L13: Conditioning Information1 Lecture 13: Conditioning Information The following topics will be covered: Conditional versus unconditional models Managed portfolios Application of conditioning information in factor pricing models Demystifying GMM Materials from Chapter 8 of JC

L13: Conditioning Information2 Conditional vs. Unconditional where E t means expectation conditional on the investor’s time-t information. Conditional expectation can also be written as: Take unconditional expectations to obtain Also:

L13: Conditioning Information3 Instruments and Managed Portfolios Suppose we multiply the payoff and price by any variable or instrument z t observed at time t. Then Instrument variable correlated with explanatory variables but uncorrelated with regression residuals Take unconditional expectation z t x t+1 are the payoffs to managed portfolios General Approach: Add managed portfolio payoffs, and proceed with unconditional moments as if conditioning information did not exist.

L13: Conditioning Information4 A Note on Managed Portfolios Managed portfolios, one invests more or less in an asset according to some signal. The price of such a strategy is the amount invested at time t, z t. The payoff is z t R t+1 Example: making investment in stocks proportional to the price-dividend ratio. We can represent this strategy as a payoff using z t =a-b(p t /d t ). With managed portfolios, the set of asset payoffs expands dramatically, potentially multiplying every asset return by every information variable Then expected price of managed portfolio show up for p instead of p=0 and p=1 if we started with basic asset returns and excess returns (see Chapter 1)

L13: Conditioning Information5 Conditional Information and Scaled Return What is meant by conditional information? What is meant by “scaled return”? – multiplying x t+1 by z t Conditional information is equal to a regression forecast of the dependent variable using every variable z t, ( z t є I). –i.e., zt is an instrument Thus E[(m t+1 x t+1 -p t )z t ]=0 implies conditional information As a result:

L13: Conditioning Information6 The Relation of Conditional and Unconditional Models Conditional model does not imply unconditional models. Unconditional models implies conditional models Implication: Hansen-Richard Critique –If CAPM predicts the market portfolio is conditionally mean-variance efficient, this does not imply that the market portfolio is unconditionally mean-variance efficient –Conditioning information of agents might not be observable, thus CAPM (in the conditional form) is not testable.

L13: Conditioning Information7 Scaled Factors The parameters of the factor pricing model mt+1=at+btft+1 may vary over time. A partial solution is to model the dependence of parameters at and bt on variables in the time-t information set Let at=a(zt) and b=b(zt). In particular, try linear model: at=a’zt; bt=b’zt Thus, in place of the 1-factor model with time-varying coefficients, we have a three factor model (zt, ft+1, ztft+1) with fixed coefficients.

L13: Conditioning Information8 More on Scaled Factors An implication is that we cannot simply use the CAPM unconditionally. We, however, can add in some scaled factors If we have many factors f and many instruments zt, we should in principle multiply every factor by every instrument

L13: Conditioning Information9 Example: Ferson-Schadt Conditional Model The 1-factor regression form in Ferson-Schadt (1996) setting is: Zt is a vector of lagged macroeconomic variables (k=4) –the yield on Treasury bills with three month maturity –the term premium, measured as the 10-year Treasury yield in excess of the yield on 3-month Treasury yield –the credit premium, measured as the average yield on Moody’s BAA- rated corporate bonds in excess of that for AAA-rated corporate bonds –the dividend yield on S&P Composite Index. Simple regression can handle the work

L13: Conditioning Information10 GMM Demystifying