Modelling the portfolio selection problem

The basic model: Markowitz mean-variance model

Markowitz’s mean-variance (MV) model [103] is concerned with a trade-off between the expected return and the risk. In this formulation, the risk of the portfolio is measured by the covariance among the selected assets. The MV formulation provides a fundamental basis for the modern portfolio selection theory in financial investment.

1 1 1 1 min (3-1) . . (3-2) 1 j n i n ij i j i j i n i i i i n i i w w s t r w R w           







(3-3) 0w_i1,i1,...n

Where n is number of assets A = { a1, …, an }. Each assetai is associated with an

expected return (per period) ri, and each pair of assets < ai, aj > has a covariance ij.

The covariance matrix _{n n} is symmetric and each diagonal element _ii represents the variance of asset ai, while the covariance ij represents the correlated risks between

pairs of assets. A positive valueRrepresents the expected return.

To obtain the expected return, rational investors should pick a combination of diversified assets, i.e. a portfolio, to reduce the risk which is measured by the covariance of the combined portfolios. A portfolio can be represented by a set W= { w1, …, wn },

where wi represents the percentage wealth invested on asset ai. The value

1 1 j n i n ij i j i j w w     



represents the variance of the portfolio, and is considered as the measure of the risk associated with the portfolio.

Variables and domains

In the basic MV model, the variables wi are real and their domain is 0wi 1,

represents the percentage wealth invested on the asset.

In practice, there are a wide range of real-world trading constraints. These include the cardinality constraint (a limit on the total number of assets hold in the portfolio), the minimum position size constraint (bounds on the amount of each asset), the minimum trade size constraint (bounds on the amount of transaction occurred on each asset) and transaction costs, etc. When such constraints are considered and added to the basic MV

model, usually integer variables are needed. These will be investigated in chapters 7 and 8.

Objective functions

In the basic MV formulations, the objective can be either to minimize the risk (3-1) (satisfying a given return), or maximize the return (not exceeding a given maximum risk), or both. In the former cases the problem is single-criterion, while in the latter case it is multi-criteria. The problem can be modeled as a multi-objective problem with two conflicting objectives: minimize the variance, denoting the risk associated with the portfolio, whilst maximizing its profits. Essentially, the optimization problem is to find portfolios amongst the n assets that satisfy these two objectives simultaneously. An optimal portfolio is one that has the maximum return with the minimum risk and the set of all these optimal portfolios will form the efficient frontier illustrating the trade-off between the conflicting objectives, as represented by the line in Fig. 3.1. This efficient frontier will be used to evaluate the quality of solutions in chapter 7 and 8.

Fig. 3.1 Efficient Frontier (EF) which defines the trade-off between returns and risk in a portfolio of assets

In this thesis, we focus on the single-criterion problem. The applications on the single- objective formulation (in which the risk has to be minimized) very often solve a

portfolio selection problem instance with given expected return R. Solving the instance for R ranging over values from a finite set can give an estimation of the efficient frontier.

In the basic MV model, covariance is applied as risk measure in the objective function. Applying which term to measure the risk associated with the portfolio, to a certain extent, determines the complexity of the model built. Besides applying covariance as the risk measure of the portfolio, several other risk measures have been investigated in the literature and practice. In [104], the authors propose to use the mean absolute deviation as a measure of risk and formulate the first Linear Programming model for the problem. They show that the mean absolute deviation model, under the assumption of a normal distribution of the return, is equivalent to the quadratic MV model. Later on, in [105] the mean absolute semi-deviation is proposed instead of the mean absolute deviation as a risk measure to reduce the constraints in the mathematic model. More recently, some researchers focus on other risk measures such as value at risk and conditional worst expectation [106, 107].

Constraints

There are two constraints in the basic MV model: return (3-2) and budget (3-3) constraints. They are the most important constraints in portfolio selection problems, because they characterize the essential part of the problems. Return constraint (3-2) presents that the expected return should be met. Budget constraint (3-3) means that all the capital must be investigated.

As we stated before, in practice, there are wide range of real-world trading constraints. These include the cardinality constraint, the minimum position size constraint, the minimum trade size constraint and the transaction costs, etc. We illustrate all of the portfolio selection model attributes (variables, objectives and constraints) that will be investigated in this thesis in Fig. 3.2.

Fig. 3.2 Variables, objective and constraints of portfolio selection problems

In document Effective integrations of constraint programming, integer programming and local search for two combinatorial optimisation problems (Page 82-86)