5.9.3 Factor-Based Allocation

Basic Prerequisites: 02 · Asset-Class Allocation and Factor Investing & Factor Timing.

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1. Intuition & Practical Objective

Factor-based allocation is the modern answer to a question asset-class allocation never asked: how many independent return sources do I actually own? Instead of asking “how much equity?”, you ask “how much value, momentum, carry, and defensive?” and then construct whatever asset-class vehicle delivers those exposures. Ang (2014) makes this the organising principle of asset management: asset classes are bundles of factors; factors are the true building blocks.

The practical objective of this page: state the factor model of returns, show how a factor allocation maps back to assets, and demonstrate numerically why the factor portfolio is far more diversified than the asset-class portfolio.

Why it is more diversified: factors are relatively uncorrelated across the block. Momentum and value are (mildly) negatively correlated - they are different bets. The TSM (equity-market) bets embedded in a US equity fund and an EM equity fund are the same bet twice. Allocating across the labels inherits the market factor’s dominance; allocating across the factors removes it.

Three steps:

1. The label is a container, not a bet. A “value equity fund” and a “momentum equity fund” are both mostly one market factor with a small tilt. The tilt is the bet; the market is the container.
2. Factors are the orthogonalised building blocks. After removing the shared market component, the residual factor premia have low mutual correlations, so a factor-equal-weight portfolio diversifies much better than an asset-equal-weight one.
3. The map goes both ways. Given factor weights $w_f$, the asset exposure is $B{w}_f$; given a target asset portfolio, you can attribute it to factors and see how concentrated the underlying bets really are.

The one-sentence essence. “Allocate to the sources (value, momentum, carry, defensive) and let the containers (equities, bonds, commodities) sort themselves out - a portfolio of labels is usually a concentrated factor bet in disguise.”

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2. Mathematical Ground Truth & Derivations

The factor model. Asset returns follow the linear model

$$r_t=\alpha +B{f}_t+{\epsilon }_t,\mathbb{E}[f_t]=0,\mathrm{Cov}(f_t)={\Sigma }_f,\mathrm{Cov}({\epsilon }_t)=D=\mathrm{diag}({\sigma }_{\epsilon ,i}^2),$$

with $\mathrm{Cov}(f_t,{\epsilon }_t)=0$. Taking covariances gives the structured covariance

$$\boxed{{\Sigma }_r=B{\Sigma }_f{B}^{\top }+D}.$$

This is the object that makes factor allocation possible: everything systematic lives in $B{\Sigma }_f{B}^{\top }$; everything asset-specific lives in the (near-diagonal) $D$.

Allocating to factors. Let $w_f$ be factor weights ($1^{\top }{w}_f=1$, allowing negative weights for a long-short factor). The induced asset exposure is

$$w_{\text{asset}}=B{w}_f,$$

and the portfolio’s systematic volatility is ${\sigma }_f^2=w_f^{\top }{\Sigma }_f{w}_f$. The portfolio’s total volatility adds the undiversified specific part,

$${\sigma }_{\text{tot}}^2=w_f^{\top }{\Sigma }_f{w}_f+w_{\text{asset}}^{\top }D{w}_{\text{asset}},$$

but for a broad portfolio the second term is small, so the factor weights essentially determine the risk. This is the theoretical reason the factor portfolio’s diversification ratio is higher: the asset portfolio’s ${\Sigma }_r$ is dominated by $B{\Sigma }_f{B}^{\top }$, which is a low-rank concentration; the factor portfolio’s ${\Sigma }_f$ has small off-diagonals.

Attribution - the reverse map. Given a proposed asset portfolio $w_{\text{asset}}$, regress it on the factors to get its exposures: ${\beta }_w=B^{\top }{w}_{\text{asset}}$, and decompose its variance as

$$w^{\top }{\Sigma }_{r}w=\underset{\text{common}}{\underbrace{{\beta }_w^{\top }{\Sigma }_f{\beta }_w}}+\underset{\text{specific}}{\underbrace{\sum _i{w}_i^2{\sigma }_{\epsilon ,i}^2}}.$$

A portfolio showing 90% “common factor” variance is a concentrated factor bet no matter how many tickers it holds.

The diversification of factors, stated. The correlation between any two pure factors is not unity (value vs momentum are negatively correlated), whereas the correlation between any two equity containers is dominated by their shared market loading and therefore high. Averaging the off-diagonal correlations makes the point in one number: the factor block averages ${\bar{\rho }}_f\approx 0.017$ here versus ${\bar{\rho }}_{\text{asset}}\approx 0.19$ for the asset block - and the resulting diversification ratios are $1.98$ vs $1.46$.

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3. Computational Implementation - factor portfolio vs asset portfolio

Runnable (numpy) contrast: build a low-correlation factor covariance matrix and a higher-correlation asset covariance matrix, then compare the diversification ratio of an equal-weight portfolio in each space. The asset blocks are the same four as pages 01–02 (equity, bond, commodity, credit), so their average pairwise correlation ($0.19$) is the one that dominates a label-based allocation; the factor block (value, momentum, carry, defensive) averages only $0.017$.

The factor portfolio’s diversification ratio is 1.98 against 1.46 for the asset portfolio, driven entirely by the difference in average pairwise correlation ($0.017$ vs $0.19$) - the asset portfolio is more correlated because every asset loads on shared factors. The last line confirms the map is consistent: an equal factor weight induces an equal asset exposure (because $B$‘s rows sum to 1).

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4. Failure Modes & First-Principles Breakdowns

1. Estimation error in $B$ and ${\Sigma }_f$. The factor covariance is only as good as the estimated loadings; when $N$ is large and $T$ small the sample ${\Sigma }_f$ is noisy (see shrinkage & RMT).
2. Two “different” factors can share a hidden risk. Value and carry both have a “short growth / long cheap” tilt; if a crash factor loads on both, the apparent diversification is illusory (page 05).
3. Factor crowding. Factors are traded. As flows crowd a factor, its premium decays and its correlation with everything else rises - the bridge to Factor Crowding & Capacity.
4. The specific-risk term is not always small. In a concentrated factor portfolio, $w_{\text{asset}}^{\top }D{w}_{\text{asset}}$ can dominate; ignoring it understates risk.

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5. References

• Ang, Asset Management (2014)
• Tsay, Analysis of Financial Time Series (3rd ed., 2010)
• Fama & French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics 33(1):3–56, 1993
• Hastie, Tibshirani & Friedman, The Elements of Statistical Learning (2nd ed., 2009)
• Qian, Hua & Sorensen, Quantitative Equity Portfolio Management (2007)

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6. Connected Graph Bridges

• Back: 02 · Asset-Class Allocation
• Forward: 04 · Carry & Styles · Index Hub
• Cross-pillar: Factor Investing & Factor Timing · Fundamental Multi-Factor Models
• Sibling: Covariance Shrinkage & RMT Denoising