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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.