Infrequent Extreme Risks

The main tools and concepts of financial and actuarial theory are designed to handle standard, or even small risks. The aim of this paper is to reconsider some selected financial problems, in a setup including infrequent extreme risks. We first consider investors maximizing the expected utility function of their future wealth, and we establish the necessary and sufficient conditions on the utility function to ensure the existence of a non degenerate demand for assets with extreme risks. This new class of utility functions, called LIRA, does not contain the classical HARA and CARA utility functions, which are not adequate in this framework. Then we discuss the corresponding asset supply-demand equilibrium model.     

On additivity of tail comonotonic risks

ABSTRACT

As perceived from daily experience together with numerous empirical studies, the multivariate risks demonstrate a strong coherence in the extremal dependence structure especially over the course of financial turmoil or industrial accidents and outbreaks. Under this motivating paradigm, we show the universal asymptotic additivity under upper tail comonotonicity, as the probability level approaching to 1, for Value-at-Risk and Conditional Tail Expectation for a portfolio of fixed number of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three max domains of attraction. Our obtained results do not require the tail equivalence assumption as needed in the existing literature. This resolves a lasting problem in quantitative risk management and covers most distributions commonly encountered in practice.     

Managing extreme risks in tranquil and volatile markets using conditional extreme value theory

Financial risk management typically deals with low-probability events in the tails of asset price distributions. To capture the behavior of these tails, one should therefore rely on models that explicitly focus on the tails. Extreme value theory (EVT)-based models do exactly that, and in this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets. We find conditional EVT models to give particularly accurate Value-at-Risk (VaR) measures, and a comparison with traditional (Generalized ARCH (GARCH)) approaches to calculate VaR demonstrates EVT as being the superior approach both for standard and more extreme VaR quantiles.     

Testing for differences in the tails of stock-market returns

In this paper, we use a database consisting of daily stock-market returns for 20 countries to test for similarities between the left and right tails of returns, as well as across countries. We estimate and test using the distribution of extreme returns over subsamples approach. Via Monte-Carlo simulations, we show that maximum-likelihood estimators are essentially unbiased, provided the size of subsamples is correctly chosen, and that the likelihood-ratio tests on parameters characterizing the behavior of extremes are correctly sized. For actual returns, we find that left and right tails behave very similarly. Across countries, we find that extremes are located at different levels and that their dispersion varies. The tail index, characterizing large extreme realizations, is found to be constant within each geographical group. We verify that the perception that left tails are heavier than right ones is not due to clustering of extremes. The failure to detect statistical significant differences is likely to be due to the relative infrequency of large extremes.     

El efecto de la volatilidad del peso mexicano en los rendimientos y riesgo de la Bolsa Mexicana de Valores

The effect of heavy tails due to rare events and different levels of asymmetry associated with high volatility clustering in the emerging financial markets requires sophisticated models for statistical modelling of such stylized facts. This article applies extreme value theory (EVT) to quantify tail risk on the daily returns of Mexican stock market under aggregation of foreign exchange rate risk from January 1971 to December 2010. This study focuses on the maximum-block method and generalized extreme value distribution (GEVD) to model the asymptotic behavior of extreme returns in US dollars. The empirical results show that EVT-Based VaR measured at high confidence levels performs better than simulation historical and delta-normal VaR models on capturing fat-tails in the returns of highly volatile stock markets. Additionally, international investors holding long positions in Mexican stock market are more prone to experience larger potential losses than investors with short positions during local currency depreciation and financial crisis periods.     

Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process

In this paper, we study the retention levels for combinations of quota-share and excess of loss reinsurance by maximizing the insurer’s adjustment coefficient, which in turn minimizes the asymptotic result of ruin probability. Assuming that the premiums are determined by the expected value principle, we consider a discrete risk model, in which a dependence structure is introduced based on Poisson MA(1) process between the claim numbers for each period. The impact of dependence parameter on the adjustment coefficient is discussed and numerical examples are provided to illustrate the results obtained in this paper.     

Are gold,USD, and Bitcoin hedge or safe haven against stock? The implication for risk management

This study investigates whether gold, USD, and Bitcoin are hedge and safe haven assets against stock and if they are useful in diversifying downside risk for international stock markets. We propose a combined GO-GARCH-EVT-copula approach to examine the hedge and safe haven properties of gold, USD, and Bitcoin. We then examine the attractiveness of these assets in reducing stock portfolio risk by using downside risk measures estimated by the proposed approach and other competing models. We also evaluate the relative performance of the proposed model in reducing downside risk with the competing models. The findings of the study indicate that the USD is the most valuable hedge and safe haven asset closely followed by gold, while Bitcoin is the least valuable. It is also observed that the proposed combined approach performs best in reducing the portfolio downside risk. The findings of this study are of significance for portfolio managers and individual investors who wish to protect the portfolio value during market turmoil.