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## Stochastic Discount Factors

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**Stochastic Discount Factors**Chapter 4**Complete markets**Section 4.1.1**Fundamental asset pricing formula**• Almost all of modern asset pricing can be written in this form**Utility**• Beta is the time rate of preference for the agent (NOT CAPM beta) • Staying consistent with Campbell notation (my goal)**Perfect risk sharing**• Marginal utilities move in the same way across states for different consumers • Consumption moves close together • For many utility functions proportions are the same (try this with power utility) • For world with same power utility functions, consumption exactly in proportion • This is at the core of justifying a representative agent**Pareto optimal**• This also generates a Pareto optimal allocation • Economy solves a weighted expected utility maximization • This gives same conditions (should be familiar from micro) • See (4.20-21)**Empirical tests**• Developed countries • Individual consumption does not move together • This is also related to how well we can insure individual risk • Developing countries • Some cases where it seems to work (Townsend(1994)) • Talk to Nidhiya about this!**Existence of a representative agent**• Doing this fast • See (4.22-4.26) • Complete markets implies existence of representative agent • A given pricing/agg consumption outcome is optimization as if this is only person in economy • Powerful theorem • Limitations: (Big) • Utility function might be weird (not look like any sensible individual function) • Depends on wealth distributions • New wealth distribution – new utility • Can’t really think about demands**Belief aggregation**• Consumers adjust consumption to their own beliefs • Less consumption when own beliefs are lower than true probabilities • Interesting facts: • For some utility functions (exponential) can build hybrid beliefs • In an evolutionary environment with complete markets, consumers closest to true probabilities survive • Not true w/o complete markets • See Blume/Easley (JET 1990, Econometrica 2006) • Interesting question: • How did true probabilities get into M, q(s)**Incomplete Markets**Section 4.2**Incomplete markets**• More realistic • Minimal assumptions • Portfolio formation (combine payouts) • Law of one price**Stochastic discount factor**• Is there an M which prices all assets (payouts)? • And satisfies the law of one price • Reminder: • For complete markets, the answer was yes, and we built it from the underlying A/D securities • Proof by construction • Build an M which is a linear combination of payouts • In the “linear space” of asset returns**SDF: Incomplete markets**• Few assumptions • Linear portfolio construction • Law of one price • Need XX’ nonsingular • May be many M(s)’s which “price” payouts**Does it satisfy law of one price?**• Easy • This is trivial, but the math is kind of deep • The existence of M is deeply related to the theory of linear operators • The pricing operator and the expectations operator are both linear • Share similar mathematical structure**Is this SDF unique?**• NO!**Absence of arbitrage**• Stronger assumption • Puts more restrictions on M • Definition: Absence of arbitrage • In words: If a security pays at least zero in all states, and pays a positive amount in some states, with probabilities strictly positive, then this security must have a positive price.**Absence of arbitrage and the SDF**Absence of arbitrage (again)**Absence of arbitrage (again)**• Big theorem • IFF we have M(s)>0 for all states s, then there is Absence of Arbitrage**Proof**• If (right to left) direction is easy • Just think about this and M(s)>0 • Only if direction is hard (will not do) • See Campbell, page 92 • Two notes: • M=X* probably does NOT satisfy M(s)>0 conditions • M(s)>0 empirical restrictions on M are: • Difficult • Haven’t really led to big insights**SDF’s and asset pricing models**Section 4.3.1**How does this theory relate to general AP ideas?**• SDF is just a stand-in for lots of AP models (almost all) • Pick your favorite model. Put into M • All AP models hold to linear pricing and absence of arbitrage • They should ALL work with a stochastic discount factor**Log form**• Two things to note: • Easy formula (will be useful in more complex models) • Jensen’s inequality adjustment**Volatility bounds**• Hansen/Jagannathon bounds • 1980’s/90’s people testing lots of different asset pricing models (we will see how soon) • All rejected • H/J wanted to divide models broadly into some that could work, and some that were hopeless • H/J bounds are their approach • Can be done with or without risk free rate**Bounds and picture**• Drop assumption of risk free • HJ (Hansen/Jagannathan(91)), and Campbell (pp96-97) provide detailed technical proof • We are doing this in an approximate way • Skip/skim 96-97 for details • Often, we just guess at a risk free rate, and look at the bounds for different rates • This really doesn’t need a big deal proof (will skip) • Generates very nice picture and some intuition • Different models will imply a given risk free or often E(M) (same thing)**Bounds without Risk Free**• Plots: • Move risk free around • Get Sharpe ratio • When R(f)>min variance return, tangent shifts to other side • Min Sharpe with R(f)=min variance return • Note: All bounds can be made tighter by imposing M>0 (absence of arb), but this is tricky**Empirical issues**• Generalized method of moments • Very, very intuitive overview • Absence of arbitrage • Quick view on how well this works**Generalized Method of Moments (GMM)**• Powerful econometric tool • Most all econometrics can be done with GMM • Asset pricing well suited for GMM**Very quick intro to GMM**• Can be lots of assets, N, (say 25) • 25 equations to set to zero • Many fewer parameters K (say 4) • Only 4 degrees of freedom • Can’t get all of these to be exactly zero • Search over parameters to minimize last sum**GMM machinery (light)**• With minimal assumptions, this gives consistent estimator for b • Does not depend on • Distributions of u() • Time correlations in u() • Pretty powerful in terms of minimal assumptions**GMM efficiency**• There is a “best W” in terms of efficiency • This makes sense. It is weighting each moment by its variance. High variance -> low weight • Think about the case where covariances are zero and S is diagonal • This also gives nice asymptotic distribution for parameters (4.85) • Goodness of fit test (4.87)**GMM implementation**• One more issue • Estimate of S depends on parameter vector, b • Solution: • Stage 1: Estimate with some initial weighting matrix, I, get initial b(0) • Stage 2: From this estimate S, then use this to get b(1) • Seems like you could keep iterating, but in practice this seems to make little difference**Instruments and FOC’s**• Z(t) is vector of instruments (I), information (anything) known at time t • This attempts to incorporate conditionals into an unconditional • Now IxN restrictions (each return, and each instrument) • This really is like traditional instrumental variables, but in GMM context**GMM properties**• Very weak assumptions • Error distributions flexible • Can allow for correlations over time in errors • Consistent • Asymptotic theory • A little tricky: • Getting variance/covariance matrix • Often estimate with identity matrix • Then put in estimated Omega • Then estimate again, repeat**GMM implementations**• See section 4.4.3 for traditional tests mapped to GMM • Problems/questions (4.4.4) • Small sample problems (T not big enough) • Bootstrap or monte-carlo approaches can be useful here • Would maximum likelihood have been better? (more assumptions, but better performance if true) • Moment choices • Instrument choices • Long run covariance estimator (S) • This can involve Newey-West correction (page 111), but this might not be perfect • Other choices for S • Always a good idea to compare first and second stage estimators • GMM is not perfect solution to all econometric problems • However, it is more intuitive in terms of a model fitting/estimation tool, than you might think • Related tests: • SMM (simulated method of moment) • EMM (efficient method of moments, tries to find optimal moments based on model)**Absence of arbitrage**• Same assets should be priced the same • Examples: • 3Com owns Palm, but Palm more valuable • Royal Dutch Shell (UK and Netherlands shares) • Government bonds (Long Term Capital Management) • Closed end fund puzzle • These exist, but not common • Limits to arbitrage • Funding • Institutional restrictions (short sales) • Participation • Solvency (Keynes) • Not correctly measuring risk (momentum, carry trades) • Goes away (McLean/Pontiff(2016)) • Behavioral patterns • Mistakes (people and machines, Berkshire/Hathaway and Anne Hathaway)