An investigation of higher order moments of empirical financial data and the implications to risk

TL;DR

This paper examines how higher order moments of financial data vary with time window size and economic conditions, revealing scaling laws and their impact on risk measures like Value-at-Risk.

Contribution

It uncovers distinct scaling relations of higher order moments in financial data and links these to risk assessment during different economic periods, including crises.

Findings

Different scaling laws for moments in short and long time windows

Significant change in moments during financial crises

Scaling relations influence Value-at-Risk levels

Abstract

Here, we analyse the behaviour of the higher order standardised moments of financial time series when we truncate a large data set into smaller and smaller subsets, referred to below as time windows. We look at the effect of the economic environment on the behaviour of higher order moments in these time windows. We observe two different scaling relations of higher order moments when the data sub sets' length decreases; one for longer time windows and another for the shorter time windows. These scaling relations drastically change when the time window encompasses a financial crisis. We also observe a qualitative change of higher order standardised moments compared to the gaussian values in response to a shrinking time window. We extend this analysis to incorporate the effects these scaling relations have upon risk. We decompose the return series within these time windows and carry out a Value-at-Risk calculation. In doing so, we observe the manifestation of the scaling relations through the change in the Value-at-Risk level. Moreover, we model the observed scaling laws by analysing the hierarchy of rare events on higher order moments.