MotoTrojan wrote: ↑Sat Oct 26, 2019 12:41 pm

Hydromod wrote: ↑Fri Oct 25, 2019 11:24 pm

MotoTrojan wrote: ↑Fri Oct 25, 2019 7:13 pm

I would focus a good bit of effort into volatility drag/decay. Moneymarathon had some great charts showing the impact of regular rebalancing with an uncorrelated asset. Another post (can't recall) posted some fascinating charts that compared annual performance of UPRO to unleveraged S&P500 as a factor of UPRO/S&P500 (daily is 3x, but it showed annual factor). This chart was extremely telling for me personally as it showed that UPRO actually outperforms 3x S&P500 when the S&P500 does exceedingly well (>3x) AND most interestly, when the S&P500 does very poorly with steep declines (<3x). Volatility decay really hurts you primarily during flatter market periods.

This thought was the primary reason for my departure of TMF in favor of unleveraged EDV; equities have a reason to go up over time, but rates could easily bounce around 2% forever while remaining uncorrelated with equities. In a situation like that, the portfolio theory (equity and uncorrelated long bonds) could work, but TMF could be crushed by volatility decay.

Just so we're clear, I think the original poster with the charts was showing that UPRO moved more than 3x S&P500 when the S&P was moving steadily, regardless of whether the change was positive or negative. So you might get +5x and -5x S&P for UPRO, say. I think the wording was something like positive reinforcement, which is easy to misinterpret. The key is that UPRO will move more strongly than 3x UPRO in the direction of change. I certainly wouldn't like that kind of outperformance when the S&P is dropping...

This is incorrect actually. When the S&P500 is steadily declining you get less than 3x exposure, outperforming a true 3x exposure (via futures or similar). Think about it; during the GFC the S&P500 went down almost 50%, which should have put a true 3x S&P500 exposure well into the negative, but UPRO only went down 97%. The only way that UPRO moves more strongly downward than 3x S&P500 is with steady ups and downs, creating volatility decay.

The chart that I referenced showed just this as well.

It depends on whether one is discussing rate of change or cumulative change. I'm claiming that, for a series of the same fractional loss per step, UPRO will cross a given threshold more than 3 times faster than the S&P 500 does. But you are correct, in the sense that the total loss for UPRO will be less than 3 times the total loss for the S&P 500. These are not inconsistent.

Let FV = 1 + f = (1+v)^n, so = ln(FV / (1+v))

Let R = ((1 + 3v)^n - 1) / ((1 + v)^n - 1)

where v is the fractional change per step, FV is the future value, f is the fractional change, n is the number of steps, and R is the ratio of f for UPRO to f for S&P.

Say S&P loses 5%/day, UPRO loses 15%/day.

The number of steps until only 10% of the portfolio is left: (i) for S&P, n = ln(0.1/0.95) = 44.89; (ii) for UPRO, n = ln(0.1/0.85) = 14.17.

So it takes 44.89/14.17 = 3.17 times as many steps for the S&P to drop 90% than UPRO. In this sense, UPRO drops more than 3 times faster.

After 10 days, say, UPRO has lost 80% of its value while S&P has lost 40%, so R is 2. In general, R < 3 whenever v < 0. In this sense, UPRO loses less than 3 times the S&P loss for a fixed number of steps.

The effect reverses when v > 0. In this case, it takes less than 3 times as many steps for S&P than UPRO to cross a given threshold (UPRO increases less than 3 times faster), but R > 3 (the amount of change for UPRO is more than 3 times the amount of change for S&P 500, given a fixed number of steps).

So relative performance is opposite depending on whether one is discussing (i) the number of steps to a reach a threshold or (ii) the amount of change given a fixed number of steps.