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

This groundbreaking publication extends conventional techniques of possibility dimension and portfolio optimization through combining distributional types with hazard or functionality measures into one framework. all through those pages, the professional authors clarify the basics of chance metrics, define new ways to portfolio optimization, and talk about a number of crucial danger measures. utilizing a variety of examples, they illustrate various purposes to optimum portfolio selection and possibility thought, in addition to purposes to the world of computational finance that could be priceless to monetary engineers.

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**Example text**

We will not provide the second-order condition for functions of n-dimensional arguments because it is rather technical and goes beyond the scope of the book. We only state it for two-dimensional functions. In the case n = 2, the following conditions hold: ■ If ∇f (x1 , x2 ) = (0, 0) at a given point (x1 , x2 ) and the determinant of the Hessian matrix evaluated at (x1 , x2 ) is positive, then the function has: — A local maximum in (x1 , x2 ) if ∂ 2 f (x1 , x2 ) <0 ∂x21 or ∂ 2 f (x1 , x2 ) < 0. ∂x22 or ∂ 2 f (x1 , x2 ) > 0.

4 . 4 1 A property of convex functions is that the sum of convex functions is a convex function. 4 The surface of a two-dimensional convex quadratic function f (x) = 12 x Cx and the corresponding contour lines. where λ > 0 and C is a positive semidefinite matrix, is a convex function as a sum of two convex functions. 9) in Chapter 8 in the discussion of the mean-variance efficient frontier.

A popular measure for the asymmetry of a distribution is called its skewness. 4 The density graphs of a positively and a negatively skewed distribution. 4). 4). 4 Concentration in Tails Additional information about a probability distribution function is provided by measuring the concentration (mass) of potential outcomes in its tails. The tails of a probability distribution function contain the extreme values. In financial applications, it is these tails that provide information about the potential for a financial fiasco or financial ruin.