Question about inverse leverage etfs

Question

https://www.math.nyu.edu/faculty/avellane/LETFSlides.pdf

This shows that when that the inverse fund grows assuming zero volatility at $e^{r(1-b)}$ where $b$ is the leverage used and $r$ is the interest rate. So a -1x, (i.e. $b=-1$), fund would gain about 5% a year in a 2.5% interest rate environment. That does not make sense but that is what the math says.

It occurred to me you can gain MORE than the current interest rate by simply going long 33% the 3x inverse fund and long 66% the 1x fund to create hedge.

So you put \$33,333 into the -3x spy fund and collect about $4,200 (assuming a 3% interest rate). The \$66,666 goes into SPY and makes zero interest. This creates roughly a perfect hedge. The \$4250 exceeds putting the \$100k in the bank. We're assuming zero volatility and zero fees.

Answer 1

You got it wrong the math says the opposite. According to the equations in the drift of the LETF is negatively impacted by the realized volatility which you completely skipped over.

The realized volatility impact on the drift is displayed on p.20 and is equal to $0.5\beta(1-\beta)\sigma^2 t$ so it is a giant drag on any LETF which is immediately visible if you look at their time series.

This is the main reason why those instruments are quite toxic for non-sophisticated investors who can see their capital decrease even when the market goes in the direction where they would expect to make a win by the mechanical effect of being heavily short gamma.