Thank you for your input! I'm very grateful for your help; I agree this stuff can be confusing. I realized I didn't explain the mechanics of the computation very well, let me see if I've understood your points.
My understanding was that LETFs don't just borrow 'factor - 1' times their capital, but hold some stock and some swaps. The fraction of swaps and the swap rate (the borrow rate + swap spread) vary by ETF and over time and are difficult to obtain. Thus their cost is "some multiple" times the borrow rate (plus some small fixed transaction fees), where we can't know the multiple; thus (per-ETF) curve fitting. Is there a better way to find this?a) 'S' should be 'factor - 1' (i.e. 2 if leverage factor is 3, etc). This is made clear by researcher articles and by reasoning (there were a few cogent explanations in this thread a while ago). There is no reason to do any curve-fitting here.
My variables are the daily simple returns; after subtracting expenses, they're all multiplied to get the cumulative returns.b) the math implementing the equation should be geometric instead of arithmetic, i.e. a+b really means (1+a)*(1+b)-1.
Just to be extra sure, because this is something I want to be 100% confident on: Daily expenses are exactly the annual expense ratio divided by 252, right? And this is subtracted from the daily return? This is the literal calculation spelled out in the prospectus that funds themselves use to adjust their NAV, yes? (I'm taking after for example DiLellio 2018, equation 3.)
So let's say you had a (magic...) fund that grosses 0.1% per day with an annual ER of 0.05%. That means the net return for one day would be 1.001 - 0.0005 / 252. The cumulative return for the year would the product of 252 of these, (1.001 - 0.0005 / 252) ^ 252 ~= 1.28579. Yes? (I'm ignoring the -1.)
Let's then incorporate a borrow rate of 2% annually. Assuming we borrow 1X for simplicity, then our net cumulative return would be (1.001 - 0.0005 / 252 - 0.02 / 252) ^ 252, yeah? Or maybe it's 0.02 / 360 for rates, but that's the basic structure.
Agreed on interest rates! My approach is to get a good fit / extrapolation with a simple formula, rather than get interpretable fit variables. By analogy, if you're fitting a line y = mx + b for variables m and b, you could fit instead y = (m/2)x + b - 0.1, and you'll get different variables, but the same fit. Similarly, we could fit a borrowing cost of something like (swap_fraction * (borrow_rate + swap_spread)) / 360, but since the swap_fraction and swap_spread (and the 360?? ) are (changing) unknowables, I just lump them together. "something" times borrow_rate. The units and scale of "something" aren't meaningful, but it gives the same fit.c) the '252' math seems strange. I use a daily index data series, so I side-stepped the issue, but as far as I remember, the right divider is 360. But then my model does the math for all days of the year, not only trading days. Interest rates are SO confusing.
d) yes, some sort of curve-fitting factor ends up being necessary to match actuals, but it is clearly not the same across index families and is likely to not be a constant over time for a given index/fund or a function of some other variable condition. Plus we have few years of actuals, so the curve-fitting process is downright dubious. This is all we have though, but this really has to be taken with a big grain of salt.
I agree that curve fitting is fraught. I do fit the parameters per-ETF, but it seems the historical best fit is the best we can do.
Yeah... I guess I'm trying to make it easy to get a pretty-good simulation automatically, without a lot of custom / proprietary digging per ETF. For most ETFs it tracks within a handful of basis points per year, which is approaching hair-splitting territory. I do worry about systematic or conceptual error, which is why it's great to have smart people give it a good once-over.e) leveraged ETFs expense ratios varied over time too... So any kind of curve-fitting would be better achieved when checking ER-free gross returns (I know, this isn't easy to retrieve from public sources).
Yes! That's the next step. Thanks again for helping me on this!f) this is the equation based on daily returns, which is kind of the easy part. What is more tricky is to go further in the past when daily values for the (total-return) index of reference were not available...