See part one

It took longer to complete this section than expected owning to the difficulty of modeling historic stock market returns. Market returns to some degree follow a normal distribution, but ,rather, markets are more characterized by regimes. This means there are long periods in which market does well, followed by long periods in which it does poorly, and there is no easy way to time this. Either you are in a propitious regime, or not. Sometimes what may seem like streak of mediocre years, such as from 20156-2016, the market can suddenly and without warning surge higher, as we saw after Trump won. Other times, the market crashes like it in 2008 following a medicare 2007. The decade from 2010-2020 was arguably one of the best ever. But the period from 2000-2010 was one of the worst. The ’80s and ’90s were superb, but the ’70s weren’t. The difference in terms of performance of good regimes versus bad ones is so abrupt and stark, that it’s hard to model a it as continuity. Using data from NYU, the S&P 5000 posts, on average historical return of 11% /year. This includes dividends. Excluding dividends, it’s more like 9%. This will be used for future calculations. So in spite of these unpredictable and high variable regimes, the market, over the long run, posts a very consistent return that has held over half a century, but the problem is the variance.

The total amount of data [1] to work with is approx 90 years. What is needed is a sample size to estimate a market cycle. This will contain a bear market and bull market yet equal to the expected annual compounded return of around 9%.

Running some analysis on the data, we can estimate a market cycle length of 13 years, which is about 1/7 of 90.

So we can guesstimate there is:

1 year (out of the sample size of 13) in which the market falls a lot (-25%)

2 years in which it falls a little (-5%)

2 years in which it gains little (5%%)

3 years in which it gains more (15%)

2 years in which it gains a lot (20%)

2 years in which it gains considerably (25%)

1 year in which it gains immensely (30%)

The second and third rows effectively cancel out, so this can be approximated as four years in which the market is flat.

Taking the product of this, however the total exceeds the average annual compounded return of 9% a year. To militate this, I decided to make the first column a variable, which is solved. This gives a decline of 31% instead, which is still in line with historic records.

So that completes the step of determining a suitable typical market cycle for which the method can be backtested against.

The next step is approximating the average annual decay of 3x leveraged ETFs.

i will defer to Zang 2010 agian and Part 3 of the 2016 guide.

There are a lot of valuables but most of them cancel out.

The 3x S&P 500 etfs, SPXL or UPRO, have a 1% expense ratio. But because the underlying index pays approx a 2% annual dividend, this is added to the NAV (net asset value) of the ETF continuously as opposed to being paid out discretely as dividends, for a total of 6%/year, which helps considerably to offset decay.

This means that the funding rate and expenses ratio can be cancelled out by dividends. For reference, the funding rate is the cost to borrow at 2x margin, which is 2 times the present interest rate. We are assuming 2% interest rates and a 2% dividend yield, and although these are obviously subject to change; the change is gradual I don't think will hurt the performance of the method.

B=3, meaning a 3x ETF. To calculate volatility, we need to compare the historic UPRO performance with the underlying index, the S&P 500. Also, we are solving for the realized volatility, not the vix-volatility. A common mistake is confusing realized volatility for expected volatility, the latter which the VIX measures. Because the former can be significantly less than the latter, using VIX-volatility will overestimate decay.

Between Jan 20th 2015 and Jan 20th 2020, the S&P index gained 62%

The 3x index, UPRO, gained 248%

So using the above equation (b=3, t=5, etc.), solving gives a realized 5-year volatility of 11.5%, or .115

It is possible to do even better. The Problem with the S&P 500 is it contains mediocre sectors such as utilities, commodity companies, and energy companies.

The The DJIA, although it is only composed of 30 stocks, has much higher quality companies but with less volatility than the S&P 500, yet the returns are even greater.

So running the above calculations for the DIA, the underlying index gained 70% between Jan 20th 2015 and Jan 20th 2020.

The 3x ETF version, UDOW, gained 330%. You can already see the difference between UDOW and UPRO is huge. Solving gives a much smaller volatility of .085. This is attributable to the huge out-performance of defense stocks and Boeing in the year following Trump’s win, but since 2018 the S&P 500 has has done somewhat better.

What about the Nasdaq 100? The 1x Version, QQQ, gained 120%. The 3x version, TQQQ, gained 560%. Solving gives a very high volatility of around 18%, which is expected given that tech stocks tend to be more volatile than defensive sectors. This is high enough that holding TQQQ is like holding 300% of the QQQ, without any secondary and third order compounding effects. However, the benefit is that unlike buying on margin, you cannot be liquidated (the worst that happens is the ETF goes asymptotically close to zero) and the funding rate is around 2.5% a year instead of 5-7%/year, which is what a broker changes. So you come out ahead. You are getting cheap leverage with no liquidation risk.

I think going with TQQQ is the superior choice. The problem is historical back-testing does not work that well since the data is so overwhelmingly skewed by the 1995-2003 tech bubble and crash.

This method , mentioned in part 1, involves a combination of cash, 3x ETFs and 3x ETF options. however, upon analysis the options last week I found they are not suitable, because the strikes do not go out far enough, the spreads are too wide, and insufficient liquidity. I cannot develop a trading for something that does not exist. So this means that far out of money SPY options will be used instead. This are easy to trade and have narrow bid/ask spreads.

[1]

market returns: positive vs. negative years – ICMA-RC

S&P 500 Historical Annual Returns

S&P 500 Total Returns by Year