Why Pricing Factors May Not Explain Return Covariances

Summary

The document examines whether every factor that explains expected returns must also explain return covariances. It presents a theoretical counterexample using two groups of stocks: each stock’s return consists of a group-specific expected return plus independent white noise. Membership in a group therefore determines expected return, while individual returns remain uncorrelated with other stocks and factors.

The example shows that a pricing distinction can exist without a corresponding common statistical factor in the covariance structure. The answer suggests that behavioral explanations for such expected-return differences may be difficult to identify in practice, because characteristics that create them could also generate statistical arbitrage opportunities. It contrasts the stylized independent-return construction with value stocks, which it says tend to move together. The example clarifies a logical possibility, but does not provide an empirical taxonomy or establish how often pricing-only factors occur in actual markets.

Key ideas

• A factor that explains expected returns need not explain covariance in every theoretical model.
• Two groups can have different expected returns while their individual return shocks are independent.
• Group membership in the example affects expected returns but does not create cross-stock correlations.
• The answer notes that real characteristics may create statistical relationships as well as pricing differences.
• The example is theoretical and does not quantify the prevalence of such factors.

Tags

# APT - Pricing Factors that are not Statistical Factors


# APT - Pricing Factors that are not Statistical Factors












Arbitrage Pricing Theory makes the implication that statistical factors (i.e. those that explain covariances) imply pricing factors (i.e. those that explain returns).

Is the reverse implication (i.e. pricing factor --> statistical factor) also true? If not, what are some common pricing factors that do not explain covariances? I can only think of transactional factors (e.g. merger arb) but didn't know if there was a broader taxonomy of pricing-but-not-statistical factors.

## Answer by fes (score 1)

https://quant.stackexchange.com/a/55857

Yes at least in theory. Assume there are lot of stocks, each being of type A or type B. Let, the return of type A stock $i$ and B stock $j$ be

$$r_{t,i}^{A}=\mu_A+\epsilon_t^{i}$$

$$r_{t,j}^{B}=\mu_B+\epsilon_t^{j}$$

where $\epsilon_t^{i}$ and $\epsilon_t^{j}$ are independent white noise. Now the return of a stock is uncorrelated with that of any other stock or any factor except its own return. However, being of type A or type B determines the expected return.

You might try to come up with some behavioral reason why e.g. A type stocks have higher expected returns. However, it might be hard to find such characteristics likely because they tend to imply statistical arbitrage opportunities. In practice for example all value stocks tend to move together.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.