# Inverse ETFs versus short selling: a misleading equivalence

### J. González

#### 09/02/2015

Many people incorrectly think that buying an inverse ETF is equivalent to having a short position in the components of the ETF.

The mistake arises from assuming that the ETF holds its open positions for days or months. But that’s not the case. Rather, this kind of ETFs is built for day trading, since they close their positions at the end of the session.

## Some calculations

### Inverse ETFs

What’s the best way to calculate the performance of an inverse ETF if you know the evolution of the underlying asset? Well, this is quite simple: you just have to accumulate “minus” the returns of the underlying asset. This is correct because the inverse ETFs close the positions each day and reopen them the next day with the accumulated previous returns. Let’s see an example:

Imagine an inverse ETF over an index (e.g. S&P 500). If the index goes down from 100 to 90, which is a return of -10%, the ETF would obtain a gain of 10% that day.

Now we can consider that the following day the index goes back to 100 from 90, which is a return of 11. 1% (=100/90-1). So, in this case the ETF shows a return of -11.1%.

The final performance of the ETF would be obtained compounding the returns:

(1+10%) . (1-11.1%) – 1 = -2.2%                 (1)

However, the performance of the index would have been:

(1-10%) . (1+11.1%) – 1 = 0%                      (2)

How is this possible if the index starts and ends in the same price!? The trick lies in recognising that the holding period of the ETF trade is only one day, which produces the actual result, which doesn’t match our intuition.

### Short selling

Is short selling the same as buying an inverse ETF? The direct answer is no.

Short selling means borrowing securities from a broker and then selling them. Sooner or later you’ll have to return these securities to the broker; if you buy them back at a lower price to that which you purchased them, you’ll make a profit.

Let’s repeat the previous example, but instead of buying an inverse ETF, we are going to make a short selling operation. We “sell” the index at 100 the first day, and buy back the index at 100 the last day (in order to give back the securities to the broker). Obviously, the performance would have been 0%. As you’ve already noticed, the inverse ETF equity formula is no longer correct in this case. Calculating the short selling performance would be done by calculating the performance of the index and changing the sign:

-[(1-10%) . (1+11.1%) – 1] = 0%

In this case, the result matches our intuition (I am not including costs of any kind which would be present in a real operation).

## More funny things about inverse positions

From the previous paragraph it is possible to draw some conclusions:

1. It is almost impossible to lose all your investment with inverse ETFs: not so when short selling during a long period of time.
2. If the underlying asset of the inverse ETF keeps a falling direction, we will gain more with an inverse ETF than selling borrowed assets, since with the ETF we are accumulating a chain of positive returns.

Next, I give two examples to clarify these points:

1. Let’s consider an asset with a series of prices in 3 consecutive days: 100, 160, 210. Therefore the index shows two returns of 60% and 31.25%. Following equation (2) the index has a performance of:

(1+60%) . (1+31.25%)-1 = 110%

If we had had an inverse ETF on this asset during these 3 days, we would have lost:

(1-60%).(1-31.25%)-1 = -72.5%

However, if we had borrowed the asset and sold it (short selling), when we close the position the third day we obtain:

-[(1+60%) . (1+31.25%)-1] = -110%

Yes, this means exactly what you think it does, dear reader; we’ve lost all our money and owe 10% more!

So managing the inverse ETF daily, instead of keeping a longer position, avoids a possible loss of more than 100%. In order for an inverse ETF to cross the barrier of a loss of -100%, the underlying assets would have to rise 100% in one day, which is quite unlikely.

2. If the same index had a series of prices 100, 80, 60, equivalent to -20% and -25% returns, an inverse ETF would obtain the following gain:

(1+20%) . (1+25%)-1 = 50%

Meanwhile, a short position would perform worse:

-[(1-20%).(1-25%)-1] = 40%

However, any sudden rise in the index would further affect the ETF. For example, if we had a final price of 68, the ETF would have a final performance of:

(1+20%) . (1+25%).(1-13.3%)-1 = 30%

While the short selling:

-[(1-20%).(1-25%).(1+13.3%)-1] = 32%

## S&P Index

Finally, here are two graphs with the evolution of the S&P 500 index, a short-selling position on the index and an inverse ETF (costs not included).

In 2000, a bearish year, we can see that short selling the index would have hedged the index decline entirely, whereas the inverse ETF falls behind, because the index evolution is not completely downwards the whole year, and we have seen how badly these corrections affect the inverse ETFs. From 2009 on, keeping a short position for enough time would have meant losing your entire investment and more. Meanwhile, the inverse ETF would have lost quite a lot, but in no case would have reached a -100% performance. 