Lecture 13: Conditioning Information

1. 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 Information

2. Conditional vs. Unconditional where Et 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 Information

3. Instruments and Managed Portfolios • Suppose we multiply the payoff and price by any variable or instrument zt observed at time t. Then • Instrument variable correlated with explanatory variables but uncorrelated with regression residuals • Take unconditional expectation • ztxt+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 Information

4. 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, zt. The payoff is ztRt+1 • Example: making investment in stocks proportional to the price-dividend ratio. We can represent this strategy as a payoff using zt=a-b(pt/dt). • 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 Information

5. Conditional Information and Scaled Return • What is meant by conditional information? • What is meant by “scaled return”? – multiplying xt+1 by zt • Conditional information is equal to a regression forecast of the dependent variable using every variable zt, (zt єI). • i.e., zt is an instrument • Thus E[(mt+1xt+1-pt)zt]=0 implies conditional information • As a result: L13: Conditioning Information

6. 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 Information

7. 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 Information

8. 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 Information

9. 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 Information

10. GMM Demystifying L13: Conditioning Information