By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA

ISBN-10: 047005316X

ISBN-13: 9780470053164

This groundbreaking ebook extends conventional methods of probability dimension and portfolio optimization through combining distributional types with hazard or functionality measures into one framework. all through those pages, the specialist authors clarify the basics of chance metrics, define new methods to portfolio optimization, and talk about quite a few crucial danger measures. utilizing various examples, they illustrate a number of purposes to optimum portfolio selection and threat concept, in addition to functions to the realm of computational finance that could be helpful to monetary engineers.

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**Extra resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)**

**Sample text**

Working paper, London School of Economics. ¨ Ruschendorf, L. (2004). ‘‘Comparison of multivariate risks and positive dependence,’’ Journal of Applied Probability 41(2): 391–406. Shiryaev, A. N. (1996). Probability, New York: Springer. Sklar, A. (1959). ‘‘Fonctions de r´epartition a` n dimensions et leurs marges,’’ Publications de l’Institut de Statistique de l’Universit´e de Paris 8: 229–231. 1 INTRODUCTION The mathematical theory of optimization has a natural application in the field of finance.

2A 43 Optimization A simple example of a convex function is the linear function, f (x) = a x, x ∈ Rn where a ∈ Rn is a vector of constants. In fact, the linear function is the only function that is both convex and concave. In finance, if we consider a portfolio of assets, then the expected portfolio return is a linear function of portfolio weights, in which the coefficients equal the expected asset returns. 8) where C = {cij }ni,j = 1 is a n × n symmetric matrix. In portfolio theory, the variance of portfolio return is a similar function of portfolio weights.

Chebyshev’s inequality is rough as demonstrated geometrically in the following way. 9 Chebyshev’s inequality, a geometric illustration. The area of the rectangle in the upper-left corner is smaller than the shaded area. which means that it equals the area closed between the distribution function and the upper limit of the distribution function. 9 as the shaded area above the distribution function. 9. 9. For an arbitrary random variable, Chebychev’s inequality takes the form 1 P(|X − EX| ≥ σX ) ≤ 2 , where σ X is the standard deviation of X and > 0.

### Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series) by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA

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