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| Project Details |
| Funding Scheme : | General Research Fund | |||||||||||||||||||||
| Project Number : | 14211316 | |||||||||||||||||||||
| Project Title(English) : | Asymptotic Analysis of Portfolio Tail Risk and the Diversification Effect under Multivariate Elliptical Distributions for Static Portfolios | |||||||||||||||||||||
| Project Title(Chinese) : | 多元橢圓概率分佈下投資組合的側尾概率以及風險分散的漸進分析 | |||||||||||||||||||||
| Principal Investigator(English) : | Dr WU, Qi | |||||||||||||||||||||
| Principal Investigator(Chinese) : | ||||||||||||||||||||||
| Department : | Department of Data Science | |||||||||||||||||||||
| Institution : | City University of Hong Kong | |||||||||||||||||||||
| E-mail Address : | qiwu55@cityu.edu.hk | |||||||||||||||||||||
| Tel : | 34427018 | |||||||||||||||||||||
| Co - Investigator(s) : |
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| Panel : | Engineering | |||||||||||||||||||||
| Subject Area : | Mechanical, Production & Industrial Engineering | |||||||||||||||||||||
| Exercise Year : | 2016 / 17 | |||||||||||||||||||||
| Fund Approved : | 482,605 | |||||||||||||||||||||
| Project Status : | Completed | |||||||||||||||||||||
| Completion Date : | 31-12-2019 | |||||||||||||||||||||
| Project Objectives : |
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| Abstract as per original application (English/Chinese): |
Following the 2007-2008 financial crisis, both the United States and the European Union have enacted legislative proposals, according to which, existing risk management framework is considered imperfect. It is concerned that financial institutions may not have sufficient capital to weather catastrophic events if market risks are measured by the ``Value-at-Risk" metric. It is also concerned that aggregated counterparty risk, if not mitigated, is a threat to the stability of financial system. On the other hand, simply increasing capital requirements for banks and charging heavy margins at central counterparties may lead to liquidity problems as well as inefficient use of capital. At the center of this intricate trade off is the assessment and management of tail risk. This motivates us to conduct a technical investigation into the mathematical behavior of portfolio tail risk under the prevailing risk metrics. In particular, we plan to consider a rich parametric family of multivariate distribution and provide accurate analytical estimates of both portfolio value-at-risk (VaR) and portfolio expected shortfall (ES). We work with these distributions based on the consideration that they not only exhibit distinct behavior of tail heaviness and tail dependence, but also have the property that, under which, both VaR and ES are coherent risk measures. Our goal is to use these analytical results to examine the diversification effect associated with portfolio tail risk. By looking at the risk reduction amount which measures the dollar difference between the total risk of a static portfolio and the sum of that of individual positions, we intend to understand the role played by the following properties on tail risk diversification: the tail heaviness of individual asset, the pair-wise tail dependence, and the cross-asset heterogeneity.
2008年的全球金融危機讓監管當局深刻意識到現有的風控體制在面對系統性金融風險時存在嚴重不足。作為應對，歐美均積極通過立法，一方面要求銀行和金融機構大幅增加資本金防範於未然，另一方面要求各證券交易市場全面實行保證金交易制度。但無論是增加資本金還是保證金交易，如果不加分析的全面實施，對整個金融市場來說，都會增加不必要的資金需求，從而降低全市場的資金使用效率，使流動性承壓。其根本原因在於，合理的資本金額度和保證金額度因當因具體資產的虧損敞口不同而不同。因此這個兩難問題背後的核心之一是如何準確的度量一個資產組合的尾部風險（小概率的大額虧損風險） 以及對尾部風險的系統管理。本課題試圖從一族具有多元重尾特徵的資產價格分佈入手，通過運用漸進分析的方法，來分析並量化一個包含有限個資產的投資組合其巨額虧損是如何依賴於各個子資產價格分佈的重尾程度和以及各個子資產價格之間的極值相關性這兩個方面。在風險度量的選取方面，我們考慮 Value-at-Risk (VaR) 和 Expected Shortfall (ES) 這兩個當下最為重要的度量標準。在價格分佈的選取方面，我們假設組合中的各子資產價格滿足一族參數化的多元橢圓分佈。一方面，我們構造的這一族分佈能夠同時產生具有代表性的 Exponential-type 輕尾分佈和 Power-type 的重尾分佈。另一方面，這一族分佈也能產生寬範圍的極值相關係數以便我們研究資產間的相關性對整體風險的影響。於此同時，選取橢圓分佈的一個考慮是其對 VaR 和 ES 均滿足次可加性，便與我們能附帶研究尾部風險在一個組合中是如何分散降低的(Diversification benefits). 本課題的最終目的是理解並量化各個子資產價格的重尾度以及他們之間的相關性對整個投資組合的全局風險的影響。 |
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| Realisation of objectives: | The three project objectives have been achieved satisfactorily with rich and comprehensive research output. The realization of each objective is described below. 1. We solved the two analytical challenges associated with the problem. The first one is solved through the specification of the elliptical distribution family where we used a geometric approach together with its intrinsic stochastic decomposition property. The second one is solved by constructing a general uniform expansion approach which works in cases where the method of saddlepoint approximation does not apply well. 2. We thoroughly analyzed the diversification effect associated with portfolio tail risk. This is possible due to the analytical methods we developed in meeting Objective 1. In particular, we used results obtained in the first objective to examine the relationship between the amount of risk reduction and the portfolio composition and the parameters controlling asset tail behavior. With this analysis, we have identified the key determinants of the diversification effect. 3. With the help of co-Is, we have conducted extensive empirical studies using real-world financial data on equities, government bonds, corporate debt, and mortgage-back securities. The empirical results are satisfactory up to the industrial standard at the risk management division of wall street banks. | |||||||||||||||||||||
| Summary of objectives addressed: |
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| Research Outcome | ||||||||||||||||||||||
| Major findings and research outcome: | The major findings are as follows: a) Tail heaviness dictates the absolute amount of initial margin charge when switching from VaR to ES; 2) Dependence structure dictates the relative magnitude of diversification benefits when merging multiple similar portfolios, irrespective of the risk metric; 3) Diversification does not reduce the tail event probability; it only mitigates the magnitude of damage. Research outcomes of this project are as follows: a) Journal paper: 1 paper submitted to the SIAM Journal of Financial Mathematics. It is currently under review. b) Conference talks: 6 presentations are accepted and delivered at all the major and most internationally recognized conferences in the area of Financial Engineering. These include the World Congress of the Bachelier Finance Society, the INFORMS Annual Meeting, the SIAM Conference on Financial Mathematics & Engineering, the European Conference on Operational Research, Federal Reserve Bank of New York Conference, and Asian Quantitative Finance Conference. 4 in North America, 1 in Europe and 1 in Asia. c) Industrial impact: Thanks to our co-I's collaboration, the models developed, together with their implementation and algorithms, are successfully tested and used at the risk management division of the New York-based Bank of Mellon. | |||||||||||||||||||||
| Potential for further development of the research and the proposed course of action: |
The setup in the current study assumes homogeneous tail heaviness among asset returns. Future work should consider the heterogeneous case. | |||||||||||||||||||||
| Layman's Summary of Completion Report: | In the current environment of financial regulation, traceability is essential in day-to-day risk management, especially portfolio tail risks which are essential to the fast calculation of margin/capital requirements. This research develops a novel asymptotic expansion method that makes the portfolio tail risk tractable when the asset returns follow the multivariate elliptical distribution family. The models and the methods we developed are well received by the wall street banks through our collaborator. The models and methods are designed to suit the real-world production environment. | |||||||||||||||||||||
| Research Output | ||||||||||||||||||||||
| Peer-reviewed journal publication(s) arising directly from this research project : (* denotes the corresponding author) |
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| Recognized international conference(s) in which paper(s) related to this research project was/were delivered : |
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| Other impact (e.g. award of patents or prizes, collaboration with other research institutions, technology transfer, etc.): |
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| SCREEN ID: SCRRM00542 |