Portfolio optimization under shortfall risk constraint

This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained problem and characterize the bi-dual relation between the respective value functions of the constrained problem and its dual. This approach applies to both complete and incomplete markets. Moreover, the extension to more complicated cases is illustrated by solving the problem with a consumption process added. Finally, we give an example of utility and loss functions in the Black–Scholes market where the solutions have explicit forms.     

Distribution assumptions and risk constraints in portfolio optimization

Empirical distributions are often claimed to be superior to parametric distributions, yet to also increase the computational complexity and are therefore hard to apply in portfolio optimization. In this paper, we approach the portfolio optimization problem under constraints on the portfolios Value at Risk and Expected Tail Loss, respectively, under empirical distributions for the Standard and Poors 100 stocks. We apply a heuristic optimization method which has been found to overcome the restrictions of traditional optimization techniques. Our results indicate that empirical distributions might turn into a Pandoras Box: Though highly reliable for predicting the assets risks, employing these distributions in the optimization process might result in severe mis-estimations of the resulting portfolios actual risk. It is found that even a simple mean-variance approach can be superior despite its known specification errors.AMS Classification: G11, C61Dietmar G. Maringer: Im grateful to two anonymous referees, Peter Winker, Manfred Gilli, Berç Rustem, Erricos Kontoghiorghes, Alfred Lehar, Josef Zechner, Suresh Sundaresan, and conference participants at Aix-en-Provence, Limassol, and Sydney for valuable discussions and comments on earlier versions of this paper.     

内部信息者的最小亏损风险策略

在一定的假设条件下,利用扩大信息流方法解决了跳扩散环境下内部信息者的最小亏损风险策略问题.首先构建了内部信息者最小亏损风险策略模型,证明了内部信息市场的完备性.然后利用风险资产价格的Markov性和鞅表示定理得到了线性损失函数下的最小亏损风险最优策略和相应的价值函数.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Easy distributions for combinatorial optimization problems with probabilistic constraints

We show how we can linearize individual probabilistic linear constraints with binary variables when all coefficients are independently distributed according to either N(μ_i,λμ_i), for some λ>0 and μ_i>0, or Γ(k_i,θ) for some θ>0 and k_i>0. The constraint can also be linearized when the coefficients are independent and identically distributed and either positive or strictly stable random variables.     

t-分布与不同风险度量下有效投资组合及其边缘的相互关系研究

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Portfolio optimization under entropic risk management

framework in the risk uniqueness In this paper, properties of the entropic risk measure are examined rigorously in a general This risk measure is then applied in a dynamic portfolio optimization problem, appearing management constraint. By considering the dual problem, we prove the existence and of the solution and obtain an analytic expression for the solution.     

Heston随机波动率市场中带VaR约束的最优投资策略

本文研究了Heston随机波动率市场下, 基于VaR约束下的动态最优投资组合问题。
假设Heston随机波动率市场由一个无风险资产和一个风险资产构成,投资者的目标为最大化其终端的期望效用。与此同时, 投资者将动态地评估其待选的投资组合的VaR风险,并将其控制在一个可接受的范围之内。本文在合理的假设下,使用动态规划的方法,来求解该问题的最优投资策略。在特定的参数范围内,利用数值方法计算出近似的最优投资策略和相应值函数, 并对结果进行了分析。     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

Credit risk optimization using factor models

We study portfolio credit risk management using factor models, with a focus on optimal portfolio selection based on the tradeoff of expected return and credit risk. We begin with a discussion of factor models and their known analytic properties, paying particular attention to the asymptotic limit of a large, finely grained portfolio. We recall prior results on the convergence of risk measures in this “large portfolio approximation” which are important for credit risk optimization. We then show how the results on the large portfolio approximation can be used to reduce significantly the computational effort required for credit risk optimization. For example, when determining the fraction of capital to be assigned to particular ratings classes, it is sufficient to solve the optimization problem for the large portfolio approximation, rather than for the actual portfolio. This dramatically reduces the dimensionality of the problem, and the amount of computation required for its solution. Numerical results illustrating the application of this principle are also presented. JEL Classification G11     

A chance-constrained portfolio selection model with risk constraints

The paper by Huang [Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation 177 (2006) 500-507] proposes a fuzzy chance-constrained portfolio selection model and presents a numerical example to illustrate the proposed model. In this note, we will show that Huang’s model produces optimal portfolio investing in only one security when candidate security returns are independent to each other no matter how many independent securities are in the market. The reason for concentrative solution is that Huang’s model does not consider the investment risk. To avoid concentrative investment, a risk constraint is added to the fuzzy chance-constrained portfolio selection model. In addition, we point out that the result of the numerical example is inaccurate.     

具有间断扩散性质的线性约束全局优化随机算法

研究带线性约束的非凸全局优化问题,在有效集算法的基础上提出了一个具有间断扩散性质的随机微分方程算法,讨论了算法的理论性质和收敛性,证明了算法以概率收敛到问题的全局最优解,最后列出了数值实验效果.     

Algorithms for optimization problems with exclusion constraints

This paper proposes algorithms for minimizing a continuously differentiable functionf(x): ⁿ subject to the constraint thatx does not lie in specified bounded subsets of ⁿ . Such problems arise in a variety of applications, such as tolerance design of electronic circuits and obstacle avoidance in the selection of trajectories for robot arms. Such constraints have the form . The function is not continuously differentiable. Algorithms based on the use of generalized gradients have considerable disadvantages because of the local concavity of at points where the set {j|g ^j (x)=(x)} has more than one element. Algorithms which avoid these disadvantrages are presented, and their convergence is established.This research was sponsored in part by the National Science Foundation under Grant ECS-81-21149, the Air Force Office of Scientific Research (AFSC), United States Air Force under Contract F49620-79-C-0178, the Office of Naval Research under Grant N00014-83-K-0602, the Air Force Office of Scientific Research under Grant AFOSR-83-0361, and the Semiconductor Research Consortium under Grant SRC-82-11-008.     

与TSD准则一致的组合证券投资分析

基于预先给定的目标收益率,利用投资者对低于目标收益率的风险损失和高于目标收益率的风险报酬之间的权衡,给出了一些非对称风险度量模型,特别其中一种风险度量是低于参考点的方差和高于参考点的方差的加权和,它利用二阶上偏矩来修正二阶下偏矩,进一步建立了在该非对称风险度量下的组合投资优化模型,并证明了该模型在三阶随机占优的意义下是有效的.此外,还给出了其它3个模型与三阶随机占优准则是否一致的结论,并对所给出的几个组合证券投资模型的求解方法及其应用进行了分析.以上研究和分析为投资者在选择投资模型时避免盲目性、任意性提供了有益的决策参考.     

Vector optimization problems with quasiconvex constraints

Let X be a real linear space, a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and and are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.      

Stochastic comparison algorithm for continuous optimization with estimation

The problem of stochastic optimization for arbitrary objective functions presents a dual challenge. First, one needs to repeatedly estimate the objective function; when no closed-form expression is available, this is only possible through simulation. Second, one has to face the possibility of determining local, rather than global, optima. In this paper, we show how the stochastic comparison approach recently proposed in Ref. 1 for discrete optimization can be used in continuous optimization. We prove that the continuous stochastic comparison algorithm converges to an -neighborhood of the global optimum for any >0. Several applications of this approach to problems with different features are provided and compared to simulated annealing and gradient descent algorithms.This work was supported in part by the National Science Foundation under Grants EID-92-12122 and ECS-88-01912, and by a Grant from United Technologies/Otis Elevator Company.     

Continuity for vector optimization problems with equilibrium constraints

The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (e) for vector-valued mapping.