# If CAPM holds, should alpha be zero for all assets?

If CAPM holds, should alpha be zero for all assets?

• quant.stackexchange.com/questions/34091/… Dec 10, 2021 at 14:06
• As you see from the responses below, it really depends on what you call alpha Dec 14, 2021 at 20:48

Yes. In fact one way to test if the CAPM is true is to apply the Gibbons-Ross-Shanken or GRS test.

See for example how to interpret the GRS F test values? and the answer by M. Gunn

How does GRS work? This is a statistical test of whether all alphas are equal to zero. If they are not, we conclude that the CAPM does not hold.

My answer would be yes. My way of looking at it is the following:

• Lets consider $$\alpha$$ is different and greater than zero. That would mean that such an asset performs better (in average) than the index when this one doesn't move. In addition, that means that it performs always better than any other asset that has $$\alpha = 0$$ but the same $$\beta$$. That would make the market buy this asset and therefore make it more expensive, reducing its $$\alpha$$.
• If you consider now $$\alpha < 0$$, then the opposite holds. The asset performs worse (in average) than the index when the index doesn't move, and performs worse than the assets with same $$\beta$$. Therefore investors would sell such an asset, to replace it for one with a higher $$\alpha$$, making it cheap, so that its $$\alpha$$ increases.

Note that the idea is that the price of the asset should always be pushed by the market so that $$\alpha$$ is close enough to zero. Otherwise, the asset with $$\alpha \neq 0$$ would be ideal for long/short selling (depending on whether $$\alpha$$ is positive or negative) and then taking the opposite position with another asset with the same $$\beta$$. With that strategy you would be beating the market in the long run.

I thought I would edit my original post to make things a bit clearer.

First, let us distinguish between the CAPM as a theoretical construction and then as a construction to be measured.

Let $$I=r_i-r_f$$ and $$M=r_m-r_f$$. The formula for the CAPM is $$E(I)=E(\beta{M}).$$ There must be no $$\alpha$$ in the formula. If there were an $$\alpha$$ in the formula, then it would imply that people do not minimize a variance subject to a chosen return, or the dual form, choose a variance and maximize return. If $$\alpha$$ exists, then people do not behave in that manner.

The CAPM is built on top of Kolmogorov’s axioms, assuming that the parameters are known. So, logically, you need not perform any calculations. I believe it was Milton Friedman that argued that although the parameters are not known, the market behaves as if they were. So there is some concept of collective but not individual knowledge here.

So, if the CAPM is true, the proper regression to run would be $$I=\beta{M}+\epsilon$$ and not $$I=\beta{M}+\alpha+\epsilon.$$ Nonetheless, if we need to test it using an $$\alpha$$ then the latter regression would be run. Since null hypothesis methods can be thought of as a probabilistic version of modus tollens we want to test either $$H_0:\alpha=0$$, or we want to show that there exists a variable $$F$$ such that $$I=\beta{M}+\gamma{F}$$, where $$\gamma\ne{0}$$. This latter case is problematic because there are an infinite number of potential sets $$F$$. It is problematic in other ways as well.

If you performed a master F test, then its assumption would be that $$\beta=\gamma=0$$ and then the subsidiary t-test that $$\gamma=0$$. Of course, that is what Fama and MacBeth did. Still, if it had not been performed and the CAPM falsified, it would hang forever over the CAPM because of its construction that there could exist a set $$F$$ that mattered.

We know a set $$F$$ exists, so we also know that the construction of the CAPM is also falsified. Since it is falsified, the idea of excess return is also not supported without a new model. Without it, then $$\alpha$$ as a mathematical construction is also not valid. It is a spurious regression.

Now let us look at the empirical side a little closer, assuming that the CAPM is true. If the CAPM is true, then you should be able to run a regression $$I=\beta{M}+\alpha+\epsilon$$ such that $$\alpha=0$$.

The CAPM is a Frequentist construction and does not hold under a Bayesian construction. It is tightly dependent upon its axioms. So, the Frequentist solution should be to minimize the squared error. However, parameter estimates $$\hat{\beta}$$ and $$\hat{\alpha}$$ are random variables because they are functions of random variables. Each is a statistic.

Because $$\alpha\in\Re$$ and the real numbers are a continuum, the chance of $$\hat{\alpha}=0$$ is a measure zero event. It can happen, but it has zero probability. You should never have a case in a countable set of experiments where you hit zero exactly.

That does not imply that the parameter, $$\alpha\ne{0}$$. For any chosen cutoff point for statistical significance and a large enough set of experiments, you are guaranteed to have some experiments falsify the null even if the parameter value is zero. Indeed, for a long enough time series, the Lindley paradox guarantees that if the null is true it will be falsified once the length of the time series becomes long enough.

The most you can do, with respect to $$\alpha$$, is test whether it is zero or not. If the CAPM were true, then the null would not be rejected much more often than the significance cutoff. In finite samples, you can get more false rejections than the cutoff.

On the assumption that the CAPM holds, then any rejections of $$\alpha=0$$ are meaningless and do not imply an actual performance difference.

If the CAPM is false, the $$I$$ and $$M$$ are no longer supported constructions and so $$\alpha$$ is also without meaning.

This is a belt-and-braces reformulation of the "No" argument. And the "NNNNOOOO NNNOOO NNOO NO" still holds.

Let's assume that CAPM does indeed work. This just defines a company's cost of capital, ie its ex-ante expected return ASSUMING THAT MARKET EXPECTATIONS ARE CORRECT. This can indeed then be precisely defined.

But there will be surprises whereby these expectations prove incorrect. There will be news, some of this security-specific, some of this market-wide, that causes outcomes to realise differently from expectations.

This means that there will be ex-post variations from these ex-ante expectations, and associated pricing. This will cause realised deviations from CAPM-projected returns. So there will be a random residual, that is "alpha" not "beta".

Sorry, but I'm not backing down here ;-) And thanks @BobJansen and @RichardHardy for the discipline of having to re-consider and re-state this one (less im)properly. Much appreciated.

best, DEM

Yes, assuming liquidity is infinite or information acquisition is frictionless.

• Can you explain why, in a CAPM world those two extra (are they extra?) conditions need to hold? Dec 14, 2021 at 19:30
• See Merton 1987 for example. Dec 15, 2021 at 21:03