Search

We conduct a decision-theoretic analysis of optimal portfolio choices and, in particular, their comparative statics under two types of entropic risk measures, the coherent entropic risk measure (CERM) and the convex entropic risk measure (ERM). Starting with the portfolio selection between a risky and a risk free asset (framework of Arrow (1965) and Pratt (1964)), we find a restrictive all-or-nothing investment decision under the CERM, while the ERM yields diversification. We then address a portfolio problem with two risky assets, and provide comparative statics with respect to the investor’s risk aversion (framework of Ross (1981)). Here, both the CERM and the ERM exhibit closely interrelated inconsistencies with respect to the interpretation of their risk parameters as a measure of risk aversion: for any two investors with different risk parameters, it may happen that the investor with the higher risk parameter invests more in the riskier one of the two assets. Finally, we analyze the portfolio problem “risky vs. risk free” in the presence of an independent background risk, and analyze the effect of changes in this background risk (framework of Gollier and Pratt (1996)). Again, we find questionable predictions: under the CERM, the optimal risky investment is always increasing instead of decreasing when a background risk is introduced, while under the ERM it remains unaffected.

Since the financial crisis of 2007–2009, there has been a renewed interest in quantifying more appropriately the risks involved in financial positions. Popular risk measures such as variance and value-at-risk have been found inadequate because we now give more importance to properties such as monotonicity, convexity, translation invariance, positive homogeneity, and law invariance. Unfortunately, the challenge remains that it is unclear how to choose a risk measure that faithfully represents a decision maker’s true risk attitude. In this work, we show that one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions. We assume that the decision maker can commit to a subset of the above properties (the use of a law invariant convex risk measure for example) and that he can provide a series of assessments comparing pairs of potential risky payoffs. Given this information, we propose to seek financial positions that perform best with respect to the most pessimistic estimation of the level of risk potentially perceived by the decision maker. We present how this preference robust risk minimization problem can be solved numerically by formulating convex optimization problems of reasonable size. Numerical experiments on a portfolio selection problem, where the problem reduces to a linear program, will illustrate the advantages of accounting for the fact that the choice of a risk measure is ambiguous. This paper was accepted by Yinyu Ye, optimization.

To estimate short-term, medium-term, and long-term financial connectedness, we propose a frequency-based approach and measure the contribution of individual financial institutions to overall systemic risk. We derive Wavelet Conditional Value at Risk (WCoVaR) – a robust market-based measure of systemic risk across financial cycles of differing length. We evaluate the systemic importance of financial institutions based on their stock returns and use wavelet framework to analyze returns in a time-frequency domain. Empirical analysis on US banking sector data between 2004 and 2013 demonstrates that wavelet decomposition can improve the forecast power of the CoVaR measure. We use panel regression to explain systemic importance of individual banks, using their objectively measurable characteristics and conclude that size, volatility and value-at-risk are the most robust determinants of systemic risk.

The last global financial crisis (2007–2008) has highlighted the weaknesses of value at risk (VaR) as a measure of market risk, as this metric by itself does not take liquidity risk into account. To address this problem, the academic literature has proposed incorporating liquidity risk into estimations of market risk by adding the VaR of the spread to the risk price. The parametric model is the standard approach used to estimate liquidity risk. As this approach does not generate reliable VaR estimates, we propose estimating liquidity risk using more sophisticated models based on extreme value theory (EVT). We find that the approach based on conditional extreme value theory outperforms the standard approach in terms of accurate VaR estimates and the market risk capital requirements of the Basel Capital Accord.

Considering the implementability and the properties that a reasonable and realistic risk measure should satisfy, we propose a new class of risk measures based on generalized lower deviation with respect to a chosen benchmark. Besides convexity and monotonicity, our new risk measure can reflect the investor's degree of risk aversion as well as the fat-tail phenomenon of the loss distribution with the help of different benchmarks and weighted functions. Based on the new risk measure, we establish a realistic portfolio selection model taking market frictions into account. To examine the influence of the benchmarks and weighted functions on the optimal portfolio and its performance, we carry out a series of empirical studies in Chinese stock markets. Our in-sample and out-of-sample results show that the new risk measure and the corresponding portfolio selection model can reflect the investor's risk averse attitude and the impact of different trading constraints. Most importantly, with the new risk measure we can obtain an optimal portfolio which is more robust and superior to the optimal portfolios obtained with the traditional expected shortfall risk measures.

Dynamic risk management requires the risk measures to adapt to information at different times, such that this dynamic framework takes into account the time consistency of risk measures interrelated at different times. Therefore, dynamic risk measures for processes can be identified as risk measures for random variables on an appropriate product space. This paper proposes a wavelet feature decomposing algorithm based on the discrete wavelet transform that optimally decomposes the time-consistent features from the product space. This approach allows us to generalize the multiple-stage risk measures of value at risk and conditional value at risk for the feature-decomposed processes, and implement them into portfolio selection using high-frequency data of U.S. DJIA stocks. The overall empirical results confirm that our proposed method significantly improves the performance of dynamic risk assessment and portfolio selection.

We develop a new risk measure using transition density for the problem of risk assessment when investors implement the stop strategy. Since the introduction of the stop strategy, the exit time when...

In this paper we study the tail risk of a properly diversified credit portfolio under a latent risk factor model. The usual perception is that if the diversification leads to asymptotic independence among the risk factors, then, because of the relatively low probability of simultaneous defaults, the tail risk of the entire portfolio is negligible. However, we point out that in fact there may be substantial tail risk hidden in this situation. We use a conditional tail probability of the portfolio loss to quantify the hidden tail risk, and then provide an asymptotic characterization for the risk under a hidden regular variation structure assumed for the risk factors. We also propose applications of the characterization to the determination and allocation of related insurance risk capital, based on the Conditional Tail Expectation risk measure. To understand the impact of dependence on the quantities of interest, we study two special cases where the risk factors have a Gaussian copula or an Archimedean copula. Numerical examples are provided to illustrate the results.

In this paper, we define the conditional risk measure under regime switching and derive a class of time consistent multi-period risk measures. To do so, we describe the information process with regime switching in a product space associated with the product of two filtrations. Moreover, we show how to establish the corresponding multi-stage portfolio selection models using the time consistent multi-period risk measure for medium-term or long-term investments. Take the conditional value-at-risk measure as an example, we demonstrate the resulting multi-stage portfolio selection problem can be transformed into a second-order cone programming problem. Finally, we carry out a series of empirical tests to illustrate the superior performance of the proposed random framework and the corresponding multi-stage portfolio selection model.

The present study seeks to examine the mutual fund performance of the open-ended selected equity schemes of UTI based on multi-index measures as well as conditional multi-index measure. It is observed from the analysis that multi-index measure is able to capture the beta and alpha effects on market adjusted basis and the estimated coefficients is a better representative as compared to the single index measure. When time lagged (lagged at 1 month, 2 months, quarterly and yearly) multi-index measures are applied then the estimated coefficients (alpha & beta) which are market adjusted and time adjusted look more representative than the multi-index measure (without lagged effect). Finally, when we extended the time lagged multi-index measure on a conditional way (conditional on public information variables) then we observe that conditional multi-index lagged measure provides much more representative results in all respects as compared to the all measures after conditioning public information effects.

In this paper, we address a basic production planning problem with price dependent demand and stochastic yield of production. We use price and target quantity as decision variables to control the risk of low production yield. The value of risk control becomes more important especially for products with short life cycle where high losses are unbearable in the short run. In this cases, optimization of a solely scalar function of profit is not sufficient to control the risk. We apply Conditional Value at Risk (CVaR) measure to model the risk preferences of the producer. The producer is interested in shaping the risk by bounding from below the means of α -tail distributions of profit for different values of α . The resulting model is nonconvex. We propose an efficient solution algorithm and present a sufficient optimality condition.

Investing in the economic world, characterized by a high level of uncertainty and volatility, entails a higher level of risk related to investment. One of the most commonly used risk measure is Value-at-Risk. However, despite the ease of calculation and interpretation, this measure suffers from a significant drawback - it is not subadditive. This property is the key issue in terms of portfolio diversification. Another risk measure, which meets this assumption, has been proposed - Conditional Value-at-Risk, defined as a conditional loss beyond Value-at-Risk. However, the choice of a risk measure is an individual decision of an investor and it is directly related to his attitudes to risk. In this paper the new risk measure is proposed - the GlueVaR risk measure, which can be defined as a linear combination of VaR and GlueVaR. It allows for calculating the level of investment loss depending on investment’s attitudes to risk. Moreover, GlueVaR meets the subadditivity property, therefore it may be used in portfolio risk assessment. The application of the GlueVaR risk measure is presented for the non-ferrous metals market.

In this paper we propose a quantile-based risk measure which is defined using the modified loss distribution according to the decision maker’s risk and loss aversion. The properties related to different classes of disutility functions are established. A portfolio selection model in the Mean-Risk framework is proposed and equivalent formulations of the model generating the same efficient frontier are given. The advantages of this approach are investigated using real world data from NYSE. The differences between the efficient frontier of the proposed model and the classical Mean-Variance and Mean-CVaR are quantified and interpreted. Extensive experiments show that the efficient portfolios obtained by using the proposed model exhibit lower risk levels and an increased satisfaction compared to the other two Mean-Risk models.

We identify new conditions ensuring risk aversion in the sense of Arrow–Pratt in a two-argument utility framework in which a financial risk is accompanied by a background risk. Our results generalize the findings of Finkelshtain et al. (1999). We consider a sequence of possible dependence among risks. We also provide an empirical example showing that second-order expectation dependence cannot be ignored in determining risk aversion with two risks.

none 2 no Published online: 06 Oct 2015 In this article we study a risk-minimizing hedge ratio with futures contracts, where the risk of the hedged portfolio is measured through a spectral risk measure (SRM), thus incorporating the degree of agent’s risk aversion. We empirically estimate the optimal hedge ratio (OHR) using a long time series of UK and US equity indices, the EURUSD and EURGBP exchange rates and four liquid commodities (Brent crude oil, corn, gold and copper), to represent different asset classes. Comparing the results with common OHRs (such as the minimum variance and the minimum expected shortfall), we find that the agent’s risk aversion has a material impact, and should not be ignored in risk management. mixed Barbi, Massimiliano; Romagnoli, Silvia Barbi, Massimiliano; Romagnoli, Silvia

The concept of risk is essential to many problems in economics and business. Usually, risk is treated in the traditional expected utility framework where it is defined only indirectly through the shape of the utility function. The purpose of utility functions, however, is to model preferences. In this paper, we review those approaches which directly model risk judgements. After a short review of naive risk measures used in earlier economic literature, we present recent theoretical and empirical developments.

Arguments on the existence of dynamic arbitrage and price manipulation strategies are often invoked to guide modeling price impacts of large trades. We revisit the concept of dynamic arbitrage in illiquid markets in the presence of time-varying stochastic price impact functions and a broad class of market price dynamics. We first establish a sufficient condition under which searching in the space of F0-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage certificate. This result simplifies identifying price impact structures that rule out dynamic arbitrage and accredits the analysis in some existing literature, where its assessment is limited to the search in the set of F0-measurable round-trip trades. For time-varying stochastic linear price impact functions, we show that this condition is necessary and sufficient for the absence of dynamic arbitrage. The present quantitative analysis implies that a trader’s opinion on the existence of dynamic arbitrage opportunities for a price impact model depends on his belief about expected future price changes and expected future price impacts, which can be revised over time by the collection of new information. This motivates us to let the existence of such arbitrage opportunities not only depend on the investor’s belief on expected price movements but also depend on the trader’s risk attitude. We thus introduce the concept of risk-averse dynamic arbitrage using a general time-consistent dynamic risk measure and a risk-aversion threshold level. Similar sufficient conditions are studied under which searching in the space of static round-trip trades enables us to conclude on no-risk-averse dynamic arbitrage.

Utilizing a specific acceptance set, we propose in this paper a general method to construct coherent risk measures called the generalized shortfall risk measure. Besides some existing coherent risk measures, several new types of coherent risk measures can be generated. We investigate the generalized shortfall risk measure’s desirable properties such as consistency with second-order stochastic dominance. By combining the performance evaluation with the risk control, we study in particular the performance ratio-based coherent risk (PRCR) measures, which is a sub-class of generalized shortfall risk measures. The PRCR measures are tractable and have a suitable financial interpretation. Based on the PRCR measure, we establish a portfolio selection model with transaction costs. Empirical results show that the optimal portfolio obtained under the PRCR measure performs much better than the corresponding optimal portfolio obtained under the higher moment coherent risk measure.

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein–Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price.

Recently Haezendonck-Goovaerts risk measure is receiving much attention in actuarial science with applications in the study of optimal portfolio and optimal reinsurance policy. Nonparametric estimation is proposed by Ahn and Shyamalkumar (Insur Math Econ 55:78–90, 2014), where the derived asymptotic limit can be employed to construct an interval for the Haezendonck-Goovaerts risk measure. In this paper, we propose an alternative empirical likelihood inference for this risk measure. A simulation study shows the good performance of the proposed method.

Model risk has a huge impact on any risk measurement procedure and its quantification is therefore a crucial step. In this paper, we introduce three quantitative measures of model risk when choosing a particular reference model within a given class: the absolute measure of model risk, the relative measure of model risk and the local measure of model risk. Each of the measures has a specific purpose and so allows for flexibility. We illustrate the various notions by studying some relevant examples, so as to emphasize the practicability and tractability of our approach., Comment: 23 pages, 6 figures

The 2007–2009 financial turmoil highlighted the need for more active management of credit portfolios. After measuring portfolio credit risk, an important step toward active risk management is to measure risk contributions of individual obligors to the overall risk of the portfolio. In practice, value-at-risk is often used as a risk measure for credit portfolios, and it can be decomposed into a sum of the risk contributions of individual obligors. Estimation of these risk contributions is computationally challenging, mainly because they are expectations conditioned on a rare event. In this paper, we tackle this computational problem by developing a restricted importance sampling (RIS) method for a class of conditional-independence credit risk models, where defaults of obligors are conditionally independent given an appropriately chosen random vector. We propose fast estimators for risk contributions and their confidence intervals. Furthermore, we study the incorporation of traditional importance sampling methods into the RIS method to further improve its efficiency for the widely used Gaussian copula model. Numerical examples show that the proposed method works well.

This paper proposes a decomposition of the cost of risk (as measured by a risk premium) across intervals/quantiles of the payoff distribution. The analysis is based on general smooth risk preferences. While this includes the expected utility model as a special case, the investigation is done under a broad class of non-expected utility models. We decompose the risk premium into additive components across quantiles. Defining downside risk as the risk associated with a lower quantile, this provides a basis to evaluate the cost of exposure to downside risk. We derive a local measure of the cost of risk associated with each quantile. It establishes linkages between the cost of risk, risk preferences and the distribution of risky prospects across quantiles (as measured by quantile variance and skewness). The analysis gives new and useful information on how risk aversion, exposure to downside risk and departures from the expected utility model interact as they affect the risk premium.

This paper introduces an investor-specific risk measure derived from the linear-exponential (linex)utility function. It combines the notions of risk perception and risk aversion. To make thismeasure interpretable and comparable with others like variance or value-at-risk, it is translatedinto an Equivalent Risky Allocation (ERA), where the risk value is matched with the one of a selectedbenchmark. We demonstrate that portfolio allocations are sensitive to risk perception. Thelinex risk measure provides more stable allocations and is closer to the target risk profile than thevariance, while it provides better consistency of risk exposures over time than the value-at-risk.Data source: none

In oil production optimization, we usually aim to maximize a deterministic scalar performance index such as the profit over the expected reservoir lifespan. However, when uncertainty in the parameters is considered, the profit results in a random variable that can assume a range of values depending on the value of the uncertain parameters. In this case, a problem reformulation is needed to properly define the optimization problem. In this paper we describe the concept of risk and we explore how to handle the risk by using appropriate risk measures. We provide a review on various risk measures reporting pro and cons for each of them. Finally, among the presented risk measures, we identify two of them as appropriate risk measures when minimizing the risk.

Purpose – The purpose of this paper is to measure a single financial institution's contribution to systemic risk by using extremal quantile regression and analyzing the influential factors of systemic risk. Design/methodology/approach – Extreme value theory is applied when measuring the systemic risk of financial institutions. Extremal quantile regression, where extreme value distribution is assumed for the tail, is used to measure the extreme risk and analyze the changes in and dependencies of risk. Furthermore, influential factors of systemic risk are analyzed using panel regression. Findings – The key findings of the paper are that value at risk and contribution to systemic risk are very different when measuring the risk of a financial institution; banks’ contributions to systemic risk are much higher; and size and leverage ratio are two significant and important factors influencing an institution's systemic risk. Practical implications – Characterizing variables of financial institutions such as size, leverage ratio and market beta should be considered together when regulating and constraining financial institutions. Originality/value – To take extreme risk into account, this paper measures systemic financial risk using extremal quantile regression for the first time.

We study the problem of optimal reinsurance as a means of risk management in the regulatory framework of Solvency II under Conditional Value-at-Risk and, as its natural extension, spectral risk measures. First, we show that stop-loss reinsurance is optimal under both Conditional Value-at-Risk and spectral risk measures. Spectral risk measures thus constitute a more general class of suitable regulatory risk measures than specific Conditional Value-at-Risk. At the same time, the established type of stop-loss reinsurance can be maintained as the optimal risk management strategy that minimizes regulatory capital. Second, we derive the optimal deductibles for stop-loss reinsurance. We show that under Conditional Value-at-Risk, the optimal deductible tends towards restrictive and counter-intuitive corner solutions or “plunging”, which is a serious objection against its use in regulatory risk management. By means of the broader class of spectral risk measures, we are able to overcome this shortcoming as optimal deductibles are now interior solutions. Especially, the recently discussed power spectral risk measures and the Wang risk measure are shown to avoid any plunging. They yield a one-to-one correspondence between the risk parameter and the optimal deductible and, thus, provide economically plausible risk management strategies.

The first portfolio risk diversification strategy was put into practice by the All Weather fund in 1996. The idea of risk diversification is related to the risk contribution of each available asset class or investment factor to the total portfolio risk. The maximum diversification or the risk parity allocation is achieved when the set of risk contributions is given by a uniform distribution. Meucci (2009) introduced the maximization of the Renyi entropy as part of a leverage constrained optimization problem to achieve such diversified risk contributions when dealing with uncorrelated investment factors. A generalization of the risk parity is the risk budgeting when there is a prior for the distribution of the risk contributions. Our contribution is the generalization of the existent optimization frameworks to be able to solve the risk budgeting problem. In addition, our framework does not possess any leverage constraint.

We address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the risk sharing problem is solved through explicit construction. Three relevant issues on optimal allocation are investigated: extra sources of randomness, comonotonicty, and model uncertainty. We show that, in general, a robust optimal allocation exists if and only if none of the underlying risk measures is a VaR. Practical implications of our main results for risk management and policy makers are discussed. In particular, in the context of the calculation of regulatory capital, we provide some general guidelines on how a regulatory risk measure can lead to certain desirable or undesirable properties of risk sharing among firms. Several novel advantages of ES over VaR from the perspective of a regulator are thereby revealed.

As we know, there is a belief in the finance literature that Value at Risk (VaR) and Conditional Value at Risk (CVaR) are new approaches to manage and control the risk. Regard to, value at risk is not a coherent risk measure and it is not sub-additive and convex, so, we have considered conditional value at risk as a risk measure by different confidence level in the Mean-CVaR and multi objective proportional change Mean-CVaR models and compared these models with our previous mean-VaR models. This paper focuses on the performance evaluation process and portfolios selection by using Data Envelopment Analysis (DEA). Conventional DEA models assume non-negative values for inputs and outputs, but many of data take the negative value. Therefore, we have used our models based on Range Directional Measure (RDM) that can take positive and negative values. Here value at risk is obtained by non-parametric methods such as historical simulation and Monte Carlo simulation. Finally, a numerical example in Iran's market is presented. Keywords: Portfolio, Data Envelopment Analysis, Negative data, Value at risk, Conditional value at risk, Efficiency

The dynamic evolution of tail–risk interdependence among institutions is of primary importance when extreme events such as financial crisis occur. In this paper we introduce two new risk measures that generalise the Conditional Value–at–Risk and the Conditional Expected Shortfall in a multiple setting. The proposed risk measures aim to capture extreme tail co–movements among several multivariate connected market participants experiencing contemporaneous distress instances. Analytical expressions for the risk measures are obtained under a parametric model that postulates a joint dynamic evolution of the underlying institutions' losses and gains. We consider a multivariate Student–t version of Markov Switching models as a robust alternative to the usual multivariate Gaussian specification, accounting for heavy–tails and time varying non–linear correlations. An empirical application to US banks is considered to show that our model–based risk measurement framework provides a better characterisation of the dynamic evolution of the overall risk of a financial system and a more complete picture of how the risk spreads among institutions.

Calculation of risk contributions of sub-portfolios to total portfolio risk is essential for risk management in insurance companies. Thanks to risk capital allocation methods and linearity of the loss model, sub-portfolio (or position) contributions can be calculated efficiently. However, factor risk contribution theory in non-linear loss models has received little interest. Our concern is the determination of factor risk contributions to total portfolio risk where portfolio risk is a non-linear function of factor risks. We employ different approximations in order to convert the non-linear loss model into a linear one. We illustrate the theory on an annuity portfolio where the main factor risks are interest-rate risk and mortality risk.

In this article, I compare credit risk models that are used for loan portfolios, both from a theoretical perspective and via simulation studies. My study is distinct from previous studies by including new models, considering sector correlation, and performing comprehensive sensitivity analysis. CreditRisk , CreditMetrics, Basel II internal rating based method, and Mercer Oliver Wyman's model are considered. Risk factor distribution and the relationship between risk components and risk factors are the key distinguishing characteristics of each model. CreditRisk , due to its extra degree of freedom, has the highest flexibility to fit various loss distributions. It turns out that sector covariance is the most important risk component for risk management in terms of risk sensitivity. Risk sensitivities not only differ among models but also depend on the input parameters and the quantile at which risk is measured. This implies that risk models can only be judged in terms of the portfolio under consideration, and banks should evaluate them based on their own portfolios.

Different from the short-term risk measure for traditional financial assets stocks, bonds, etc., the key to illiquid inventory portfolio traded in the over-the-counter markets is to estimate the long-term extreme price risk with time varying volatility. In this article, a new long-term extreme price risk value at risk and conditional value at risk measure method for inventory portfolio and an application to dynamic impawn rate interval are proposed. To realize this, we first establish AutoRegressive Moving Average-Exponential Generalized Autoregressive Conditional Heteroskedasticity-Extreme Value Theory model and multivariatet-Copula to depict the autocorrelation, fat tails, and volatility clustering of returns of inventories and the nonlinear dependence structure of inventories. Furthermore, we obtain the long-term extreme price risk with time varying volatility via Monte Carlo simulation instead of square-root-of time rule. The results show that, first, benefits from risk diversification is significant; second, long-term extreme price risk measure of inventory portfolio via Monte Carlo method outperforms the square-root-of time rule; the last is that the dynamic rate interval based on the long-term price risk is superior to the crude rules of thumb in terms of reducing efficiency loss and improving risk coverage. In summary, this article provides a new quantitative framework for managing the risk of portfolio in inventory financing practice for banks constrained by risk limitation. © 2014 Wiley Periodicals, Inc. Complexity 20: 17-34, 2015

This paper presents a dynamic portfolio credit model following the regulatory framework, using macroeconomic and latent risk factors to predict the aggregate loan portfolio loss in a banking system. The latent risk factors have three levels: global across the entire banking system, parent-sectoral for the intermediate loan sectors and sector-specific for the individual loan sectors. The aggregate credit loss distribution of the banking system over a risk horizon is generated by Monte Carlo simulation, and a quantile estimator is used to produce the aggregate risk measure and economic capital. The risk contributions of the individual sectors and risk factors are measured by combining the Hoeffding decomposition with the Euler capital allocation rule. For the U.S. banking system, we find that the real GDP growth rate, the global and sector-wide frailty risk factors and their spillovers significantly affect loan defaults, and the impacts of the frailty factors are not only economy-wide but also sector-specific. We also find that the frailty risk factors make more significant risk contributions to the aggregate portfolio risk than the macroeconomic factors, while the macroeconomic factors help to improve the accuracy and efficiency of the credit risk forecasts.

The aim of this paper is to highlight and illustrate the use of some quantitative techniques for risk estimation in finance and insurance. The first component involved in risk assessment concerns the risk measure used and the second one is based on the estimation technique. We will study the theoretical properties, the accuracy of modeling the economic phenomena and the computational performances of the risk measures Value-at-Risk, Conditional Tail Expectation, Conditional Value-at-Risk and Limited Value-at-Risk in the case of logistic distribution. We also investigate the most important statistical estimation methods for risk measure evaluation and we will compare their theoretical and empirical behavior. The quality of the risk estimation process corresponding to the quantitative techniques discussed will be tested for both real and simulated data. Numerical results will be provided.

In this study we empirically explore the capacity of historical VaR to correctly predict the future risk of a financial institution. We observe that rolling samples are better able to capture the dynamics of future risks. We thus introduce another risk measure, the Sample Quantile Process, which is a generalization of the VaR calculated on a rolling sample, and study its behavior as a predictor by varying its parameters. Moreover, we study the behavior of the future risk as a function of past volatility. We show that if the past volatility is low, the historical computation of the risk measure underestimates the future risk, while in period of high volatility, the risk measure overestimates the risk, confirming that the current way financial institutions measure their risk is highly procyclical.

This paper studies a coherent acceptability measure which is a negative coherent risk measure, in a multi-period model. When a coherent acceptability measure changes according to new information in the market, a time consistency plays an important role. The usual strong time consistency gives too severe a multi-period Tail Value at Risk (Tail VaR) from a practical viewpoint. We study a weak type of time consistency and propose new multi-period Tail VaR measures.

The Fourier Transform Monte Carlo (FTMC) method, a powerful algorithm for robust computation of marginal risk contributions and capital allocations for credit portfolios in the framework of mixture models, is presented. The method outperforms results obtained from simple Monte Carlo simulations which are flawed by high variances if expected values conditional on rare events are calculated. The FTMC method exploits the conditional independence property of the underlying latent variable model and, in addition, makes use of the Fast Fourier Transform technique for risk aggregation. Marginal risk contributions for expected shortfall, value at risk and capital at risk are presented for a synthetic but realistic credit portfolio.

In this paper, we provide a new measure for evaluation of risk in financial markets. This measure is based on the return interval of critical events in financial markets or other investment situations. Our main goal was to devise a model like Value at Risk (VaR). As VaR, for a given financial asset, probability level and time horizon, gives a critical value such that the likelihood of loss on the asset over the time horizon exceeds this value is equal to the given probability level, our concept of Time at Risk (TaR), using a probability distribution function of return intervals, provides a critical time such that the probability that the return interval of a critical event exceeds this time equals the given probability level. As an empirical application, we applied our model to data from the Tehran Stock Exchange Price Index (TEPIX) as a financial asset (market portfolio) and reported the results.

Risk aversion is a prevalent phenomenon when sufficiently large amounts are at risk. In this paper, we introduce a new prescriptive approach for coping with risk in sequential decision problems with discrete scenario space. We use Conditional Value-at-Risk (CVaR) risk measure as optimization criterion and prove that there is an explicit linear representation of the proposed model for the problem.

Trading portfolios at nancial institutions are typically driven by a large number of nancial variables. These variables are often correlated with each other and exhibit by time-varying volatilities. We propose a computationally ecient Value-at-Risk (VaR) methodology based on Dynamic Factor Models (DFM) that can be applied to portfolios with time-varying weights, and that, unlike the popular Historical Simulation (HS) and Filtered Historical Simulation (FHS) methodologies, can handle time-varying volatilities and correlations for a large set of nancial variables. We test the DFM-VaR on three stock

Choosing a proper external risk measure is of great regulatory importance, as exemplified in the Basel II and Basel III Accords, which use value-at-risk with scenario analysis as the risk measures for setting capital requirements. We argue that a good external risk measure should be robust with respect to model misspecification and small changes in the data. A new class of data-based risk measures called natural risk statistics is proposed to incorporate robustness. Natural risk statistics are characterized by a new set of axioms. They include the Basel II and III risk measures and a subclass of robust risk measures as special cases; therefore, they provide a theoretical framework for understanding and, if necessary, extending the Basel Accords.

Portfolio risk can be decomposed into two parts, the systematic risk and the nonsystematic risk. It is well known that the nonsystematic risk can be eliminated by diversification, while the systematic risk cannot. Thus, the portfolio risk, except for that of undiversified small portfolios, is always dominated by the systematic risk. In this paper, under the mean–variance framework, we propose a model for actively allocating the systematic risk in portfolio optimization, which can also be interpreted as a model of controlling risk sensitivity in portfolio selection. Although the resulting problem is, in general, a notorious non-convex quadratically constrained quadratic program, the problem formulation is of some special structures due to the features of the defined marginal systematic risk contribution and the way to model the systematic risk via a factor model. By exploiting such special problem characteristics, we design an efficient and globally convergent branch-and-bound solution algorithm, based on a second-order cone relaxation. While empirical study demonstrates that the proposed model is a preferred tool for active portfolio risk management, numerical experiments also show that the proposed solution method is more efficient when compared to the commercial software BARON.

Recent theoretical works have found a link between return sign forecastability and conditional volatility. This paper compares the predictive performance of the conditional country risk and the conditional residual risk in forecasting the direction of change in the return on the UK stock market index. The conditional country risk and the conditional residual risk are estimated using the bivariate BEKK-GARCH technique and the direction of change in the UK stock market index is modelled using the binary logit approach. Both the in-sample and the out-of-sample predictions suggest that, as a predictor, the conditional residual risk is superior to the conditional country risk. Our findings support the residual risk model while contradicting the traditional capital asset pricing model (CAPM). Moreover, our tactical asset allocation simulations show that when the conditional residual risk is used in conjunction with multiple-threshold trading strategies to guide the investment decisions, the actively managed portfolio achieves greater returns than the return on a buy and hold portfolio.

Operational risk is increasingly being recognised as a significant area of risk and regulation, yet there exists relatively little research on it. In this paper we show that operational risk represents a fundamental risk to option hedging and investigate it by proposing a new theoretical model. We derive an exposure indicator for the operational risk of option hedging and the resulting operational risk distribution. We obtain analytical results for various risk measures including the Value at Risk equation; this includes deriving a new analytical result for the quantile function of the half-normal distributions (which will be of interest to Statisticians in general). We determine an analytical solution to the price of options under operational risk. We conduct numerical experiments on empirical option data to validate our model and estimate the operational Value at Risk for option hedging.

Innovative transition matrix techniques are used to compare extreme credit risk for Australian and US companies both prior to and during the global financial crisis (GFC). Transition matrix methodology is traditionally used to measure Value at Risk (VaR), a measure of risk below a specified threshold. We use it to measure Conditional Value at Risk (CVaR) which is the risk beyond VaR. We find significant differences in VaR and CVaR measurements in both the US and the Australian markets. We also find a greater differential between VaR and CVaR for the US as compared to Australia, reflecting the more extreme credit risk that was experienced in the US during the GFC. Traditional transition matrix methodology assumes that all borrowers of the same credit rating transition equally, whereas we incorporate an adjustment based on industry share price fluctuations to allow for unequal transition among industries. Our revised model shows greater change between Pre-GFC and GFC total credit risk than the traditional model, meaning that those industries that were riskiest during the GFC are not the same industries that were riskiest Pre-GFC. Overall, our analysis finds that our innovative modelling techniques are better able to account for the impact of extreme risk circumstances and industry composition than traditional transition matrix techniques.

PurposeThe purpose of this paper is to illustrate the importance of modeling parameter risk in premium risk, especially when data are scarce and a multi‐year projection horizon is considered. Internal risk models often integrate both process and parameter risks in modeling reserve risk, whereas parameter risk is typically omitted in premium risk, the modeling of which considers only process risk.Design/methodology/approachThe authors present a variety of methods for modeling parameter risk (asymptotic normality, bootstrap, Bayesian) with different statistical properties. They then integrate these different modeling approaches in an internal risk model and compare their results with those from modeling approaches that measure only process risk in premium risk.FindingsThe authors show that parameter risk is substantial, especially when a multi‐year projection horizon is considered and when there is only limited historical data available for parameterization (as is often the case in practice). The authors' results also demonstrate that parameter risk substantially influences risk‐based capital and strategic management decisions, such as reinsurance.Practical implicationsThe authors' findings emphasize that it is necessary to integrate parameter risk in risk modeling. Their findings are thus not only of interest to academics, but of high relevance to practitioners and regulators working toward appropriate risk modeling in an enterprise risk management and solvency context.Originality/valueTo the authors' knowledge, there are no model approaches or studies on parameter uncertainty for projection periods of not just one, but several, accident years; however, consideration of multiple years is crucial when thinking strategically about enterprise risk management.

Adequately specifying the parameters of a financial or actuarial model is challenging. In case of historical estimation, uncertainty arises through the estimator’s volatility and possible bias. In case of market implied parameters, the solution of a calibration to market data might not be unique or the numerical routine returns a local instead of a global minimum. This paper provides a new method based on convex risk measures to quantify parameter risk and to translate it into prices, extending results in Cont (Math Finance 16(3):519–547, 2006) and Lindstrom (Adv Decision Sci, 2010). We introduce the notion of risk-capturing functionals and prices, provided a distribution on the parameter (or model) set is available, and present explicit examples where the Average-Value-at-Risk and the entropic risk measure are used. For some classes of risk-capturing functionals, the risk-captured price preserves weak convergence of the distributions. In particular, the risk-captured price generated by the distributions of a consistent sequence of estimators converges to the true price. For asymptotically normally distributed estimators we provide large sample approximations for risk-captured prices. Following Bion-Nadal (J Math Econ 45(11):738–750, 2009); Carr et al. (J Financ Econ 62:131–167, 2001); Cherny and Madan (Int J Theor Appl Finance 13(8):1149–1177, 2010); Xu (Ann Finance 2:51–71, 2006), we interpret the risk-captured price as an ask price, reflecting aversion towards parameter risk. To acknowledge parameter risk in case of calibration to market prices, we create a parameter distribution from the pricing error function, allowing us to compare the intrinsic parameter risk of the stochastic volatility models of Heston and Barndorff-Nielsen and Shephard as well as the Variance Gamma option pricing model by pricing different exotics.

This article assesses credit risk by using a parameterized model under risk-neutral measure, elaborating the assumption of Bystrom and Kwon (2007) by using interpolation to estimate the risk-free yield curve. The required data are minimal; the proposed model only necessitates information regarding loans, such as loan rates, and risk-free rates that can avoid shortcomings of rating data. The default probabilities are estimated under risk-neutral measure though few studies have done so. The empirical results show that default probabilities of financial distress are higher compared to those of normal firms. Furthermore, the proposed model is also closely associated with the economic state.