**By Coin-Toss Investor :**

Investment writers turn out large numbers of articles on leveraged ETFs . These are generally modeled on PSAs about sexually transmitted diseases. Leveraged ETFs are to be avoided. If you do find yourself owning one, the worst thing you can do is to ignore it. Leveraged decay, rebalancing, and the risk of catastrophic loss in a crash make them toxic to any long-term portfolio.

True? Well, when it comes to leveraged *inverse* ETFs, absolutely. Holding one those long-term would clearly be very silly. For long funds, it's not so obvious.

There are, certainly, a few awkward facts about the kind of leverage these funds supply. The best-known and most-feared of these is a mathematical beast with several names, including "leveraged decay" and "volatility drag".

Leveraged funds try to multiply an index's daily movements in percentage terms; if the index is up 1%, the fund will try to return 2%. In a flat market, this produces losses. If the index goes from 100 to 105 and back to 100, it's lost nothing. But with 2x leverage, you would go to 110 - and then down to 99. The leverage has conjured a loss out of thin air. Over time - especially in a more volatile market - this sort of thing adds up. It's not hard to show that expected returns from a fund based on an index that bounces back and forth around its starting value drift ever downwards with time.

Obviously, though, real markets don't behave this way. In the long run, they tend to go up. And the extra gains will be compounded. Can that offset this decay? And maybe other problems and fund expenses, too?

Well, maybe. One way to try to work this out is to use some simple simulations. Simulations are incredibly useful in two situations: 1) When the process being modeled is complex; and 2) when you don't want to do math. Both of those apply here, so let's get to it.

The simulation we'll use is the famous random walk. Random walks are very, very simple. We start with a number. On the first "day", we move it up or down by a random amount. On the second day, we do it again. Eventually, we end up with something that looks kind of like a price series. Here are four random walks with a thousand steps each:

In itself, this is fun, but not wildly useful. What we can do, though, is to set up walks with interesting assumptions, run a whole lot of them, and then see how often they end up in different places. If we've run enough of them, then these frequencies give us the probability that a future run will produce a given outcome. (In the same way, we might - if we were idiots with a lot of free time - flip a coin a thousand times to get an idea of how likely future tosses are to come up heads.) If the way we've set our random walk up is like the real world in any meaningful way - and as long as we don't kid ourselves too much about that - then these probabilities tell us something about what our expectations should be.

A simple example. Here, our random walk represents a major stock index. There's a leveraged fund based on this index that, conveniently, works perfectly and costs nothing. To simulate different holding periods, we'll run walks with different numbers of steps. We'll look at holding periods of a week, a month, six months, a year, 3 years, 5 years, 10 years, and 25 years. We'll run 10,000 walks for each of these periods. To generate the steps, we will - for the sake of this example - pull random numbers from a normal distribution, with an average change of 0 and a standard deviation of about 1%. (We're using a normal distribution so that small changes are more common than large ones. Later, we'll ditch distributions entirely.)

The table below shows the results arranged by percentile. It includes the middle 90% of all outcomes for each holding period. At the left are the returns you'll get if you're very unlucky. The middle values represent "typical" returns. The value in each cell is the ratio of the final values reached by the leveraged and unleveraged funds. If the index went from 100 to 150, and the leveraged fund went from 100 to 300, then the number shown will be 300/150 = 2. There are several ways to express this difference. The important thing is that 1 is par, and that values above that favour the leveraged fund.

Unlucky | Typical | Lucky | |||||
---|---|---|---|---|---|---|---|

5% | 10% | 25% | 50% | 75% | 90% | 95% | |

One week | 0.96 | 0.97 | 0.98 | 1.00 | 1.02 | 1.03 | 1.04 |

One month | 0.92 | 0.93 | 0.96 | 1.00 | 1.03 | 1.06 | 1.08 |

Six months | 0.80 | 0.83 | 0.90 | 0.98 | 1.06 | 1.15 | 1.20 |

One year | 0.72 | 0.76 | 0.85 | 0.96 | 1.08 | 1.20 | 1.28 |

3 years | 0.53 | 0.59 | 0.71 | 0.87 | 1.06 | 1.28 | 1.43 |

5 years | 0.42 | 0.48 | 0.62 | 0.80 | 1.03 | 1.30 | 1.49 |

10 years | 0.25 | 0.31 | 0.43 | 0.63 | 0.92 | 1.29 | 1.57 |

25 years | 0.08 | 0.11 | 0.18 | 0.32 | 0.57 | 0.98 | 1.33 |

What this horror-show illustrates is volatility drag. Our walks wandered up and down at random and, in the end, mostly delivered nice fat losses. You can clearly see how the down-side of leverage manifests itself over time. After 1 year, the "leveraged fund" outperformed the index around 40% of the time. After 5 years, it was under 30%. After 25 years, 8%.

As we've said, though, equity markets don't behave this way. They go up. From January 1950 to July 2013, the average daily change in the S&P500 wasn't zero, but rather 0.033%. What would have happened if a (free, perfectly-functioning, zero-dividend) leveraged fund had started with the index in January, 1950?

The answer is that it would have returned 204,000%. It would have produced annualized gains of 12.9%, compared to 11.2% for the index with dividends (and 7.5% without). And this period includes several crashes and recessions, with single-day declines up to 20% (on Black Friday, 1987).

Of course, leveraged funds don't work perfectly, and the leverage, rebalancing, and management aren't free. Is there any real-world evidence of these things doing well over long periods? Well, consider [[SSO]] again. SSO tries to return twice the daily movement in the S&P500. It's been around since 2006. Over its lifetime, it's underperformed [[SPY]]. But its short lifetime includes a pretty major financial catastrophe; and, from an ugly low, it's clawed its way almost back to parity with SPY. Thanks to compounding, it's returned more than twice what SPY has since the low-point in March, 2009. This was a period of rapidly rising prices, which doesn't happen that often, but it does demonstrate that this form of leverage isn't instantly fatal in the real world.

So, let's go back to the simulations and try a more realistic model. We want a walk that tends to drift upwards, and we want to take expenses, errors, and dividends into account. Instead of taking random numbers from a mathematical distribution - which ordinarily won't give us enough extreme changes - we'll take them from the list of actual daily percentage index changes in the Dow Jones Industrial Average since 1896. We'll basically put all 32,000 daily changes into a hat. For each "day", we'll pull one out and apply it to the index. Then, we'll put it back in the hat.

Why the DJIA? The S&P500 only goes back to 1950, and markets have done a little too well since then. It's an optimist's dataset. The DJIA series is uglier. It has higher volatility and lower average returns, including, as it does, the Great Depression and some other grim periods. And we still have Black Friday and a handful of other one-day declines between 10% and 20%. So, we're trying to be a little conservative, here.

We're going to stick in the current S&P dividend yield of 2%. It's slightly odd to use returns as far back as 1896 with dividends from the present moment, but if we wanted to change the dividend level, we'd also have to think about how much higher our leveraged fund's dividends would be (they should vary mostly with short-term interest rates), we might want to worry about background price levels, etc. Anyway, this shouldn't be fatal our goal of trying to understand the relative long-term expected returns of leveraged and unleveraged funds.

SPY's expense ratio is absurdly low, so we're going to ignore it (and any tracking error, as well). SSO's is 0.9%. A leveraged fund's business also involves mucking about with swaps and futures, so there may be other worries, here, too. Rather than working through that in detail, let's just briefly consider its performance to date.

Since it started up, perfect performance would have seen SSO at 82.1 on June 19, 2013 (its 7th birthday). Its NAV was actually 79.1. If we were lucky, we might look more closely and discover a straightforward case of erosion due to skimming of management fees or something to do with the derivatives strategy. We are not lucky. SSO has lagged badly at some periods, only to then deliver better-than-advertised returns at others. For the sake of a simple life, we're going to simply set its dividend to zero (rather than 0.6%). This gives us a hypothetical fund with *overall* performance roughly comparable to SSO's, but with less volatility.

Here are the results. Again, we're pulling daily changes at random from DJIA historical returns, and we're allowing the index a yield of 2% and the leveraged fund a yield of zilch. Our leverage consists of multiplying all percentage changes in the underlying by 2.

Unlucky | Typical | Lucky | |||||
---|---|---|---|---|---|---|---|

5% | 10% | 25% | 50% | 75% | 90% | 95% | |

One week | 0.97 | 0.98 | 0.99 | 1.00 | 1.01 | 1.03 | 1.04 |

One month | 0.93 | 0.95 | 0.97 | 1.00 | 1.03 | 1.06 | 1.07 |

Six months | 0.84 | 0.88 | 0.94 | 1.01 | 1.09 | 1.16 | 1.21 |

One year | 0.78 | 0.83 | 0.91 | 1.01 | 1.12 | 1.23 | 1.31 |

3 years | 0.64 | 0.71 | 0.84 | 1.01 | 1.21 | 1.44 | 1.58 |

5 years | 0.57 | 0.65 | 0.81 | 1.02 | 1.30 | 1.62 | 1.83 |

10 years | 0.46 | 0.55 | 0.74 | 1.04 | 1.46 | 1.96 | 2.33 |

25 years | 0.30 | 0.41 | 0.65 | 1.11 | 1.88 | 3.05 | 4.04 |

So, leveraged ETFs basically "work" whether you're holding them for a day or a century. The median returns are quite close to those of the underlying "index". The average returns are higher, because if you do outperform, you can outperform by a lot. Really severe losses are actually fairly rare. (Bear in mind that these numbers reflect relative performance; over 25-year periods, the underlying almost always has positive returns. The 0.57 at the 20th percentile, for example, doesn't mean we lost money in absolute terms. The "index" returned 166%, and the leveraged fund 51%; and 151/266=0.57.)

What leveraged ETFs offer, in fact, looks suspiciously like the standard investment deal: Risk for return. In fact, they might reasonably be compared to individual stocks. Ultimately, it may be that the invisible hand of market efficiency - well, it's not invisible, exactly, but its math is hard to follow - will arrange things in such a way that the risk-adjusted return of leveraged ETFs falls neatly in line with those from other investments.

**Problems**

What are the invisible risks, here? Well, our approach doesn't allow daily declines greater than 22%. An emerging-market index or a sector-specific one might be capable of worse things, which could wipe out a leveraged fund. One-day catastrophes aren't the only worry, either; extremely heavy losses over a longer period can also produce more or less irrecoverable losses (consider the example of [[UYG]] in 2008-2009). This suggests that, if you were going to seriously consider buying and holding a leveraged fund, you'd want to pick a broad, developed-market one.

Looking at end-points also, of course, obscures the fact that returns would be terrifyingly volatile. You'd have to be prepared to occasionally take absolutely stomach-churning losses.

Finally, and perhaps most importantly, the ability of ETFs to truly return 2x daily returns over long periods is uncertain. SSO has done a reasonable job most of the time, but the variation has sometimes been troubling. This makes long-term investments in leveraged funds a gamble on the ability of the managers to produce, and the willingness of the relevant markets to provide, the right leverage at the right price.

**See also**

The fact that leveraged funds won't go to zero seems to be well-known to academics, as you'd expect, and probably to lots of other people. There's an excellent site here , for example, that describes the expected behaviour of these funds in a much more rigorous and cleaner way (I've tried to think about this only as a prospective investor). It notes that the optimal amount of leverage for most markets has historically been about 2x.

**Limitations**

These simulations stumble randomly on regardless of their current or past values; real markets may tend to return to a linear trend. So, some extreme results may be less likely than the simulations suggest. There's also a tiny bit of autocorrelation in the index series, which isn't reflected in the modeling. Finally, we've assumed no tracking error or other problems associated with the unleveraged alternatives, and have done some hand-waving over issues that may be significant over the long term, including risks associated with the derivatives trading used to produce leverage.

**Conclusion**

Whatever the holding period, leveraged ETFs seem to offer pretty much a standard investment deal: Risk for return. The tradeoff isn't nearly as bad as many people think.

**Disclosure:** I 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. I have no business relationship with any company whose stock is mentioned in this article.

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*The views and opinions expressed herein are the views and opinions of the author and do not necessarily reflect those of Nasdaq, Inc.*

The views and opinions expressed herein are the views and opinions of the author and do not necessarily reflect those of Nasdaq, Inc.