Portfolio Selection II – Schema

The discrete portfolio investment vs return plot can be used to build executive intuition for project interdependencies and the optimum mix of projects. An interactive dashboard is a good way to ‘get to know’ the model – this can be done directly in Excel using ‘Form’ controls or using custom dashboard software e.g. Crystal Xcelcius that sits on top of the Excel model.

Navigating a Portfolio – download the excel model here

An executive can use drop down boxes to select a particular combination of projects and navigate around the portfolio


‘Fit Schema’ in a Portfolio

Genetic Algorithms (GA) are a powerful optimisation technique particularly suited for highly non-linear problems. The variables that make up the function to be optimised are coded as binary numbers, which ‘evolve’ over time through crossover and mutation functions. There is a lot of material elsewhere on the technique – we don’t need to use a GA to optimise the portfolio because the optimum points (pareto efficient frontier) can be read directly off the portfolio plot with the available capital acting as a hard constraint. We can however make us of the useful concept of a schema. Here is a description from Wikipedia

In mathematics, in the field of genetic algorithms, a schema, schemata or Holland schemata is a template that identifies a subset of strings with similarities at certain string positions. For example, consider binary strings of length 6. The schema 1**0*1 describes the set of all strings of length 6 with 1’s at positions 1 and 6 and a 0 at position 4. The * is a wildcard symbol, which means that positions 2, 3 and 5 can have a value of either 1 or 0. The order of a schema is defined as the number of fixed positions in the template, while the defining length δ(H) is the distance between the first and last specific positions. The order of 1**0*1 is 3 and its defining length is 5. The fitness of a schema is the average fitness of all strings matching the schema. The fitness of a string is a measure of the value of the encoded problem solution, as computed by a problem-specific evaluation function.


Above is a particular portfolio combination (ADE), which would correspond to a single point in the portfolio return plot. We may be more interested in seeing all combinations in the plot that contain a particular project, for example if we wanted to see the portfolio combinations that contain project F


As we can see the highlighted points, which contain project F perform poorly – this is not surprising as this project makes a significant standalone loss. We can write the schema for this as follows

Where the * symbol means ‘don’t care’ – yes or no etc

As another example if we wanted to see the portfolio combinations that contain project A we would get the following –


The points are clustered at the high end of required capital investment as this project requires a minimum of $4m of investment so there are no possible portfolio combinations that contain A below this amount. The schema is


An executive can get a feel for what individual projects contribute heavily to the overall portfolio ‘fitness’. Things become more interesting when we start to include correlations and dependencies between the projects. For example if we had a project whose investment requirement was lower if another project was executed first we may find a fit schema that corresponds to the selection of both projects.

An optimal fit schema may be to always select a particular project and not select another.

Genetic Algorithm Optimisation

With the simple representation above there is no need for algorithmic optimisation. However we when we begin to introduce other parameters e.g. risk we increase the dimensionality, which makes it more difficult to perform multi-objective optimisation. Our binary representation of portfolio selection is particularly well suited to the GA encoding and subsequent optimisation.

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