Is there a robust way to calculate stock beta or factor exposure that's specific to crashes?

Question

Commonly known factors like market, value, momentum etc. have positive expected returns because they draw-down unexpectedly and investors require a risk premium for holding them. This idea is extended to single name stocks and suggests that stock returns (on average) are compensation for their exposure to the aforementioned factors.

A typical way to quantify these factor exposures is via the covariance of the stock's and factor's returns. The logic is that the more a stock co-moves with a factor, the higher its "similarity" to said factor which would result in a proportional risk premium.

My question is two-fold:

1. Why is the factor exposure calculated via covariance using trailing time-series returns instead of covariance during periods of the factor's drawdown, since that's what is being compensated?
2. Since the drawdown periods are likely to be a significant time ago and contain much fewer data points, are there any ways to reduce the estimation error of this "stressed period" covariance?

Any links to papers that explore this would be appreciated.

Answer 1

Within the framework you are proposing, it would make no sense. It would be failing to distinguish noise from signal. Extreme events are rarely triggered by measures of central tendency. It is like flipping heads 20 times in a row, what caused that?

From a physicist's, magician or con man's perspective that is a valid question, but from a Frequentist fair coin perspective, it is not.

If you drop the normal model though and go to a truncated Cauchy model, you no longer have a covariance matrix. Assets would still comove, but they would no longer covary. For a standard Cauchy, not truncated, 50% of events would be $\pm{1}$, but 99.99% would be $\pm 636.62$ units. That is the source of your jumps and crashes.

You end up with a copula model. If you included liquidity costs, then you could discuss the events leading to crashes. Nothing covaries anyway now, but you can discuss comovement in this manner.

If $$Y=1.02X+\epsilon$$ where $$X,Y\sim\mathcal{C}\left(\left[\begin{array}[c]\ \mu_X\\ \mu_Y\end{array}\right],\sigma\right)$$, then 50% of the time $Y$ will be 1.02X or greater and 50% of the time it will be less. In this case, $X$ and $Y$ would be returns. The Cauchy distribution lacks a mean, so there is no point to converge to, but it does have a median which in can swing around like a pendulum. In fact, the math is intimately related to pendulums.

The argument for this is that returns are product of the ratio of prices by the ratio of quantities traded. That implies that a return is a function of prices and quantities. If we ignore mergers, bankruptcies, and dividends, then we can at least temporarily drop the quantities by setting the ratio to unity.

Since stocks are traded in a double auction the winner's curse does not apply, and the rational behavior is to bid your expectation. As the number of actors becomes large enough, the distribution of reservation prices should become normally distributed. Technically this is "overkill" for a proof, but this post would go on for thirty-five pages with small populations, and the result would be the same.

Since the reservation prices at time t would be normally distributed, or at least would converge to it at the limit, and this would be true at time t+1, the distribution of prices, but not returns should be normal. If we assume independence, mostly because the headache of not assuming independence of pricing errors between times t and t+1, and if we assume that an equilibrium price exists, then by well-known theorems in probability and statistics the distribution of returns must converge to a Cauchy distribution.

If you need a formal derivation, either see the citations below or go to section two of the article linked at the end of this blog post. https://www.datasciencecentral.com/profiles/blogs/data-science-common-stocks-and-v-amp-v

Now you have automatic jumps built right into the distribution. The same thing can be shown for the multivariate case, but not without me typing for days.

Models like the CAPM assume that returns are a primitive element of the model and the rules of calculus in use assume the parameters are known. If they are not known and treated as data instead, then they will converge to a Cauchy distribution.

Because the Cauchy distribution has no sufficient statistic, you will need to use a Bayesian method. However, you can condition on any variable you like. Furthermore, as the method of inverse probability, your question is not overly difficult.
$$\Pr(\theta|\text{data and crash})\propto\Pr(\text{data and crash}|\theta)\Pr(\theta),\forall\theta.$$

I would formally model liquidity.

Your factors would no longer be considered covariances, but they would be scale parameters.

If you did decide to use a Frequentist method, you should either use Theil's median regression or quantile regression. There will be more noise, but rank based statistics are sufficient statistics for the quantile they map to.

Curtiss, J. H. (1941). On the distribution of the quotient of two chance variables. Annals of Mathematical Statistics, 12:409-421.

Gurland, J. (1948). Inversion formulae for the distribution of ratios. The Annals of Mathematical Statistics, 19(2):228-237.

Harris, D. E. (2017). The distribution of returns. The Journal of Mathematical Finance, 7(3):769-804.

Jaynes, E. T. (2003). Probability Theory: The Language of Science. Cambridge University Press, Cambridge.

Koenker, Roger (2005). Quantile Regression. Cambridge: Cambridge University Press.

Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309):193-204.

Sen, P. K. (1968). Estimates of the regression coefficient based on Kendall's tau. Journal of the American Statistical Association, 63(324):1379-1389.

Theil, H. (1950), "A rank-invariant method of linear and polynomial regression analysis. I, II, III", Nederl. Akad. Wetensch., Proc., 53: 386–392, 521–525, 1397–1412