Sets the utility function which defines an investors preferred risk - expected return profile.
The Role of the Utility Function
The construction of a portfolio with regard to an investors risk - reward profile lies at the core of portfolio theory. The utility function assigns a value judgment to the various risk - reward combinations which will be available to the investor. If we where not aware of a given investors risk preferences then it would not be possible to select a preferred portfolio on the Efficient Frontier. However, if we know information such as the maximum amount of risk which the investor will accept then we can select the portfolio on the Efficient Frontier which corresponds to this level of risk. The utility function offers much more fine grained information concerning this investors risk preferences and allows a value judgment to be assign to various risk-rewards combinations.
Providing the Investors Utility Function
As mentioned above the utility function is generated by interpolating a tabulated
function which is provided as two arrays. The first array corresponds to an ordered sequence
of the various total risk levels of the portfolio and is denote by x[0..,n - 1]
(with x[0] < x[1] < ... < x[n - 1]
). The first term of the second array
corresponds to the expected return for the total risk x[0]
. The second term of the
second array corresponds to the expected return for the total risk x[1]
. The third
term is defined in a similar fashion and so on. This provides n
coordinate points
or equivalently a tabulated function which we can interpolate in order to provide a unique utility
function which expresses the investors risk-reward profile.
Finding the investors risk-reward profile in practice
The investors risk-reward utility function is the locus of points at which the investor gets a particular level of satisfaction or utility from a combination of expected return and risk. Clearly, each investor will have their own utility function depending on their individual trade-off between expected return and risk.
Below we give three examples of utility functions interpolation points which correspond to investors
which exhibits a relatively low, medium and high risk tolerance. Each utility function is given on
five points represented a pairs of the form (reward, risk)
:
(1, 0.1), (2, 0.25, (3, 0.3), (4, 0.45), (5, 0.5)
(1, 0.3), (2, 0.5), (3, 1), (4, 2), (5, 2.2)
(1, 0.75), (2, 1), (3, 2), (4, 3), (5, 4)
The above returns and risk correspond to monthly expected return and corresponding risk with the three investors are prepared to accept.
The relationship between the Interpolation and Polynomial methods of setting the Utility Function
The utility function besides being given as a set of points which are then interpolations can also be given as a polynomial expression, see SetUtilityFunctionPoly. Below we detail the relationship between these two means of setting the investors utility function.
Note that the two means of setting the utility function namely the interpolation and polynomial procedures are closely related. Moreover we are able to roughly map between these two means of representing the investors utility function in the follow way.
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 is general used 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 the 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.
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