>> Decomposition of Stochastic Discount Factor and their Volatility Bounds 2012 11 21 >> 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 >> 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 >> 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 >> Decomposition of SDF So risk-free return: Long term bond return: 2012/12/17 Asset Pricing 4 >> 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 >> Decomposition of SDF Unique Decomposition (Alvarez & Jermann,2005) and: with: 2012/12/17 Asset Pricing 6 >> Decomposition of SDF How to link transitory component to Long term bond? No cash flow uncertainty 2012/12/17 Asset Pricing 7 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 8 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 9 >> 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 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 11 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 12 >> 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 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 14 >> Permanent and Transitory Bounds 2012/12/17 Asset Pricing 15 >> 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 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 17 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 18 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 19 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 20 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 21 >> Comparisons with Alvarez & Jermann (2005) 2012/12/17 Asset Pricing 22 >> 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 >> 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 >> So How to Decompose? Whats ? 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 >> 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 >> 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 >> Eigenfunction and Eigenvalue Method 2012/12/17 Asset Pricing 28 >> 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 >> 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 >> Eigenfunction and Eigenvalue Method Example: consider a multiplicative process : And : Guess an eigenfunction of the form 2012/12/17 Asset Pricing 31 >> Eigenfunction and Eigenvalue Method 2012/12/17 Asset Pricing 32 >> Eigenfunction and Eigenvalue Method define a new probability measure, resulting distorted drift of : 2012/12/17 Asset Pricing 33 >> 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 >> 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 >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 36 >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 37 >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 38 >> Asset Pricing Models Representation 2012/12/17 Asset Pricing 39 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 40 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 41 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 42 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 43 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 44 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 45 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 46 >> Empirical Application to Asset Pricing Models 2012/12/17 Asset Pricing 47 >> Thank you for listening and Comments are welcome 11 21