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Decomposition of Stochastic Discount Factor and their Volatility Bounds

Decomposition of Stochastic Discount Factor and their Volatility Bounds. 2012 年 11 月 21 日. Framework. Motivation Decomposition of SDF Permanent and Transitory Bounds Comparisons with Alvarez & Jermann (2005) Eigenfunction and Eigenvalue Method Asset Pricing Models Representation

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Decomposition of Stochastic Discount Factor and their Volatility Bounds

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  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 Asset Pricing

  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 Asset Pricing

  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 Asset Pricing

  5. Decomposition of SDF • So risk-free return: • Long term bond return: Asset Pricing

  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 Asset Pricing

  7. Decomposition of SDF • Unique Decomposition (Alvarez & Jermann,2005) and: with: Asset Pricing

  8. Decomposition of SDF • How to link transitory component to Long term bond? • No cash flow uncertainty Asset Pricing

  9. Permanent and Transitory Bounds Asset Pricing

  10. Permanent and Transitory Bounds Asset Pricing

  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 Asset Pricing

  12. Permanent and Transitory Bounds Asset Pricing

  13. Permanent and Transitory Bounds Asset Pricing

  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. Asset Pricing

  15. Permanent and Transitory Bounds Asset Pricing

  16. Permanent and Transitory Bounds Asset Pricing

  17. Comparisons with Alvarez & Jermann (2005) • In Alvarez & Jermann, L-measure (entropy) a random variableu 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. Asset Pricing

  18. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  19. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  20. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  21. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  22. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  23. Comparisons with Alvarez & Jermann (2005) Asset Pricing

  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 Asset Pricing

  25. Eigenfunction and Eigenvalue Method • 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: Asset Pricing

  26. Eigenfunction and Eigenvalue Method • 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. Asset Pricing

  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 functionif 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 Asset Pricing

  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 Asset Pricing

  29. Eigenfunction and Eigenvalue Method Asset Pricing

  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 Asset Pricing

  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 Asset Pricing

  32. Eigenfunction and Eigenvalue Method • Example: consider a multiplicative process : • And : • Guess an eigenfunction of the form Asset Pricing

  33. Eigenfunction and Eigenvalue Method Asset Pricing

  34. Eigenfunction and Eigenvalue Method • define a new probability measure, resulting distorted drift of : Asset Pricing

  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 . Asset Pricing

  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. Asset Pricing

  37. Asset Pricing Models Representation Asset Pricing

  38. Asset Pricing Models Representation Asset Pricing

  39. Asset Pricing Models Representation Asset Pricing

  40. Asset Pricing Models Representation Asset Pricing

  41. Empirical Application to Asset Pricing Models Asset Pricing

  42. Empirical Application to Asset Pricing Models Asset Pricing

  43. Empirical Application to Asset Pricing Models Asset Pricing

  44. Empirical Application to Asset Pricing Models Asset Pricing

  45. Empirical Application to Asset Pricing Models Asset Pricing

  46. Empirical Application to Asset Pricing Models Asset Pricing

  47. Empirical Application to Asset Pricing Models Asset Pricing

  48. Empirical Application to Asset Pricing Models Asset Pricing

  49. Thank you for listening andComments are welcome. 2012年11月21日

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