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Mathematical Breakdown Of Leverage Decay And Compounding


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...