Sets the utility function which defines the investors preferred risk-return profile.
This is an alternative means to SetUtilityFunctionInterp by which the utility function can be given. We refer to the polynomial approach as an alternative approach since for most investors it will be more natural to give the utility function as a set of risk-rewards sweet spots (around which the utility function is interpolated).
Structure of the polynomial which defines the Utility Function
The polynomial which defines the investors utility function takes the following form:p(x) = coefficient[0] + (coefficient[1] * xi) + ... + (coefficient[n-1] * xn-1)
where the coefficients[i], i = 0, ..., n-1
are given as a parameter. Now within this representation
the values at the variable x
, corresponds to the risk level of a polynomial and the corresponding
value of p(x)
, is the value of the expected return which to investors demands in order to be
exposed to the chosen level of risk.
The relationship between the Interpolation and Polynomial methods of setting the Utility Function
Note that the two means of setting the utility function namely the interpolation and polynomial procedures are closely related and move over we are able to roughly map between these two means of representing the investors utility function.
If the interpolation points are known at (xi, yi)
, for i = 0, 1, ..., n-1
then we can construct the utility function given as a polynomial of order n
, by solving the
following n
polynomial expressions which will allow us to deduce to values of the coefficients
of the polynomial which takes the same values at the interpolation points:p(xi) = coefficient[0] + (coefficient[1] * xi) + ... + (coefficient[n-1] * xin-1)
where i = 0, ..., n
. Alternatively, if we are given a polynomial p(x) = y
, which
represents the utility function of the investor then we are able to read off the interpolation points
at xi, i= 0, ..., n-1
, by which the utility function can be defined in accordance with
the interpolation approach. That is, the interpolation points (xi, yi)
, for
i = 0, 1, ..., n-1
, for some xi, i= 0, ..., n-1
, are given by:
p(xi) = yi
where i= 0, ..., n-1
.
The above procedure illustrates that there is a close relationship between the polynomial and interpolation
ways of defining the utility function. However care should be taken to point out that though they are closely
related they are not equivalent. The reason for this is that the interpolation method uses cubic spline
interpolation in order to construct the utility function from the interpolation points. The polynomial method
in general uses an n
degree polynomial in order to define the utility function. Therefore, except
in the case of the polynomial method using a cubic polynomial these two approaches can not represent a utility
curve which is identical for all points. However, in practice to above procedure will result in a polynomial
and interpolation representation which agrees on the interpolation points and is generally qualitatively and
quantitatively very close for non-interpolation points.
Markowitz Class | WebCab.Libraries.Finance.Portfolio Namespace