Friday, 26 July 2013

Driving Complexity-To-Target: Application to Portfolio Design

Driving the complexity of a given system to a prescribed target value has numerous applications, ranging from engineering (who wouldn't want a simpler design that performs according to specs?) to management, advanced portfolio design, wealth management or investment strategy.

But more than just complexity it is also the robustness of systems that is of most concern. When considering portfolios both diversification and volatility are of concern

We know that in system design (and this applies to portfolios) the mini-max principle, whereby you maximise something (e.g. the expected return) while minimising at the same time something else (e.g. risk) leads to inherently fragile solutions. Taking simultaneously many things to the limit is of course possible but the price one pays is a rigid and fragile solution: you basically push yourself into a very tight corner of  the design space where you have little margin of manoeuvre in case things go wrong. And things do go wrong. Especially if you think that most things in life are linear and follow a Gaussian distribution you should prepare yourself for a handful of surprises.

Portfolio diversification and design can be accomplished differently based on complexity and, in particular, on these two simple facts:

  • High complexity increases exposure - a less complex portfolio is better than a more complex one.
  • A less complex portfolio accomplishes better diversification (more or along the lines of the MPT and Markowitz logic).
Let us see an example. Suppose you want to build a portfolio based on the Dow Jones Industrial Average Index and its components. Without going into unnecessary technicalities, below is an example of our first portfolio. We observe that:

Its complexity is 64.3 (pretty close to the critical value of 68.75)
Entropy is 823
Robustness is 66.8%
Rating: 2 stars

Nothing to celebrate.

Suppose now that you wish to increase the robustness to, say, 85%. Using our Complexity-To-Target Technology it is possible to "force" the robustness of the portfolio to this target value. Since robustness and complexity are linked it is possible to do this either for robustness or complexity or even both. The new portfolio is illustrated below.

Complexity is now 50.9
Entropy is 542 - this tells us that the behaviour of the portfolio is substantially more predictable
Robustness is 84.9%
Rating: 4 stars

The hubs of the portfolio (red discs) have now changed but that is another matter.