Structure of the effective front, Method of critical...

Structure of the effective front

In the previous paragraph, the optimization problem was solved for each specific value of the slope angle of the tangent to the effective front. To restore the entire structure of the front, we will prove that the functions x * (k) are piecewise-linear and, therefore, it suffices to solve the problem for those values ​​ k , for which these functions have a break. Such points are called angular points. We give a rigorous definition of such points. For this we need the Kuhn-Tucker theorem, which we immediately state in applying to our problem.

The problem of quadratic programming is considered

(16.14)

under constraints

where is a positive definite symmetric matrix.

The Lagrange function has the form

The Kuhn-Tucker theorem. An N-dimensional vector x is a solution of the problem (16.14), if and only if there exists an m-dimensional vector λ, satisfying the conditions

(16.15)

Let's use the img src="images/image2004.jpg"> solution of the problem (16.14) for a particular value of to. Divide the indices with the following

way: the index/is called internal, if the condition is met, and external, if the condition

We call the point to the non-corner, if there exists a value of ε, such that in the interval (), the internal and external indices are the same for all values k. Otherwise, the point to 0 will be called angular When passing through the corner point, the structure of the optimal portfolio changes - some assets are removed from it or are added to it.

THEOREM. If there are no corner points in the , the functions are linear in this space.

For the proof it suffices to write down the conditions of the Kuno-Tucker theorem in the form of a system of linear equations with a coefficient matrix independent of ⅜ and a column of free terms that linearly depend on k. Then the assertion of the theorem will follow from well known facts of linear algebra.

Thus, on a gap not containing angular points, the solution of the problem (16.14) has the form

(16.16)

Therefore, to restore the entire structure of the effective front, it is sufficient to find angular points.

Critical Lines Method

Let k - not an angular point and find a solution x (k) of the form (16.16). The method of critical lines of Markowitz is based on a very simple idea. Define the numbers a, b as follows: a is the left one, and b - the right boundary of the range of values ​​for which the inequalities for all internal indices /,

for all external indices i.

Then a and b are the corner points. If there is an appropriate value a nc, then we should set a = 0.

Thus, you can find all the corner points, if their number is finite. To implement this procedure, various numerical algorithms have been developed.

Consider for illustration a simple model example. Suppose that there are three assets on the market, while the yield vector and the covariance matrix have the following form:

The Lagrange function has the form

Taking for the initial value ⅛ = 0,1, we obtain the following solution

(all indices are internal). Using the method of critical lines, we obtain the angular point k = 0.2. When you go through this point, the portfolio structure changes, the third asset is deleted, and the third index becomes external. Choosing the value of k = 0.3, we get a solution

The next corner point is . With the solution has the form

Note that the effective front (the boundary of the set of allowable portfolios) does not have kinks at the corner points, although its functional form is changing. It consists of two different hyperbolas and one boundary point. A break in the effective front occurs if the functions Xj are constant on the interval between the corner points.

In general, an effective front can consist of hyperbolas and straight segments.

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