# Reducing pricing errors (Alpha) in the CAPM with Bitcoin

I have been trying to examine, using the CAPM, if Bitcoin belongs in the market portfolio or not.

With 10 industry portfolios from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html I have performed a GRS-test for jointly significant alphas. The resultat was significant.

The proxy for the market portfolio was found on the same website, and is just a large portfolio of stocks.

However, by adding bitcoin to this market portfolio with a weight of 2.1% i was able to make the alphas jointly insignificant at 5% significance level.

However i am having trouble interpreting these results. Does it mean that Bitcoin should have a weight of 2.1% in the market portfolio? If the S&P 500 was the entire market, all of cryptocurrency would have a weight of less than 0.5%

Am i actually improving the CAPM by adding Bitcoin? Or how should i interpret this?

• It's rather hard to believe that the bitcoin price is a major hedging concern to investors: securities have higher and lower expected returns because of their correlation to Bitcoin. My guess/concern of what's going on is that by adding Bitcoin to the market portfolio, you've added noise, thereby decreasing market betas of assets and increasing the magnitude of the residuals. With larger residuals, your standard errors are larger so the alphas are no longer significant. That's just a guess of what's happening, something mechanical/statistical rather than of economic significance. Commented May 18, 2019 at 17:18

The CAPM has been falsified repeatedly. See

Fama, Eugene F.; MacBeth, James D. (1973). "Risk, Return, and Equilibrium: Empirical Tests". Journal of Political Economy. 81 (3): 607–636

The CAPM is still taught because nothing has replaced it. The CAPM shouldn't be used for any real purpose. It is uncorrelated with reality.

EDIT

You cannot back into the appropriate weighting in this manner for two reasons. First, the appropriate weighting is $$\sum_{f=1}^Fn_fp_f$$ where $$f$$ is the index of assets and $$n$$ is the number of units. The appropriate weighting is the relative market capitalization of the assets. Second, you cannot use Frequentist methods that way.

Frequentist methods assume you know the true model, you cannot test your way into the true model. You cannot use a Bayesian method because if you would solve the CAPM from scratch using a Bayesian method, the integral for the expectation would diverge. You would not end up with $$(r_i-r_f)=\beta(r_m-r_f).$$

In fact, that in itself should be a gigantic warning. All Bayesian estimators are admissible estimators, but Frequentist estimators are admissible only to the extent they either map to a Bayesian estimator at every sample, or to a Bayesian estimator at the limit.

The CAPM fails in the Bayesian framework because the likelihood function for $$R$$ in the equation $$\tilde{w}_i=R\bar{w}_i+\epsilon_i$$ is $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r+\mu)^2}$$ which has no expectation. That is because it is really an AR(1) function and the likelihood is known since $$R$$ must greater than unity or no one would invest.

The proper weight of one Bitcoin is its capitalization relative to all other capital. However, within the model, it shouldn't be capital as it is a currency. It should be the case that $$x_{t+1}=x_t+\varepsilon_{t+1}.$$

Since the CAPM depends on equilibrium pricing and if cryptocurrencies do not collapse to zero, then in equilibrium they should provide a zero percent rate of return. They are a strictly dominated asset.

• Hi, thanks for the answer. I do realize that the CAPM has been falsified. However, i am trying to interpret what this would mean in a world where the CAPM is true. Commented May 18, 2019 at 11:29