Is there a modern theory for the probability distribution of stock returns? It is relatively easy to deduce that under idealized conditions stock returns follow a log normal distribution. One arrives at this by considering the product of ratios of prices ("stock returns"), applying a natural logarithm to convert the product into a sum and then applying the Central Limit Theorem under the condition that the ratios are iid (independent and identically distributed) and have finite mean and variance.

The problem is of course that we cannot just assume that returns are iid or that they have finite variance. So I am seeking alternative theories that try to address these shortcomings.

I am aware of Mandelbrot's Multifractal Model of Asset Returns. Is this considered SOA in the field? Is there something else that is considered a better model or easier to work with?

  • $\begingroup$ To what end?....... $\endgroup$
    – Arshdeep
    Mar 30 at 22:52
  • $\begingroup$ Better modeling of stock returns. $\endgroup$ Mar 30 at 23:09
  • $\begingroup$ For what? Usually people do "modelling" to produce an output $\endgroup$
    – Arshdeep
    Mar 30 at 23:15
  • $\begingroup$ What is your interest in what my interest is? $\endgroup$ Mar 30 at 23:21
  • 1
    $\begingroup$ If you are calculating VaR, you can be sufficiently confident with something that takes into account fat tails and need not go beyond. For portfolio optimization, needs are stronger. So what is sufficient depends on what you are trying to do. My interest is in having completely specified questions on this forum. $\endgroup$
    – Arshdeep
    Mar 30 at 23:23

1 Answer 1


Order of importance of inputs in portfolio optimization are:

  1. Expected returns (how you clean them)
  2. Covariances (how you clean them)
  3. Variances

If you don't want normal distributions, you introduce more variables like kurtosis etc. explicitly into your objective. You then specify limits of exposure to kurtosis.

At this point, most of your risk is coming from the above 3 points as well as the having local solutions. Depending on asset universe, you will have multiple solutions close to each other.

Sorting out these is NOT inside a statistical paradigm anymore. It requires considerations in your portfolio.

In other words, you will see that the lack of performance of your portfolio is because realized means and variances are different from your estimation. What will you do by refining the stock distribution?

  • $\begingroup$ Thanks for your answer. It is relatively easy to graph returns and eyeball a distribution, then add terms to it until it "works". However I seek a deeper theoretical understanding of the underlying principles and I like to work from first principles to the extent that is possible. The ideal for me would be to arrive at a theoretical answer that then gets corroborated by the data. My practical goal is to design a loss function for an automated process and having a better understanding of the distribution will inform the design of this function. $\endgroup$ Mar 31 at 0:33
  • $\begingroup$ At the risk of beating a dead horse, let's say you get a magical distribution that captures all subtleties perfectly. You still don't know mean and covariances. Which are the main factors you are exposed to - no matter how you define your objective $\endgroup$
    – Arshdeep
    Mar 31 at 3:09
  • $\begingroup$ mandelbrot spent a lot of time trying to find a distribution so I would read his papers ( he has books also ) if you want to avoid re-inventing the wheel. And I would also check if he was rich because, if he was not, then he probably didn't find one that worked very well. $\endgroup$
    – mark leeds
    Mar 31 at 4:53

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