8. Conditioning Information

Why not model conditional distributions directly? Because letting the conditional mean, variance, covariance of \(N\) returns depend flexibly on \(M\) information variables quickly needs more parameters than observations; and worse, it typically requires assuming investors use the same model of conditioning information that we do. But the beauty of prices is that they summarize an enormous amount of information only individuals see (Hayek 1945) — we observe, and can include, only a fraction of it. So whenever possible our treatment should let agents see more than we do. The way out is to turn to unconditional moments (or moments conditioned on less than agents see).

8.1 Scaled Payoffs

We can incorporate conditioning information by adding scaled payoffs and doing everything unconditionally. The simplest unconditional implication is "conditioning down": take unconditional expectations of \(p_t=\mathbb E_t(m_{t+1}x_{t+1})\),

This uses the law of iterated expectations: your best forecast today of your best forecast tomorrow equals your best forecast today — \(E(E_t(x))=E(x)\), and more generally \(E[E(x|\Omega)\,|\,I\subset\Omega]=E[x|I]\). It lets us condition down to the coarser information sets we observe, and is the key to later arguments. But we can do more. Multiply price and payoff by any instrument \(z_t\) observed at time \(t\):

This is an additional implication of the conditional model, not captured by conditioning down (8.1). The name "instruments" comes from GMM's instrumental-variables heritage. The key: regard \((x_{t+1}z_t)\) as a payoff \(x=x_{t+1}z_t\) with price \(p=E(p_tz_t)\); then (8.2) reads \(p=E(mx)\) again — and all the asset pricing theory applies directly.

The linearity of \(xz\) is no restriction: a nonlinear (measurable) transform \(z_t^2=2+3z_t^2\) is still a random variable and a valid instrument.

8.2 Sufficiency of Adding Scaled Returns

Does adding scaled returns miss any implications of conditioning information? In principle no. The fact: the conditional expectation \(E(y_{t+1}|I_t)\) equals the regression forecast of \(y_{t+1}\) using every variable \(z_t\in I_t\) (and every nonlinear measurable transform). Taking \(y_{t+1}=m_{t+1}x_{t+1}-p_t\), "\(E[(m_{t+1}x_{t+1}-p_t)z_t]=0\) for every \(z_t\in I_t\)" implies \(0=E(m_{t+1}x_{t+1}-p_t\,|\,I_t)\). So checking the expected prices of all managed portfolios is, in principle, sufficient to check all the implications of conditioning information.

In practice the number of instruments is limited: only variables that forecast returns or \(m\) (or their higher moments) add information. Choosing a few instruments = choosing a few assets/portfolios — exactly the choice one makes in any asset pricing test by forming portfolios and omitting many assets. Most studies merely check whether a model prices 10–25 stock and a few bond portfolios, implicitly trusting these span the available risk loadings or mean returns. Excluding instruments is exactly excluding assets — no better founded, but a common sin one worries less about. Unscaled returns are not special: a mutual fund following the managed strategy would have unscaled returns equal to a scaled return.

8.3 Conditional and Unconditional Models

For explicit discount factor models (parameters constant over time), the conditional/unconditional distinction is immaterial: \(p_t=\mathbb E_t(m_{t+1}x_{t+1})\Rightarrow E(p_t)=E(m_{t+1}x_{t+1})\), and that's it (e.g. power utility \(m=\beta(c_{t+1}/c_t)^{-\gamma}\), log-utility CAPM \(m=1/R^W\)). But linear factor models have parameters that may vary with conditioning information. For the CAPM \(m=a-bR^W\), forcing it to price \(R^W\) and a risk-free rate gives

(8.3) shows \(a_t,b_t\) must vary as \(\mathbb E_t(R^W),\sigma^2_t(R^W),R^f_t\) vary — so the conditional CAPM must be a time-varying-weight linear factor model \(m_{t+1}=a_t-b_tR^W_{t+1}\). This means we cannot transparently condition down: taking unconditional expectations throws up covariance terms \(\operatorname{cov}(a_t,R_{t+1})\), \(\operatorname{cov}(b_t,R^W_{t+1}R_{t+1})\), generally nonzero. In beta language likewise: \(\mathbb E_t(R^i_{t+1})=R^f_t+\beta^i_t\lambda_t\) does not imply \(E(R^i_{t+1})=\gamma+\beta^i\lambda\) (conditioning down leaves a covariance of \(\beta^i_t\) and \(\lambda_t\)).

Special cases (that do condition down). Since \(\lambda_t=\operatorname{var}_t(f)b_t\): ① \(b_t=b\) constant ⟺ market prices of risk move proportionally to conditional variance; ② constant conditional covariance of returns with factors (even if \(b_t\) varies); ③ constant conditional betas — each is sufficient.

A precise statement. Let \(X\) be the space of all portfolios of primitive assets (including managed portfolios whose weights depend on conditioning information — scaled returns). A conditional factor pricing model \(m_{t+1}=a_t+b_t'f_{t+1}\) satisfies \(p_t=\mathbb E_t(m_{t+1}x)\) for all \(x\in X\); an unconditional (better, "fixed-weight") model \(m_{t+1}=a+b'f_{t+1}\) satisfies \(E(p_t)=E(m_{t+1}x)\) for all \(x\in X\). So the unconditional is just a special case of the conditional (fixed weights): hence conditional does not imply unconditional, but unconditional implies conditional. The subtlety: both share the payoff space \(X\) containing all managed portfolios — if the static CAPM cannot price assets scaled by instruments, it is not a valid factor pricing model at all.

Mean-variance frontiers. Define the conditional frontier as the returns minimizing \(\operatorname{var}_t(R)\) given \(\mathbb E_t(R)\); the unconditional frontier as the returns (including managed portfolios) minimizing \(\operatorname{var}(R)\) given \(E(R)\). Then:

Figure 8.1　Return A is on the conditional mean-variance frontiers but not the unconditional frontier; return B is on both (B has zero unconditional variance). The two dashed lines are conditional frontiers under two different information sets.

Note the "unconditional frontier" includes managed-portfolio returns. What many call the "unconditional frontier" is actually the fixed-weight frontier — it lies inside the other two, may touch but need not, and has no connection to them. For example, the risk-free rate is on the unconditional frontier only if it is a constant (\(R^f=R^*+R^fR^{e*}\) needs \(w=R^f\) constant), but it is always on the conditional frontier.

8.4 Scaled Factors: A Partial Solution

Since the conditional model is not testable, can we step back? A partial solution: model the parameters \(a_t,b_t\) as functions of time-\(t\) variables \(z_t\), in particular linear forms \(a_t=a'z_t\), \(b_t=b'z_t\) (linearity is no restriction, \(z_t^2\) is just another instrument). This yields a beautiful "scaled factors" reading; with a single factor and instrument,

So a single-factor model with time-varying coefficients becomes a three-factor model with fixed coefficients (factors \(z_t,f_{t+1},z_tf_{t+1}\)), usable with unconditional moments. E.g. in the standard CAPM the market is conditionally efficient, \(m_{t+1}=a_t-b_tR^W_{t+1}\); if the dividend/price ratio and term premium summarize the variation in conditional moments well, the conditional CAPM implies an unconditional five-factor (plus constant) model: a constant, the market return, the dividend/price ratio, the term premium, and the market return times those two instruments (Jagannathan and Wang 1996 test whether such factors explain CAPM anomalies). With many factors and instruments, multiply every factor by every instrument, written compactly with the Kronecker product

Summary

Conditioning information sounds scary at first, but this chapter gives a simple, beautiful treatment: to express a model's conditional implications, just add a few well-chosen scaled/managed-portfolio returns, then pretend you never heard of conditioning information. Some factor models are conditional with respect to investors' information and generally untestable; but if you assume the relevant conditioning information is summarized by a few variables, multiply the old factors by those variables to get new factors and again forget conditioning information. Keep it as a diagnostic or for economic interpretation (plot conditional \(b\)'s, betas, risk premia), but don't worry much about it in estimation and testing. The central asymmetry — unconditional ⟹ conditional, not vice versa — and the Hansen-Richard critique are the two things to remember.

Problems

1. If there is a risk-free asset, is it on the (a) conditional, (b) unconditional frontier, or (c) both?
2. If there is a conditionally risk-free asset (a claim to 1 traded each date), does that mean there is an unconditionally risk-free asset? (Define the latter first!) And vice versa?
3. If you take the unconditional population moments \(E(R),E(RR')\) and build a frontier, does it correspond to the conditional frontier, the unconditional frontier, or neither?
4. (a) Show \(\sigma^2(x_{t+1})=E[\sigma^2_t(x_{t+1})]+\sigma^2[\mathbb E_t(x_{t+1})]\). When do variances condition down (unconditional variance equals average conditional variance)? (b) Find the analogous decomposition for covariances.
5. Using the \(E(R)\)-\(\beta\) and \(m=a+bf\) representations, show each of the following is sufficient for a model to condition down (consider only excess returns \(0=\mathbb E_t(mR^e)\), normalize \(m=1+b'f\), \(\mathbb E_t(f)=0\)): (a) \(b_t=b\) constant ⟺ \(\lambda_t=-\operatorname{var}_t(ff')b\); (b) \(\operatorname{cov}_t(R^e,f')\) constant (even if \(b_t\) varies); (c) constant conditional betas.