Tail Asymptotics of Deflated Risks

Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X=RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated riskComment: 2

Extreme risk in Asian equity markets

Extreme price movements associated with tail returns are catastrophic for all investors and it is necessary to make accurate predictions of the severity of these events. Choosing a time frame associated with large financial booms and crises this paper investigates the tail behaviour of Asian equity market returns and quantifies two risk measures, quantiles and average losses, along with their associated average waiting periods. Extreme value theory using the Peaks over Threshold method generates the risk measures where tail returns are modelled with a fat-tailed Generalised Pareto Distribution. We find that lower tail risk measures are more severe than upper tail realisations at the lowest probability levels. Moreover, the Kuala Lumpar Composite exhibits the largest risk measures.

Modelling Censored Losses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme Value Distributions

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement

Comparative Analyses of Expected Shortfall and Value-at-Risk (2): Expected Utility Maximization and Tail Risk

We compare expected shortfall and value-at-risk (VaR) in terms of consistency with expected utility maximization and elimination of tail risk. We use the concept of stochastic dominance in studying these two aspects of risk measures. We conclude that expected shortfall is more applicable than VaR in those two aspects. Expected shortfall is consistent with expected utility maximization and is free of tail risk, under more lenient conditions than VaR.

Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure

Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distribution-driven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty. In this paper we propose an analytic approach to problem-driven scenario generation. This approach applies to stochastic programs where a tail risk measure, such as conditional value-at-risk, is applied to a loss function. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread their scenarios evenly across the support of the random vector, struggle to adequately represent tail risk. Our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions. We provide conditions under which our approach is consistent with sampling, and as proof-of-concept demonstrate how our approach could be applied to two classes of problem, namely network design and portfolio selection. Numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard Monte Carlo sampling

The impact of heavy tails and comovements in downside-risk diversification

This paper uncovers the factors influencing optimal asset allocation for downside-risk averse investors. These are comovements between assets, the product of marginal tail probabilities, and the tail index of the optimal portfolio. We measure these factors by using the Clayton copula to model comovements and extreme value theory to estimate shortfall probabilities. These techniques allow us to identify useless diversification strategies based on assets with different tail behaviour, and show that in case of financial distress the asset with heavier tail drives the return on the overall portfolio down. An application to financial indexes of UK and US shows that mean-variance and downside-risk averse investors construct different efficient portfolios.

Adaptive Premiums for Evolutionary Claims in Non-Life Insurance

Rapid growth in heavy-tailed claim severity in commercial liability insurance requires insurer response by way of flexible mechanisms to update premiums. To this end in this paper a new premium principle is established for heavy-tailed claims, and its properties investigated. Risk-neutral premiums for heavy-tailed claims are consistently and unbiasedly estimated by the ratio of the first two extremes of the claims distribution. That is, the heavy-tailed risk-neutral premium has a Pareto distribution with the same tail-index as the claims distribution. Insurers must predicate premiums on larger tail-index values, if solvency is to be maintained. Additionally, the structure of heavy-tailed premiums is shown to lead to a natural model for tail-index imprecision (demonstrably inescapable in the sample sizes with which we deal). Premiums which compensate for tail-index uncertainty preserve the ratio structure of risk-neutral premiums, but make a 'prudent' adjustment which reflects the insurer's risk-profile. An example using Swiss Re's (1999) major disaster data is used to illustrate application of the methodology to the largest claims in any insurance class.Insurance Claims, Premiums, Tail-Index, Extreme Values

Tail Conditional Expectation for vector-valued Risks

In his paper we introduce a quantile-based risk measure for multivariate financial positions "the vector-valued Tail-conditional-expectation (TCE)". We adopt the framework proposed by Jouini, Meddeb, and Touzi [9] to deal with multi-assets portfolios when one accounts for frictions in the financial market. In this framework, the space of risks formed by essentially bounded random vectors, is endowed with some partial vector preorder >= accounting for market frictions. In a first step we provide a definition for quantiles of vector-valued risks which is compatible with the preorder >=. The TCE is then introduced as a natural extension of the "classical" real-valued tail-conditional-expectation. Our main result states that for continuous distributions TCE is equal to a coherent vector-valued risk measure. We also provide a numerical algorithm for computing vector-valued quantiles and TCE.Risk measures, vector-valued risk measures, coherent risk-measures, quantiles, tail-conditional-expectation