# Leveraged ETF Math: This May Smell Bad, Kid

The other day I was asked why I’m short SSO as opposed to just long SDS. The answer is that there is natural drag on leveraged ETF prices. Part of this is due to the decay factor in futures pricing. But the bigger factor is just the math of multiplication and time linked returns.

Bear in mind these are my personal investments, not anything I’m doing professionally.

OK, let’s use a product like SSO, which is the 2x leveraged S&P 500. The way its supposed to work is that every day, you get 2x whatever percentage return is on the S&P 500. On Monday, SSO closed at \$25.73. If the S&P 500 were to finish up 1% today, SSO should be up 2%, or \$0.51 to \$26.24. If its then down 1% the next day, SSO should be down 2%, or -\$0.52 to \$25.72.

Over time, the math of total return (in percentage) looks like this (just in general, not of these ETF’s specifically):

(1 + X0) * (1 + X1) * (1 + X2) … (1 + Xn) -1

Where X is day n’s return.

Now you’ll notice that if we get the exact same percentage return, but in opposite directions, on two separate days, it doesn’t mean your total return will be zero. For example, say you lose 1% on day 1, but gain 1% on day 2.

(1 – 0.01) * (1+ 0.01) – 1 = -0.0001, or -1bps.

The more severe the return, the more severe the result. Say its -10% and +10%. The result would be …

(1 – 0.1) * (1 + 0.1) – 1 = -0.01, or -1%.

It doesn’t matter what order these occur in, because multiplication is commutative.

(1 + 0.1) * (1 – 0.1) – 1 = -0.01, or -1%.

It would therefore seem like there is a natural negative drift in security prices. But in the normal world, we assume security prices aren’t dependent on previous percentage gains, but on some fundamental valuation. For example, if I buy a bond at \$100 but it subsequently has some credit problem that results in it falling to \$90, I will have lost 10%. But if the credit problem is resolved and it gets back to par, I realize a 11% gain. I’m not limited to getting back the opposite of what I lost.

But the multiplication factor of ETFs sort of mess with this. Let’s say the S&P 500 drops by 2% on day 1, then rises by 2.0408% on day 2 (which puts you back to where you started), and repeats this pattern for 6 days.

Now let’s say you own the double long ETF, and we’ll assume the ETF works as its supposed to. On day 1, you’d lose 4% (-2% * 2) and on day 2 you’d make 4.0816% (2.0408% * 2). But do the math…

(1 – 0.04) * (1 + 0.040816) * (1 – 0.04) * (1 + 0.040816) * (1 – 0.04) * (1 + 0.040816) – 1 = -0.24%

Why? Think about the pay back formula, i.e., percentage return you need in period 2 to go back to zero given a loss in period 1. Its …

1 / (1+x) – 1

Now if the S&P return was x, then the ETF return is going to be 2x. But notice that…

2 [1 / (1+x) -1] <> [1 / (1+2x) -1]

See? In fact, the left equation is always going to be larger than the right equation if x is negative, and always smaller than if x is positive.

Now I can’t say there is a real arbitrage here, because if the market moves higher or lower decisively, that will dominate all these pretty equations. But if you are short-term trading, it seems to me you’re better off shorting the opposite ETF than going long. So I’m making a bearish play by going short SSO as opposed to owning SDS.