I am exploring leveraged/inverse ETF's and came across a paper on how the leverage + daily rebalancing effect works. https://www.math.nyu.edu/faculty/avellane/LeveragedETF20090515.pdf. I have a math question I need some help on. It's how to distinguish the break evens for a long L(everaged)ETF vs a short LETF. For a portfolio where I am long a 3x LETF and short 3*regular ETF. My break-even over a 21 day(one month) period would be: St/S0 = exp(sigmaS^2*(21/252))*(1+/- sqrt(1-exp(-sigmaS^2*(21/252)) Where S is the non-levered ETF. However, I am really confused about the portfolio where we are long a single 2x inverse ETF and long 2x regular etf. The author of the above paper states it as: Where Vt is I am not too sure why we are finding the roots of the cubic equation. I would like to understand the math here rather than plug it into a calculator (starting from bottom-up). Thanks for your time
You got it: the idea is to compare two portfolios. First one: one dollar invested in a 2x inverse ETF Second one: two dollars invested in the regular ETF Over a period of time which one is the better? The values for which the first portfolio equals the second portfolio are the solutions of equation (15). X+ and X- are the positive roots. If X > X+ or X < X- it is better to have invested in the first portfolio. If X is inside ]X-, X+[ it is better to have invested in the second portfolio.