Asymptotic Analysis of Portfolio Tail Risk and the Diversification Effect under Multivariate Elliptical Distributions for Static Portfolios

Description

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 bythe "Value-at-Risk" metric. It is also concerned that aggregated counterparty risk, ifnot mitigated, is a threat to the stability of financial system. On the other hand, simplyincreasing capital requirements for banks and charging heavy margins at centralcounterparties may lead to liquidity problems as well as inefficient use of capital. At thecenter of this intricate trade off is the assessment and management of tail risk. Thismotivates us to conduct a technical investigation into the mathematical behavior ofportfolio tail risk under the prevailing risk metrics. In particular, we plan to consider arich parametric family of multivariate distribution and provide accurate analyticalestimates of both portfolio value-at-risk (VaR) and portfolio expected shortfall (ES). Wework with these distributions based on the consideration that they not only exhibitdistinct 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 theseanalytical results to examine the diversification effect associated with portfolio tail risk.By looking at the risk reduction amount which measures the dollar difference betweenthe total risk of a static portfolio and the sum of that of individual positions, we intendto understand the role played by the following properties on tail risk diversification: thetail heaviness of individual asset, the pair-wise tail dependence, and the cross-assetheterogeneity.