Stoyan V. Stoyanov

Different Approaches to Risk Estimation in Portfolio Theory

Some new performance measures may be regarded as alternatives to the most popular criterion for portfolio optimization, the Sharpe ratio. Analysis of some allocation problems here takes into consideration portfolio selection models based on different risk perceptions and sample paths of the final wealth process for each allocation problem. One new performance ratio seems to be suitable for some optimization problems, but we need a thorough classification of the set of performance measures that would be ideal for large classes of financial optimization problems.

Desirable Properties of an Ideal Risk Measure in Portfolio Theory

This paper examines the properties that a risk measure should satisfy in order to characterize an investors preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investors risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investors preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics, underlying the computational advantage and disadvantage of different approaches.

Optimal Financial Portfolios

The classes of reward‐risk optimization problems that arise from different choices of reward and risk measures are considered. In certain examples the generic problem reduces to linear or quadratic programming problems. An algorithm based on a sequence of convex feasibility problems is given for the general quasi‐concave ratio problem. Reward‐risk ratios that are appropriate in particular for non‐normal assets return distributions and are not quasi‐concave are also considered.

THE PROPER USE OF RISK MEASURES IN PORTFOLIO THEORY

This paper discusses and analyzes risk measure properties in order to understand how a risk measure has to be used to optimize the investors portfolio choices. In particular, we distinguish between two admissible classes of risk measures proposed in the portfolio literature: safety-risk measures and dispersion measures. We study and describe how the risk could depend on other distributional parameters. Then, we examine and discuss the differences between statistical parametric models and linear fund separation ones. Finally, we propose an empirical comparison among three different portfolio choice models which depend on the mean, on a risk measure, and on a skewness parameter. Thus, we assess and value the impact on the investors preferences of three different risk measures even considering some derivative assets among the possible choices.

The Methods of Distances in the Theory of Probability and Statistics

Main directions in the theory of probability metrics.- Probability distances and probability metrics: Definitions.- Primary, simple and compound probability distances, and minimal and maximal distances and norms.- A structural classification of probability distances.-Monge-Kantorovich mass transference problem, minimal distances and minimal norms.- Quantitative relationships between minimal distances and minimal norms.- K-Minimal metrics.- Relations between minimal and maximal distances.- Moment problems related to the theory of probability metrics: Relations between compound and primary distances.- Moment distances.- Uniformity in weak and vague convergence.- Glivenko-Cantelli theorem and Bernstein-Kantorovich invariance principle.- Stability of queueing systems.-Optimal quality usage.- Ideal metrics with respect to summation scheme for i.i.d. random variables.- Ideal metrics and rate of convergence in the CLT for random motions.- Applications of ideal metrics for sums of i.i.d. random variables to the problems of stability and approximation in risk theory.- How close are the individual and collective models in risk theory?- Ideal metric with respect to maxima scheme of i.i.d. random elements.- Ideal metrics and stability of characterizations of probability distributions.- Positive and negative de nite kernels and their properties.- Negative definite kernels and metrics: Recovering measures from potential.- Statistical estimates obtained by the minimal distances method.- Some statistical tests based on N-distances.- Distances defined by zonoids.- N-distance tests of uniformity on the hypersphere.-

Multivariate Skewed Student's t Copula in the Analysis of Nonlinear and Asymmetric Dependence in the German Equity Market

Analyzing comovements in equity markets is important for risk diversification in portfolio management. Copulas have several advantages compared to the linear correlation measure in modeling comovement. This paper introduces a copula ARMA-GARCH model for analyzing the comovement of indexes in German equity markets. The model is implemented with an ARMA-GARCH model for the marginal distributions and a copula for the joint distribution. After goodness-of-fit testing, we find that the skewed Students t copula ARMA(1,1)-GARCH(1,1) model with Lévy fractional stable noise is superior to alternative models investigated in our study where we model the simultaneous comovement of six German equity market indexes. This model is also suitable for capturing the long-range dependence, tail dependence, and asymmetric correlation observed in German equity markets.

Stochastic Models for Risk Estimation in Volatile Markets: A Survey

Portfolio risk estimation in volatile markets requires employing fat-tailed models for financial returns combined with copula functions to capture asymmetries in dependence and an appropriate downside risk measure. In this survey, we discuss how these three essential components can be combined together in a Monte Carlo based framework for risk estimation and risk capital allocation with the average value-at-risk measure (AVaR). AVaR is the average loss provided that the loss is larger than a predefined value-at-risk level. We consider in some detail the AVaR calculation and estimation and investigate the stochastic stability.

Distortion Risk Measures in Portfolio Optimization

Distortion risk measures are perspective risk measures because they allow an asset manager to reflect a client’s attitude toward risk by choosing the appropriate distortion function. In this paper, the idea of asymmetry was applied to the standard construction of distortion risk measures. The new asymmetric distortion risk measures are derived based on the quadratic distortion function with different risk-averse parameters.

Sensitivity of portfolio VaR and CVaR to portfolio return characteristics

Risk management through marginal rebalancing is important for institutional investors due to the size of their portfolios. We consider the problem of improving marginally portfolio VaR and CVaR through a marginal change in the portfolio return characteristics. We study the relative significance of standard deviation, mean, tail thickness, and skewness in a parametric setting assuming a Student’s t or a stable distribution for portfolio returns. We also carry out an empirical study with the constituents of DAX30, CAC40, and SMI. Our analysis leads to practical implications for institutional investors and regulators.

An Empirical Examination of Daily Stock Return Distributions for U.S. Stocks

This article investigates whether the Gaussian distribution hypothesis holds 382 U.S. stocks and compares it to the stable Paretian hypothesis. The daily returns are examined in the framework of two probability models - the homoskedastic independent, identical distributed model and the conditional heteroskedastic ARMA-GARCH model. Consistent with other studies, we strongly reject the Gaussian hypothesis for both models. We find out that the stable Paretian hypothesis better explains the tails and the central part of the return distribution.