A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

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Optimization of conditional value-at-risk

Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with significant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. CVaR is able to quantify dangers beyond VaR and moreover it is coherent. It provides optimization short-cuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.

/pdf/conditional-value-at-risk-for-general-loss-distributions-4789km634z.pdf

Conditional value-at-risk for general loss distributions

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

Introduction to linear and nonlinear programming

The purpose of this paper is to demonstrate that a portfolio optimization model using the L1 risk (mean absolute deviation risk) function can remove most of the difficulties associated with the classical Markowitz's model while maintaining its advantages over equilibrium models In particular, the L1 risk model leads to a linear program instead of a quadratic program, so that a large-scale optimization problem consisting of more than 1,000 stocks may be solved on a real time basis Numerical experiments using the historical data of NIKKEI 225 stocks show that the L1 risk model generates a portfolio quite similar to that of the Markowitz's model within a fraction of time required to solve the latter

https://www.bertrand-maillet.net/bibliography/KonnoYamazaki_1991_Preview.pdf

Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market

It is assumed in the standard portfolio analysis that an investor is risk averse and that his utility is a function of the mean and variance of the rate of the return of the portfolio or can be approximated as such. It turns out, however, that the third moment (skewness) plays an important role if the distribution of the rate of return of assets is asymmetric around the mean. In particular, an investor would prefer a portfolio with larger third moment if the mean and variance are the same. In this paper, we propose a practical scheme to obtain a portfolio with a large third moment under the constraints on the first and second moment. The problem we need to solve is a linear programming problem, so that a large scale model can be optimized without difficulty. It is demonstrated that this model generates a portfolio with a large third moment very quickly.

A mean-absolute deviation-skewness portfolio optimization model

We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.

Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints

A new portfolio optimization model using a piecewise linear risk function is proposed. This model is similar to, but has several advantages over the classical Markowitz's quadratic risk model. First, it is much easier to generate an optimal portfolio since the problem to be solved is a linear program instead of a quadratic program. Second, integer constraints associated with real transaction can be incorporated without making the problem intractable. Third, it enables us to distinguish two distributions with the same first and second moment but with different third moment. Fourth, we can generate the capital-market line and derive CAPM type equilibrium relations. We compared the piecewise linear risk model with the quadratic risk model using historical data of Tokyo Stock Market, whose results partly support the claims stated above.

https://www.orsj.or.jp/~archive/pdf/e_mag/Vol.33_02_139.pdf

Piecewise linear risk function and portfolio optimization

Preface. Part I: Foundations. 1. Scope of Global Optimization. 2. Quasi-Convexity. 3. D.C. Functions and D.C. Sets. 4. Duality. 5. Low-Rank Nonconvex Structures. 6. Global Search Methods and Basic D.C. Optimization Algorithms. Part II: Methods and Algorithms. 7. Parametric Approaches in Global Optimization. 8. Multiplicative Programming Problems. 9. Monotonic Problems. 10. Decomposition Methods by Prices. 11. Dynamic Programming Algorithms in Global Optimization. Part III: Selected Applications. 12. Low Rank Nonconvex Quadratic Programming. 13. Continuous Location. 14. Design Centering and Related Geometric Problems. 15. Multiobjective and Bilevel Programming. References. Index.

http://www.gbv.de/dms/goettingen/220568715.pdf

Hiroshi Konno

Papers

Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market

A mean-absolute deviation-skewness portfolio optimization model

Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints

Piecewise linear risk function and portfolio optimization

Optimization on Low Rank Nonconvex Structures