CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

2019 ◽  
Vol 34 (2) ◽  
pp. 297-315
Author(s):  
Linxiao Wei ◽  
Yijun Hu
Keyword(s):  
Standard Deviation ◽  
Portfolio Management ◽  
Risk Measures ◽  
Risk Measure ◽  
Axiomatic Approach ◽  
Capital Allocation ◽  
Axiom System ◽  
Multivariate Risk ◽  
Multivariate Risk Measures ◽  
Multivariate Risk Measure

AbstractCapital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

scholarly journals Time consistency for scalar multivariate risk measures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zachary Feinstein ◽  
Birgit Rudloff
Keyword(s):  
Transaction Costs ◽  
Systemic Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Time Consistency ◽  
Entire Family ◽  
Dual Representations ◽  
Multivariate Risk ◽  
Usual Scalar ◽  
Multivariate Risk Measures

Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

scholarly journals Sharing Risk – An Economic Perspective

2009 ◽  
Vol 39 (2) ◽  
pp. 591-613 ◽  
Author(s):  
Andreas Kull
Keyword(s):  
Value At Risk ◽  
Risk Sharing ◽  
Risk Measures ◽  
Insurance Industry ◽  
Risk Measure ◽  
Capital Allocation ◽  
Risk Capital ◽  
Risk Class ◽  
Capital Costs ◽  
Risk Transfer

AbstractWe revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (‘risk-based supervision’) pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structures.

CAPITAL ALLOCATION FOR SET-VALUED RISK MEASURES

2020 ◽  
Vol 23 (01) ◽  
pp. 2050009
Author(s):  
FRANCESCA CENTRONE ◽  
EMANUELA ROSAZZA GIANIN
Keyword(s):  
Convex Set ◽  
Risk Measures ◽  
Risk Measure ◽  
Capital Allocation ◽  
Allocation Rule ◽  
Scalar Case ◽  
Representation Theorems ◽  
Allocation Rules ◽  
The One ◽  
Definition Of

We introduce the definition of set-valued capital allocation rule, in the context of set-valued risk measures. In analogy to some well known methods for the scalar case based on the idea of marginal contribution and hence on the notion of gradient and sub-gradient of a risk measure, and under some reasonable assumptions, we define some set-valued capital allocation rules relying on the representation theorems for coherent and convex set-valued risk measures and investigate their link with the notion of sub-differential for set-valued functions. We also introduce and study the set-valued analogous of some properties of classical capital allocation rules, such as the one of no undercut. Furthermore, we compare these rules with some of those mostly used for univariate (single-valued) risk measures. Examples and comparisons with the scalar case are provided at the end.

scholarly journals Marginal Decomposition of Risk Measures

2006 ◽  
Vol 36 (2) ◽  
pp. 375-413
Author(s):  
Gary G. Venter ◽  
John A. Major ◽  
Rodney E. Kreps
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Directional Derivative ◽  
Component Composition ◽  
Capital Allocation ◽  
Directional Derivatives ◽  
Total Risk ◽  
Business Decision ◽  
Marginal Risk

The marginal approach to risk and return analysis compares the marginal return from a business decision to the marginal risk imposed. Allocation distributes the total company risk to business units and compares the profit/risk ratio of the units. These approaches coincide when the allocation actually assigns the marginal risk to each business unit, i.e., when the marginal impacts add up to the total risk measure. This is possible for one class of risk measures (scalable measures) under the assumption of homogeneous growth and by a subclass (transformed probability measures) otherwise. For homogeneous growth, the allocation of scalable measures can be accomplished by the directional derivative. The first well known additive marginal allocations were the Myers-Read method from Myers and Read (2001) and co-Tail Value at Risk, discussed in Tasche (2000). Now we see that there are many others, which allows the choice of risk measure to be based on economic meaning rather than the availability of an allocation method. We prefer the term “decomposition” to “allocation” here because of the use of the method of co-measures, which quantifies the component composition of a risk measure rather than allocating it proportionally to something.Risk adjusted profitability calculations that do not rely on capital allocation still may involve decomposition of risk measures. Such a case is discussed. Calculation issues for directional derivatives are also explored.

scholarly journals Generalized Systematic Risk

2016 ◽  
Vol 8 (2) ◽  
pp. 86-127 ◽  
Author(s):  
Ohad Kadan ◽  
Fang Liu ◽  
Suying Liu
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Broad Class ◽  
Systematic Risk ◽  
Axiomatic Approach ◽  
Security Market ◽  
Risk Allocation ◽  
Multiple Dimensions ◽  
Rare Disasters ◽  
Distribution Moments

We generalize the concept of “systematic risk” to a broad class of risk measures potentially accounting for high distribution moments, downside risk, rare disasters, as well as other risk attributes. We offer two different approaches. First is an equilibrium framework generalizing the Capital Asset Pricing Model, two-fund separation, and the security market line. Second is an axiomatic approach resulting in a systematic risk measure as the unique solution to a risk allocation problem. Both approaches lead to similar results extending the traditional beta to capture multiple dimensions of risk. The results lend themselves naturally to empirical investigation. (JEL D81, G11, G12)

scholarly journals Marginal Decomposition of Risk Measures

2006 ◽  
Vol 36 (02) ◽  
pp. 375-413 ◽  
Author(s):  
Gary G. Venter ◽  
John A. Major ◽  
Rodney E. Kreps
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Directional Derivative ◽  
Component Composition ◽  
Capital Allocation ◽  
Directional Derivatives ◽  
Total Risk ◽  
Business Decision ◽  
Marginal Risk

The marginal approach to risk and return analysis compares the marginal return from a business decision to the marginal risk imposed. Allocation distributes the total company risk to business units and compares the profit/risk ratio of the units. These approaches coincide when the allocation actually assigns the marginal risk to each business unit, i.e., when the marginal impacts add up to the total risk measure. This is possible for one class of risk measures (scalable measures) under the assumption of homogeneous growth and by a subclass (transformed probability measures) otherwise. For homogeneous growth, the allocation of scalable measures can be accomplished by the directional derivative. The first well known additive marginal allocations were the Myers-Read method from Myers and Read (2001) and co-Tail Value at Risk, discussed in Tasche (2000). Now we see that there are many others, which allows the choice of risk measure to be based on economic meaning rather than the availability of an allocation method. We prefer the term “decomposition” to “allocation” here because of the use of the method of co-measures, which quantifies the component composition of a risk measure rather than allocating it proportionally to something. Risk adjusted profitability calculations that do not rely on capital allocation still may involve decomposition of risk measures. Such a case is discussed. Calculation issues for directional derivatives are also explored.

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