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This paper introduces the minCluster portfolio, which is a portfolio optimization method combining the optimization of downside risk measures, hierarchical clustering and cellwise robustness. Using cellwise robust association measures, the minCluster portfolio is able to retrieve the underlying hierarchical structure in the data. Furthermore, it provides downside protection by using tail risk measures for portfolio optimization. We show through simulation studies and a real data example that the minCluster portfolio produces better out-of-sample results than meanvariances or other hierarchical clustering based approaches. Cellwise outlier robustness makes the minCluster method particularly suitable for stable optimization of portfolios in highly volatile markets, such as portfolios containing cryptocurrencies. [ABSTRACT FROM AUTHOR]

This paper introduces the minCluster portfolio, which is a portfolio optimization method combining the optimization of downside risk measures, hierarchical clustering and cellwise robustness. Using cellwise robust association measures, the minCluster portfolio is able to retrieve the underlying hierarchical structure in the data. Furthermore, it provides downside protection by using tail risk measures for portfolio optimization. We show through simulation studies and a real data example that the minCluster portfolio produces better out-of-sample results than mean-variances or other hierarchical clustering based approaches. Cellwise outlier robustness makes the minCluster method particularly suitable for stable optimization of portfolios in highly volatile markets, such as portfolios containing cryptocurrencies.

Risk contributions of portfolios form an indispensable part of risk-adjusted performance measurement. The risk contribution of a portfolio, e.g. in the Euler or Aumann-Shapley framework, is given by the partial derivatives of a risk measure applied to the portfolio profit and loss in the direction of the asset units. For risk measures that are not positively homogeneous of degree 1, however, known capital allocation principles do not apply. We study the class of lambda quantile risk measures that includes the well-known Value-at-Risk as a special case but for which no known allocation rule is applicable. We prove differentiability and derive explicit formulae of the derivatives of lambda quantiles with respect to their portfolio composition, that is, their risk contribution. For this purpose, we define lambda quantiles on the space of portfolio compositions and consider generic (also non-linear) portfolio operators. We further derive the Euler decomposition of lambda quantiles for generic portfolios and show that lambda quantiles are homogeneous in the space of portfolio compositions, with a homogeneity degree that depends on the portfolio composition and the lambda function. This result is in stark contrast to the positive homogeneity properties of risk measures defined in the space of random variables, which admit a constant homogeneity degree. We introduce a generalised version of Euler contributions and Euler allocation rule, which are compatible with risk measures of any homogeneity degree and non-linear but homogeneous portfolios. These concepts are illustrated by a non-linear portfolio using financial market data. [ABSTRACT FROM AUTHOR]

We present a framework focused on the interdependence of high‐dimensional tail events. This framework allows us to analyse and quantify tail interdependence at different levels of extremity, decompose it into systemic and residual part and to measure the contribution of a constituent to the interdependence of a system. In particular, tail interdependence can capture simultaneous distress of the constituents of a (financial or economic) system and measure its systemic risk. We investigate systemic distress in several financial datasets confirming some known stylized facts and discovering some new findings. Further, we devise statistical tests of interdependence in the tails and outline some additional extensions. [ABSTRACT FROM AUTHOR]

Portfolio decisions are affected by the volatility of financial markets and investors’ risk tolerance levels. To better allocate portfolios; we introduce risk tolerance into the portfolio management problem by considering the risk contribution of portfolio components. In this paper, portfolio weights are allocated to two stages. In the first stage, the portfolio risks and the risk contribution of each share are forecasted. In the second stage, we put forward three weighting techniques—“aggressive”, “moderate” and “conservative”, according to three standard levels of risk tolerance. In addition, a new risk measure called “joint extreme risk probability” (JERP), with risk tolerance taken into account, is proposed. A case study of the Chinese financial industry is conducted to verify the performance of our methods. The empirical results demonstrate that weighting techniques constrained by risk tolerance lead to higher gains in a normal market and less loss when a market is risky. Compared with risk-tolerance-adjusted strategies, the relationship between the performance of the traditional conditional value at risk (CVaR) minimization method and the market risk level is less obviously demonstrated. Viewed from the results, JERP functions as an effective signal that helps investors to deal with potential market risks.

To quantify the annual probability of earthquake ground motion (GM) exceeding a given threshold, the extensively used method named by probabilistic seismic hazard analysis (PSHA) can be adopted. The PSHA software made this method more effortless for estimating earthquake hazards for a seismic site. The main motivation of the PSHRisk-tool is to evaluate the PSHA by a user-friendly graphical interface as well as identify the intensities of GM, which will contribute to the most vulnerable condition for the infrastructure. This python-code based tool can demonstrate the source identification, probability distribution plot of magnitude and distance, formulate the hazard curve according to almost all ground motion prediction equations (GMPEs). The deaggregation for each intensity measure (IM) and the effect of seismic parameters in each GMPE can also be determined. Alongside this, the combination of the failure frequency and the hazard analysis for identifying risk assessment separates this tool from the other existing PSHA software. Accurate verification with analytical and existing test models and a case study inspires its acceptance rate. However, with the quickest and easiest way users can determine the seismic hazard analysis for any location. Failure risk analysis can be evaluated simply based on the structural failure parameters. [ABSTRACT FROM AUTHOR]

Diversification is one of the major components of investment decision-making under risk or uncertainty. However, paradoxically, as the 2007–2009 financial crisis revealed, the concept remains misunderstood. Our goal in writing this paper is to correct this issue by reviewing the concept in portfolio theory. The core of our review focuses on the following diversification principles: law of large numbers, correlation, capital asset pricing model and risk contribution or risk parity diversification principles. These four diversification principles are the DNA of the existing portfolio selection rules and asset pricing theories and are instrumental to the understanding of diversification in portfolio theory. We review their definition. We also review their optimality, with respect to expected utility theory, and their usefulness. Finally, we explore their measurement. [ABSTRACT FROM AUTHOR]

This paper proposes a method of estimating Value-at-Risk by combining asym- metric multivariate GARCH models and filtered historical simulation (Barone-Adesi et al., 1999). Next, incremental VaR is implemented in order to decompose the portfolio and as- sess the risk of every individual component. Ten competitive models were estimated and subsequently back tested using five techniques. All methodologies were applied on a sample of 11 financial assets from Bucharest Stock Exchange between 2014-07-08 and 2019-10-04. The results indicate that the method using filtered historical simulation in combination with multivariate GARCH models that account for asymmetry of financial returns lead to good VaR estimates. The methods discussed in this paper could help an investor to create a better risk-optimized portfolio, but could also be used by a regulatory authority in order to impose restrictions regarding risk. [ABSTRACT FROM AUTHOR]

This paper tests whether the financial interpretation of risk contributions (Qian, 2006), as measured by marginal change in volatility, holds when accounting for fat tails in the asset return distributions. This important result is the theoretical foundation of risk-based portfolios, but relies on the assumption of normality. If the result does not hold, more sophisticated techniques are required to estimate risk-based portfolios. A simulation study is conducted to replicate the stressed environment required by Qian (2006). The Quantile Simulation method (Alexander, 2013) is used to simulate asset return distributions that are reasonable replicates of the empirical samples. Given the relative novelty of the simulation method, this paper also reports the extent to which the simulated samples can approximate the empirical sample of each asset. [ABSTRACT FROM AUTHOR]

To quantify the annual probability of earthquake ground motion (GM) exceeding a given threshold, the extensively used method named by probabilistic seismic hazard analysis (PSHA) can be adopted. The PSHA software made this method more effortless for estimating earthquake hazards for a seismic site. The main motivation of the PSHRisk-tool is to evaluate the PSHA by a user-friendly graphical interface as well as identify the intensities of GM, which will contribute to the most vulnerable condition for the infrastructure. This python-code based tool can demonstrate the source identification, probability distribution plot of magnitude and distance, formulate the hazard curve according to almost all ground motion prediction equations (GMPEs). The deaggregation for each intensity measure (IM) and the effect of seismic parameters in each GMPE can also be determined. Alongside this, the combination of the failure frequency and the hazard analysis for identifying risk assessment separates this tool from the other existing PSHA software. Accurate verification with analytical and existing test models and a case study inspires its acceptance rate. However, with the quickest and easiest way users can determine the seismic hazard analysis for any location. Failure risk analysis can be evaluated simply based on the structural failure parameters.

Given a decomposition of a portfolio P as a sum of K components, practitioners commonly decompose the risk of P as a corresponding sum of risk contributions. In this paper, we prove two theorems about risk contributions. The first theorem concerns a form of duality identified in Grinold [J. Portfolio Manage. 2011, 37(2), 15-30], which may be described as follows. When we view a portfolio decomposition as a coordinate representation of the portfolio with respect to a given vector-space basis, then there is a natural dual basis with respect to which there is an alternative decomposition, here referred to as the dual decomposition. The dual decomposition gives the same contributions to risk as the original decomposition. The first theorem gives necessary and sufficient conditions for a change of basis to preserve risk contributions, and shows that all such changes of basis can be explained in terms of dual decompositions. The second theorem explores sensitivity of portfolio risk to a risk regime change and indicates that large risk contributions or large risks of the components of a decomposition may be harbingers of high sensitivity. This provides a motivation for the practice of reporting both the risk contributions and the risks of the components in a decomposition. [ABSTRACT FROM AUTHOR]

China sped up its progress toward the opening of its stock market in the post-crisis period after 2010. This study aims to investigate the risk contribution of the Chinese stock market to four representative developed markets. The significance and dominance of the risk contribution are tested with the extended Kolmogorov-Smirnov statistic by a bootstrap strategy. The results show a significant risk contribution of China to all the four developed countries. The dominance testing result shows clear regional effect in the risk contribution. The determinants of the risk contribution by macroeconomic variables are also identified in a forward-looking way. [ABSTRACT FROM AUTHOR]

In its October 2013 consultative paper for a revised market risk framework (Fundamental Review of the Trading Book, FRTB) and subsequent versions published thereafter, the Basel committee suggested new ways of dealing with market risk in banks' trading and banking books. The Basel committee estimates that the new rules will result in an approximate median capital increase of 22 per cent and a weighted average capital increase of 40 per cent, compared with the current framework. Key changes are found in the internal model approach (IMA), the standard rules and the scope/approval process. Among the significant changes that are being introduced by the FRTB is a stricter separation of the trading book and banking book. Regardless of whether they use standardised or internal models, banks will need to review their portfolios to determine if existing classifications of instruments and desks as trading book or banking book are still applicable or whether a revision of desk structure is needed. In this article, the theoretical foundations of the IMA are explained, which are the stressed expected shortfall, liquidity adjustments, default and migration risk, and non-modellable risk factors. The various approaches for IMA and the introduction of a standardised floor, the sensitivity based approach (SBA) with delta, vega and curvature, shock scenarios and the aggregation with asymmetric correlation and reflection of basis/default risk are elaborated. [ABSTRACT FROM AUTHOR]

This article unravels the fundamentally different roles of correlation when building risk-based portfolios by means of either risk control or risk contribution. We focus on the case of a portfolio manager who aligns the riskiness of the portfolio with the risk profile of the investor through a varying combination of equity and bonds. Risk control is shown to reduce exposure to equity in case of poor asset class diversification, while risk contribution does the opposite. Portfolio managers who consider building their balanced portfolios on either of these risk-based techniques will find this insight valuable. [ABSTRACT FROM AUTHOR]

This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for [ABSTRACT FROM AUTHOR]

Current cost allocation methods require generating companies (GENCOs) to afford spinning reserve (SR) costs according to their energy production rather than the impact on grid stabilization. The differences in generator reliability and forecast accuracy of renewables cause difficulty in quantifying the contribution of individual factors on the SR requirements (SRRs). First, this paper employs a reliability-constrained unit commitment (RCUC) model to determine the SRR and SR costs according to the grid reliability. Then, a cost allocation method is proposed to allocate these SR costs based on risk contribution theories. The risk contribution theories, marginal contribution, and stand-alone contribution are employed to measure the effect of individual risk factors on grid safety. The cost allocation method is demonstrated and discussed in the IEEE-RTS. The proposed risk contribution method can quantify the impacts of risk factors on grid safety and allocate SR costs into them according to their contributions. Additionally, this risk-based cost allocation method can encourage GENCOs to enhance the reliability level of generators and continuously improve the forecast accuracy of renewables to lower SR costs. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

This article reviews the literature on techniques of credit risk models, multi-period risk measurement, and capital allocation, and gives a tutorial on applying these techniques to credit portfolios with a focus on practical aspects. The effects of the choice of considered loss process concerning the handling of write-offs and matured assets or rating migration are displayed, and the impact on portfolio optimization decisions is discussed. We highlight the trade-off between short-term and long-term profitability and allude to the practical challenges of an application of multi-period risk measurement. [ABSTRACT FROM AUTHOR]

In order to protect stakeholders of insurance companies and financial institutions against adverse outcomes of risky businesses, regulators and senior management use capital allocation techniques. For enterprise-wide risk management, it has become important to calculate the contribution of each risk within a portfolio. For that purpose, bivariate lower and upper orthant tail value-at-risk can be used for capital allocation. In this paper, we present multivariate value-at-risk and tail-value-at-risk for d ≥ 2 , and we focus on three different methods to calculate optimal values for the contribution of each risk within the sums of random vectors to the overall portfolio, which could particularly apply to insurance and financial portfolios.

Risk capital allocation is based on the assumption that the risk of a homogeneous portfolio is scaled up and down with the portfolio size. In this article we show that this assumption is true for large portfolios, but has to be revised for small ones. On basis of numerical examples we calculate the minimum portfolio size that is necessary to limit the error of gradient risk capital allocation and the resulting error in a portfolio optimization algorithm or pricing strategy. We show the dependency of this minimum portfolio size on different parameters like the probability of default and on the credit risk model that is used. [ABSTRACT FROM AUTHOR]

According to the principle of coherent capital allocation and the Shapley value of game theory, this paper makes an essential improvement to Denault's method. On this basis, we develop a suitable method to allocate economic capital for commercial banks. By introducing the Yasumitsu condition, we find a sufficient and necessary condition to judge whether the Shapley value satisfies the coherence principle. With this method which is consistent with the requirements of the bank's overall best, the commercial banks can stand the perspective of general optimality to calculate risk contribution of its sub branches and determine their annual economic capital limits. [ABSTRACT FROM AUTHOR]

We consider an enhancement of the credit risk+ model to incorporate correlations between sectors. We model the sector default rates as linear combinations of a common set of independent variables that represent macro-economic variables or risk factors. We also derive the formula for exact VaR contributions at the obligor level. [ABSTRACT FROM AUTHOR]

We define the capital allocation and the risk contribution for discrete-time coherent risk measures and provide several equivalent representations of these objects. The formulations and the proofs are based on two instruments introduced in the paper: a probabilistic notion of the extreme system and a geometric notion of the generator. These notions are also of interest on their own and are important for other applications of coherent risk measures. All the concepts and results are illustrated by JP Morgan's Risk Metrics model. [ABSTRACT FROM AUTHOR]

This paper deals with multidimensional coherent and convex risk measures. The approach described takes into account risks of changing currency exchange rates and transaction costs. Representation theorems for multidimensional risk measures are proved. The important examples of multidimensional coherent risk measures such as tail V@R and weighted V@R are investigated. Two applications of multidimensional coherent risk measures are considered, i.e., application to the capital allocation problem and to the problem of risk contribution. [ABSTRACT FROM AUTHOR]

This is the first of two papers dealing with applications of coherent risk measures to basic problems of financial mathematics. In this paper, we study applications to pricing in incomplete markets. We prove the fundamental theorem of asset pricing for the no good deals (NGD) pricing technique based on coherent risks. The model considered includes static and dynamic models as well as models with infinitely many assets, and models with transaction costs. In particular, we prove that in a dynamic model with proportional transaction costs the fair price interval converges to the fair price interval in the frictionless model as the coefficient of transaction costs tends to zero. Moreover, we study some problems in the "pure" theory of risk measures. In particular, we introduce the notion of a generator that opens the way for geometric constructions. Based on this notion, we give a simple geometric solution of the capital allocation problem. [ABSTRACT FROM AUTHOR]