Michael Zabarankin

Generalized Deviations in Risk Analysis

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large assortment of examples and assessing which may exhibit superior behavior. Connections are shown with coherent risk measures in the sense of Artzner, Delbaen, Eber and Heath, when those are applied to the difference between a random variable and its expectation, instead of to the random variable itself. However, the correspondence is only one-to-one when both classes are restricted by properties called lower range dominance, on the one hand, and strict expectation boundedness on the other. Dual characterizations in terms of sets called risk envelopes are fully provided.

Drawdown Measure in Portfolio Optimization

A new one-parameter family of risk measures called Conditional Drawdown (CDD) has been proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter α, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 - α) * 100% drawdowns. The CDD measure generalizes the notion of the drawdown functional to a multi-scenario case and can be considered as a generalization of deviation measure to a dynamic case. The CDD measure includes the Maximal Drawdown and Average Drawdown as its limiting cases. Mathematical properties of the CDD measure have been studied and efficient optimization techniques for CDD computation and solving asset-allocation problems with a CDD measure have been developed. The CDD family of risk functionals is similar to Conditional Value-at-Risk (CVaR), which is also called Mean Shortfall, Mean Excess Loss, or Tail Value-at-Risk. Some recommendations on how to select the optimal risk functionals for getting practically stable portfolios have been provided. A real-life asset-allocation problem has been solved using the proposed measures. For this particular example, the optimal portfolios for cases of Maximal Drawdown, Average Drawdown, and several intermediate cases between these two have been found.

Optimal Risk Path Algorithms

Analytical and discrete optimization approaches for routing an aircraft in a threat environment have been developed. Using these approaches, an aircraft’s optimal risk trajectory with a constraint on the path length can be efficiently calculated. The analytical approach based on calculus of variations reduces the original risk optimization problem to the system of nonlinear differential equations. In the case of a single radarinstallation, the solution of such a system is expressed by the elliptic sine. The discrete optimization approach reformulates the problem as the Weight Constrained Shortest Path Problem (WCSPP) for a grid undirected graph. The WCSPP is efficiently solved by the Modified Label Setting Algorithm (MLSA). Both approaches have been tested with several numerical examples. Discrete nonsmooth solutions with high precision coincide with exact continuous solutions. For the same graph, time in which the discrete optimization algorithm computes the optimal trajectory is independent of the number of radars. The discrete approach is also efficient for solving the problem using different risk functions.

Risk Tuning with Generalized Linear Regression

A framework is set up in which linear regression, as a way of approximating a random variable by other random variables, can be carried out in a variety of ways, which, moreover, can be tuned to the needs of a particular model in finance, or operations research more broadly. Although the idea of adapting the form of regression to the circumstances at hand has already found advocates in promoting quantile regression as an alternative to classical least-squares approaches, it is carried here much farther than that. Axiomatic concepts of error measure, deviation measure, and risk measure are coordinated with certain “statistics” that likewise say something about a random variable. Problems of regression utilizing these concepts are analyzed and the character of their solutions is explored in a range of examples. Special attention is paid to parametric forms of regression which arise in connection with factor models. It is argued that when different aspects of risk enter an optimization problem, different forms of regression ought to be invoked for each of those aspects.

Equilibrium With Investors Using a Diversity of Deviation Measures

It has been argued that investors who optimize their portfolios with attention paid only to mean and standard deviation will all end up choosing some multiple of a certain master fund portfolio. Justification for the capital asset pricing model of classical portfolio theory, which relates individual assets to such a master fund, has come from this direction in particular. Attempts have been made to provide solid mathematical support by showing that the imputed behavior of investors is a consequence of price equilibrium in a market in which assets are traded subject to budget constraints, and optimization is carried out with respect to utility functions that depend only on mean and standard deviation. In recent years, reliance on standard deviation has come under increasing criticism because of inconsistencies in its effect on portfolio references. One response has been to introduce generalized measures of deviation which lead to alternative master funds. The market implications of such extensions of theory have hitherto been unclear, but in this paper the existence of equilibrium is established in circumstances where nonstandard deviations are admitted. Equilibrium is guaranteed even when different investors use different measures of deviation and thereby end up with portfolios scaled from different master funds. Whether they employ the same measure or not, they may impose caps on deviation, which likewise may be different.

Capital Asset Pricing Model (CAPM) with drawdown measure

The notion of drawdown is central to active portfolio management. Conditional Drawdown-at-Risk (CDaR) is defined as the average of a specified percentage of the largest drawdowns over an investment horizon and includes maximum and average drawdowns as particular cases. The necessary optimality conditions for a portfolio optimization problem with CDaR yield the capital asset pricing model (CAPM) stated in both single and multiple sample-path settings. The drawdown beta in the CAPM has a simple interpretation and is evaluated for hedge fund indices from the HFRX database in the single sample-path setting. Drawdown alpha is introduced similarly to the alpha in the classical CAPM and is evaluated for the same hedge fund indices. Both drawdown beta and drawdown alpha are used to prioritize hedge fund strategies and to identify instruments for hedging against market drawdowns.

Maximum Entropy Principle with General Deviation Measures

An approach to the Shannon and Renyi entropy maximization problems with constraints on the mean and law-invariant deviation measure for a random variable has been developed. The approach is based on the representation of law-invariant deviation measures through corresponding convex compact sets of nonnegative concave functions. A solution to the problem has been shown to have an alpha-concave distribution (log-concave for Shannon entropy), for which in the case of comonotone deviation measures, an explicit formula has been obtained. As an illustration, the problem has been solved for several deviation measures, including mean absolute deviation (MAD), conditional value-at-risk (CVaR) deviation, and mixed CVaR-deviation. Also, it has been shown that the maximum entropy principle establishes a one-to-one correspondence between the class of alpha-concave distributions and the class of comonotone deviation measures. This fact has been used to solve the inverse problem of finding a corresponding comonotone deviation measure for a given alpha-concave distribution.

Value-at-risk support vector machine: stability to outliers

A support vector machine (SVM) stable to data outliers is proposed in three closely related formulations, and relationships between those formulations are established. The SVM is based on the value-at-risk (VaR) measure, which discards a specified percentage of data viewed as outliers (extreme samples), and is referred to as

Mean-Deviation Analysis in the Theory of Choice

\mathrm{VaR}