Leveraged and inverse ETFs, 2

8 Feb 2010 by Jim Fickett

Despite alarming examples in various articles, leveraged and inverse ETFs perform more or less as expected over short periods in normal markets.

This post, and the next one or two, will be a bit heavy on the algebra. But for those who don't want to wade through the equations, there will also be example results.

Setting the stage

Starting with an investment of unit value, an investment that gives a return of a on the first day and b on the second day would give a return of

(1+a)(1+b) - 1 = a + b + ab

after two days. Similarly, after three days of returns of a, b, and c, the returns on the original index, on the 2x leveraged fund, the -x inverse fund, and the -2x inverse leveraged fund would be


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

Worst case examples

The Financial Times gave the following hypothetical example in a 1 Jun 2009 article (“Investors warned on niche ETFs”, by Steve Johnson; the example appeared in the paper, but not in the on-line version):

Day 1 Day 2 Day 3 Day 4 Day 5 Overall change, %
% change -10 +10 -10 +15
Index 100 90 99 89.1 102.5 +2.5
Leveraged, 2x 100 80 96 76.8 99.8 -0.2
Inverse, -x 100 110 99 108.9 92.6 -7.4
Lev. inverse, -2x 100 120 96 115.2 80.6 -19.4

This looks very scary – the leveraged fund ends down when it “should” be up, and the leveraged and inverse fund is down almost 20% when the index is up only 2.5%! Similar examples have been shown in articles warning that one should never hold these funds for more than a day. However such examples, while relevant to the worst case, give the wrong impression for average use. Even in a very volatile market very few indices would show four days in a row with reversing 10% (or more) moves. Clearly the author was exaggerating to make a point. The problem is that exaggerating, in this case, does more than magnify – it distorts.

Look back at the “RETURNS” box above. After three days, twice the primary index would contain terms 2ab, 2ac, 2bc and 2abc, but the 2x leveraged fund contains the terms 4ab, 4ac, 4bc and 8abc. This is how the difference between the actual returns of the leveraged fund, and what one might naively expect, comes about.

So what happens when one exaggerates for effect? A more typical daily market move might be more like 1%. If the daily moves a, b, and c are exaggerated by a factor of 10, then the terms ab, ac, and bc are exaggerated by a factor of 100, and the term abc is exaggerated by a factor of 1000. In other words, if the market moves are exaggerated by a factor of 10, then the way these funds differ from naive expectations may be exaggerated by a factor of 100 or more.

Look at more typical market moves:

Day 1 Day 2 Day 3 Day 4 Day 5 Overall change, %
% change -1 +1 -1 +1.5
Index 100 99 99.99 98.99 100.47 +0.47
Leveraged, 2x 100 98 99.96 97.96 100.90 +0.90
Inverse, -x 100 101 99.99 100.99 99.48 -0.42
Lev. inverse, -2x 100 102 99.96 101.96 98.90 -1.10

With more normal market moves, the funds do pretty much what one would hope.

There is nothing wrong with the FT's arithmetic, and if an index really does show such marked moves, the leveraged and inverse ETFs will perform very poorly. So in certain cases the warning may be in order. But for most indices the example above, and many others like it, give the wrong impression.

In real life

According to ProShares, for the S&P 500 the leveraged fund usually perform more or less as expected over periods of 30 days:

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.)