Mathematical Breakdown Of Leverage Decay And Compounding

| About: VelocityShares 3x (UWTI)


Leveraged funds create an illusion that is initially abstract and enticing to investors but mathematically eliminates holding as a long-term objective.

In simplified analysis, I will review the long-term consequences of holding a leveraged fund and analyze exactly how much decay can be expected and the minimum return required.

Regardless of growth in the underlying asset, long-term investments in leveraged funds will depreciate an investor's funds to zero if held indefinitely.

Leveraged Funds, including ETFs and ETNs, are investments that seek to replicate some multiple (usually 2-3x) of the daily performance of the underlying fund. These funds use derivatives trading, such as options and futures, to generate a daily return of some multiple for investors. Consequently, these funds typically have large expense and management fees, as compared to strictly holding the underlying asset. However, these expenses are trivial compared to the decay and compounding effects we will later discuss.

Some popular leveraged funds are:

  • Direxion Daily Gold Miners Index Bull 3x Shares ETF (NYSEARCA:NUGT)
  • Direxion Daily Gold Miners Index Bear 3x Shares ETF (NYSEARCA:DUST)
  • VelocityShares 3x Long Crude Oil ETN (NYSEARCA:UWTI)
  • VelocityShares 3x Inverse Crude Oil ETN (NYSEARCA:DWTI)
  • ProShares Ultra S&P 500 ETF (NYSEARCA:SSO)
  • ProShares Ultra VIX Short-Term Futures ETF (NYSEARCA:UVXY)
  • ProShares Ultra Bloomberg Crude Oil ETF (NYSEARCA:UCO)
  • ProShares UltraShort S&P 500 ETF (NYSEARCA:SDS)

For example, UWTI is a fund that replicates 3x the daily performance of crude oil. It is very closely tied to the trend of the United States Oil ETF (USO), the 1:1 based commodity investment in crude oil. In reality, as with any leveraged fund, UWTI will only replicate 3x the returns of the underlying funds during an intra-day position. Although I use UWTI as an example in this article, the concept of leverage decay and compounding apply to ALL leveraged funds.

Due to compounding and decay, or beta-slippage, leveraged funds are meant to be extremely short-term investments and many brokers highly discourage holding them for long-term investments. Intra-day trading does not affect decay or compounding, as it takes more than one trading session to occur. However, many investors have been fooled into using these leveraged funds to avoid trading on margin (as it appears at first to have the same effect without high interest and margin calls) and to provide extra leverage to their portfolio.

These types of funds have become widely popular, especially among millennials, looking to find an easy way to riches. The popular subreddit, Wallstreetbets, mentions leveraged funds more often than any other type of fund due to the investment's "all or nothing" strategy. Unfortunately, many young investors do not know the consequences and do not realize that the odds are stacked against them and they would probably be better off in a casino. Again, these types of funds are meant for day traders and experienced professionals that know the risks and are looking to benefit from a large swing in daily volatility.

While maximizing potential gains, investors in these funds are simultaneously increasing their risk considerably. Unlike typical investments, these types of funds do not grow with time; they will always underperform the underlying asset over the long term. Even if the underlying fund grows with time (such as the S&P 500, which historically grows over time), eventually, volatility will cause the leveraged fund to underperform, closely examined later.

Conclusively, these funds should only be held for longer than one day when the investor strongly believes that there is a massive potential upside within the short term. Depending on the volatility of the fund, and the analysis we will perform below, I predict that in order to counteract negative compounding and inherent decay, a minimum expected return required in the underlying asset could range from 3-15% over 31 days or 20-50% over the course of 365 trading days.

The only way this decay could have been counteracted is if an investor created 3x leverage through borrowing, or trading on margin, at a virtually 0% interest rate, which is impossible due to time-value of money.

Notice the graph below and the performance of UWTI (the 3x leveraged fund) versus USO, the basic 1:1 ETF for oil investment.

If the above 3-year graph of USO in blue, UWTI in yellow, (found here using the compare feature) is not convincing enough, perhaps look at UWTI's price per share. In 2012, a share of UWTI was over $5,000 and is currently trading under $20 a share. That's a 99.6% loss due to decay and compounding.

More recently, on January 20th, 2016, USO closed at $8.24 a share, while UWTI closed at $15.90 a share. While this is not the bottom of bear territory, it is close enough to notice the decay. On August 3rd, 2016, USO closed at $9.69 and UWTI at $18.01. If you do the math, you will find that an investment in USO in January returned 17.6% while UWTI returned 13.3% to date. This return in UWTI excludes the investor's fee. What happened to the 3x leverage? Shouldn't UWTI have made 52.8% by definition? Yes, only if USO rose 17.6% in one day, not over six months.

What's even more shocking is that USO also has decay, even though it is supposed to exactly match the daily performance of crude. The historical close of a barrel of crude oil on January 20, 2016 was $26.44, compared to $41.14 today, representing a 54.9% gain, opposed to 17.6% with USO. The only way to safely invest in a commodity seems to be through purchasing the physical commodity and storing it in your backyard. Hmm…

The above graph (found here) shows the 1-year relationship between crude oil in blue, USO in orange, and UWTI in purple. Thus, the focus remains on UWTI, the leveraged fund, as it has decayed the most.

Now let's assume an investor is determined to invest in that 3x leveraged fund. How long can they hold it? Later on, we will do some analysis to see just how long an investor can hold a leveraged fund.

Let's assume for theoretical analysis that 1 barrel of crude oil is worth exactly $50.00 and UWTI is trading at exactly $40.00. This approximates a relationship that occurred in early June of 2016, when oil touched its year-to-date highs.

Let's see what happens when oil falls 2% on a given day.

Day 1

Crude Oil (per barrel)


Starting price



Ending price

$49.00 (-2%)

$37.60 (-6%)

Here, an investor would have lost 6% of their investment in UWTI for the single day, as opposed to an investor in 1:1 crude oil that lost 2%.

It makes sense that since oil decreases in price, rising $1.00 is a higher percentage of $49 as it is of $50. To compute that percentage, $1/$49 = ~2.04%

Now, assume that oil recovers its $1.00 decline and rises 2.04%.

Day 2

Crude Oil (per barrel)


Starting price



Ending price

$50.00 (+2.04%)

$39.90 (+6.12%)

Here, Crude oil recovered its initial loss, but UWTI is still $0.10 or .25% short of its full recovery.

Any investor that held onto UWTI during the decline AND reversal of losses, missed out on .25% of the rise due to decay.

However, if an investor purchased UWTI at $37.60 before it began to rise, they would outperform crude oil by exactly 300%, or 3x for that single day.

Now, one may ask, if there is compounding on the way down, wouldn't compounding on the way up negate this effect and act in an investor's favor? Well… let's go back in time.

Day 1

Crude Oil (per barrel)


Starting price



Ending price

$51.00 (+2%)

$42.40 (+6%)

In this case, UWTI imposed the same 6% increase of $2.40 as it did during the decline, evident in the previous example.

Let's see what happens when crude oil declines and returns to its $50.00 per barrel level.

Day 2

Crude Oil (per barrel)


Starting price



Ending price

$50.00 (-1.96%)

$39.90 (-5.88%)

Woah! In the previous example, oil declined first and then recovered; now oil rose first and then declined. In both scenarios, $0.10 of the underlying value in UWTI ceased to exist. Shocking.

So, basically, because a certain percent of a higher number is a higher number, and a certain percent of a smaller number is a smaller number, the leveraged fund will always lose out whenever the market fluctuates.

Watch oil fluctuate over a 30-day period at a constant volatility of +/-2%

Crude Oil (per barrel)


Starting Price



Day 1 +2%, 6% respectively



Day 2 -2%, 6% respectively



Day 3 +2%, 6% respectively



Day 4 -2%, 6% respectively



Day 5 +2%, 6% respectively



Day 6 -2%, 6% respectively



Day 7 +2%, 6% respectively



Day 8 -2%, 6% respectively



Day 9 +2%, 6% respectively



Day 10 -2%, 6% respectively



Day 11 +2%, 6% respectively



Day 12 -2%, 6% respectively



Day 13 +2%, 6% respectively



Day 14 -2%, 6% respectively



Day 15 +2%, 6% respectively



Day 16 -2%, 6% respectively



Day 17 +2%, 6% respectively



Day 18 -2%, 6% respectively



Day 19 +2%, 6% respectively



Day 20 -2%, 6% respectively



Day 21 +2%, 6% respectively



Day 22 -2%, 6% respectively



Day 23 +2%, 6% respectively



Day 24 -2%, 6% respectively



Day 25 +2%, 6% respectively



Day 26 -2%, 6% respectively



Day 27 +2%, 6% respectively



Day 28 -2%, 6% respectively



Day 29 +2%, 6% respectively



Day 30 -2%, 6% respectively



*may be slightly off due to rounding

Now, assuming oil's intrinsic value is $50.00, the barrel of oil, due to compounding, is 0.6% from its intrinsic value. The valuation, and whether oil will return to $50 or if it actually trended negative for the month is all irrelevant, but the discrepancy between $50 and $49.70 becomes very much relevant when compared to UWTI.

UWTI went from $40 to $37.89, losing ~5.275% of its intrinsic value due to compounding over time while a barrel of crude oil changed by .6%. Had this variance been over a single day, the .6% could be multiplied times the leverage of 3x to get an adjusted expected loss of 1.8% due to the by definition 3x loss/return risk in oil of the ETF. Thus, the net decay is 5.275% minus 1.8%, which equals 3.475%. Conclusively, UWTI will never be able to trade at a $40 equivalent for a $50 barrel of oil, ever again. This example is based on a period of time where the price of oil remains stagnant, while more volatility would cause a much more dramatic effect on UWTI's relationship with crude.

If you look at UWTI and the daily decay, you will see that it loses about $0.14 to $0.15 of value every oscillation (i.e.: start minus day 2 close: $40.00 minus $39.86 = $0.14) and that averages to be about $0.075 a day.

Now, assuming this pattern with UWTI continued for 365 trading days from the purchase date at a price of $40, we can calculate an expected price after 365 days. You can calculate this yourself using excel and inputting $50 in Cell A1 and formula "=1.06*A1" in Cell A2 and formula "=.94*A2" in cell A3 and then highlighting cells A2 and A3 and dragging them down to cell A366. The same concept can be used to replicate the change in price of a barrel of oil. The result is that UWTI will decay to $21.99, as compared to $47.42 per barrel. Again the net decay is calculated as (percent change of UWTI) minus (percent change of a barrel of oil times the leverage factor) = 45.025% minus (5.16%*3) = 29.55%.

Let's increase the volatility on crude oil from 2% per day to 3% per day, or 6% to 9% in the case of UWTI.

Using excel, we come up with the following results:


Crude Oil (per barrel)


Starting Price



After 2 Days



After 30 Days



After 365 Days



Here, the decay factor is 75.1925% (net change in UWTI) minus 37.68% (3x change in crude) = 37.51%

In other words, the total loss is 75.1925% of the initial $40 per UWTI, while the loss due to decay is 37.51% and the loss due to an investment in oil is 37.68%.

Now, assume on the 366th day crude oil rises 14.38% to the beginning $50.00 per barrel. UWTI rises 43.1% to $14.20. An investment in crude oil broke even, while an investment in UWTI lost 64.5% on the initial investment, despite the one day gain.

This method for calculating expected leverage decay ONLY works if volatility is estimated, which is, unfortunately unpredictable. There would be extreme variation in the resulting decay if volatility was larger and less if it was smaller. It would also be wrong to assume that "if the average volatility in oil is X, then the decay will be Y," as larger volatility can dramatically change the outcome and cannot be depicted through averages.

For example, let's take an average of the following volatility in the daily change of the price of oil (or, in this case, percent changes)

Theoretical daily volatility: -33.3%, +1%, +2%, -4%, +3%, +20%

Arithmetic mean of the six days = -1.888%

Using the model in the previous example yields an AVERAGE that indicates a small decay over 6 days, of little significance. Also, due to compounding returns, an average would not suffice. However, it would be inaccurate as on Day 1, oil fell 33.3% and UWTI collapsed to zero by definition of its 3x leverage. Although extremely unlikely, the odds of it happening are not zero and we cannot ignore a collapse or large swings due to volatility. Thus, we cannot rely on an average if we expect large swings in volatility, as we will greatly lose accuracy. Hence. We must use this idea as an approximation only.

Here's how estimating volatility can be useful:

Assume: Crude oil is $50/barrel, UWTI trading at $40, +/- 2% oscillations

At what point would an investment in UWTI break even over 365 days?

The easiest way I see to calculate this would be by looking at our first example, and the daily decay. As we previously mentioned, the decay for UWTI appears to be 14 to 15 cents per +/-2% oscillation (at least initially). This is enough information to compute the percent decay on day 1. $0.15/$40 = .375% Therefore, we can conclude that UWTI has to rise in excess of .375% every other day or .1875% per day to counteract the expected decay, assuming the price of oil oscillates +/-2% every day, consistently. This means that .1875% per day, compounded over 365 days is an effective rate of return of 98.13%. That means that UWTI needs to have an overall return of 98.13% to stay at $40.00 per share after 365 days. More intuitively, a barrel of oil needs to rise in price by 32.71% just for UWTI to break even.

In conclusion, it is important for investors to be aware of the consequences of holding leveraged funds for a long period of time. They are ideal for investors looking to benefit from short-term spikes, but are typically not suitable for more than a week. Furthermore, leveraged funds will result in the disappearance or elimination of an investor's funds if held indefinitely.

Important Note: While I use 30 days and 365 days as time measures throughout the article, it is important to remind readers that the market is not open every day, and thus 30 and 365 represent 30 and 365 trading days, respectively. 30 trading days is about 40 calendar days and thus 365 trading days is about 1.3 years, with some variance due to holidays and weekends in which a calendar month includes.

Disclosure: I am aware that there is more in depth analysis found here, which utilizes calculus and highly detailed mathematics that better derive leverage decay and perhaps approximate a formula to calculate it. However, as a self-taught 20 year old, my intention was to give a basic understanding to others that had the questions that I had about a year ago and want to know the consequences of investing in a leveraged fund without hours of derivation.

As I am still learning myself, I encourage readers to correct me if I went wrong somewhere and post their insight as well. In addition, I welcome readers to share any simplified formulas they've found that calculate decay over time more efficiently and accurately than I have shown. I would also be interested in knowing about whether a 100% directly correlated fund in oil would be possible, i.e. set up a fund with a per share price identical to the price in oil. Let me know if this is possible, without decay.

For the purpose of this article, I have used USO as an example of a non-leveraged investment in oil. I am aware there is time decay in USO and a discrepancy between the rolling costs in futures contracts since they are negotiated on a monthly basis.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

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