By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA
This groundbreaking ebook extends conventional methods of chance dimension and portfolio optimization by means of combining distributional types with possibility or functionality measures into one framework. all through those pages, the professional authors clarify the basics of chance metrics, define new methods to portfolio optimization, and talk about numerous crucial danger measures. utilizing quite a few examples, they illustrate quite a number purposes to optimum portfolio selection and hazard conception, in addition to purposes to the realm of computational finance which may be priceless to monetary engineers.
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Extra info for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)
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.
A local minimum may not be global as there may be vectors outside the small neighborhood of x0 for which the objective function attains a smaller value than f (x0 ). 2 shows the graph of a function with two local maxima, one of which is the global maximum. There is a connection between minimization and maximization. Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value, maxn f (x) = − minn [−f (x)]. 1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa.
Since copulas are essentially probability distributions defined on the unit hypercube, Fr´echet-Hoeffding inequality holds for them as well. In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fr´echet bounds equal W(u1 , . . , un ) = max(u1 + · · · + un + 1 − n, 0) and M(u1 , . . , un ) = min(u1 , . . , un ) respectively. Fr´echet-Hoeffding inequality is given by W(u1 , . . , un ) ≤ C(u1 , . . , un ) ≤ M(u1 , . . , un ). In the two-dimensional case, the inequality reduces to max(u1 + u2 − 1, 0) ≤ C(u1 , u2 ) ≤ min(u1 , u2 ).