Entropic risk measures and their comparative statics in portfolio selection: Coherence vs. convexity

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.     

一致风险度量和锥优化分析

一致风险理论的公理系统为风险分析建立了坚实的基础,然而它背后的数学却和凸优化理论思想密切相关,特别是对偶理论. 本文在有限维空间中,利用锥优化的对偶定理给出了一致风险度量的一般表达式的简单证明. 分析了可接受集的概念在一致风险度量中的中心作用,根据锥优化的对偶关系,探索了常用风险度量的性质. 尽管可接受集的大小能够表达风险控制的强弱,但是我们不知道如何定量地表示. 本文提出用相对熵控制风险度量松紧度的方法和意义. 另外,根据一致风险度量的灵活的结构,给出了无套利条件的一种放松,这一结果可用于不完全市场中的期权定价问题.     

Bounding Contingent Claim Prices via Hedging Strategy with Coherent Risk Measures

We generalize the notion of arbitrage based on the coherent risk measure, and investigate a mathematical optimization approach for tightening the lower and upper bounds of the price of contingent claims in incomplete markets. Due to the dual representation of coherent risk measures, the lower and upper bounds of price are located by solving a pair of semi-infinite linear optimization problems, which further reduce to linear optimization when conditional value-at-risk (CVaR) is used as risk measure. We also show that the hedging portfolio problem is viewed as a robust optimization problem. Tuning the parameter of the risk measure, we demonstrate by numerical examples that the two bounds approach to each other and converge to a price that is fair in the sense that seller and buyer face the same amount of risk.     

Multivariate coherent risk measures induced by multivariate convex risk measures

In this paper, we study the close relationship between multivariate coherent and convex risk measures. Namely, starting from a multivariate convex risk measure, we propose a family of multivariate coherent risk measures induced by it. In return, the convex risk measure can be represented by its induced coherent risk measures. The representation result for the induced coherent risk measures is given in terms of the minimal penalty function of the convex risk measure. Finally, an example is also given.

     

Risk measuring under liquidity risk

We present a general framework for measuring the liquidity risk. The theoretical framework defines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a security or a portfolio of securities cannot be sold or bought without causing changes in prices. The risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinate security are increasing monotonic and convex cash sub-additive on long positions. The contrary, in certain situations, holds for the sell positions. By using convex risk measures, we apply our framework to the situation in which large trades are broken into many small ones. Dual representation results are obtained for both positions in securities and portfolios. We give many examples of risk measures and derive for each of them the respective capital requirement. In particular, we discuss the VaR measure.     

Liquidity Risk with Coherent Risk Measures

This paper concerns questions related to the regulation of liquidity risk, and proposes a definition of an acceptable portfolio. Because the concern is with risk management, the paper considers processes under the physical (rather than the martingale) measure. Basically, a portfolio is ‘acceptable’ provided there is a trading strategy (satisfying some limitations on market liquidity) which, at some fixed date in the future, produces a cash‐only position, (possibly) having positive future cash flows, which is required to satisfy a ‘convex risk measure constraint’.     

Portfolio insurance under a risk-measure constraint

We study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor’s utility function subject to the risk-measure constraint. We give a full solution to this non-convex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists) and for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases).     

Portfolio insurance under a risk-measure constraint

We study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor’s utility function subject to the risk-measure constraint. We give a full solution to this non-convex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists) and for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases).     

Coherent and convex risk measures for portfolios with applications

In this paper, we propose a framework of risk measures for portfolio vectors, which is an extension of the ones introduced by Burgert and Rüschendorf (2006) and Rüschendorf (2013). Representation results for coherent and convex risk measures for portfolio vectors are provided. Applications to the multi-period risk measures are also given.     

Extended Gini-Type Measures of Risk and Variability

ABSTRACT

The aim of this paper is to introduce a risk measure, Extended Gini Shortfall (EGS), that extends the Gini-type measures of risk and variability by taking risk aversion into consideration. Our risk measure is coherent and catches variability, an important concept for risk management. The analysis is made under the Choquet integral representations framework. We expose results for analytic computation under well-known distribution functions. Furthermore, we provide a practical application.     

均值方差偏好和期望损失风险约束下的动态投资组合

本文在均值方差框架下,研究了期望损失风险约束下的连续时间动态投资组合问题。运用鞅理论和凸对偶方法,分别给出了最优财富和最优投资策略的解析式,而且两基金分离定理仍然成立。最后通过数值例子分析了风险约束对最优投资策略的影响。     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

Entropic Value-at-Risk: A New Coherent Risk Measure

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure.     

A new class of coherent risk measures based on p‐norms and their applications

To exercise better control on the lower tail of the loss distribution and to easily describe the investor's risk attitude, a new class of coherent risk measures is proposed in this paper by taking the minimization of p‐norms of lower losses with respect to some reference point. We demonstrate that the new risk measure has satisfactory mathematical properties such as convexity, continuity with respect to parameters included in its definition, the relations between two new risk measures are also examined. The application of the new risk measures for optimal portfolio selection is illustrated by using trade data from the Chinese stock markets. Empirical results not only support our theoretical conclusions, but also show the practicability of the portfolio selection model with our new risk measures. Copyright © 2006 John Wiley & Sons, Ltd.     

On efficient portfolio selection using convex risk measures

In this article we systematically revisit the classic portfolio selection theory in both of its branches, the determination of the efficient financial positions among such a choice set and the selection of the financial position which maximizes some utility function whose functional form involves some ‘measure of risk’. We study these problems by considering certain classes of convex risk measures and we show that for these classes the solution of the utility maximization problems in reflexive spaces take the form of a zero-sum game between the investor and the market.     

Composite time-consistent multi-period risk measure and its application in optimal portfolio selection

Through the composition of two real-valued functions, we propose a new class of multi-period risk measure which is time consistent. The new multi-period risk measure is monotonous and convex when the two real-valued functions satisfy monotonicity and convexity. Based on this generic framework, we construct a specific class of time-consistent multi-period risk measure by considering the lower partial moment between the realized wealth and the target wealth at individual periods. With the new multi-period risk measure as the objective function, we formulate a multi-period portfolio selection model by considering transaction costs at individual investment periods. Furthermore, this stochastic programming model is transformed into a deterministic programming problem using the scenario tree technology. Finally, we show through empirical tests and comparisons the rationality, practicality and efficiency of our new multi-period risk measure and the corresponding portfolio selection model.     

Risk measurement in the presence of background risk

A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolio’s re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measure’s aggregation properties are obtained.     

基于凸函数的风险测度

利用凸函数构造了一种新的风险测度,发现它是包含了损失概率、损失期望值、绝对离差、绝对半离差,下偏矩、(α,t)模型、ES等常见方法的更为广泛的风险测度.对其性质的研究发现该风险测度满足凸性和协调性,考虑到凸性以及协调性在投资组合以及风险管理中的重要意义,因此对该风险测度的研究就具有一定的实践和学术价值.