Непараметрический метод вычисления величины условной напряженности при наличии риска

We consider the Value at Risk (VaR) of a portfolio under stressed conditions. In practice, the stressed VaR (sVaR) is commonly calculated using the data set that includes the stressed period. It tells us how much the risk amount increases if we use the stressed data set. In this paper, we consider the VaR under stress scenarios. Technically, this can be done by deriving the distribution of profit or loss conditioned on the value of risk factors. We use two methods; the one that uses the linear model and the one that uses the Hermite expansion discussed by Marumo and Wolff (2013, 2016). Numerical examples shows that the method using the Hermite expansion is capable of capturing the non-linear effects such as correlation collapse and volatility clustering, which are often observed in the markets.Рассматривается оценка рисковой стоимости в условиях напряженности. На практике напряженная величина риска обычно рассчитывается с использованием набора данных, включающего напряженный период. Это говорит о том, насколько возрастает риск, если мы используем данные в условиях напряженности. В данной работе мы рассматриваем величину риска (VaR) при напряженных сценариях. Технически это можно сделать, получив распределение прибыли или убытка, обусловленное величиной факторов риска. Мы используем два метода: один, который использует линейную модель, и другой, который использует распределение по Эрмиту, рассмотренный Марумо и Вольфом (2013, 2016). Численные примеры показывают, что метод распределения по Эрмету способен фиксировать нелинейные эффекты, такие как корреляционный коллапс и кластеризация волатильности, которые часто наблюдаются на рынках

On optimal smoothing of density estimators obtained from orthogonal polynomial expansion methods

We discuss the application of orthogonal polynomials to the estimation of probability density functions, particularly with regard to accessing features of a portfolio’s profit/loss distribution. Such expansions are given by the sum of known orthogonal polynomials multiplied by an associated weight function. However, naive applications of expansion methods are flawed. The shape of the estimator’s tail can undulate under the influence of the constituent polynomials in the expansion, and it can even exhibit regions of negative density. This paper presents techniques to remedy these flaws and improve the quality of risk estimation.We show that by targeting a smooth density that is sufficiently close to the target density, we can obtain expansion-based estimators that do not have the shortcomings of equivalent naive estimators. In particular, we apply optimization and smoothing techniques that place greater weight on the tails than on the body of the distribution. Numerical examples using both real and simulated data illustrate our approach. We further outline how our techniques can apply to a wide class of expansion methods and indicate opportunities to extend to the multivariate case, where distributions of individual component risk factors in a portfolio can be accessed for the purpose of risk management

Association between Catechol-O-Methyltrasferase Val108/158Met Genotype and Prefrontal Hemodynamic Response in Schizophrenia

BACKGROUND:"Imaging genetics" studies have shown that brain function by neuroimaging is a sensitive intermediate phenotype that bridges the gap between genes and psychiatric conditions. Although the evidence of association between functional val108/158met polymorphism of the catechol-O-methyltransferase gene (COMT) and increasing risk for developing schizophrenia from genetic association studies remains to be elucidated, one of the most topical findings from imaging genetics studies is the association between COMT genotype and prefrontal function in schizophrenia. The next important step in the translational approach is to establish a useful neuroimaging tool in clinical settings that is sensitive to COMT variation, so that the clinician could use the index to predict clinical response such as improvement in cognitive dysfunction by medication. Here, we investigated spatiotemporal characteristics of the association between prefrontal hemodynamic activation and the COMT genotype using a noninvasive neuroimaging technique, near-infrared spectroscopy (NIRS). METHODOLOGY/PRINCIPAL FINDINGS:Study participants included 45 patients with schizophrenia and 60 healthy controls matched for age and gender. Signals that are assumed to reflect regional cerebral blood volume were monitored over prefrontal regions from 52-channel NIRS and compared between two COMT genotype subgroups (Met carriers and Val/Val individuals) matched for age, gender, premorbid IQ, and task performance. The [oxy-Hb] increase in the Met carriers during the verbal fluency task was significantly greater than that in the Val/Val individuals in the frontopolar prefrontal cortex of patients with schizophrenia, although neither medication nor clinical symptoms differed significantly between the two subgroups. These differences were not found to be significant in healthy controls. CONCLUSIONS/SIGNIFICANCE:These data suggest that the prefrontal NIRS signals can noninvasively detect the impact of COMT variation in patients with schizophrenia. NIRS may be a promising candidate translational approach in psychiatric neuroimaging

Expansion methods applied to distributions and risk measurement in financial markets

Obtaining the distribution of the profit and loss (PL) of a portfolio is a key problem in market risk measurement. However, existing methods, such as those based on the Normal distribution, and historical simulation methods, which use empirical distribution of risk factors, face difficulties in dealing with at least one of the following three problems: describing the distributional properties of risk factors appropriately (description problem); deriving distributions of risk factors with time horizon longer than one day (time aggregation problem); and deriving the distribution of the PL given the distributional properties of the risk factors (risk aggregation problem). Here, we show that expansion methods can provide reasonable solutions to all three problems. Expansion methods approximate a probability density function by a sum of orthogonal polynomials multiplied by an associated weight function. One of the most important advantages of expansion methods is that they only require moments of the target distribution up to some order to obtain an approximation. Therefore they have the potential to be applied in a wide range of situations, including in attempts to solve the three problems listed above. On the other hand, it is also known that expansions lack robustness: they often exhibit unignorable negative density and their approximation quality can be extremely poor. This limits applications of expansion methods in existing studies. In this thesis, we firstly develop techniques to provide robustness, with which expansion methods result in a practical approximation quality in a wider range of examples than investigated to date. Specifically, we investigate three techniques: standardisation, use of Laguerre expansion and optimisation. Standardisation applies expansion methods to a variable which is transformed so that its first and second moments are the same as those of the weight function. Use of Laguerre expansions applies those expansions to a risk factor so that heavy tails can be captured better. Optimisation considers expansions with coefficients of polynomials optimised so that the difference between the approximation and the target distribution is minimised with respect to mean integrated squared error. We show, by numerical examples using data sets of stock index returns and log differences of implied volatility, and GARCH models, that expansions with our techniques are more robust than conventional expansion methods. As such, marginal distributions of risk factors can be approximated by expansion methods. This solves a part of the description problem: the information on the marginal distributions of risk factors can be summarised by their moments. Then we show that the dependence structure among risk factors can be summarised in terms of their cross-moments. This solves the other part of the description problem. We also use the fact that moments of risk factors can be aggregated using their moments and cross-moments, to show that expansion methods can be applied to both the time and risk aggregation problems. Furthermore, we introduce expansion methods for multivariate distributions, which can also be used to approximate conditional expectations and copula densities by rational functions

A Non-parametric Method for Calculating Conditional Stressed Value at Risk

We consider the Value at Risk (VaR) of a portfolio under stressed conditions. In practice, the stressed VaR (sVaR) is commonly calculated using the data set that includes the stressed period. It tells us how much the risk amount increases if we use the stressed data set. In this paper, we consider the VaR under stress scenarios. Technically, this can be done by deriving the distribution of profit or loss conditioned on the value of risk factors. We use two methods; the one that uses the linear model and the one that uses the Hermite expansion discussed by Marumo and Wolff (2013, 2016). Numerical examples shows that the method using the Hermite expansion is capable of capturing the non-linear effects such as correlation collapse and volatility clustering, which are often observed in the markets

A Non-parametric Method for Calculating Conditional Stressed Value at Risk

We consider the Value at Risk (VaR) of a portfolio under stressed conditions. In practice, the stressed VaR (sVaR) is commonly calculated using the data set that includes the stressed period. It tells us how much the risk amount increases if we use the stressed data set. In this paper, we consider the VaR under stress scenarios. Technically, this can be done by deriving the distribution of profit or loss conditioned on the value of risk factors. We use two methods; the one that uses the linear model and the one that uses the Hermite expansion discussed by Marumo and Wolff (2013, 2016). Numerical examples shows that the method using the Hermite expansion is capable of capturing the non-linear effects such as correlation collapse and volatility clustering, which are often observed in the markets

Expansion methods applied to asset return distributions

In this paper we attempt to apply the Laguerre expansion to asset return distributions and compare its performance with that of the Hermite expansion, which is most commonly employed in studies in finance. Numerical examples using market data show that the Laguerre expansion can perform better and can be used to calculate the value-at-risk. We also apply this method to the time aggregation problem, which is concerned with value-at-risk over a longer time horizon than one day