Mimicking Portfolios: Tracking Factors and Estimating Risk Premia
Summary
The document compares two ways to construct portfolios that track observable macroeconomic factors. Its naive proposal regresses each factor’s time series on returns from a base universe of assets; the resulting coefficients define weights that minimize in-sample squared tracking error. The author contrasts this direct fit with an asset-pricing approach that first estimates each asset’s factor betas, then uses cross-sectional regressions to estimate risk premia over time.
The question is whether the two-stage procedure is necessary and how it yields portfolio weights, given that it optimizes a different objective from direct return matching. The post reports that the two-stage results seem reasonable but supplies no data, derivation, or answer. It therefore highlights a useful distinction between portfolios designed to replicate realized factor returns and portfolios associated with factor risk premia in an asset-pricing model. The appropriate construction depends on the intended interpretation and constraints; the document leaves that issue unresolved.
Key ideas
- Direct regression of factor returns on asset returns produces weights aimed at minimizing in-sample tracking error.
- The asset-pricing procedure estimates asset betas first and then estimates factor risk premia cross-sectionally.
- A portfolio that tracks a factor’s realized returns and one tied to estimated risk premia may represent different objectives.
- The document poses the relationship between the procedures but does not provide a derivation or resolution.
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Full text
# Methods for Constructing Mimicking Portfolios for Observable Factors # Methods for Constructing Mimicking Portfolios for Observable Factors I've created some macroeconomic factors (e.g. analogs of real GDP growth) that I believe have explanatory power for asset returns. By a factor here, I mean a stationary time-series of real numbers. I'd like to create so-called "mimicking portfolios" for these factors - e.g. portfolios of assets in my base universe whose returns closely match the returns of factors I've constructed. Naive Method: At first, this seemed simple enough: for some period, take the constructed factors, and take the returns of the base universe. Run a regression (to keep it simple, OLS) for each factor, with the factor as the $y$ variable, and asset returns as the $X$. This gives the weights of a portfolio that mimic the factor (minimize a tracking error over the time-period of the regression). It seems like if one would want to hedge exposure to this factor, the best way to do that would be to buy/sell this mimicking portfolio in appropriate amounts. What I've seen in Asset Pricing Literature: So far so good, however, when I read asset pricing literature, it seems like this is not done anywhere. It seems like the reverse regressions are run (e.g., collect all factors on the right hand side, and run regressions for each asset in the universe to estimate asset betas to factors), and then cross-sectional regressions with these estimated betas on the RHS are run to get risk premia (Fama-Macbeth type procedure). The time-series of risk-premia estimates are then supposed to be the returns of the mimicking portfolio. I have implemented this method, and the results seem reasonable, but I'm having difficulty understanding why this two-step procedure is necessary, and more importantly, how one can extract weights from it. It seems like the above procedure is actually the solution to an optimization problem, but I can't understand why anyone would consider an optimization problem that is different from that of the Naive Method (minimize tracking error). My question is - why is this second method necessary, and why is it used? The first method seems like the most obvious one to employ and minimizes the tracking error (mean squared residual return). Am I misunderstanding what a "mimicking portfolio" means in the context of asset pricing?
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