Leveraged and inverse ETFs, 3

17 May 2010 by Jim Fickett.

Leveraged and inverse ETFs return, each day, a multiple or the inverse of what an underlying stock market index returns. As described earlier, these show, in addition to the naively expected results, some distortion. In this post we give some rules-of-thumb on when the distortion is greatest and what sorts of market conditions require special vigilance.

To review:

  • In Leveraged and inverse ETFs, 1, we introduced the topic, and showed how the most basic tracking error comes about – a simple matter of arithmetic.
  • In Leveraged and inverse ETFs, 2, we showed that the made-up examples often used in the news do more than exaggerate any tracking error, rather they give an unrealistic impression, due to nonlinear effects. The truth is, leveraged and inverse ETFs normally track more or less as you would expect, at least over a period of a few weeks.

In this post we will show how to simplify the algebra in a way that covers most real-life examples and then, using the simplified algebra, show exactly where the volatility danger comes from.

At the end we give a few general rules about what to expect from these funds and when to take special precautions. If you are not comfortable with algebra, just skip straight to the end.

Here, reproduced from the second post, are the equations for returns over three days, where the underlying index changes by a on the first day, b on the second, and c on the third.


Primary index:  (1+a)(1+b)(1+c) - 1     = a + b + c + ab + ac + bc + abc

2x leveraged:   (1+2a)(1+2b)(1+2c) - 1  = 2a + 2b + 2c + 4ab + 4ac + 4bc + 8abc

-x inverse:     (1-a)(1-b)(1-c) - 1     = -a - b - c + ab + ac + bc - abc

-2x lev & inv   (1-2a)(1-2b)(1-2c) -1   = -2a - 2b - 2c + 4ab + 4ac + 4bc - 8abc

For most indices and most days, the change in the market falls between -2% and +2%. Because of this, the first order terms above (a, b, c) carry much more weight than the second order terms (ab, ac, bc), which in turn carry more weight than the third order term (abc). For example, with a daily move of 2%, (0.02), we have a equal to 0.02, ab equal to 0.0004, and abc equal to 0.000008.

For longer periods the equations get messier, but we can illustrate the same point with some sample calculations. Again suppose daily movements are at most 2%, and you want to hold one of these funds for one, two, or four weeks. Then the largest possible aggregate values for each class of summands are as follows. (The calculation here is the number of possible terms times the maximum size of each one. So in the 7 day calculation, there are 21 second order terms, each of maximum size 0.0004, giving the 0.0084 in the second cell of the top row.)

1st order (e.g. 'a') 2nd order (e.g. 'ab') 3rd 4th 5th
7 days 0.14 0.0084 0.00028 0.0000056 0.0000000672
14 days 0.28 0.036 0.0029 0.00016 0.0000064
28 days 0.56 0.15 0.026 0.0033 0.00031

From this it is pretty clear that for holding periods up to about a month, and in normal markets where daily moves are under 2%, it suffices to look at the first and second order terms.

With that simplification, we can see pretty clearly where the distortion comes in for holding periods longer than a day or two.

In the three day case, and ignoring the triple terms, one might naively expect the 2x fund to return twice the returns of the primary index, i.e. 2a + 2b + 2c + 2ab + 2ac + 2bc, but it returns instead this plus 2(ab+ac+bc). So the 2x fund has a distortion of 2(ab+ac+bc). The -x fund also returns what might naively expect plus 2(ab+ac+bc). The -2x fund gives the naive return plus a distortion of 6(ab+ac+bc).


2x leveraged:   2(ab + bc + ac)

-x inverse:     2(ab + bc + ac)

-2x lev & inv   6(ab + bc + ac)

Hopefully it is clear that this same analysis generalizes to longer periods, with the distortion being 2 or 6 times the sum of all the pairwise cross-terms.

Combining this with the table above shows that with a one week holding period the distortion will typically be quite small.

Note that if all the daily moves are in one direction, then all the cross-terms are positive, and so the distortion (in this simplified algebra) is also positive. So although the distortion does cause the funds to deviate from naive expectations, that is not always a bad thing. If the market is steadily going down, for example, and you buy the -2x leveraged and inverse fund, the distortion of six times all the cross terms is a bonus return which could be fairly substantial. Similarly, if the market moves steadily upwards, the -x and -2x funds will not drop quite as badly as one might expect.

However the distortion can certainly be negative, which means a loss. Just to get some feel for how this works out, look at the following table for results over 10 days:

Up days Down days Negative cross terms Positive cross terms
0 10 0 45
1 9 9 36
2 8 16 29
3 7 21 24
4 6 24 21
5 5 25 20
6 4 24 21
7 3 21 24
8 2 16 29
9 1 9 36
10 0 0 45

Note that even in the case of highest volatility – 5 up days and 5 down days – there are almost as many positive second-order terms as negative. So in a market with fairly uniform moves, volatility will not, in fact, hurt very much. Of course it is quite possible to find examples where the negative terms dominate, as shown by the example from the Financial Times in the second post of this series.

At this point it is probably good to repeat the results of simulating the leveraged and inverse fund returns using real historical data, from ProShares:

Joanne Hill, PhD and George Foster, CFA conducted a historical study that showed a high likelihood of approximating the daily target over short periods. The shorter the period, and the lower the volatility of the underlying index, the more likely returns were to approximate the daily target. Longer and more volatile periods tended to show a greater deviation from the daily target. Using historical data, a model based on 2x the daily return of the S&P 500 index showed a 90% likelihood of producing a return between 1.75x and 2.25x the index return over any 30-day period over the last 50 years. (Models based on an index with higher volatility would have deviated more.)

Note that the 2x fund, used in the simulation, has only 1/3 the (simplified) distortion of the -2x fund. So for the -2x fund the “return between 1.75x and 2.25x the index return” would probably be replaced by a somewhat wider band.

In conclusion, then, here are a few general rules about what to expect from these funds and when to take special precautions:

  • There is distortion in the leveraged and inverse ETFs, that is, the returns over more than one day do not follow the same rule as returns over one day
  • If there is a market move of over 2%, or you are at all uncomfortable with the mathematics, or if you are using these funds for the first time, check on them daily
  • The major component of distortion for the -2x (leveraged and inverse) fund is triple that for either the -x (inverse), fund or the 2x (leveraged) fund
  • When the market is going in one direction, be it either up or down, the major component of distortion is always a positive
  • For periods of up to a week, with normal market moves of under 2%, the distortion is not of much concern
  • For fairly normal markets (not post-Lehman!), these ETFs give results fairly close to that expected for periods up to about a month
  • The worst losses from distortion can typically be expected in very volatile markets, with a similar number of up and down days; even in such cases, distortion may or may not be severe; check your investment often