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This groundbreaking ebook extends conventional methods of chance size and portfolio optimization through combining distributional types with possibility or functionality measures into one framework. all through those pages, the professional authors clarify the basics of likelihood metrics, define new techniques to portfolio optimization, and speak about quite a few crucial chance measures. utilizing a variety of examples, they illustrate quite a number purposes to optimum portfolio selection and hazard idea, in addition to purposes to the world of computational finance which may be helpful to monetary engineers.
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Additional info for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
8 provides an illustration is the two-dimensional case. 6. All points that have an elevation above 1 have a local dependence implying that the events Y 1 ∈ (y1 , y1 + ) and Y 2 ∈ (y2 , y2 + ) for a small > 0 are likely to occur jointly. This means that in a large sample of observations, we observe the two events happening together more often than implied by the independence assumption. In contrast, all points with an elevation below 1 have a local dependence implying that the events Y 1 ∈ (y1 , y1 + ) and Y 2 ∈ (y2 , y2 + ) for a small > 0 are likely to occur disjointly.
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. We use Chebyshev’s inequality in Chapter 6 in the discussion of dispersion measures. 2 ´ Fr echet-Hoeffding Inequality Consider an n-dimensional random vector Y with a distribution function FY (y1 , . . , yn ). Denote by W(y1 , . . , yn ) = max(FY1 (y1 ) + · · · + FYn (yn ) + 1 − n, 0) 32 ADVANCED STOCHASTIC MODELS and by M(y1 , . . , yn ) = min(FY1 (y1 ), .
For example, the portfolio choice problem, which we consider in Chapter 8, concerns the optimal trade-off between risk and reward. We say that a portfolio is optimal in the sense that it is the best portfolio among many alternative ones. The criterion that measures the ‘‘quality’’ of a portfolio relative to the others is known as the objective function in optimization theory. The set of portfolios among which we are choosing is called the set of feasible solutions or the set of feasible points.