Seeing in n-Dimensions

Two dimensional video games were my bread and butter.

Then along came three dimensions and games became a lot less interesting.

Complications

The brain can compare seven pieces of information at any one time when doing things like shopping.

A 2D game character comprises of at least 3 dimensions (up, down, left, right plus speed).

Enemies add to the 'dimensions' I need to worry about (each adds another 3 dimensions).

Let's assume I only care about those enemies that are an immediate threat (and don't forget about moving spikey walls!).

A gamer can process more pieces of information or dimensions at once than 7 (by using a more efficient instinctive part of their brain) but add a third ('depth') dimension to the screen and the number of things you have to worry about during play grows quickly.

3D gaming gets hard very quickly.

Shooters like GoldenEye were successful, because while being 3D they limited other aspects - no need to worry too much about running, jumping and odd camera angles - unlike Mario 64.

Multi Dimensional Elevator Pitch

Financial analysts have the same problem. Dumbing down in a complicated world.

Show your client a two dimensional picture and you can sell him on an idea. Three dimensions and the chart gets more difficult to draw and interpret. Four dimensions is only for the elite amongst you!

And yet, any realistic decision will be based upon three or more dimensions.

Let's walk through an example problem.

I want to stress test a portfolio which contains a short equity call option.

First thing to understand, there are four dimensions which I care about. Movements of the price, volatility, interest rate and dividend yield.

Second step is to identify what I am interested in explaining (aka be smart like GoldenEye not Mario 64).

I want to show situations where I lose more than 30% - perhaps that's the point where a margin call occurs or when my portfolio goes bust. That's another dimension, though binary.

n-Dimensional Object? Or Just Happy to See Me?

Usually we plot using zero dimensional points.

If we plot with circles we can add another dimension (radius length or circle area).

Add a little colour or change shapes and we have four dimensions.

Understanding then deteriorates when we need to compare spacial dimensions with colour or shape dimensions.

We can exchange one dimension of space for a dimension from an object.

They're the same currency.

Traditionally, for n dimensions, this is shown in parallel coordinates charts.

Four parallel axes with one line linking each point.

Plots containing a space and objects are also additive.

So for our example, let's use two plots with points on a 2D space (2 * (2 + 0) = 4).

I dispense with the lines linking each pair of points, as they are superfluous.

Each square represents a shock.

We start with shocking each risk factor by -10% and check whether our losses are more than -30%.

They are below and add a shaded square (or 'point') to each diagram.

Every four dimensional shock that causes a loss below our threshold of -30% is recorded in this way. Any shock that is not below is not recorded, as we are only interested in bankruptcy events.

We do not show the severity, as once the loss is below the threshold level we are bankrupted (no such thing as very bankrupted!).

A second shock is recorded in a similar way.

Perhaps this shock is volatility up 10%; equity price up 10%; dividend rate down 10%; and interest rates down by 15%.

All the shock combinations (in this case 3^4 = 81) are cycled through.

If a bankruptcy shock hits the same part of the same plot twice or more, it is shaded again.

We do care about frequency of bankrupties occuring on any one part of the plots.

As we cycle through the combinations of shocks we see a picture emerging.

The interest and dividend risk shock bankruptcy scenarios are evenly distributed. They are not a driver of bankruptcy, as their movement doesn't matter a jot.

The equity and volatility risk factors, are of course extremely important.

Now we know to focus on hedging these risk factors in order to avoid bankruptcy during extreme market movements.

[Check out Stress Map for more]

n-D Glasses

A dislike of 3D-movies, -games and lack of ability as a programmer stops me from breaking down n dimensional plots into 3D plot spaces.

(Then again, I do have a third dimension on each plot - frequency of bankruptcy.)

For even the most complicated portfolios three or four 2D plots (6-8 risk factors) should be more than enough to understand what's going on. In my experience there are really only ever a handful of important risk factors to look out for.

We have really created a form of frequency plot; and the lack of linking lines between each pair of points may be a draw back in a few cases. Nevertheless, it's been interesting to explore how plot space and object dimensionality is related.

Another day another duality.