How does one analyze diversification if stock prices follow a Cauchy distribution?

Question

How does diversification actually lead to less variance in a portfolio? I'm looking for a formal reason why this is the case. There are a number of explanations I have been able to find, but they make the assumption that stocks are wiener processes - that is they assume that the daily changes in a stock's price are normally distributed. In this model of stocks, diversification leads to less variance in portfolio prices because the average of some normally distributed random variables is always a normal variable with less variance.

For example, if a and b are distributed normally with mean 0 and variance 1, then their average is normally distributed with mean 0 and variance 1/Sqrt(2). If a and b represented daily price differences for stocks, then this means we could reduce the variance of the daily price difference by spreading our investment out among multiple such stocks.

This is essentially what is shown on the Wikipedia page for the theory of diversification.

Changes in stock price are Cauchy Distributed - that is stocks are better modeled as Levy processes. The Cauchy distribution doesn't exhibit the behavior above which is used to justify diversification. Instead, averaging out a number of Cauchy random variables does nothing to reduce the amount of spread in the distribution. Put more formally, if a and b are identically distributed Cauchy random variables, then their average is a Cauchy random variable with the same parameters. In this way, spreading out your investment across many different levy processes will not result in a levy process with a less variation and there is no reason then to diversify.

I might be missing something pretty simple, however it seems kinda strange to me how mischaracterized diversification seems to be in everything I've read. Is there a paper or something which covers the topic better and doesn't assume that stocks are Wiener processes?

As a bit or warning, I'm clearly not a quant. I just happen to know math and was kinda curious about stocks.

Answer 1

I think this is a very good and valid question. I will try to give a more general answer here.

It is by now a well known fact that much of the classical stuff won't work in the manner it was thought and supposed to work.

One of the points that are not clear when it comes to real financial time series is how they are distributed (they are obviously not normal) and by far the more important question: Do they exhibit finite variance? By definition you can't find out by just looking at finite samples but of course the validity of such cornerstones of classical statistics (like the central limit theorem) rest on the assumption of a finite variation in the population.

Another thing is that financial times series are not i.i.d. - you have all kinds of dependencies, like e.g. autocorrelation, vol clustering etc.

On the other hand some of the results (like basic option pricing, but I would also count in diversification) seem to be pretty robust (but of course not perfect in a mathematical sense) when it comes to using them although many of the assumptions are not met.

As a starting point I would recommend google'ling for articles/books from Taleb and Mandelbrot for a very critical stance on the classical results, but also e.g. for Wilmott for a more balanced view. A very good author is also Meucci who writes about robust methods under non-normality assumptions.

Answer 2

There is, in fact, a large literature on this subject, but you may have been searching for the wrong terms. This issue is broadly explored within the literature regarding a preference for higher moments. Cauchy distributions, themselves, are very hard to work with because they don't have any well-defined moments. One possible solution to this is to use what is termed the "truncated Levy flight" in a paper by Xiong (2010). Modeling stocks using a Levy distribution has also been discussed in a previous question.

The specific issue of how a preference for third moments (particularly skewness) leads to underdiversification is addressed by Mitton and Vorkink (2007). Their goal, however, is more to explain the fact that people do not "diversify sufficiently" in a mean-variance sense than to replace the theory of diversification.

More broadly, however, examining the implications of a preference for higher moments has been analyzed by:

• Xiong and Idzorek (2011)
• Barberis and Huang (2008)
• Boyer, Mitton, and Vorkink (2009)
• Harvey and Siddique (2000)
• Arditti (1967)

These references should definitely be enough to get you started. They can explain it much better in their own words than I can paraphrasing.

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On a related side note, a different set of papers, much more empirical in nature, have shown that the higher moments of measured equity returns do converge towards a normal distribution (albeit very slowly) as the horizon lengthens. Hence diversification may not work in the short run (days to years) but is still likely to work over decades. This fact may or may not provide much comfort.

Answer 3

As a start. it is easy to prove that diversification always helps whenever the variances are all finite. To see this, consider two stocks A and B with variances var(A) and var(B). Then the variance of the portfolio where you mix them in equal parts is: $$var((A+B)/2) = var(A)/4+var(B)/4+cov(A,B)/2 =\\ =var(A)/2 + cov(A,B)/2.$$ The largest that $cov(A,B)$ can be is $var(A)/2$. So the largest this expression can be is $var(A)$, but when $A$ and $B$ are less correlated, this expression will give less than $var(A)$, meaning that diversification helped.

Overall, we see that when all variances are not infinite then diversification doesn't hurt, and it always helps unless the stocks are perfectly correlated.

But what about the case where the variances are undefined, like in processes based on Cauchy distributions? Well, the math breaks down, but you see that the intuition and conceptual reasoning does not break down: mixing stocks reduces the influence of extreme movements in either individual stock. This is still true, so I'd expect that no matter how we choose to re-define our concepts, we'll still get a proof that diversification helps.

For example, one might try to define volatility for wild processes (like Levy processes) not by measuring the standard deviation of the price change, but by measuring the chance that our loss after $T$ time units would be more than $L$. Then you get a two-parameter measures that is always correctly defined. You can then look at variables $A$ and $B$, assume the measure gives the same result for both, and try to compute this measure for their average. I'm pretty sure that you'll get that this measure comes out smaller for their average than it was for $A$ and $B$. Likewise, I expect any reasonable definition of volatility to exhibit the same behavior.