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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.

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  • $\begingroup$ I guess it depends on your exact definition of "explain returns". Two random variables can be related (i.e. not independent) yet uncorrelated. Also a behavioral finance type view is that correlations with aggregate level variables don't explain returns that well. $\endgroup$
    – fes
    Jul 23, 2020 at 13:57

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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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