factors,Betas,and Mean-Variance Frontiers 19:45 1 Asset Pricing Zheng Zhenlong Main contents we will draw the connection between discount factors,mean- variance frontiers, and beta representations,then we will show how they transform between each other,because these three representations are equivalent. Asset Pricing Zheng Zhenlong Transformation between the three representations Asset Pricing Zheng Zhenlong Transformation between the three representations(2). If we have an expected return-beta model with factors f, then linear in the factors satisfies. If a return is on the mean-variance fron-tier,then there is an expected return-beta model with that return as reference variable. Asset Pricing Zheng Zhenlong Transformation between the three representations(2) Asset Pricing Zheng Zhenlong 6.1 From Discount Factors to Beta Representations 19:45 6 Asset Pricing Zheng Zhenlong Beta representation using m Multiply and divide by var(m),define,we get: Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Theorem Asset Pricing Zheng Zhenlong Proof Asset Pricing Zheng Zhenlong P=0( RfRf P=1( 1 2 R* 1 R e* x* pc Asset Pricing Zheng Zhenlong P=0( RfRf P=1( 1 2 R* 1 R e* x* pc Asset Pricing Zheng Zhenlong Special case Asset Pricing Zheng Zhenlong 6.2 From Mean-Variance Frontier to a Discount Factor and beta Representation 19:45 14 Asset Pricing Zheng Zhenlong Theorem Asset Pricing Zheng Zhenlong Proof Asset Pricing Zheng Zhenlong Proof(2) Asset Pricing Zheng Zhenlong Proof(3) n Asset Pricing Zheng Zhenlong Note If the denominator is zero, i.e., if,this construction cannot work. If there is a risk-free rate, we are ruling out the case If there is no risk-free rate, we must rule out the case (the constant- mimicking portfolio return). Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong 6.3Factor Models and Discount Factors 19:45 21 Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Theorem Asset Pricing Zheng Zhenlong Proof From (6.7), Here we get (6.8) where Asset Pricing Zheng Zhenlong Theorem Asset Pricing Zheng Zhenlong Proof Asset Pricing Zheng Zhenlong Proof(2) Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Factor-mimicking porfolios Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong 6.4 Discount Factors and Beta Models to Mean-Variance Frontier 19:45 33 Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong 6.5 Three Risk-free Rate Analogues 19:45 36 Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong =E(R *2 )/E(R * ) Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Minimum-Variance Return The risk-free rate obviously is the minimum -variance return when it exists. When there is no risk-free rate, the minimum- variance return is (6.15) Taking expectations, Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Constant-Mimicking Portfolio Return Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong Risk-Free Rate Here we will show that if there exists a risk-free rate,then all the zero-beta return, minimum-variance return,and constant-mimicking portfolio return reduce to the risk-free rate. These other rates are: Constant-mimicking: Asset Pricing Zheng Zhenlong Minimum-variance: Zero-beta: And the risk-free rate: (6.19) To establish that there are all the same when there is a risk- free rate, we need to show that: Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong 6.6 Mean-Variance Special Cases with No Risk-Free Rate 19:45 51 Asset Pricing Zheng Zhenlong There exist special cases for the equivalence theorems,that is,when the expected discount factor,price of a unit payoff,or risk-free rate is zero or infinity. If risk-free rate is traded or the market is complete,then it won t be a problem; however,in an incomplete market in which no risk free rate is traded,we must pay attention to it and make it sure that Asset Pricing Zheng Zhenlong The special case for a mean- variance frontier to a discount factor Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong The special case for mean- variance frontier to a beta model We can use any return on the mean-variance frontier as the reference return for a single-beta representation,except the minimum-variance return. Asset Pricing Zheng Zhenlong Theorem: 19:45 56 Asset Pricing Zheng Zhenlong Asset Pricing Zheng Zhenlong