Measuring Tail Risks
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909649342562304 |
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| author | Chen, Kan Cheng, Tuoyuan |
| author_facet | Chen, Kan Cheng, Tuoyuan |
| contents | Value at risk (VaR) and expected shortfall (ES) are common high quantile-based risk measures adopted in financial regulations and risk management. In this paper, we propose a tail risk measure based on the most probable maximum size of risk events (MPMR) that can occur over a length of time. MPMR underscores the dependence of the tail risk on the risk management time frame. Unlike VaR and ES, MPMR does not require specifying a confidence level. We derive the risk measure analytically for several well-known distributions. In particular, for the case where the size of the risk event follows a power law or Pareto distribution, we show that MPMR also scales with the number of observations $n$ (or equivalently the length of the time interval) by a power law, $\text{MPMR}(n) \propto n^η$, where $η$ is the scaling exponent. The scale invariance allows for reasonable estimations of long-term risks based on the extrapolation of more reliable estimations of short-term risks. The scaling relationship also gives rise to a robust and low-bias estimator of the tail index (TI) $ξ$ of the size distribution, $ξ= 1/η$. We demonstrate the use of this risk measure for describing the tail risks in financial markets as well as the risks associated with natural hazards (earthquakes, tsunamis, and excessive rainfall). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07092 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Measuring Tail Risks Chen, Kan Cheng, Tuoyuan Risk Management Statistical Finance Value at risk (VaR) and expected shortfall (ES) are common high quantile-based risk measures adopted in financial regulations and risk management. In this paper, we propose a tail risk measure based on the most probable maximum size of risk events (MPMR) that can occur over a length of time. MPMR underscores the dependence of the tail risk on the risk management time frame. Unlike VaR and ES, MPMR does not require specifying a confidence level. We derive the risk measure analytically for several well-known distributions. In particular, for the case where the size of the risk event follows a power law or Pareto distribution, we show that MPMR also scales with the number of observations $n$ (or equivalently the length of the time interval) by a power law, $\text{MPMR}(n) \propto n^η$, where $η$ is the scaling exponent. The scale invariance allows for reasonable estimations of long-term risks based on the extrapolation of more reliable estimations of short-term risks. The scaling relationship also gives rise to a robust and low-bias estimator of the tail index (TI) $ξ$ of the size distribution, $ξ= 1/η$. We demonstrate the use of this risk measure for describing the tail risks in financial markets as well as the risks associated with natural hazards (earthquakes, tsunamis, and excessive rainfall). |
| title | Measuring Tail Risks |
| topic | Risk Management Statistical Finance |
| url | https://arxiv.org/abs/2209.07092 |