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Abstract: A new one-parameter family of risk measures, which is called Conditional Drawdown-at-Risk (CDaR), is proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in an active portfolio management. For some value of the tolerance parameter beta, the CDaR is defined as the mean of the worst (1-beta)*100% drawdowns. The CDaR risk measure contains the Maximal Drawdown and Average Drawdown as its limiting cases. For a particular example, the optimal portfolios for a case of Maximal Drawdown, a case of Average Drawdown, and several intermediate cases between these two were found. The CDaR family of risk measures is similar to Conditional Value-at-Risk (CVaR), which is also called Mean Shortfall, Mean Access loss, or Tail Value-at-Risk. Some recommendations on how to select the optimal risk measure for getting practically stable portfolios are provided. A real life portfolio allocation problem using the proposed measures was solved.
Abstract: 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 Alpha, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 - Alpha) 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.
Abstract: General deviation measures, which include standard deviation as a special case but need not be symmetric with respect to ups and downs, are defined and shown to correspond to 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. A property called expectation-boundedness of the risk measure is uncovered as essential for this correspondence. It is shown to be satisfied by conditional value-at-risk and by worst-case risk, as well as various mixtures, although not by ordinary value-at-risk. Interpretations are developed in which inequalities that are "acceptably sure", relative to a designated acceptance set, replace inequalities that are "almost sure" in the usual sense being violated only with probability zero. Acceptably sure inequalities fix the standard for a particular choice of a deviation measure. This is explored in examples that rely on duality with an associated risk envelope, comprised of alternative probability densities. The role of deviation measures and risk measures in optimization is analyzed, and the possible influence of "acceptably free lunches" is thereby brought out. Optimality conditions based on concepts of convex analysis, but relying on the special features of risk envelopes, are derived in support of a variety of potential applications, such as portfolio optimization and variants of linear regression in statistics. measures, value-at-risk, conditional value-at-risk, portfolio optimization, convex analysis
risk management, deviation measures, coherent risk
Abstract: Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, associated for example with conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems as well as to covariance relations which resemble those commonly used in capital asset pricing models (CAPM), but have wider interpretations. A more customized version of portfolio optimization is the aim, rather than the idea that a single "master fund" might arise from market equilibrium and serve the interests of all investors. The results cover discrete distributions along with continuous distributions, and therefore are applicable in particular to financial models involving finitely many future states, whether introduced directly or for purposes of numerical approximation. Through techniques of convex analysis, they deal rigorously with a number of features that have not been given much attention in this subject, such as solution nonuniqueness, or nonexistence, and a potential lack of differentiability of the deviation expression with respect to the portfolio weights. Moreover they address in detail the previously neglected phenomenon that, if the risk-free rate lies above a certain threshold, a master fund of the usual type will fail to exist and need to be replaced by one of an alternative type, representing a "net short position" instead of a "net long position" in the risky instruments.
deviation measures, risk measures, value-at-risk, conditional value-at-risk, portfolio optimization, one-fund theorems, master funds, efficient frontiers, CAPM, convex analysis
Abstract: 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.
Risk management, deviation measures, coherent risk measures
Abstract: Optimality conditions are derived for problems of minimizing a generalized measure of deviation of a random variable, with special attention to situations where the random variable could be the rate of return from a portfolio of financial instruments. Generalized measures of deviation go beyond standard deviation in satisfying axioms that do not demand symmetry between ups and downs. The optimality conditions are applied to characterize the generalized master funds which elsewhere have been developed in extending classical portfolio theory beyond the case of standard deviation. The consequences are worked out for deviation based on conditional value-at-risk and its variants, in particular.
Generalized deviation measures, portfolio analysis, generalized master funds, CAPM-like relations, optimality conditions, risk envelopes, risk identifiers, conditional value-at-risk, risk management, stochastic optimization
Abstract: Generalized measures of deviation are considered as substitutes for standard deviation in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, derived for example from conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems in which a more customized version of portfolio optimization is the aim, rather than the idea that a single "master fund" might arise from market equilibrium and serve the interests of all investors. The results that are obtained cover discrete distributions along with continuous distributions. They are applicable therefore to portfolios involving derivatives, which create jumps in distribution functions at specific gain or loss values, well as to financial models involving finitely many scenarios. Furthermore, they deal rigorously with issues that come up at that level of generality, but have not received adequate attention, including possible lack of differentiability of the deviation expression with respect to the portfolio weights, and the potential nonuniqueness of optimal weights. The results also address in detail the phenomenon that if the risk-free rate lies above a certain threshold, the usually envisioned master fund must be replaced by one of alternative type, representing a "net short position" instead of a "net long position" in the risky instruments. For nonsymmetric deviation measures, the second type need not just be the reverse of the first type, and there can sometimes even be an interval for the risk-free rate in which no master fund of either type exists. A notion of basic fund, in place of master fund, is brought in to get around this difficulty and serve as a single guide to optimality regardless of such circumstances.
Deviation measures, risk measures, value-at-risk, conditional value-at-risk, portfolio optimization, one-fund theorem, master fund, basic fund, efficient frontier, convex analysis
Abstract: 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.
linear regression, error measures, deviation measures, risk measures
Abstract: 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.
financial equilibrium, portfolio analysis, generalized deviation measures
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