By Giorgio P. Szegö, Karl Shell

ISBN-10: 0126807809

ISBN-13: 9780126807806

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**Extra info for Portfolio Theory. With Application to Bank Asset Management**

**Example text**

22) holds, cov(x*, x 2 ) changes its sign. Before further analyzing the problem of the covariance between boundary portfolios, we shall first conclude the discussion on orthogonal portfolios. 25) χ(π 0 ) = ny — β . 26) nn0y - πβ - π0β + α = 0. 24). 25). 30) + cc yv{oty - ß2) = . 31) Corollary Assume that x = χ(π) is F-orthogonal to x = χ(π 0 ). If π < ß/y, then π 0 > ß/y, while if π > ß/y, then π 0 < ß/y. 22) and proves the statement . We shall next prove two theorems that allow an immediate graphical construction of the boundary portfolio orthogonal to any given boundary portfolio.

66), to these intersection points there correspond two values of x3. 2 The independence of xv = V le/y from r has some deep consequences. 69) 0 2 0 » ^0 0 4_ r= 2 _2_ One could argue that the investment / 3 (v = 4,π = 2) is totally dominated by the investment l2 (v = 2, π = 2) and therefore could be taken out from the given set of the three investments. 71) _ 4 _ 12. » — 7 — 21 corresponds. 71), corresponds. Thus the third investment contributes to reducing the variance of the portfolio. Indeed, it can be proved that each time an uncorrelated investment is added to a set of investments, it will always be present (in a positive amount) in the minimum variance portfolio.

48). 49) and will again for x = 0 define a parabola in the plane (v, π). Hence an iteration of the procedure will always generate a parabola as the locus of points of the plane (v, π) to which there corresponds a unique vector x. 7) must coincide. 7) are satisfied for some i, i = 1 , . . , n, it means that the corresponding ith investment is identically zero on the boundary of the region of admissible portfolios. This property of the ith investment with respect to the remaining investments will be further investigated in Chapter 5, where an investment with such a property will be said to be dominated by the other investments.

### Portfolio Theory. With Application to Bank Asset Management by Giorgio P. Szegö, Karl Shell

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