HARA utility based convex risk measures

Top PDF HARA utility based convex risk measures:

Conditional and Dynamic Convex Risk Measures

Conditional and Dynamic Convex Risk Measures

This paper aims at giving a possible axiomatic foundation to the risk assess- ment of final payoffs when additional information is available. This is the case, for example, when the riskiness of a payoff occurring at time T is quantified at an intermediate date t ∈ (0, T ). We define conditional convex risk measures as maps, satisfying some natural axioms, which associate to every payoff, rep- resented by a random variable X, its riskiness ρ(X) which is itself a random variable, depending on the available information. Furthermore, under a mild technical assumption, we give a characterization of these maps as worst condi- tional expected loss with respect to a given set of probabilistic models, maybe corrected by some random penalty function. A new regularity property is in- troduced and several equivalent formulations are presented; this property, which is economically plain, states that ρ(X) should not depend on that part of the future which is ruled out by the additional information. As an example for con- ditional convex risk measures, the class of entropic risk measures, as defined in [7], is generalized to the conditional setting. These risk measures are first defined as capital requirements with respect to an utility-based acceptability criterion. Then their penalty functions are identified as the conditional relative entropy between the considered probabilistic models and a reference model. The last part of the paper is devoted to a study of dynamic convex risk measures, i.e. families of conditional convex risk measures, describing the risk assessment of a final payoff at successive dates. We introduce two economically motivated prop- erties of time consistency that relate different components of a dynamic convex risk measure. Finally, we provide some characterizations of these properties in terms of the family of penalty functions of their components.
Show more

23 Read more

Multivariate risk measures in the non-convex setting

Multivariate risk measures in the non-convex setting

https://doi.org/10.1515/strm-2019-0002 Received February 2, 2019; accepted June 21, 2019 Abstract: The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.
Show more

11 Read more

Entropy Coherent and Entropy Convex Measures of Risk

Entropy Coherent and Entropy Convex Measures of Risk

E Q [−X] . The contribution of this paper is twofold. First we derive precise connections between risk measurement under the theories of variational, homothetic and multiple priors preferences — (1.4) — and risk measurement using convex measures of risk — (1.5). In particular, we identify two subclasses of convex risk measures that we call entropy coherent and entropy convex measures of risk, and that include all coherent risk measures. We show that, under technical conditions, negative certainty equivalents under variational, homothetic, and multiple priors preferences are translation invariant if and only if they are convex, entropy convex, and entropy coherent measures of risk, respectively. It entails that convex, entropy convex and entropy coherent measures of risk induce linear or exponential utility functions in the theories of variational, homothetic and multiple priors preferences. We show further that, under a normalization condition, this characterization remains valid when the condition of translation invariance is replaced by requiring convexity. The mathematical details in the proofs of these characterization results are delicate.
Show more

42 Read more

Disparity, Shortfall, and Twice Endogenous HARA Utility

Disparity, Shortfall, and Twice Endogenous HARA Utility

1 Introduction The purpose of this short paper is to expose an extremely interesting relationship between Hyperbolic Absolute Risk Aversion (HARA) utility, disparity minimization, and shortfall. Specifically, we show that the entire family of HARA utility functions has a minimium- divergence, shortfall-based representation, which means that HARA utility can be under- stood through the simple notion that the decision maker seeks the allocation that minimizes the probability of realizing an outcome below some pre-determined reference level. This result bridges the behavioral notion of shortfall minimization, first espoused by Roy (1952), with the now-familiar expected utility idea in a broad way. Specifically, we extend the en- dogenous utility arguments of Stutzer (2000, 2003), showing that his findings are special cases of a much more expansive relationship between shortfall, disparity, and conventional expected utility.
Show more

19 Read more

Performance evaluation, portfolio selection, and HARA utility

Performance evaluation, portfolio selection, and HARA utility

combines a fund f with riskless lending or borrowing based on a quadratic utility function. The quadratic Jensen measure as well as the quadratic Treynor measure can be interpreted as situations with a restriction y = ε > 0, but small, and – once again – quadratic utility. Cubic Sharpe, Jensen and Treynor measure describe decision situations with corresponding settings for y but cubic utility. For all 25 portfolio selection problems from July 1997 to July 1999 we determine optimal portfolios based on the rules just described and compute resulting certainty equivalents for an investor whose utility function is actually cubic (and of the HARA type). Certainly, there are greater certainty equivalents achievable by portfolio selection according to the optimized cubic performance measure and with a restriction y ∈ [0, 1] instead of y = 1 or y = ε so that we express all resulting certainty equivalents as percentages of this attainable maximum value. Besides quadratic and cubic Sharpe, Jensen, and Treynor measure we also consider the optimized quadratic measure for which we assume portfolio selection with a re- striction y ∈ [0, 1] based on quadratic utility. Furthermore, we consider portfolio selection based on expected excess returns. Since risk neutrality would not lead to an inner solution for an investor’s riskless lending or borrowing we assume risk neutral fund selection and the choice y = 1 but a quadratic utility for the determination of the amount of the riskless invest- ment.
Show more

56 Read more

Convex risk measures for portfolio optimization and concepts of flexibility

Convex risk measures for portfolio optimization and concepts of flexibility

1. Introduction In their seminal paper Artzner et.al. [ADEH99] presented an axiomatic foundation of coherent risk measures to quantify and compare uncertain future cash-flows within financial institutions. Recently, F¨ollmer et.al. [FS02a] extended the notion of coher- ent risk measures to convex risk measures. It is evident from the axiomatic structure referred above that convex analysis plays a crucial role. Indeed, as we will outline in this paper, convex optimization with its embedded duality is the underlying operational and computational technique for its applications. This was implicitly addressed by Artzner et.al. [ADEH99], Delbaen [De00] and F¨ollmer et.al. [FS02a] in their proofs of the fundamental representation theorems. But to our knowledge, the fundamental concepts have never been presented explicitly under, at least from a computational point of view, “natural" perspective of convex optimization as we will do in this paper for the restricted but important finite dimensional case. Independently Ruszczy´nski et.al. [RS04] studied the intimate relation between convex risk functions and duality structure in topological vector spaces of measurable functions. Indeed, the spirit of their work is very similar to ours but they use advanced theory of convex analysis of measurable functions. Restrict- ing the analysis to the finite dimesional case, the representation results can be derived by applying elementary concepts of convex analysis based on the impressive work of R. T. Rockafellar. At the same time, this perspective opens up the computational aspects
Show more

19 Read more

On Dynamic Coherent and Convex Risk Measures : Risk Optimal Behavior and Information Gains

On Dynamic Coherent and Convex Risk Measures : Risk Optimal Behavior and Information Gains

A sensible axiomatic approach to quantify risk was first mentioned in [Artzner et al., 99] for a static setting: The authors introduced the notion of coherent risk measures assessing risk of projects considered as real valued random variables. Several other references as [Delbaen, 02] advanced upon this approach for more general probability spaces. The approach to coherent risk measures is based on four quite intuitive axioms and leads to a simple and hence applicable robust representation that we encounter later. We will rigorously introduce the underlying notion of risk measures in the respective chapters of this thesis. However, for the sake of completeness and an intuitive understanding at this stage, the four axioms for a risk measure to be coherent are given by monotonicity, cash invariance, sub-additivity and positive homo- geneity of degree one. The major advantage of coherent risk measures is their simple and intuitive robust representation in terms of maximized expected loss as elaborated below. Furthermore, coherent risk measures do not ne- cessitate a specific probabilistic model and hence help to significantly reduce model risk in applications. However, coherent risk measures have two ma- jor shortcomings: First, they overestimate risk as they lead to a worst-case approach by virtue of robust representation: An issue that has to be scoped with from point of view of financial institutions having an intrinsic interest in assessing risk not too conservatively when calculating minimal capital re- quirements. Secondly, due to the assumption of homogeneity, coherent risk measures do not take into account liquidity risk as one of the major problems in the current financial crisis.
Show more

209 Read more

Social choice of convex risk measures through Arrovian aggregation of variational preferences

Social choice of convex risk measures through Arrovian aggregation of variational preferences

2010 Mathematics Subject Classication: 91B14, 91B16, 03C20, 03C98 Journal of Economic Literature classication: D71, G11 ∗ This work has been partially supported by a German Research Foundation (DFG) grant while the author visited the Mathematics Department of Princeton University. A talk based on this paper was presented at the 10th Society for the Advancement of Economic Theory (SAET) Conference on Current Trends in Economics in Singapore, August 2010. I would like to thank Daniel Eckert, Edward Nelson, Konrad Podczeck and Frank Riedel for discussions and comments.

22 Read more

Investment Strategies for HARA Utility Function : A General Algebraic Approximated Solution

Investment Strategies for HARA Utility Function : A General Algebraic Approximated Solution

Furthermore, the values of the financial assets are supposed to depend on a set of stochastic state variables and a stochastic inflation risk is considered. In the literature about the optimal portfolio rules two main fields of research can be found. On the one hand some authors concentrate on establishing the ex- istence (and uniqueness) of a viscosity solution for the Hamilton-Jacobi-Bellman equation deriving from the stochastic optimal control approach (see for instance Crandall et al., 1992; and Buckdahn and Ma, 2001a, 2001b). On the other, some authors offer an algebraic closed form solution to the optimal portfolio compo- sition. In particular, we refer to the works of Kim and Omberg (1996), Wachter (1998), Chacko and Viceira (1999), Deelstra et al. (2000), Boulier et al. (2001), Zariphopoulou (2001) and Menoncin (2002). The two last works use a solution approach based on the Feynman-Kaˇc theorem, 1 in an incomplete market and in a complete market with a background risk respectively.
Show more

18 Read more

The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures

The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures

2 . In this paper we consider a slightly more general, and completely nat- ural, concept of stochastic precedence and analyze its relations with the notions of stochastic ordering. Motivations for our study arise from differ- ent fields, in particular from the frame of Target-Based Approach in deci- sions under risk. Although this approach has been mainly developed under the assumption of stochastic independence between Targets and Prospects, our analysis concerns the case of stochastic dependence, that we model by means of a special class of copulas, introduced for the purpose. Examples are provided to better explain the behavior of the target-based model un- der changes in the connecting copulas of the random variables, especially regarding their properties of symmetry and dependence.
Show more

120 Read more

Optimization of Convex Risk Functions

Optimization of Convex Risk Functions

Key words: Convex analysis, stochastic optimization, risk measures, mean-variance models, duality. 1 Introduction Comparison of uncertain outcomes is central for decision theory. If the outcomes have a probabilistic description, a wealth of concepts and techniques from the theory of probability can be employed. We can mention here the expected utility theory, stochastic ordering, and various mean–risk models. Our main objective is to con- tribute to this direction of research, by exploiting relations between risk models and optimization theory.

26 Read more

Representing Risk Preferences in Expected Utility Based Decision Models

Representing Risk Preferences in Expected Utility Based Decision Models

The paper is organized as follows. First, the literature concerning functional forms for the utility function is briefly reviewed. The CARA, CRRA, HARA, EP, PRT, and FTP functional forms are described. Following this, the procedure used to obtain a utility function from a risk aversion measure is discussed in some detail. There are three main steps in this procedure, with the final step being the transformation of the marginal utility function into the utility function. It is observed that this final step is the one that most severely restricts the risk preferences that can be represented by a utility function. Section 4 observes that in theory marginal utility is a complete representation of the risk preferences of an EU decision maker. In addition, however, this section presents a series of EU based decisions, and indicates how marginal utility is sufficient to determine and analyze the particular decision being discussed.
Show more

21 Read more

The fundamental nature of HARA utility

The fundamental nature of HARA utility

determines the relevant linear parameters of optimal behavior. The HARA form itself implies lin- ear risk tolerance. We discuss the connections between linear scale transformations, linear optimal solutions and linear risk tolerance. It needs to be stressed at the outset that the idea is not just to re-derive a well-known set of results. Rather, the aim is to show that the HARA form is inherent to the economic optimization problem, one which is a fundamental one, appearing in many Macro and Finance contexts. Hence, the analysis does not aim to provide another solution method to a problem that had been solved, but rather to show what restrictions on preferences are embodied in economic reasoning about this fundamental economic problem.
Show more

28 Read more

Conditional and dynamic convex risk measures

Conditional and dynamic convex risk measures

we provide a representation for these risk measures as worst conditional loss with respect to a set of probabilistic models and a penalty function. The main difference in comparison with the unconditional setting is provided by the ran- dom nature of these two objects. This is natural, since they describe, in some sense, the degree of trustworthiness towards different models, which depends on available information and thus may change in time. In the representation we propose, additional information is reflected both in the conditional nature of the expectations and in the penalty function. This issue is particularly important when successive risk measurements of the same payoff are performed or, in our terminology, when a dynamic risk measure has to be constructed. In this case, a penalty process has to be chosen, describing how the degree of trustworthiness of different models evolves through time. In the last section it is shown how this choice is constrained by some basic natural consistency properties. Notwith- standing, in our opinion the class of penalty processes is still too large from an economic viewpoint, so that other consistency properties have to be discussed even in connection with the theory of updating information. Finally, a complete economic interpretation of the penalty term still lacks, even in the classical set- ting. This interpretation could be related to some sort of preference structure in the dual space, that of probabilistic models. We leave this important issue to further investigation.
Show more

23 Read more

To split or not to split: capital allocation with convex risk measures

To split or not to split: capital allocation with convex risk measures

This particular argument demonstrates some difficulties in the use of con- vex risk measures in risk management. In the case of distortion-exponential measures, using the coherent measure ρ g,a=0 creates a stable portfolio. How- ever the introduction of even a slight dependence on the the scale of losses, by using ρ g,a even with an arbitrarily small a, produces an incentive for infinite fragmentation of portfolios. Note that this still happens when we start with a particular pair of sub-portfolios (e.g. business lines) X 1 , X 2 such that X 1 + X 2 = X and ρ(X) ≤ ρ(X 1 ) + ρ(X 1 ). That is, even if the initial configuration of the portfolio is such that benefits from pooling risks occur, once splitting without any constraints is allowed, fragmentation of the portfolio is inevitable.
Show more

29 Read more

To split or not to split: Capital allocation with convex risk measures

To split or not to split: Capital allocation with convex risk measures

This particular argument demonstrates some difficulties in the use of con- vex risk measures in risk management. In the case of distortion-exponential measures, using the coherent measure ρ g,a=0 creates a stable portfolio. How- ever the introduction of even a slight dependence on the the scale of losses, by using ρ g,a even with an arbitrarily small a, produces an incentive for infinite fragmentation of portfolios. Note that this still happens when we start with a particular pair of sub-portfolios (e.g. business lines) X 1 , X 2 such that X 1 + X 2 = X and ρ(X) ≤ ρ(X 1 ) + ρ(X 1 ). That is, even if the initial configuration of the portfolio is such that benefits from pooling risks occur, once splitting without any constraints is allowed, fragmentation of the portfolio is inevitable.
Show more

37 Read more

Subgradients of Law-Invariant Convex Risk Measures on L1

Subgradients of Law-Invariant Convex Risk Measures on L1

Key words: equilibria, generalised subgradients, law-invariant convex risk measures, optimal capital and risk allocations. 1 Introduction In [17] we established that every law-invariant convex risk measure on L ∞ is σ(L ∞ , L ∞ )-lower semi-continuous and thus canonically extended to a law- invariant closed convex risk measure on L 1 . There are several advantages of the model space L 1 : in contrast to L ∞ , the model space L 1 includes important risk models such as normally distributed. Moreover, L 1 is in some sense maximal amongst the law-invariant model spaces bearing a locally convex topology and thus allowing for convex duality. Other attempts to extending the model space beyond L ∞ suggest spaces which depend on the risk measure, in terms of being chosen such that some given risk measure stays real valued. But when studying optimal risk allocations and equilibria which involves more than one risk mea- sure, the model space should preferably be independent of these risk measures.
Show more

32 Read more

Real-valued conditional convex risk measures in
            Lp(ℱ, R)

Real-valued conditional convex risk measures in Lp(ℱ, R)

Abstract. The numerical representation of convex risk measures beyond essentially bounded financial positions is an important topic which has been the theme of recent literature. In other direction, it has been discussed the assessment of essentially bounded risks taking explicitly new information into account, i.e., conditional convex risk measures. In this paper we combine these two lines of research. We discuss the numerical representation of conditional convex risk measures which are defined in a space L p (F, R), for p ≥ 1, and take values in L 1 (G, R) (in this sense, real-valued ).
Show more

15 Read more

Curvature Measures of Convex Bodies (*).

Curvature Measures of Convex Bodies (*).

Concerning the possibility of (~ localizing >> the classical integral-geometric for- mulae for convex bodies by introducing certain locally defined measures, the [r]

34 Read more

Loss-Based Risk Measures

Loss-Based Risk Measures

under this metric can be characterized by the following: for any G n , G ∈ Q, G n → G if and only if G n (z) → G(z) at any continuity points of G. Most of the time, we work with quantile functions that are continuous on (0, 1) in order to avoid irregularities due to the presence of atoms. In practice, it is not restrictive to focus on continuous quantile functions. Indeed, people do assume the continuity of quantile functions in many applications, e.g., when computing the Value at Risk. The study of discontinuous quantile functions is more technical and of little interest, so we choose not to pursue in this direction. In the following, we denote by Q c the set of all continuous quantile functions. We also denote by Q ∞ the set of all bounded quantile functions, and Q ∞ c the set of all bounded continuous quantile functions.
Show more

29 Read more

Show all 10000 documents...