Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions

Abstract

The aim of this work was to test how returns are distributed across multiple asset classes, markets and sampling frequency. We examine returns of swaps, equity and bond indices as well as the rescaling by their volatilities over different horizons (since inception to Q2-2020). Contrarily to some literature, we find that the realized distributions of logarithmic returns, scaled or not by the standard deviations, are skewed and that they may be better fitted by t-skew distributions. Our finding holds true across asset classes, maturity and developed and developing markets. This may explain why models based on dynamic conditional score (DCS) have superior performance when the underlying distribution belongs to the t-skew family. Finally, we show how sampling and distribution of returns are strictly connected. This is of great importance as, for example, extrapolating yearly scenarios from daily performances may prove not to be correct.

Forecasting portfolio returns with skew‐geometric Brownian motions

“…As empirically investigated in Orlando and Bufalo, 3 returns, either standardized or not, do not seem to be unconditionally normally distributed. They often show a significant amount of skewness and extra‐kurtosis 4‐7 .…”

“…Azzalini and Capitanio 11 provide a general account of the statistical and mathematical properties of the skew‐ class of densities. What is noteworthy about ‐skew distribution, apart from the ability to fit heavy‐tailed and skewed data, is “their mathematical tractability, the inclusion of the normal law as a proper member of the family, and the shape parameter regulating the skewness.” 3 Such an extension allows for a continuous variation from normality to non‐normality 10 . For an overview, see Azzalini 12 …”

“…Furthermore, Maxwell and van Vuuren 20 build “TE‐constrained portfolios which are maximally diversified, exhibit risk parity, and have minimal intra‐correlation between constituents.” In this work, a key part of the optimization scheme is played by the ex‐ante tracking error which depends on distributions of asset returns. As mentioned, we will follow theoretical 8‐10,12,27 and empirical work 3 and, for the rest of the article, we will assume skewed return distributions.…”

“…In this article, we follow the existing literature such as Mahoney 1 where the idea is to model the cross dependency between assets and, thus, to minimize the tracking error of a portfolio. However, it is well known that returns of financial assets are not normally distributed and more flexible models are necessary (see, e.g., References 2 and 3). For this reason, given an index and its subindices, we assume that their (log‐)returns follow skew‐geometric Brownian motions which are correlated among each other.…”

Straightening skewed markets with an index tracking optimizationless portfolio

“…Indeed, some scholars claim that standardized daily returns "are approximately unconditionally normally distributed" (Andersen et al, 2001) or that "they are IID Gaussian, with variance equal to 1" (Rogers, 2018). In real life, instead, returns, either standardized or not, do not conform to those hypotheses (see, e.g., Cont, 2001;Orlando and Bufalo, 2021). To account for skewness and extra-kurtosis, one may consider skew-normal distributions as first introduced by Azzalini (1985) and Henze (1986).…”