Analysing Statistical Robustness of Cross-Sectional Volatility
Felix Goltz, Lionel Martellini, Stoyan Stoyanov: Volatility is a statistical measure of the dispersion of returns for a given security or market index.
Head of applied research at EDHEC-Risk Institute and research director at ERI Scientific Beta.
Professor of finance at EDHEC Business School and scientific director of EDHEC-Risk Institute.
Professor of finance at EDHEC Business School and head of research at EDHEC Risk Institute–Asia.
It refers to the amount of uncertainty or risk about the size of changes in an underlying security or index value. Higher volatility means that the price of the security can change dramatically in either direction over a given interval of time.
A key distinction exists between systematic and specific volatility: total volatility can be decomposed into systematic volatility, driven by the stock exposure with respect to systematic risk factors, and specific volatility, which is driven by the uncertainty impacting a particular company. The recent financial literature has paid considerable attention to idiosyncratic volatility indicating that idiosyncratic volatility can have forecasting power for future excess returns.
Garcia, Mantilla-Garcia and Martellini (2013) discuss a link between idiosyncratic volatility and cross-sectional volatility (CSV). They show that the cross-sectional variance of stock returns can be regarded as an efficient estimator for the average idiosyncratic variance of stocks within the universe under consideration. Goltz et al. (2011) find this measure of variance to be highly correlated to standard measures of systematic risk when they exist, which further justifies its use in the context of equity volatility measurement. Key advantages of this measure over currently available measures such as sampledependent historical volatility measures or option-based implied volatility measures are: its observability at any frequency; its model-free nature; and its availability for every region, sector and style of the world equity markets, without the need to resort to any auxiliary option market.
A methodology of CSV estimation has to be based on a statistically robust estimation method because, empirically, cross-sections of returns are polluted by outliers – abnormal returns in the cross-section that deviate significantly from the rest of the sample. The classical method of estimation is sensitive to the outliers in the sample that the resulting noise can hide the dynamics of CSV in the time domain. An estimator is considered robust in a statistical sense if a small proportion of big outliers does not lead to big deviations in the estimated values. The outliers are usually regarded as a contaminant and the goal of any robust approach in statistics is to limit or completely neutralise their influence on the estimated quantity. By construction, the notion of statistical robustness is related to a notion of continuity of the estimator with respect to small deviations from a given reference model.
In this paper, we propose a robust method of estimation which is intuitive and is functionally similar to the weighted average estimator studied by Garcia, Mantilla-Garcia and Martellini (2013) in the context of one-factor and multifactor regression models. To meet this objective, we adopt a statistical technique called M-estimation.
Although the interpretation of CSV as the average idiosyncratic volatility of the universe follows from asymptotic arguments, the CSV is computable for any universe. We study the relevance of robust estimation for universes of different size such as the top 50, 100 and 200 US stocks listed on NYSE and confirm that the properties of robustness become more important for universes of smaller size. Finally, we consider cross-sections of returns of different frequency and conclude that cross-sections of returns of lower frequencies are characterised by fewer outliers. In the case of monthly frequency and the S&P500 universe, the classical estimator combined with mild trimming seems to have a satisfactory performance. The daily and weekly frequencies, however, are characterised by a higher percentage of outliers in the cross-section which justifies the use of a more sophisticated robust estimation technique.
Analysing Statistical Robustness of Cross-Sectional Volatility...
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