>Given the apparently inability of active funds to routinely outperform passive funds[0], "not much" would be my response.

Renaissance Technologies and Two Sigmas are just two examples of firms that constantly beat the market using math, algorithms and machine learning. It's doable, but not easy.

It's silly to say Two Sigma consistently beat the market. They have multiple funds and only one of their numerous funds (Two Sigma Enhanced Compass) has managed to outperform the S&P 500 over a 5 year period, the rest have significantly underperformed.

Until the day they don't.

This kind of thing is inherently un-provable. You can have a method that works stupidly well for decades, and then it stops working. Building a model that provenly (in the mathematical sense) works with X% efficiency would be akin to building a free energy device.

It's very likely you put your life on the line every day with things that can't be proven. That is engineering after all.

> your life on the line

Your life at 0.00000x% risk is one thing. I'm fairly relaxed about that.

Your capital at - say - 50% risk of total loss is another. I don't play that game.

This is just... wrong, to put it mildly.

When it comes to serious financial modeling in big firms, everything from PDEs and stochastic calculus to cutting edge forecasting models and deep learning is fair game.

> fair game

Ummm, not really. If it were readily predictable, then <ooops> it actually couldn't be predictable.

We simply can't all be making money all the time.