Why a factor model
The form
Every asset return is driven by the same set of factors, plus a specific return unique to that asset. Loadings differ by asset and change over time.
Ri(t) = Σj βij Fj(t) + εi(t)
| Term | Meaning |
|---|---|
| Ri(t) | Return of asset i over period t. |
| Fj(t) | Return of factor j over period t. |
| βij | Loading — the sensitivity of asset i to factor j. |
| εi(t) | Specific, or idiosyncratic, return of asset i. |
Estimation
The modeller has to decide what is known and what is to be estimated. That choice separates the two families of model.
Time series
Each factor return is known for the period. Loadings and specific return are derived through a time-series regression, asset by asset.
The factors may be latent portfolios, as in principal-component approaches, or pre-identified thematic portfolios. Both require choices about how to maintain stability and interpretability through time.
Cross-sectional / ARC
Loadings are observed for each contract. Factor and specific returns are derived through a regression across all contracts, for each period.
Exposures are built from observable data, normalised and expressed as z-scores. Factor returns are then determined endogenously by the returns of the contracts and their level of exposure.
Both approaches produce exposures, a factor covariance matrix and factor returns—from which follow portfolio variance, volatility, value at risk and attribution.
Commodities
Commodities span agriculture, energy and metals. The futures contract standardises the physical market enough to make those diverse exposures jointly modellable.
| Feature | Consequence |
|---|---|
| Standardisation | The contract standardises what is often a heterogeneous physical product, introducing delivery optionality. |
| Fixed expirations | Standardised expirations coordinate market behaviour and allow the maturity structure to be observed directly. |
| Two populations | Hedgers and speculators are both present and active in the same contracts. |
| Volatility premium | Commodities show periods of extreme volatility, with evidence of a premium in excess of compensation for that volatility. |
| Common factors | Commodity futures prices are influenced by a small number of common factors, including momentum, basis, liquidity and open interest. |
| Correlation structure | The asset class has historically exhibited return relationships that differ from other markets. |
Consequence
01
A book can be sector-neutral and still carry substantial systematic risk. In the published worked example, the long and short legs offset across Energy, Metals and Agriculture while the same book ran a basis exposure close to one standard deviation.
02
Representing positions through factor exposures lets a researcher separate common drivers from contract-specific maturity effects.
03
A factor model separates systematic from specific risk. In the worked example the two contribute in roughly equal measure to total variance, a split a correlation matrix would not surface.
04
Portfolio return can be represented as a linear combination of exposures and factor returns. Applying historical factor returns to a defined set of exposures produces an illustrative stress scenario, not a forecast.
The ARC model
Sub-sectors are nested inside sectors so that risk and factor performance are unchanged whether the model is read at sector or sub-sector level.
References
| 1 | Grinold, R. and Kahn, R. (1999). Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk. McGraw-Hill. |
| 2 | Petram, L. (2014). The World's First Stock Exchange. Columbia Business School Publishing. |
| 3 | Ferris, W. G. (1988). The Grain Traders: The Story of the Chicago Board of Trade. Michigan State University Press. |
| 4 | Sakkas, A. and Tessaromatis, N. (2018). Factor based commodity investing. EDHEC Business School. |
| 5 | Asness, C. S., Moskowitz, T. J. and Pedersen, L. H. (2013). Value and momentum everywhere. The Journal of Finance, 68(3). |