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Are there any classical, major theories/models that consider positive return due to idiosyncratic risk? For example, CAPM only considers return due to systemic risk but not idiosyncratic risk. If there are no major theories/models that fit the criterion, are there perhaps unorthodox ones worth mentioning? I would be grateful for references.
(I am a novice in asset pricing, so pardon if the question is too basic.)
There are no major models that come to the conclusion that idiosyncratic risk can drive positive returns. The reason is that idiosyncratic risk is diversifiable. The models, however, do in fact consider idiosyncractic risk. The theories don't forget or ignore them. It just turns out that this risk doesn't matter.
In this sense, CAPM actually does consider idiosyncratic risk. In this model, if you are perfectly rational, you will diversify your portfolio to minimize risk. This leads you to hold the whole market portfolio, in which case idiosyncratic risks are diversified away in your portfolio and therefore no longer matter. In simple terms, market risk then is not diversifiable, because once you hold the market portfolio, there's no way to diversify further. What drives returns are risk, but rational investors can remove the idiosyncratic risk through diversification. Hence, what matters for returns in the CAPM therefore is just the market risk (which you refered to as systemic risk, but that term actually means something else). That's why under CAPM, the returns are driven by the asset correlation with the market.
EDIT:
The CAPM model does not put any explicit assumptions on the number of assets available, hence the model does not require an infinite number of assets. However, the derivation is based on the decision to add an asset to an already well-diversified portfolio, i.e. adding an asset to the market portfolio.
Nevertheless, if only a handful of assets exist, then it might not be possible to diversify away all idiosyncratic risk. In that case, the CAPM does not work, because it implicity assumes enough assets exist, since it assumes a well-diversified portfolio exists to which you can add an asset. In such a case idiosyncratic risk may become relevant.
Such considerations are unlikely to be relevant in practice though. It is difficult to put a number on how many assets are necessary for idiosyncratic risk to be diversified away completely. Often, a number that is mentioned is 30-40 assets. The number required in practice is usually very realistic and far below infinity. The number of assets required for idiosyncratic risk to be completely diversified away becomes larger if assets are more correlated with each other or very volatile.
Lastly, an asset with such high idiosyncratic risk that no investor would want to add it to their portfolio cannot exist in equilibrium, because supply of the asset could never equal demand.
Idiosyncratic risk is, by definition, impossible to predict so it is not really useful to model it. The probability of positive return due to idiosyncratic risk is the exact same as the probability of negative return due to idiosyncratic risk, so simple random walk models should suffice.
I have some more time so I will elaborate on my answer. Any risk that is systematic should have a factor corresponding to it in a corresponding factor pricing model. For example, CAPM fails with small stocks, so we added a small firm factor. At the end of the day, in theory, investors should not be able to trade on idiosyncratic risk, which is the point of factor models.
