The techniques described in this article are aimed at reasonably sophisticated investors possessing a suitable comfort level with options, or those who care to get there. If you feel you might benefit, please read on.

Since I’ll be discussing leveraged ETFs, you might want to read the risks and characteristics described by FINRA and the SEC as linked below:

## Preface

Wikipedia defines tail risk this way, and it’s easy to find a dozen similar definitions on the internet.

*“Tail risk, sometimes called “fat tail risk,” is the financial risk of an asset or portfolio of assets moving more than three standard deviations from its current price, above the risk of a normal distribution. Tail risks include low-probability events arising at both ends of a normal distribution curve, also known as tail events. However, as investors are generally more concerned with unexpected losses rather than gains, a debate about tail risk is focused on the left tail. Prudent asset managers are typically cautious with the tail involving losses which could damage or ruin portfolios, and not the beneficial tail of outsized gains.”*

I won’t quibble about the “normal’ distribution reference, but aside from that, I think the definition is misleading in its presumptions and tone. It leads to fallacious thinking.

Why focus on a bell curve which emphasizes the high-probability small impacts when the biggest impacts occur at the left and right tails. And why ignore the right tail when it provides the lion’s share of returns. The top 1% of daily returns is equivalent to 80% of all returns for the S&P. And the top 1% occurs once every 100 days on average. The same is true of the bottom 1% of days but obviously in the opposite direction – much pain every 100 days.

Rather than a bell curve, consider a graph of the impacts. This graph shows the sum of uncompounded daily returns over the last 15 years sorted in 100 equal-probability buckets. The “outlier” buckets are just as likely to occur as the middle buckets; each occurred about 39 times out of 3,906 days.

And who decided that a “tail” is limited to 3-sigma outcomes? The worst 5% of daily S&P returns equate to about 2.7 times all returns for the S&P, and the top 5% is an occurrence that pops up every 20 days. Tails matter!

**Background – Tail Risk of Unleveraged S&P**

Most investors tend to be under the mistaken impression that normal day-to-day gains and losses accumulate to long-term results. They focus on what’s normal and consider outliers to be anomalies. The truth is that big days swamp the middle 80% of daily returns.

Over 15 years, the sum of daily S&P returns without compounding was 235%. The best 2% of days alone beat that, summing to 326%. The following table shows how daily outlier results for SPX have a dramatic impact on investment results.

Nassim Taleb had a colorful way to illustrate a normalcy bias:

“Consider a turkey that is fed every day. Every single feeding will firm up the bird’s belief that it is the general rule of life to be fed every day by friendly members of the human race “looking out for its best interests,” as a politician would say. On the afternoon of the Wednesday before Thanksgiving, something unexpected will happen to the turkey. It will incur a revision of belief.”

― Nassim Nicholas Taleb

And here is a related quote:

“The fragile wants tranquility, the antifragile grows from disorder, and the robust doesn’t care too much.”– Nassim Nicholas Taleb

Now, we’ll discuss how to implement a “robust” investment approach.

**Tail-Risk Hedges**

My focus here will be using UPRO, a 3X-leveraged S&P ETF, along with weekly XSP puts to eliminate the worst weekly losses while preserving the best weekly gains. XSP tracks 1/10^{th} S&P prices and returns.

Since the discussion will focus on UPRO with XSP hedges, I should explain the reasons for using a leveraged ETF. There are four reasons for this preference:

- Risks can be constrained using two tactics described here, and
- 20% portfolio commitments produce comparable returns to a 60% S&P commitment while freeing up large amounts of capital for yield holdings, and
- When exiting on risky intervals, as measured by changes in the VIX term structure, any volatility decay normally associated with a leveraged ETF is eliminated, and
- This factor might be most important of all, but it’s only enabled by the risk approach. Once tail risk is constrained with certainty by the option approach described below, an investor can hold leveraged positions with absolute confidence that the craziest of days can be tolerated. So, it eliminates those irrational exits driven by news cycles and fundamental fears and can even facilitate expanded commitments beyond the nominal 20% referenced in #2 above.

In a few paragraphs we’ll briefly discuss the algorithm underlying that third point, but first let’s focus on tail-risk hedges.

I’ve recently decided to use UPRO rather than SPXL; its more faithful tracking of 3X S&P returns makes it superior to SPXL for this purpose. Tracking errors are important because while UPRO provides the underlying position, XSP options will be used to tailor the risk profile. XSP trades at 1/10^{th} S&P levels and options provide daily expirations and a useful spectrum of strikes. The options also settle in cash which is helpful for more complex structures not discussed here.

That UPRO/XSP combination dramatically constrains risk while enhancing returns. This graphic illustrates why that is true.

Note by the way, that the horizontal axis in this graph is not linear because it spans prices at equally distributed confidence intervals from 1% to 99%. Also, the Monte Carlo analysis underlying the graph considers ‘In’ periods only in accordance with the 15-year data of the EZV algorithm.

XSP trades at about 9 times UPRO quotes, so even though UPRO is 3X-leveraged, in the current volatility environment, XSP options can provide heavy protection for about 1.2% of the UPRO investment.

To really understand the dynamics of this structure it will take some effort, but it could change your investment approach dramatically. A few reference points might be helpful. Last Friday, at the time of the analysis, UPRO traded at $51.33. Over the average week (average of gains and losses), UPRO sees a gain of 0.78%, but the average upsides-only week sees a gain of 5.2%, so the 1.2% put protection cost is about 23% of the *average* upside outcome.

But once again, it’s not about averages. Look at both tails. On the right, as prices rise, the put-option cost is fully depleted while UPRO gains continue to run. In fact, very little gain or loss is realizable in the space between the UPRO starting price and the point at which the puts cease being any further cost burden. On the left, the deltas* of those XSP puts approach one and since XSP changes are multiples of UPRO changes, the losses turn to gains at the lowest of price outcomes.

In summary, gains are preserved while losses are curtailed, and at the worst potential-loss outcomes, the tail risk is flipped to create additional rewards.

** Note: Option deltas reflect the change in option value for any change in XSP. *

For those willing to put in the effort, this table shows many input and output details of the tail-risk hedge strategy’s Monte Carlo analysis. It happens to be the Monte Carlo that produced the above chart.

To point out a few details, the 98%-confidence loss potential is (2.8%), so I’d expect that once every 50 weeks on average. The worst single week of the 63,000 sample weeks in the Monte Carlo simulation produced a loss of $15,510 or (3.8%); I don’t expect to see that in my lifetime, but one never knows. Meanwhile, the average gain is expected to be 1.7% and one-in-fifty weeks should see a $54,148 gain. The downside is constrained, and the upside runs. Repeating the strategy every week would produce an 84% return, although with implied volatilities so low, that represents a better-than-normal opportunity, so I don’t expect sustained annual returns that high.

Multiple alternative structures like this one are produced weekly within the EZV service so that anyone can choose in accordance with any individual’s risk appetite. We’ve been running these structures since April and results have been outstanding.

**What About Sustained Downturns?**

The Monte Carlo analysis underlying the above graph and discussion considers ‘In’ periods only, in accordance with the EZV algorithm. The tail-risk hedges would be very useful on their own, but when markets cycle downward for multiple weeks, even constrained weekly losses can add up. So, with or without the hedges, it is very helpful to just exit occasionally when risk is elevated. Then the question becomes what sort of metrics can successfully identify elevated risk. News cycles and fundamental analysis are terrible indicators. Most investors, including me, have been expecting a recession for a year now, but if acted upon, that fear could easily have sidestepped 20% gains this year. And how could anyone hope to succeed by reacting to every Fed pronouncement or CPI release?

The truth is that the most prevalent effect of following news cycles and fundamentals is to undermine confidence. The result is under committing to positions or needlessly riding out gainful periods from the sidelines.

The best risk indicator I have found relies on measuring changes in the VIX term structure. This graphic shows the first four months of VIX futures (“VX”) as of the Friday, July 28^{th} settlement; it’s the solid black line. The shape is contango, meaning near-dated quotes are lower than later-dated quotes. Taken alone, contango shapes indicate low risk. The opposite, backwardated, would slant downward from left to right and would correspond to more risky times.

Think of it this way. Far-dated quotes gravitate toward a normal condition because no one knows what the future holds, but near-dated quotes reflect the market’s more volatile assessment of current conditions. Since the VIX and VX futures reflect S&P options’ implied volatilities, lower-than-long-term quotes reflect calm, and higher-than-long-term quotes reflect stress. Hence the shape of the curve is indicative of VX traders’ consensus assessment, the wisdom of the crowd so to speak.

But the shape itself is only a start at making a risk assessment. Just as prices do not usually drop 20% in a single day, the VX curve does not shift from a strong contango shape to a backwardated shape quickly. So, the rate of change in that shape is critical to a valid assessment of conditions. This chart shows three metrics: the SHAPE – a quantified metric derived from the colloquially stated ‘shape’ of the curve, a short-lookback slope, and a longer-lookback slope.

The red dotted line is a breach line. The full algorithm consists of two core models, EZV-1 and EZV-2, with interpretive protocols for merging them into signals. And while I won’t go into how these above-the-breach lines are interpreted, I will say current conditions are fairly healthy. In fact, the algorithm has been ‘In’ the market for the last 90 trading days, accumulating gains of 22% when aimed at unleveraged SPY positions. My personal fear-and-greed instincts would have never been so well invested.

If you’d care to see all the signals, as-modelled since 2008 and as-called since inception in 2019, feel free to peruse Signals of Record. They are updated on a lag, so you won’t see current or even near-recent data.

**Closing**

These two risk tactics have made a big difference for me personally. While the algorithm alone has produced a doubling of S&P returns with a quarter of the drawdowns, my fear of black swans and a general discomfort around Fed actions and recession probabilities since 2022 had left me too cautious. I have traditionally targeted a minimum commitment to 3X-leveraged positions (during ‘Buy/In’ intervals) equal to 20% of my portfolio. In the first two months of 2023, I maintained a variable 10% to 20% commitment, preferring preservation of capital, carrying over my 2022 posture. Today, I’m running a 20% commitment to 3X ETFs consistently, UPRO this week, subject to occasional sell signals, and I plan to move up toward a 30% commitment on the next opportunity.

I hope you found this helpful. My approach is clearly unconventional and looks nothing like a traditional 60-40 buy-and-hold portfolio. But it works for me, and some of this just might set you to thinking on adjustments to your own approach.