CHAPTER 6 Relation between Discount Factors,Betas,and Mean-Variance Frontiers

1. CHAPTER 6Relation between Discount Factors,Betas,and Mean-Variance Frontiers

2. 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.

3. Transformation between the three representations

4. 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.

5. Transformation between the three representations(2)

6. 6.1 From Discount Factors to Beta Representations

7. Beta representation using m Multiply and divide by var(m),define ,we get:

9. Theorem

10. Proof

11. 状态2回报 Rf R* 1 x* P=1(收益率） pc 状态1回报 Re* P=0(超额收益率）

12. 状态2回报 Rf R* 1 x* P=1(收益率） pc 状态1回报 Re* P=0(超额收益率）

13. Special case

14. 6.2 From Mean-Variance Frontier to a Discount Factor and beta Representation

15. Theorem

16. Proof

17. Proof(2)

18. Proof(3) n

19. 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”). • 证毕。

21. 6.3Factor Models and Discount Factors

23. Theorem

24. Proof • From (6.7), • Here we get (6.8) • where

25. Theorem

26. Proof

27. Proof(2)

30. Factor-mimicking porfolios

33. 6.4 Discount Factors and Beta Models to Mean-Variance Frontier

36. 6.5 Three Risk-free Rate Analogues

40. =E(R*2)/E(R*) 利用相似三角形 其长度为

44. 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,

46. Constant-Mimicking Portfolio Return

48. 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:

49. 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: