Multi-period Value-at-Risk Scaling Rules: Calculations and Approximations

摘要

The thesis firstly introduces a commonly used risk measure in the financial market---Value-at-Risk (VaR) and then the research about multi-period risk management is proposed. A general tool for multi-period Value-at-Risk (VaR), proposed by the Basel Committee on Banking Supervision (1996), is the square-root-of-time rule (SQRT rule), which is derived based on the Gaussian distributional assumption. Owing to the theoretical limitations of Gaussian and the lessons from the financial catastrophe, this thesis develops new scaling rules based on the distributions that belong to the so-called convolution equivalent class and the semi-heavy tailed distribution class in which the tails of distribution seem adequate and comply with the empirical tail property of real financial data. In this thesis, under some regularity conditions, a result about multi-period VaR scaling approach based on convolution equivalence assumption (CE rule) is derived and proved, which may provide a conservative risk value to regulators. Furthermore, this thesis provides a precise numerical multi-period VaR scaling approach based on the semi-heavy tail assumption (SH rule), which is a numerical method that can be considered as an alternative to the SQRT rule and an internal scaling model for risk managers. Based on the assurnption of a specific the semi-heavy form in the tail, we devise a semi-parametric estimation for single-period VaR calculation. The steps for using the two rules (Denoted by the SP-CE rule and the SP-SH rule) are summarized. For the whole parametric distributional assumption such as the example of the Normal Inverse Gaussian (NIG) distribution, we give specific scaling rules: the NIG-CE rule and the NIG-SH rule. The thesis also derives the saddlepoint approximation to the NIG model's multi-period VaR for internal risk management. Simulations and real data analysis evaluated and verified the feasibility of the CE rule, the SH rule and the approximated VaR method. It is found that, unlike the SQRT rule, the newly derived scaling rule has the advantage that captures the long horizon risk in a feasible way that can help regulators and risk managers. Specifically, the CE rule proposed under convolution equivalence assumption is highly recommended to regulators. About the internal supervision, the semiparametric estimation of the single-period VaR combined with the semi-heavy rule (denoted by SP-SH) would be the preferred choice. In the parametric modeling, the NIG fitting combined with the semi-heavy rule (denoted by NIG-SH) is reasonable. The saddlepoint approximation provides a fast and accurate VaR when the assumption is close to the true one.