Here we allow constraints to be placed on the weights of the assets from which the portfolios within the Efficient Frontier will be constructed.
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) of the i-th asset from the collection of assets from which the portfolios can be constructed. Note that the length of this array must equal the number of assets from which the portfolios can be constructed.[0,1]
) of the i-th asset from the collection of assets from which the portfolios can be constructed. Note that the length of this array must equal the number of assets from which the portfolios can be constructed.Methods effected by the setting of constraints on the Asset Weights
The methods within this XML Web service which are effected and how they are effected by the setting of constraints on the weights of the assets from which the portfolios are constructed is given below:
Nature of the Constraints
We illustrate the nature of the constraints with the following example. Say an
investor requires a portfolio selected from n
asset which has the
lowest risk for a given expected return but also has the requirement that all of
the assets must have a weight between 0.05
and 0.1
(i.e. between 5 and 10 percent). In this instance we would set the constraints
on the assets to be:
lowerBounds = {0.05, 0.05, 0.05,...., 0.05}
upperBounds = {0.1, 0.1, 0.1, ......, 0.1}
where each of the arrays above has the same number of terms of the number of assets.
Constraints have Default Values
If the constraints are not set then they will take there default values which are
0
and 1
, for the lower and upper bound respectively for
each asset weight.
Performance Issues
The introduction of constraints on the weights of the portfolios which form the Efficient Frontier will have the following consequences with regards to overall performance:
1
(i.e. there default value), then they will not effect the computational demands
required to evaluate the points on the Efficient Frontier. This allows lower bounds to be set
on the asset weight without reducing the performance of this XML Web service.
Remarks:
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.
Motivation and Definition of Consistent Asset Constraints
Say that we wish to use place the following `constraints' on a portfolio which can be constructed from 3 assets:
lower bounds = {0.4, 0.4, 0.4}
upper bounds = {0.5, 0.5, 0.5}
1
, there is no `consistent'
portfolio which can be constructed which satisfies these constraints and the
definition of the weight. Since even if we take the lower bounds for each of the
three assets the sum of the weights is greater than 1
(i.e.
0.4 + 0.4 + 0.4 = 1.2 > 1
).Similarly, say we wish to use the following set of constraints:
lower bounds = {0.2, 0.2, 0.2}
upper bounds = {0.3, 0.3, 0.3}
1
. Since even if we take the upper bounds for each
of the three assets the sum of the weights is less than 1
(i.e.
0.3 + 0.3 + 0.3 = 0.9 < 1
).For these reason reasons we introduce the following property of `consistent constraints':
Definition: A set of upper and lower bound asset constraints of a portfolio are said to be
consistent if the sum of the lower bounds is less than or equal to 1
, and the sum
of the upper bounds is greater than or equal to 1
.
Without the `consistent constraint' condition there will not exist any possible selections of the asset weights which satisfy the constraints and the definition of the asset weights, i.e. the domain of possible portfolios will be empty. Therefore this condition of consistency of the constraints is mandatory for any set of constraints used within the construction of the constrained Portfolios on the constrained Efficient Frontier.
Constraints on the Asset Weights effect on the range of the Expected Returns for which the Efficient Frontier exists
The placing of constraints on the weights of the assets effects the range of expected returns for which the resulting portfolios can be constructed. Since the (constrained) Efficient Frontier is just a collection of portfolios subject also subject to the constraints which minimize the risk for a given level of the expected return. The range of values over which the Efficient Frontier exists must correspond to the range of expected returns of the possible constructed portfolios.
Within the methods MaxFrontierReturn, and MinFrontierReturn we allow the maximum and respectively minimum values of the expected return over which the (possibly constrained) Efficient Frontier exists. We also offer two associated methods MaxFrontierReturnWeights and MinFrontierReturnWeights, which evaluate the assets weights of the portfolio at these two ends points. These methods which construct the Portfolios on the Efficient Frontier at its end points have the significant advantage of having almost no computational overhead, unlike the construction of the portfolios on the Efficient Frontier at other points.
Markowitz Class | Portfolio Namespace