I don't think it makes sense to equate risk with volatility. Say I have $1M. If I buy a cap weighted S&P 500 fund, I have about $40,000 in Apple, because it's about 4%. If I then switch to an equal weighted S&P 500, that $40K will be split up between 20 stocks with about $2K each. That is clearly more diversified with respect to company specific risk. It may or may not be worth it given increased fees, but you can't just dogmatically say it's equivalent to cap-weighted plus leverage. It's qualitatively different.

>you can't just dogmatically say it's equivalent to cap-weighted plus leverage

I'm saying it's strictly better to do cap-weighted plus leverage. That's what financial theory tells us anyway, that the best portfolio is whatever mix between risk free cash and the same mix as all the assets in the world. Deviating from that brings you farther away from the efficiency frontier. Now for this to hold EMH must hold and we know it's not true in the stronger forms which was why I was asking for a risk adjusted benchmark.

I don't think there's a theory that tells you risk is volatility. Isn't that more of an axiom? And I have the distinct impression it was chosen because it leads to tractable math. I'm not endorsing a specific alternative definition of risk, but I was suggesting risk is clearly multidimensional and intimately related to predicting what will be correlated in the future.