By
CoinToss 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 longterm portfolio.
True? Well, when it comes to leveraged
inverse
ETFs, absolutely. Holding one those longterm 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 bestknown and mostfeared 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:
(click to enlarge)
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 horrorshow 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 downside 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, perfectlyfunctioning, zerodividend)
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 singleday 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
realworld 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
lowpoint 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 oneday 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
shortterm 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 longterm 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
betterthanadvertised 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 25year 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 riskadjusted 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 emergingmarket index or
a sectorspecific one might be capable of worse things, which could
wipe out a leveraged fund. Oneday 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 20082009). This suggests that, if you were going to
seriously consider buying and holding a leveraged fund, you'd want
to pick a broad, developedmarket one.
Looking at endpoints also, of course, obscures the fact that
returns would be terrifyingly volatile. You'd have to be prepared
to occasionally take absolutely stomachchurning 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 longterm 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
wellknown 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 handwaving 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.
See also
Intel Looks More Attractive At $22
on seekingalpha.com