Calculating variance of process with time-varying variance

This is a question stemming off a previous post I had regarding calculating portfolio volatility.

For a portfolio consisting of multiple assets, I understand that there are multiple ways to calculate its volatility:

• The simplest method is to simply acquire the portfolio returns and calculate their variance. Due to the individual assets within the portfolio having their own returns, the weights for each of those assets will naturally change. Because of this, the variance of the portfolio varies with time (a time-varying variance as was referenced by @Taylor in the prior post), so calculating the variance with just the returns doesn't do the trick.
• The typical method is to calculate the covariance matrix of all the comprising assets' returns and multiply it by the weights vector (both transposed and not).

My question is whether the covariance matrix method adequately calculates the variance of such a process with a time-varying variance. If it does, an explanation would be much appreciated. If it doesn't, an explanation would still be much appreciated.

There is something here you might not know as most people do not think about it. The math involved has assumptions that people do not think about. The first assumption is that the parameters are known. Of course, they are not, but the calculus is performed as if the parameters were. In most cases, there is no consequence from this. For many standard processes, the impact of not knowing the parameters has no consequence in expectation.

The stock market is not one of them. Imagine a stock whose value could be understood with the relationship $$\tilde{w}=\beta\bar{w}+\varepsilon$$. For a stock, it must be the case that $$\beta>1$$. After all, who would want to systematically lose money or break even?

Mann and Wald showed the maximum likelihood estimator for this equation is OLS, which also makes it the Frequentist parametric solution. We will ignore median based solutions such as Koenker's quantile regression or Theil's median regression for linear or polynomial models. After, it is mean-variance finance, not median-interquartile range finance. So far, so good. OLS doesn't depend on the distribution of $$\varepsilon$$ either, so easy-peasy.

Except it isn't. If $$\beta$$ is known, then everything ends up well behaved. If $$\beta$$ is unknown, but $$\beta>1$$ then White in 1958 showed that the sampling distribution for $$\hat{\beta}$$ was the Cauchy distribution. Since the Cauchy distribution has no population mean, it implies that $$\beta$$ estimates are meaningless. In fact, $$\hat{\beta}$$ should time-vary because it is a random number generator that doesn't converge to anything regardless of the sample size. Having a sample size of one event or one million events adds no information and doesn't reduce the uncertainty.

If you do not have a mean, then you cannot have a variance or covariance.

The second way to look at this is one security at a time. If a holding period return is a future value divided by a present value, then a return is a function of the data. It is a statistic. Its distribution shouldn't be assumed to be normal or log-normal; it should be derived.

Since value is price times quantity, we should first deal with the easy part, quantity. If we exclude dividends, including liquidating dividends, mergers, bankruptcies and liquidity costs, then we have the standard case where the future quantity and the current quantity are equal and so the second ratio can be ignored for now as it simplifies to unity. In the real world, you cannot ignore it, but I don't want to type for thirty-five pages.

That makes it $$r_t=\frac{p_{t+1}}{p_t}.$$

That is a ratio distribution. The question is "what is the distribution of prices?" If you assume normality, a dangerous assumption that happens to work out for reasons later shown but won't work for many classes of assets, then it is the ratio of two normal distributions. If you integrate around the equilibrium prices rather than $$(0,0)$$ you will get a Cauchy distribution. Again, it has no mean, variance or covariance matrix. If you try and calculate it, you will get time-varying means and time-varying variances, even if the center of location and scale parameters are a constant.

The reason normality is a safe assumption, in equilibrium only, is that stocks are sold in a double auction, so the winner's curse does not apply. The rational behavior is to bid your expectation, which makes the distribution of the limit book the distribution of expectations. If there are many potential buyers and sellers, it will become normal at the limit. The results hold under much weaker conditions. Out of equilibrium, such as during the 2008 financial crisis, this result does not hold.

Of course, I dropped out quite a few things, but they don't make the problem better behaved. There is a body of literature since the 1960s beginning with Benoit Mandelbrot and Eugene Fama that runs to today on heavy-tailed distributions. If you want to estimate the scale parameter, you could crudely estimate it with the interquartile range, or exactly estimate it with the Cauchy for regression or the truncate Cauchy if you use ratios using a Bayesian estimator. You cannot use a maximum likelihood estimator in the truncated case as the solution as to how to solve it is unknown. The formula is known, but there isn't a strategy that is clear to determine the roots.

I do not fully comprehend the social forces that keep mean-variance finance alive in finance. It is relatively easy to understand but has never passed a single validation study. It does solve an important publish-or-perish problem though. By pretending its real, thousands of articles can be published on anomalies. I found 3800 articles on one anomaly alone. I am afraid that it isn't merely a standard case of perverse incentives.

Mean-variance finance has been written into European Union law and Uniform Prudent Investor Acts of the fifty states. Further, because Nobels were given out, it grants safe harbor to financial institutions from suit in places they may bear tremendous liability.

Still, there is more going on than that in the field. The standard assumption in economics is that if you build a better mouse trap everyone will copy it. There is value in unworkable mouse traps in finance. I don't get how this has been out since 1963 and the inertia is unreal.

If you had "time-varing variance," then it may be possible to estimate it. In fact, I am sure it would be possible. However, you do not have a heteroskedastic process, you have an askedastic process.

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

Fama, E.F. (1963) Mandelbrot and the Stable Paretian Hypothesis. Journal of Business, 36, 420-429.

Fama, E. (1965) The Behavior of Stock Market Prices. Journal of Business , 38, 34-105.

Fama, E.F. and Roll, R. (1968) Some Properties of Symmetric Stable Distributions. Journal of the American Statistical Association, 63, 817-836.

Mandelbrot, B. (1963). The variation of certain speculative prices. The Journal of Business, 36(4):394-419.

Mann H, Wald A (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11:173–200

White JS (1958) The limiting distribution of the serial correlation coefficient in the explosive case. The Annals of Mathematical Statistics 29(4):1188–1197