Abstract
We suggest a new, parsimonious, method to fit financial data with a stable distribution. As a result of a stable fitting via maximum likelihood estimation (MLE), we find that some assets have similar values as stability indices, independently of the time interval considered. This fact can be exploited to pool the assets in groups and to choose a parameter \(\alpha \) as an ex ante stability index, valid for every asset in the pool sector. With this fixed parameter, MLE is used again to obtain the other stable parameters. We discuss an innovative risk measure, based on the Expected Shortfall, which exploits the above procedure. We show that it gives a good estimation of risk even when only short time series are available. Finally, we introduce the notion of Risk Class, which allows us to classify assets according to their risk exposition and to compare different methods for the computation of the Expected Shortfall.
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Notes
Nowadays, the concept of a black swan fully belongs to the usual language of the “financial culture”, also because of the success of the controversial bestseller of Taleb (2010).
A very preliminary step of this research can be found in Donati and Corazza (2014).
\({\mathrm{RedES}}^{\mathrm {TM}}\) has been developed and introduced in 2004 by Redexe and has, since then, been used for risk management by several Italian financial institutions. The patent is currently pending.
We remark that c does not represent the standard deviation, since for \( \alpha <2\) it is infinite. For \(\alpha =2\), namely when X is normally distributed, the standard deviation \(\sigma \) is related to c by the equation \(\sigma =\sqrt{2}c\).
Datastream is one of the most important international financial time series database, due to Thomson Reuters (Datastream 2019).
In our time series, we observe the presence of null returns for some assets. Most likely, these values correspond to days where, for some reasons, there was no trading on some particular asset, though there have been movements in the market (namely, if there had been some trading on that asset, then its price would have changed). In our opinion, these zeroes do not reflect adequately the real behavior of the price of the asset, and therefore we decided to eliminate them introducing a filter on zero (null filter), obtaining time filtered series.
Golden Ocean Group Limited is a leading international dry bulk shipping company, founded in 1996, based in Bermuda and listed on NASDAQ and on the Oslo Stock Exchange.
For the meaning of rare event, see also the interesting discussion on the notion of extremal event in the Reader Guidelines of the book by Embrechts et al. (1996).
An alternative choice for the stability parameter could be the mean of the group. We performed all the computations, assuming the mean as stability parameter: the final results, in terms of estimation of risk, are very similar. Moreover, in most cases, the median provides a better fit in terms of distributions.
For a portfolio with a random payoff X and distribution function \(F_{X}\), the Value at Risk at confidence level \(q\in (0,1)\) is given by
$$\begin{aligned} {\text {VaR}}_{q}(X)=-\inf \{x\in \mathbb {R}:F_{X}(x)\ge q\}, \end{aligned}$$(7)which coincides with \(F_{X}^{-1}(1)\) when \(F_{X}\) is strictly increasing.
The Expected Shortfall at confidence level q of a portfolio with payoff X is defined as
$$\begin{aligned} {\text {ES}}_{q}(X)=\frac{1}{q}\int _{0}^{q}{\text {VaR}}_{r}(X)\mathrm {d}r. \end{aligned}$$(8)Think of the analogy between the classical expression \(\delta \) = \(\ln (1+i)\) , which connects the force of interest with the annual compound interest rate i, and the expression \({\text {rc}}({\text {ES}})=\log _{2}(1+|{\text {ES}}|);\)\({\text {ES}}<0\).
Briefly, when computing the \(\hbox {R}^{*}\) measure for a 2-years time series from 2017 to 2019, we use the \(\alpha \)-pool parameter calculated on 20-years time series from 1997 to 2017, which does not “overlap” the short series. The first submitted version of this paper, based on data collected in September 2011, used a different approach: the \(\alpha \)-pool parameter was calculated on 1991–2011 time series and then used to calculate \(\hbox {R}^{*}\) of 2009–2011 time series, entirely contained in the long one. It is noteworthy that both approaches yield remarkably similar results, and we thank the anonymous referee that suggested such a check for leading us to such a double confirmation of our method.
In Harmantzis et al. (2006), the performance of different models (empirical, Gaussian, generalized Pareto, and stable distributions) is empirically tested in evaluating Value at Risk and Expected Shortfall and similar results are obtained.
See Mattel, Annual Reports (2019).
Currently, Basel III asks to use ES to estimate capital requirements for market risk and VaR for backtesting (BCBS 2013, Appendix B).
The data had been collected for a previous version of this paper. In that case, the sector stability index was computed on the 20 years 1991–2011, while \(\hbox {R}^{*}\) was calculated on the 2 years 2009–2011.
The \(\hbox {R}^{*}\) Risk class for JPM estimated in the period 2009–2011 is 3.917. The empirical Risk Class for the period 1997–2017 is 3.086.
The \(\hbox {R}^{*}\) Risk class for ANF estimated in the period 2009–2011 is 3.589. The empirical Risk Class for the period 1997–2017 is 3.512.
For instance, between September and October 2008, the Standard & Poor’s 500 index decreased by \(25.9\%\) and, on August 2015, the Shanghai Composite Index has fallen by \(8.49\%\) just on one day.
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Acknowledgements
We gratefully acknowledge the editor Markus Schmid and two anonymous referees whose comments and suggestions helped to considerably improve our manuscript. We also thank Giulio Campanini and Alice Pisani for their fundamental contribution in data collection. Our special thoughts go to Erio Castagnoli who, as always, supported us during the writing of this paper and whose invaluable suggestions will be missed.
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De Donno, M., Donati, R., Favero, G. et al. Risk estimation for short-term financial data through pooling of stable fits. Financ Mark Portf Manag 33, 447–470 (2019). https://doi.org/10.1007/s11408-019-00340-5
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DOI: https://doi.org/10.1007/s11408-019-00340-5