1. >> Decomposition of Stochastic Discount Factor and their Volatility Bounds 2012 年 11 月 21 日

2. >> Framework Motivation Decomposition of SDF Permanent and Transitory Bounds Comparisons with Alvarez & Jermann (2005) Eigenfunction and Eigenvalue Method Asset Pricing Models Representation Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 1

3. >> Motivation Economic Intuitions: Explanation Inability of Equilibrium Asset Pricing Model -Various Puzzles (Return, Volatility) -Frequency Mismatch (Daniel & Marshall,1997) -Features of Investor Preference: Local Durability, Habit Persistence or Long Run Risk Unit Root Contributions of Macroeconomic Variables Econometric Similarity: -Beveridge-Nelson Decomposition 2012/12/17 Asset Pricing 2

4. >> Decomposition of SDF No Arbitrage Opportunities in Frictionless Market if and only if a strictly positive Pricing Kernel exists: So SDF for any gross return on a generic portfolio held from to Define as the gross return from holding from time to a claim to one unit of the numeraire to be delivered at time 2012/12/17 Asset Pricing 3

5. >> Decomposition of SDF So risk-free return: Long term bond return: 2012/12/17 Asset Pricing 4

6. >> Decomposition of SDF Assumptions: -SDF and Return Jointly Stationary and Ergodic -There is a number such that -For each there is a random variable such that with finite for all 2012/12/17 Asset Pricing 5

7. >> Decomposition of SDF Unique Decomposition (Alvarez & Jermann,2005) and: with: 2012/12/17 Asset Pricing 6

8. >> Decomposition of SDF How to link transitory component to Long term bond? No cash flow uncertainty 2012/12/17 Asset Pricing 7

9. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 8

10. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 9

11. >> Permanent and Transitory Bounds Inequality (6) bounds the variance of the permanent component of the SDF, useful for understanding what time-series assumptions are necessary to achieve consistent risk pricing across multiple asset markets is receptive to an interpretation as in Hansen & Jagannathan (1991) bound: So can be interpreted as the maximum Sharpe ratio, but relative to the long-term bond 2012/12/17 Asset Pricing 10

12. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 11

13. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 12

14. >> Permanent and Transitory Bounds The transitory component equals the inverse of the gross return of an infinite-maturity discount bond and governs the behavior of interest rates The quantity on the right-hand side of equation (9) is tractable and computable from the return data. And the bound in (9) is a parabola in space. is positively associated with the square of the Sharpe ratio of the long-term bound. (9) to assess the bound market implications of asset pricing models. 2012/12/17 Asset Pricing 13

15. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 14

16. >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 15

17. >> Comparisons with Alvarez & Jermann (2005) In Alvarez & Jermann, L-measure (entropy) a random variable u as a measure of volatility: One-to-one correspondence exists between variance and L-measure when is log-normally distributed Such discrepancies between the two measures can get magnified under departures from log-normality. 2012/12/17 Asset Pricing 16

18. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 17

19. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 18

20. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 19

21. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 20

22. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 21

23. >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 22

24. >> Eigenfunction and Eigenvalue Method Continuous Time Version (Luttmer,2003): Consider State-Price Process: Suppose: For Any, and is bounded for all, the dominated convergence theorem implies that 2012/12/17 Asset Pricing 23

25. >> The process is referred to as the permanent component of SDF Define to be the residual, So: And suppose: As we all know, it also can be decomposed: 2012/12/17 Asset Pricing 24 Eigenfunction and Eigenvalue Method

26. >> So How to Decompose? What’s ? Hansen & Scheinkman (2009, Econometrica) Let be a Banach space, and let be a family of operators on. If: 1, for all 2, Positive if for any whenever 3, For each, Then is a semi-group. 2012/12/17 Asset Pricing 25

27. >> Eigenfunction and Eigenvalue Method Consider General Multiplicative Semi-group: Extended Generator: a Boral function belongs to the domain of the extended generator of the multiplica- tive function if there exists a Boral function such that is a local martingale wrt. filtration. In this case, the extended generator assigns function to and write Associates to function a function such that is the expected time derivative of 2012/12/17 Asset Pricing 26

28. >> Eigenfunction and Eigenvalue Method A Borel function is an eigenfunction of the extended generator with eigenvalue if. Intuitively, So if is an eigenfunction, the expected time derivative of is. Hence, the expected time derivative of is zero. How to get ? Expected time derivative is zero Local Martingale 2012/12/17 Asset Pricing 27

29. >> Eigenfunction and Eigenvalue Method 2012/12/17 Asset Pricing 28

30. >> Eigenfunction and Eigenvalue Method 6.1Proof: let, so: And: Interpretation: - : Growth rate of multiplicative functional - : Transient or Stationary Component - : Martingale Component, Distort the drift 2012/12/17 Asset Pricing 29

31. >> Eigenfunction and Eigenvalue Method Further more: If we treat as a numeraire, similar to the risk-neutral pricing in finance. Decomposition Existence and Uniqueness is given in Proposition 7.2 (Hansen & Scheinkman,2009) Congruity of Bakshi & Chabi-Yo Decomposition 2012/12/17 Asset Pricing 30

32. >> Eigenfunction and Eigenvalue Method Example: consider a multiplicative process : And : Guess an eigenfunction of the form 2012/12/17 Asset Pricing 31

33. >> Eigenfunction and Eigenvalue Method 2012/12/17 Asset Pricing 32

34. >> Eigenfunction and Eigenvalue Method define a new probability measure, resulting distorted drift of : 2012/12/17 Asset Pricing 33

35. >> Asset Pricing Models Representation Consider the modification of the long-run risk model proposed in Kelly (2009). The distinguishing attribute: the model incorporates heavy-tailed shocks to the evolution of nondurable consumption growth (log), governed by a tail risk state variable. 2012/12/17 Asset Pricing 34

36. >> Asset Pricing Models Representation While the transitory component of SDF is distributed log-normally, the permanent component of SDF and SDF itself are not log-normally distributed. The non-gaussian shock are meant to amplify the tails of the permanent component of SDF and SDF. 2012/12/17 Asset Pricing 35

37. >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 36

38. >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 37

39. >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 38

40. >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 39

41. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 40

42. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 41

43. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 42

44. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 43

45. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 44

46. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 45

47. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 46

48. >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 47

49. >> Thank you for listening and Comments are welcome. 2012 年 11 月 21 日