# Risk Adjusted Performance Spaces

It's my final week in Singapore.

Contrary to hearsay the 'Red Dot' is not zero dimensional; it's as multi-layered as the time you take to investigate it.

A bunch of 'lasts' in Singapore will soon be followed by many 'firsts-in-a-long-time' back in Germany.

Times like these lead to sentimentality. Thinking about past lasts and whether they may come around again.

Visits to the Brooklyn Academy of Music for example.

It took me a while to get into the swing of in-the-flesh theatrical-performances, you need to adjust your brain and expectations, but once you do, their shows are electric.

The Moth was also breath taking. The physicality of people retelling intimate experiences, only a few feet away, has an impact like little else.

My Singapore memories are of colour drenched streets during the many holidays; incense filled temples; tropical forests and getting knocked about on football pitches.

It's been good to me.

External and internal spaces are connected.

Just as the internal machinations of financial calculations can be connected with explicit graphical representations; the explicit can have a bearing on the implicit.

Let's have a quick look at a *financial* performance space.

If we normalise returns to begin and end at the same place, we can extract more intuition about how strategies work.

This is a graph of the S&P 500 and the 13 week t-bill index.

The S&P has a Sharpe of 0.42 over the last 25 years and the t-bill has a Sharpe of 0.39, but from this graph you can't discern why one would have an advantage over the other, t-bills are pretty darn flat.

After leveraging t-bills to have the same average monthly return we can see that they do in fact jump around inefficiently towards the final return.

Finally we demean, as any good statistician should do, so that returns are comparable over time.

Obviously both Sharpe Trajectories waste a lot of 'energy' on their way to the final destination.

Perhaps the t-bill's trajectory is slightly more inefficient due to its pointless sojourn into negative territory in the 2000's (paying for the Greenspan Put!).

Now for a more obvious example.

The FXI ETF is compared against returns after applying Simplest Mean Reversion to it.

Simplest Mean Reversion is brutally efficient in comparison to the FXI itself (reflecting a Sharpe of 1.2 versus 0.2).

This approach throws up a bunch of related ideas.

Thinking about strategies in terms of geometries and spaces; a straight line is the most efficient way to go from point to point in Euclidean space, but not in curved space - and the ideal trajectory here is a curved arc (constrained by volatility - zero volatility is not a viable solution).

I wonder how higher dimensions such as skewness could be included?

Sharpe Trajectories also look strikingly similar to Brownian bridges, which might lead to more insights.

And as my mind wanders to all things Brownian, how about looking into Brownian ratchets and Parrondo's paradox using Sharpe Trajectories?

Finally, the normalisation process reminds me very much of calculating rescaled ranges when estimating the Hurst exponent. Uncannily similar.

Plotting time series is rarely a good idea, but Sharpe Trajectories have a nice intuitive feel to them; and reframing Sharpe in terms of path efficiency also feels right, i.e. it's equally intuitive in both a graphical and analytical sense.