<p>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.</p>

<p>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.</p>

<p><strong>EDIT:</strong></p>

<p>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.</p>

<p>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.</p>

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2017-10-31T17:25:02.290Z
+1 Thanks! Suppose the market contains only a handful of assets, all more or less alike, except for one which has huge idiosyncratic risk. Wouldn't you be better off by excluding that asset from the portfolio? Or shouldn't that asset's return be relatively higher for you to include it into the portfolio? To drive the example to the edge, what if you only have two assets, only one of which has huge idiosyncratic risk? I guess CAPM must assume an infinite number of assets or something like that to avoid scenarios like the above.
Richard Hardy
CC BY-SA 3.0
2017-11-01T14:22:14.037Z
May I ask, are you not responding because you do not have time (but would respond when you found time) or because you do not see a need for that (so I should not expect an answer)?
Richard Hardy
CC BY-SA 3.0
2017-11-02T21:47:38.790Z
I have edited the answer, I hope it addresses your concerns now.
BB King
CC BY-SA 3.0
2017-11-03T06:31:56.050Z
thank you, it works for me much better now. Your last sentence is interesting, because it could be understood to suggest that the return on that asset should be increased to equalize supply with demand, hence, there should be return on idiosyncratic risk.
Richard Hardy
CC BY-SA 3.0
2017-10-30T18:54:19.560Z
2017-11-02T21:45:16.370Z
2017-11-02T21:45:16.370Z
BB King
<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2017-10-30T16:36:40.197Z
What do you mean by "Idiosyncratic risk is, by definition, impossible to predict"? You can estimate risk of individual stocks from past observations. That is what you do for systemic risk as well.
Giskard
CC BY-SA 3.0
2017-10-30T16:37:48.420Z
What about the following argument: you can choose between two otherwise equivalent assets, one with high idiosyncratic risk and the other with low. Why would you ever choose the one with the high risk when you can choose the one with low risk? This hints that the asset with the high risk should offer something to compensate for this, i.e. a higher expected return.
Richard Hardy
CC BY-SA 3.0
2017-10-30T17:11:39.390Z
The real question is, how would an investor know what the "idiosyncratic risk" is ahead of time? I am not sure how this is possible with the typical definitions of idiosyncratic risk.
MathStudent
CC BY-SA 3.0
2017-10-30T18:24:54.587Z
@MathStudent it almost sounds like there’s a burgeoning literature on insider trading that deals with the questions you ask...
Theoretical Economist
CC BY-SA 3.0
2017-10-30T18:32:20.410Z
The question was about idiosyncratic risk in the context of a CAPM world(at least that's how I interpreted it). However, any systematic risk that can be traded on has a market factor that can express it. True idiosyncratic risk cannot be traded on. Maybe there is an insider trading factor?
MathStudent
CC BY-SA 3.0
2017-10-30T19:01:29.643Z
@MathStudent, this does not have to be in the context of the CAPM world. Any major framework is fine. CAPM was just the most obvious example to include.
Richard Hardy
CC BY-SA 3.0
2017-10-30T19:25:27.670Z
Anyway, I do not see how idiosyncratic risk can drive any kind of nonzero expected returns. Idiosyncratic risk is unpredictable so it should not affect demand or supply equations, so it does not impact prices. Another way of saying this is that p=E(mx) where is x is the payoff and m is the stochastic discount factor. So p=E(mx1)+E(mx2) where x2 is idiosyncratic. Since x2 is idiosyncratic it does not correlate with m, so E(mx2)=0. The second term does not impact prices.
MathStudent
CC BY-SA 3.0
2017-10-30T19:44:13.553Z
Of course, since the market portfolio in practice usually does not span the entire stochastic discount fact, then the idiosyncratic risk with respect to a certain pricing model may impact prices, but that just means you have the wrong pricing model. I think the theory is still clear.
MathStudent
CC BY-SA 3.0
2017-10-31T08:10:15.510Z
@MathStudent, I might not have expressed myself clearly, because your arguments do not seem to address my question. Sorry. Here are two points to think about: (1) idiosyncratic risk may be predictable (as noted by denesp above), and indeed I am interested in such a case; (2) *I think the theory is still clear* -- I am not framing the question within any particular theory. On the contrary, I am looking for a theory that makes the question natural and valid. Thank you for your help! P.S. Please include [at]RichardHardy, otherwise I do not get notified of your comments.
Richard Hardy
CC BY-SA 3.0
2017-10-31T13:42:05.570Z
@RichardHardy Ok so we can back out of the theory, but I think we might be suffering merely from a difference in definitions. What is your definition of idiosyncratic risk?
MathStudent
CC BY-SA 3.0
2017-10-31T14:29:01.347Z
@MathStudent, mmm, idiosyncratic risk would be a risk that is specific to the particular asset and uncorrelated with the market risk. Is that a sensible definition?
Richard Hardy
CC BY-SA 3.0
2017-11-01T01:32:00.930Z
@RichardHardy Sorry for the late response. OK so if that is your definition then I would say that any multifactor model such as Fama French is attempting to answer your question. However, in the sense that you are talking about, idiosyncratic risk is predictable and your logic for trading on it is valid. My definition of idiosyncratic risk is that it is a risk that cannot be traded on, i.e. it is an i.i.d. sort of risk i.e. it is not predictable and the variance of it is constant across investors and over time. I think we just have a difference of definitions.
MathStudent
CC BY-SA 3.0
2017-11-01T01:39:38.370Z
However, under asymmetric information like insider trading it gets more complicated. We can talk about idiosyncratic risk as an broad economic notion when information is symmetric across investors because everyone has the same stochastic discount factor is the same across investors. But if information is different then sdfs will be different. We have to talk about idiosyncratic risk as a pointwise phenomenon, and so we must move to a broader theory to characterize it
MathStudent
CC BY-SA 3.0
2017-11-01T01:39:45.990Z
In my opinion that is.
MathStudent
CC BY-SA 3.0
2017-11-01T05:43:49.560Z
@MathStudent, Perhaps you are overcomplicating things. My definition was taken from Investopedia, so I think it must be pretty standard. In other words, I did not even have to depart from standard finance for you to get my point (to appreciate my logic). But maybe I am missing something here... As I said, I am no specialist in asset pricing.
Richard Hardy
CC BY-SA 3.0
2017-11-01T13:47:02.657Z
@RichardHardy  I see, I think it just depends on your definition. Your kind of idiosyncratic risk is merely a statistical phenomenon, you run a regressions and look at that regression's alpha and trade based on that. Personally I would call that "idiosyncratic risk with respect to a pricing model," but that is merely semantics. I am an academic, so we approach these things with different languages then practitioners.
MathStudent
CC BY-SA 3.0
2017-11-01T14:20:34.563Z
@MathStudent, I am educated as a statistician but also have some hands-on experience from the financial industry. With regards to the former, I am quite careful when thinking about regressions; I do not define terms such as "idiosyncratic risk" based on regressions but rather on true underlying probability distributions and, if needed, loss functions. But that was more of a side note.
Richard Hardy
CC BY-SA 3.0
2017-10-30T16:32:36.717Z
2017-10-30T17:41:14.477Z
2017-10-30T17:41:14.477Z
MathStudent
<p>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. </p>

<p>(I am a novice in asset pricing, so pardon if the question is too basic.)</p>

<reference-request><finance><asset-pricing>
Any major theory/model that considers return due to idiosyncratic risk?
Richard Hardy
# Any major theory/model that considers return due to idiosyncratic risk?

(Asked by: username_0 on 2017-10-30)

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.)

**Tags:** reference-request, finance, asset-pricing

---

## 2 answers

### Answer by username_1 on 2017-10-30. Score: 4

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.

> **Comments:**
> **username_0**: +1 Thanks! Suppose the market contains only a handful of assets, all more or less alike, except for one which has huge idiosyncratic risk. Wouldn't you be better off by excluding that asset from the portfolio? Or shouldn't that asset's return be relatively higher for you to include it into the portfolio? To drive the example to the edge, what if you only have two assets, only one of which has huge idiosyncratic risk? I guess CAPM must assume an infinite number of assets or something like that to avoid scenarios like the above.  
> **username_0**: May I ask, are you not responding because you do not have time (but would respond when you found time) or because you do not see a need for that (so I should not expect an answer)?  
> **username_1**: I have edited the answer, I hope it addresses your concerns now.  
> **username_0**: thank you, it works for me much better now. Your last sentence is interesting, because it could be understood to suggest that the return on that asset should be increased to equalize supply with demand, hence, there should be return on idiosyncratic risk.  

---

### Answer by username_2 on 2017-10-30. Score: 0

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.

> **Comments:**
> **username_3**: What do you mean by "Idiosyncratic risk is, by definition, impossible to predict"? You can estimate risk of individual stocks from past observations. That is what you do for systemic risk as well.  
> **username_0**: What about the following argument: you can choose between two otherwise equivalent assets, one with high idiosyncratic risk and the other with low. Why would you ever choose the one with the high risk when you can choose the one with low risk? This hints that the asset with the high risk should offer something to compensate for this, i.e. a higher expected return.  
> **username_2**: The real question is, how would an investor know what the "idiosyncratic risk" is ahead of time? I am not sure how this is possible with the typical definitions of idiosyncratic risk.  
> **username_4**: @MathStudent it almost sounds like there’s a burgeoning literature on insider trading that deals with the questions you ask...  
> **username_2**: The question was about idiosyncratic risk in the context of a CAPM world(at least that's how I interpreted it). However, any systematic risk that can be traded on has a market factor that can express it. True idiosyncratic risk cannot be traded on. Maybe there is an insider trading factor?  
> **username_0**: @MathStudent, this does not have to be in the context of the CAPM world. Any major framework is fine. CAPM was just the most obvious example to include.  
> **username_2**: Anyway, I do not see how idiosyncratic risk can drive any kind of nonzero expected returns. Idiosyncratic risk is unpredictable so it should not affect demand or supply equations, so it does not impact prices. Another way of saying this is that p=E(mx) where is x is the payoff and m is the stochastic discount factor. So p=E(mx1)+E(mx2) where x2 is idiosyncratic. Since x2 is idiosyncratic it does not correlate with m, so E(mx2)=0. The second term does not impact prices.  
> **username_2**: Of course, since the market portfolio in practice usually does not span the entire stochastic discount fact, then the idiosyncratic risk with respect to a certain pricing model may impact prices, but that just means you have the wrong pricing model. I think the theory is still clear.  
> **username_0**: @MathStudent, I might not have expressed myself clearly, because your arguments do not seem to address my question. Sorry. Here are two points to think about: (1) idiosyncratic risk may be predictable (as noted by denesp above), and indeed I am interested in such a case; (2) *I think the theory is still clear* -- I am not framing the question within any particular theory. On the contrary, I am looking for a theory that makes the question natural and valid. Thank you for your help! P.S. Please include [at]RichardHardy, otherwise I do not get notified of your comments.  
> **username_2**: @RichardHardy Ok so we can back out of the theory, but I think we might be suffering merely from a difference in definitions. What is your definition of idiosyncratic risk?  
> **username_0**: @MathStudent, mmm, idiosyncratic risk would be a risk that is specific to the particular asset and uncorrelated with the market risk. Is that a sensible definition?  
> **username_2**: @RichardHardy Sorry for the late response. OK so if that is your definition then I would say that any multifactor model such as Fama French is attempting to answer your question. However, in the sense that you are talking about, idiosyncratic risk is predictable and your logic for trading on it is valid. My definition of idiosyncratic risk is that it is a risk that cannot be traded on, i.e. it is an i.i.d. sort of risk i.e. it is not predictable and the variance of it is constant across investors and over time. I think we just have a difference of definitions.  
> **username_2**: However, under asymmetric information like insider trading it gets more complicated. We can talk about idiosyncratic risk as an broad economic notion when information is symmetric across investors because everyone has the same stochastic discount factor is the same across investors. But if information is different then sdfs will be different. We have to talk about idiosyncratic risk as a pointwise phenomenon, and so we must move to a broader theory to characterize it  
> **username_2**: In my opinion that is.  
> **username_0**: @MathStudent, Perhaps you are overcomplicating things. My definition was taken from Investopedia, so I think it must be pretty standard. In other words, I did not even have to depart from standard finance for you to get my point (to appreciate my logic). But maybe I am missing something here... As I said, I am no specialist in asset pricing.  
> **username_2**: @RichardHardy  I see, I think it just depends on your definition. Your kind of idiosyncratic risk is merely a statistical phenomenon, you run a regressions and look at that regression's alpha and trade based on that. Personally I would call that "idiosyncratic risk with respect to a pricing model," but that is merely semantics. I am an academic, so we approach these things with different languages then practitioners.  
> **username_0**: @MathStudent, I am educated as a statistician but also have some hands-on experience from the financial industry. With regards to the former, I am quite careful when thinking about regressions; I do not define terms such as "idiosyncratic risk" based on regressions but rather on true underlying probability distributions and, if needed, loss functions. But that was more of a side note.  

---

