1. CHAPTER 6 Relation 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. Special case

12. 6.2 From Mean-Variance Frontier to a Discount Factor and beta Representation

13. Theorem
+𝜖, 其中𝜖与回报空间正交

14. Proof

15. Proof(2)

16. Proof(3) n

17. 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。

18. 问题
下列说法是否成立：There is a discount factor of the form 𝑥 ∗ =𝑎+𝑏 𝑅 𝑚𝑣 if and only if 𝑅 𝑚𝑣 is on the mean-variance frontier, and 𝑅 𝑚𝑣 is not the risk-free rate.
P13的定理有何问题？

19. 当且仅当存在无风险
证券时，该式才成立。

20. 6.3Factor Models and Discount Factors

21. An expected return beta model is equivalent to a discount factor that is a linear function of the factors in the beta model.
It is easiest to fold means of the factors in to the constant, and write 𝑚=𝑎+ 𝑏 ′ 𝑓 with 𝐸 𝑓 =0 and hence 𝐸 𝑚 =𝑎.

22. Theorem

23. Proof

24. Proof(2)

27. Factor-mimicking porfolios

30. 6.4 Discount Factors and Beta Models to Mean-Variance Frontier

33. 6.5 Three Risk-free Rate Analogues

37. =E(R*2)/E(R*)
利用相似三角形
其长度为

41. 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,

43. Constant-Mimicking Portfolio Return

45. 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:

46. 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: