Interpreting Bitcoin’s Effect on CAPM Pricing Errors
Summary
The document considers whether adding Bitcoin to a stock-market proxy improves a CAPM test. The questioner reports applying a GRS test to ten industry portfolios and finding jointly significant alphas with a stock-based market proxy. Adding Bitcoin at a 2.1% weight makes the joint significance disappear at the stated 5% level. They ask whether that result implies Bitcoin belongs in the market portfolio at that weight, particularly given the difference between the experimental proxy and the broader market.
The response cautions that a change in a model test does not establish the correct asset weight. It argues that market weights should reflect relative capitalization and questions using repeated frequentist model testing to infer the true specification. It also presents strong criticisms of CAPM and an argument that Bitcoin may be treated as currency rather than capital. These claims are the respondent’s position; the exchange supplies no independent comparison, robustness checks, or evidence that the modified test improves out-of-sample pricing.
Key ideas
- Making estimated alphas jointly insignificant does not establish an asset’s correct market weight.
- The question reports that adding Bitcoin changed the outcome of a GRS test.
- The response says market-portfolio weights should be based on relative market capitalization.
- The exchange does not provide robustness or out-of-sample evidence for the modified CAPM.
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Full text
# Reducing pricing errors (Alpha) in the CAPM with Bitcoin
# 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?
## Answer by Dave Harris (score 3, accepted)
https://quant.stackexchange.com/a/45662
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.