Economists are obsessed by equilibria
Finding an analytical solution in economics usually means finding an equilibrium; then being able to go home for the night and make love to your partner with a clean conscience.
Economic systems jump about a lot, but when in equilibrium they settle into an easier pattern of behaviour which leads to far cleaner answers.
You see equilibria mentioned a lot in the classic academic financial economics literature, which generally say that as long as traders are smart and follow a set of prescribed rules, they will converge to a 'correct' theoretical price.
Even Black & Scholes provide an option pricing proof using this framework in the 70s, but soon this type of pricing dies out in favour of arbitrage free pricing models, which prices relatively rather than absolutely.
Perhaps a bit more realistic, definitely less ambitious!
Let's take a quick pictorial sojourn.
This shows two strategy's demeaned cumulative returns over time.
Notice anything odd about either one of them?
How about if I told you they were equity returns?
Let's look at two more charts.
Which chart looks more like equity as we know it?
If we saw both A and B evolve over time, we'd be able to differentiate one from the other by the difference in skewness over time.
Oh, and both their average returns differ over time also.
A's is positive and B's is negative.
In fact A is just the S&P 500's cumulative returns since the 50's and B is the hypothetical return on the S&P 500 by going back in time.
Yes, it's how your returns would look if you, well, if you reversed time!
What do we learn? Obviously both 'strategies' are substantially different, and it tells us that we are dealing with a non-equilibrium system.
Just as a video of a plate smashing on a floor looks different played forwards and backwards - the 'plate system' is obviously in disequilibrium.
Which... comes as a surprise to nobody right?
All those feel good economist theories cannot be applied as stock prices never reach equilibrium, they continue evolving over time.
Such a pity.
But what's interesting, is that by paper trading, you could replicate strategy B by going forward in time and shorting the S&P 500.
Shorting and reversing time is identical in this toy model.
In any case, in a very speculative manner, long / short portfolios could be treated as petri dish systems where time reversibility is achievable (i.e. you couldn't spot 'time travelling' returns) and in which equilibria or steady states might be found.
The upshot is, while we cannot apply equilibrium models to capital markets, perhaps we can construct portfolios which lend themselves to such ambitious models.
Interestingly no-arbitrage ideas revolve around self financing portfolios, which would probably be required for our 'petri dish' portfolio to achieve an equilibrium system - i.e. ensuring that the system is self contained.
Going long the S&P 500 while travelling back in time may be the most idiotic investment strategy ever, however it also throws up some interesting ideas in the process, and perhaps new practical mathematical tools from economics and physics.