Leveraged Portfolios – Quantitative Analysis

In the past decade, a specialized type of fund gained increased popularity, funds implementing leverage over a given index. A previous article explored funds leveraging the S&P 500 index. This article will focus on a portfolio-level approach mixing stocks and treasury bonds, to quantitatively explore the outcome of leveraging 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 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 tracking US stocks (S&P 500), intermediate-term treasuries (ITTs) and long-term treasuries (LTTs):

Leveraged fundIndex
ProShares Ultra S&P500 (SSO) S&P 500 2x
ProFunds UltraBull (ULPIX)S&P 500 2x
ProShares UltraPro S&P500 (UPRO)S&P 500 3x
ProShares Ultra 7-10 Year Treasury (UST)ITT (2x)
Direxion Daily 7-10 Year Treasury Bull 3X Shares (TYD)ITT (3x)
ProShares Ultra 20+ Year Treasury (UBT)LTT (2x)
Direxion Daily 20+ Year Treasury Bull 3X Shares (TMF) LTT (3x)

The ITT index is the ICE U.S. Treasury 7-10 Year Bond Index. This index is relatively new (it started in 2016); a longer-lived and very similar index is the Bloomberg Barclays US Treasury 7-10 Yr Index. Note that the maturity target (7-10 Year) is somewhat unusual for ITTs, regular passive funds tend to track a 5-10Y index or a 3-10Y index.

The LTT index is the ICE U.S. Treasury 20+ Year Bond Index. This index is relatively new (it started in 2016); a much longer-lived and very similar index is the Bloomberg Barclays US Treasury 20+ Year Index.

This article assumes that the reader is fairly familiar with the concept of leveraged ETFs. The point 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. This article will not focus much on individual funds, it will primarily focus on portfolio-level analysis of a combination of such funds. Note that we will only study positive leverage, not negative leverage (although both types of leveraged fund exist).

Real-life leveraging over past decade

Most of those leveraged funds started in 2009 or a few days within 2010, so we can look at the past decade (2010-2019) using simple growth charts, based on annual (total) returns at the end of calendar years. Note that intra-year ups and downs are not captured on such charts. We can compare:

  • The performance of real-life leveraged funds (ProShares SSO and UPRO) against an S&P 500 index fund (using Vanguard VFIAX).
  • The performance of real-life leveraged funds (ProShares UST and Direxion TYD) against an ITT index fund tracking the same index (using iShares IEF).
  • The performance of real-life leveraged funds (ProShares UBT and Direxion TMF) against an LTT index fund tracking the same index (using iShares TLT).
S&P 500 leveraged funds growth between 2010 and 2019
ITT leveraged funds growth between 2010 and 2019
LTT leveraged funds growth between 2010 and 2019

Those charts certainly looks attractive and it is easy to see that a combination of such assets in a portfolio would have delivered impressive results. 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, combined with decreasing interest rates. 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 S&P 500 graph or 2012/13 shows on the ITT/LTT graphs). Unfortunately, the history of most of those funds doesn’t go far enough in time to include any major stock market crisis and analyze the consequences.

More details about the performance of S&P 500 leveraged funds can be found in this previous article. The findings would be broadly applicable to leveraged treasuries, notably long-term treasuries.

Simulating a longer history

Trying to judge the performance of a given investment style over a single decade 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. When full daily data isn’t available, it turns out that, based on monthly total-returns and intra-month daily 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 indices 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 reliable enough to use as a basis for historical analysis.

Combining assets in portfolios

The analysis performed in the article focusing on S&P 500 leveraging clearly indicated that a single leveraged asset would present major risks as a long-term buy and hold investment and provide little rewards in return for such risks. 

A legitimate question though is to ponder about possible use of Modern Portfolio Theory and Risk Parity ideas. Models combining leveraged stocks and leveraged treasuries over recent decades can be shown to display an attractive performance and a more bearable risk/reward profile, thanks to (mostly) negative correlation between combined assets.

Aggressive investors (e.g. in the accumulation phase) may then be tempted to replace a 100% stocks position by a leveraged portfolio displaying a somewhat similar risk profile, or (more reasonably) to dedicate a contained partition of their savings to make such a risky bet.

In the rest of this article, we will explore various portfolios using a simple asset allocation (AA) combining leveraged stocks and treasuries, comparing to a common benchmark (a passive fund tracking the S&P 500).

Note 1: risk parity concepts call for determining the asset allocation starting from a targeted risk and return level (e.g. derived from the benchmark’s historical record). Trouble is ‘risk’ is usually defined in a dubious manner (e.g. volatility) and both risk and return are moving targets. The approach in this article will be more pragmatic, simply exploring a few predefined asset allocations.

Note 2: all historical numbers will come from the simulated model previously described, assuming a fixed 1% expense ratio for leveraged funds and a 0.1% expense ratio for a regular passive fund.

A quick test with 20 years periods

To gain a 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) of an initial investment for time periods of 20 years, comparing the S&P 500 benchmark with a 2x or 3x leveraged portfolio. All periods of 20 years have to fit in the 1955-2019 timeframe. The leveraged portfolio asset allocation is very simple, using a 50% stocks, 50% treasuries (intermediate or long term), rebalanced annually.

Periods of 20 years (starting year on horizontal axis) and CAGR — Intermediate Treasuries
Periods of 20 years (starting year on horizontal axis) and CAGR — Long Treasuries

The results are quite striking, there is a clear inflection point around 1975 (as a starting year). Before this inflection point, leveraged strategies always lost against the benchmark. After the inflection point, leveraged strategies decidedly won the race, often displaying impressive premiums. Tweaking the asset allocation (from 30/70 to 70/30) won’t change this inflection point.

When observing such change in hindsight, one can come up with no shortage of “this time is different because…” theories trying to explain the change in one simple stroke, or trying to guess if the coming decades will resemble one past time period (e.g. the 60s) or another (e.g. post 80s).

Such theories tend to overly focus on one technicality or another, e.g. increasing vs. declining interest rates (see chart here), specific Fed policies, bonds callability, etc. Let’s not forget that, as times and technology evolve, the fundamental nature of human beings doesn’t change (notably when it comes to greed, power and foolishness). Fact is stock (and bond) market crises have existed for multiple centuries and will occur again and again, for this simple reason (the frailty of human nature). As the saying goes, those who forget the past are destined to relive it.

In any case, this article will stick to providing historical data points and we will let the reader judge how relevant the results of a given time period are.

Historical returns – drawdowns

Academics tend to focus on volatility as a measure of risk, if only because it appears to provide a useful framework to attempt to explain returns. This is fascinating research, but in practice, volatility isn’t necessarily such a meaningful measure of risk for long-term investors.

Another form of risk, more stressful and more directly palatable 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 analyzing simple annual returns to more detailed monthly returns to fully appreciate corresponding drawdowns.

Drawdowns (sudden loss of value) will get amplified by leveraging in both depth and duration. As we’ve seen in the S&P 500 article, the results can be quite dramatic, with drawdowns deeper than 80% and stretched in time over a decade or more. In other words, something seemingly impossible to bear for regular investors. Now, if we look at things from the portfolio level, things can become quite different.

Let’s try a 60/40 leveraged portfolio mixing stocks and long-term treasuries. The following chart presents a monthly simulation of such portfolio, leveraged 2x or 3x, assessed against our S&P 500 benchmark.

As you can see, the already quite disturbing drawdowns of the regular S&P 500 index fund (blue line, 40% to 50%) would have been fairly similar to the 2x leveraged portfolio during recent crises, but would have been significantly worse during the oil crisis in the mid-70s. As to 3x leveraging, the green line trajectory would have severely tested the mettle of hardened investors, during both the oil crisis as well as during the recent ‘double whammy’ (the Internet and financial crises).

Let’s try to move to a more bonds-oriented asset allocation, e.g. 40/60, still using long-term treasuries – see below. In this case, the oil crisis remains nearly unbearable while risk (drawdowns) becomes much more similar to the S&P 500 benchmark for more recent crises — a form of risk parity IF one elects to ignore the giant drawdown of the 70s.

Finally, let’s try to use intermediate-term treasuries (60%) instead of long-term treasuries (30%) while keeping 40% stocks – see below. This would have mitigated the oil crisis to a large extent, although 3x leverage would still have displayed quite a long drawdown in the late 70s.

Historical returns: monthly accumulation

Now, what about the scenario of an accumulator who keeps contributing to a leveraged portfolio (e.g. thanks to steady employment)? In such a scenario, drawdowns could be viewed as opportunities to buy shares at a low price. Let’s quantify such accumulation scenarios.

Let’s assume that the accumulator saves and invests $1000 every month, rain or shine, for long periods of time, e.g. 20 years in a row (hence a total of $240,000 invested over time). Every month, the portfolio is rebalanced to its target allocation. The computations are performed in inflation-adjusted (real) terms to compare apple to apple. Varying the starting year of such time periods (horizontal axis), we can study the outcomes for the regular S&P 500 benchmark vs. a leveraged (2x or 3x) portfolio. The vertical axis is the (inflation-adjusted) value of the portfolio at the end of the 20 years time period.

Let’s start by the 60/40 asset allocation (even if the drawdowns were quite severe), using leveraged stocks and long treasuries. Note that we have a few time periods (e.g. starting in the late 70s) where the 3x red dots ended up off chart (that is, above the $1.6M arbitrary limit set on the vertical axis for readability’s sake). Click on the image to see a larger version.

Unsurprisingly, the outcomes would have been quite dire when starting in the 50s or 60s (e.g. less than the $240k saved over time!), then become quite attractive (notably with 3x leverage) when starting in following decades. Again, the inflection point is quite striking.

Let’s now move to a 40/60 asset allocation, still based on leveraged stocks and long treasuries. Outcomes would have been a little milder – see below.

Finally, let’s try to be more neutral when it comes to bond maturities (hence less sensitive to rising or declining interest rates patterns) and use leveraged intermediate treasuries in a 40/60 allocation – see below.

We observe again the same inflection point, while seeing less extreme outcomes for losses and gains (3x leveraging post 70s remaining quite attractive). In a time of low interest rates, not knowing the future, it might seem more reasonable to use such an allocation instead of betting the farm on long treasuries. One might also consider mixing both types of (leveraged) treasury bonds, say half in intermediate treasuries, half in long-term treasuries.

Investment period duration

Using again the historical simulation, we can also study outcomes of the $1000/month accumulation strategy while varying the duration of investment periods.

Both forms of 40/60 AA are quantified in the table (using either intermediate or long treasuries). The average portfolio balance at the end of the investment period is provided, as well as the 10% percentile (really bad outcomes) and the 90% percentile (really good outcomes). All the computations and values are expressed in inflation-adjusted (real) terms. Click on the image below for a larger display.

One thing is clear from this table, there is no ‘return to the mean’ phenomenon happening. In other words, by no measure can one say that investing for a longer period of time would reduce risk in this context.

Back to individual assets

All this portfolio-level analysis is certainly showing interesting properties, but one should not forget that a portfolio is made of individual assets (leveraged funds in this case) and it can be really difficult for the investor to ignore what’s happening to a given individual asset. The temptation to drop ‘losers’ or to switch to another investment strategy can become very high after several years of drought and nearly impossible to avoid after a full decade or so.

Checking the drawdown chart below (comparing the three types of 2x leveraged funds), it is hard to fathom how an investor would have stayed the course with a leveraged S&P 50 fund dropping more than 70% early 2000 and not recovering for nearly 15 years. As to long-term treasuries, the 1955 to 1985 time period is quite eye-popping, to say the least.

S&P 500 ; Intermediate Treasuries; Long Treasuries; 2x leverage

The same chart using 3x leveraged funds is of course even more terrifying. One would need to have an incredibly steely belief in portfolio-level dynamics (while deliberately ignoring the history from the 50s to 70s) to stay the course. This just doesn’t seem realistic. Furthermore, it is hard to shake the idea that, at the time of writing (January 2020), after a long bull market, we are not in a situation possibly reminiscent of the late 90s (then check the blue line below, the S&P 500 3x leverage trajectory).

S&P 500 ; Intermediate Treasuries; Long Treasuries; 3x leverage


IF (and that’s a really big IF) one believes that the coming decades will be much more similar to the past few decades than to the 50s to 70s, a portfolio-level strategy of combining leveraged funds may be perceived as having a good chance of significantly exceeding the risk/return profile of a 100% stocks strategy.

In a time of low interest rates, a more prudent approach than exclusively using leveraged long-term treasuries would then probably be to diversify maturities or simply settle on leveraged intermediate-term treasuries.

In any case, it remains a very risky approach that would require extreme steeliness to navigate deep and long drawdowns, notably at the asset level. Furthermore, it really should be contained in a ‘risky bet’ portfolio bucket, clearly partitioned from regular savings invested in a more regular manner. When assumptions are tenuous, hedging your bets is a well-proven strategy.

Now if one believes that a situation remotely similar to what happened in the 60s and 70s can happen again, or if one views the past few decades as rather atypical (e.g. declining interest rates), or if one is not ready to deal with gut wrenching drawdowns for individual assets, then such a strategy is definitely NOT appropriate. The great majority of passive investors would undoubtedly fall in one of those categories.

Posted in Asset allocation, Investing, Market history, Market statistics, Portfolios

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