# Leveraging S&P 500 – Quantitative Analysis

*(blog article updated in January 2020 to fully factor in returns from 2019)*

In the past decade, a specialized type of fund gained increased popularity, funds implementing leverage over a given benchmark. The most popular benchmark being the S&P 500, this article will quantitatively explore the outcome of such leveraging technique in the context of a long-term buy and hold approach.

The most common leveraged funds are implemented as ETFs, using a 2x or 3x amount of negative or positive leverage on a daily basis and rebalancing at the end of the day. Such funds can be viewed as passive as their implementation is pretty much mechanical and good providers do indeed track the daily index in quite a rigorous manner. Here are some examples of such funds:

**ProShares Ultra S&P500 (SSO)**and**ProFunds UltraBull (ULPIX)**seek daily investment results, before fees and expenses, that correspond to two times (2x) the daily performance of the S&P 500.**ProShares UltraPro S&P500 (UPRO)**seeks daily investment results, before fees and expenses, that correspond to three times (3x) the daily performance of the S&P 500.

This article assumes that the reader is fairly familiar with the concept of leveraged ETFs. The focus of this article is to provide a set of graphs and statistics about the historical performance of such vehicles and to analyze risks and rewards of such investments. Note that we will only focus on positive leverage, not negative leverage (although both types of funds exist). We will also only focus on individual assets, not on portfolio-level consequences of a combination of assets.

## Real-life leveraging over past decade

Let’s compare the performance of real-life leveraged funds (ProShares UPRO and SSO) against a regular S&P 500 index fund (Vanguard VFIAX) and a regular bond fund (Vanguard VBTLX). Since UPRO was launched mid-2009, we will look at the performance of the 2010-2019 period. Here is a simple growth chart, based on annual (total) returns at the end of calendar years. Note that intra-year ups and downs are not captured on this chart.

This chart certainly looks impressive. Real-life funds would have provided attractive annualized growth (CAGR) from 2010 to 2019 (VFIAX 13.5%, SSO 23.6%, UPRO 32.8% in nominal terms).

Of course, the reality is that such time period was exceptionally favorable to leveraged funds, thanks to a fairly steady bull market which lasted a full decade. In the same way that a bull market would typically magnify leveraged returns, a bear market would magnify losses (as the 2018 bump shows on the previous graph), but UPRO’s history doesn’t go far enough in time to include any major stock market crisis.

Note that even with such very favorable time period, SSO and UPRO delivered less than 2 times or 3 times the CAGR of the regular index fund. It is not the goal of this article to enter in the underlying mechanics of a leverage fund, but two main factors are at play here:

- volatility decay: leveraging magnifies daily volatility, leveraged funds are then rebalanced on a daily basis, the combination of both factors hurts the regular effect of compounding (notably in periods of high volatility).
- cost of borrowing: positive leverage is primarily implemented through the use of swap agreements with investment banks. The daily cost of such contracts is roughly equivalent to LIBOR rates (the cost of borrowing between banks) and further magnified by leveraging. Such daily costs come on top of the regular expense ratio and some additional friction costs, hurting the performance of leveraged funds (notably in times of high interest rates).

## Real-life leveraging through crises

Most leveraged funds are fairly new, but ProFunds UltraBull (S&P 500 2x) ULPIX annual returns are available since 1998, hence spanning the last two major stock market crises which occurred in the US.

Note that ULPIX charges a pretty hefty expense ratio (1.5%), significantly higher than the ~1% charged by the leveraged funds we discussed so far, but this has only a small role to play in the results illustrated below.

Here is the same type of growth chart, starting in 1998, before the Internet crisis and the Financial crisis struck. Clearly, the outcome is much less attractive. ULPIX only caught up to the performance of good old bonds by the end of 2019, after going through two dizzying drawdowns. After more than a decade of funk, a historical bull market was still not enough for leveraged funds to catch up with the corresponding index fund. Note that we are speaking of 2x leverage here, not 3x, which would have been worse.

## Simulating a longer history

Trying to judge the performance of a given investment style over a couple of decades is a fool’s errand, subject to all sorts of issues, notably recency bias, sensitivity to start/end dates and lack of exposure to diverse enough market conditions and human emotions/behaviors.

Fortunately, the implementation of a leveraged fund is pretty much mechanical and can be modeled in a relatively simple manner when daily index data and borrowing rates are available. Daily total returns are NOT available besides fairly recent history though, but S&P 500 price information has been recorded on a daily basis for a long time. It turns out that, based on monthly total-returns and intra-month price volatility, one can approximate a daily leverage model in quite a remarkably accurate manner.

Sparing the reader the details (check this thread on Bogleheads if interested by a deep dive), a simulation model was assembled starting in 1955, leveraging benchmarks like S&P 500, Intermediate-Term Treasuries and Long-Term Treasuries. Testing such model against known actuals delivered excellent results, after accounting for expense ratios and borrowing costs, plus some extra adjustment for other friction costs. In other words, the model seems fairly reliable.

## Historical returns – 20 years periods

To gain a more balanced historical perspective, it is useful to look at periods of time while varying the start date and checking the various outcomes. The following chart illustrates the annualized growth (CAGR, nominal) for investment periods of 20 years, comparing a regular passive index fund with a simulated 2x or 3x leveraged fund. All periods of 20 years have to fit in the 1955-2019 timeframe.

The results seem pretty clear, most 20 years investment periods ended up with the regular fund benefitting from a higher or similar CAGR. When the leveraged funds did win the race, this was for particularly rosy starting points (e.g. late 70s) where it could be argued that the outcomes for regular funds would have been more than good enough. Finally, when leveraged funds lost the race, they often did it in rather dramatic fashion (notably with 3x leverage).

This point is reinforced by a comparison of volatility over the years, where it can be observed that the standard-deviation of annual returns is consistently multiplied by the amount of leverage (or close). The bottomline is that extra risk (volatility) associated with leveraged funds just wouldn’t have provided extra rewards (returns) over such investment period of 20 years.

## Historical returns – drawdowns

Another form of risk, more stressful than simple volatility, is captured by investment drawdowns (the depth of a drop since the latest market high and the duration of recovery). As leveraged funds quickly magnify such drops, it is useful to move from simple annual returns to more detailed monthly returns to fully appreciate corresponding drawdowns.

Drawdowns (sudden loss of value) will get amplified in both depth and duration. The following chart presents a monthly simulation of a regular passive index fund, a 2x leveraged fund and a 3x leveraged fund, all tracking the S&P 500 index from 1955 to 2019.

As you can see, the already very disturbing drawdowns of the regular S&P 500 index fund (blue line, 40% to 50%) would have been magnified by leveraged funds and would have become drawdowns of 70% to 90% or more. Imagine starting with $1M and witnessing your investment fall to $100,000…

Furthermore, the time to recovery was considerably stretched out. The oil crisis in the 70s would have been magnified with dizzying drops and drawdowns lasting more than a decade. The more recent Internet and financial crises would have led to nearly two decades of misery, finally followed by a steep recovery. Staying the course with a long term investment in leveraged funds during such crises would seem extremely challenging, to say the least.

## Historical returns – risks and rewards over time

As illustrated by the charts in previous sections, leveraged ETFs proved much more volatile than corresponding indices (and index funds).

The question becomes the following. If a steely investor had been able to cope with such increased volatility (including dizzying drawdowns), what would have been the rewards for staying the course? We studied 20 years periods in a previous section, what about other period durations?

Coming back to real-life funds, between Jan-10 and Dec-19, the standard deviation of monthly returns was amplified by approximately the amount of leverage:

- VFIAX (regular S&P 500 passive index fund from Vanguard): 12.5%
- SSO (ProShares Ultra S&P500, 2x leverage): 25.3%
- UPRO (ProShares UltraPro S&P500, 3x leverage): 38.4%

The historical simulation of leveraged funds over the 1955 to 2019 time frame showed similar (aggregate) characteristics for the volatility of monthly returns:

- Regular S&P 500 passive fund: 14.4%
- S&P 500 2x leverage: 29.0%
- S&P 500 3x leverage: 43.6%

The usual risk (monthly volatility) vs. return (CAGR) chart summarizes those findings, displaying a clear lack of rewards for the risk being taken.

Using again the historical simulation, we can also study rolling (annualized) returns while varying the duration of rolling periods.

As the table indicates, over short periods of time, leveraging can deliver stronger returns, albeit with a lot of risk. Over longer periods, the average return premium tends to disappear while the risk (dispersion of outcomes) remains very acute.

Overall, the case for rewards (i.e. improved annualized returns) seems remarkably weak for long-term buy and hold investments, compared to volatility and drawdown risks as previously quantified.

## Historical returns: monthly accumulation

Now, what about the scenario of an accumulator who keeps contributing to a leveraged fund (e.g. thanks to steady employment)? In such a scenario, drawdowns could be viewed as opportunities to buy shares at a low price. It still stretches belief that one would have the nerves of steel required to stay the course for 20 years though extreme volatility and protracted drawdowns, but let’s quantify such accumulation scenarios.

Let’s assume that the accumulator saves and invests $1000 every month, rain or shine, for long time periods, e.g. 20 years in a row (hence a total of $240,000 invested over time). The following computation is performed in inflation-adjusted (real) terms to compare apple to apple. Varying the starting year of such time periods (horizontal axis), here are the outcomes for a regular vs. leveraged (2x or 3x) fund. The vertical axis is the (inflation-adjusted) value of the portfolio at the end of the 20 years time period.

It takes a bit of unpacking to interpret those results. At the first glance, the case for leveraging doesn’t seem as bad as in the previous sections. There is a decent number of time periods where the red dots (leveraging) ended up above the blue dots (regular fund), actually roughly half of the time for 2x leveraging and roughly one third of the time for 3x leveraging. We even have a few time periods (e.g. starting in the late 70s) where the red dots ended up off chart (that is, above the $1.6M arbitrary limit set on the vertical axis for readability’s sake). The whole picture is by no mean terribly attractive, but some might possibly be tempted to take a gamble.

Unfortunately, there is no way of predicting in advance if rosy scenarios will unfold. And when scenarios turn out dire, then we can see on the graphs that leveraged funds would make things much worse than a regular passive fund would. Overall, unless one has a crystal ball, it would seem foolish to tempt fate in such a way instead of just taking what a regular index fund would provide.

## Conclusion

In short, leveraged ETFs are specialized products, which present major risks as long-term buy and hold investments and little rewards in return for such risks. The clear warnings from SEC and FINRA about such products are undoubtedly based on verifiable facts.

## Future work

In follow-ups to this article, the author plans to:

- Provide a portfolio-level analysis, combining leveraged funds of a different nature (e.g. stocks and bonds) in a portfolio and see if (lack of) correlation between such assets can lead to better results at the portfolio level than at the asset level.
- Provide a full description of the logic and data sources used for the historical simulation of leveraged funds.