Time Scaling Normal Inverse Gaussian Return Distributions

Summary

The document considers return distributions for portfolios containing multiple asset classes, with the goal of representing skewness and kurtosis as well as mean and variance. It asks for a non-Gaussian distribution whose parameters can describe returns over different time horizons, and focuses on the Normal Inverse Gaussian distribution because of its convolution property. It also raises the possibility of multivariate extensions and alternative distributions, but does not resolve those questions.

The added discussion states that the NIG density can be scaled through convolution and summarizes how its moments vary with time: mean and variance increase, while skewness and kurtosis decline, at rates tied to the square root of time and time respectively. The excerpt supplies no derivation, empirical comparison, or portfolio dependence model. Its claims therefore sketch a useful scaling property but leave implementation details and suitability for multivariate portfolio returns open.

Key ideas

• The Normal Inverse Gaussian distribution can represent non-Gaussian returns with higher moments.
• Its convolution property supports constructing distributions for longer time horizons.
• The document states that mean and variance grow with time under its scaling discussion.
• It reports that skewness declines with the square root of time and kurtosis declines with time.
• The excerpt does not establish a multivariate dependence model or compare the approach empirically.

Tags

# non gaussian distributions with higher moments and time scaling properties?


# non gaussian distributions with higher moments and time scaling properties?












If we assume a portfolio comprised of n asset classes, whose log returns can be modeled with a distribution. I am interested in finding a distribution that:

- incorporates higher moments (skewness and kurtosis)

- (ideally but not necessarily multivariate as I have n asset classes)

- that can be time scaled, i.e., the distribution of returns at time T can be expressed parametrically too (the same way assuming normal distribution and GBM, we have

- I came across the NIG distribution that has nice convolution property. Is there a way to scale it up in time? and if so how?

- are there any other non gaussian distributions or approximations that you would recommend?

Many thanks

EDIT

Building on @Kermittfrog's answer, more details about how to scale up the NIG and the impact on the moments:

From the convolution property of the NIG it follows that the density at time T is given by:

And the moments evolve as follow:

i.e. mean and variance increase with t and the skewness and kurtosis decrease over time with √t and t respectively.

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.