Jensen’s Inequality for returns on short positions

Question

this is puzzling me. Say you have an asset A, that on day t+1 returns 1%, and then on day t+2 returns 1% again.

If you invest $1 in A on day t (take a long position), then on day t+2 you have earned:

1*(1+0.01)*(1+0.01)-1 = 1.0201 - 1 = 2.01%

Now, a short position in asset A should return -1% on day t+1 and -1% on day t+2. Thus, if you have a $1 short position in A on day t, then on day t+2 you will have earned:

1*(1-0.01)*(1-0.01)-1 = 0.9801 - 1 = -1.99%

The difference in magnitude between these two returns is Jensen’s inequality. But what’s really confusing me is that I think the profits earned from the long position should be exactly offset by the losses from the short position, otherwise you are creating money out of thin air. What am I missing?

Answer 1

In a nutshell, this is the "variance drag" problem. The mechanics of how you short something matter, and it's relevant to the discussion of levered/inverse ETFs that behave differently from classic/vanilla positions.

Consider an XYZ future at 100. A day later it's 1% up, at 101. Two days later, it's up 1% again, at 102.1.

If I go long, I make 2.1 profit. If I go short, I have to buy it back at 102.1, which is a loss of 2.1. It's equal and opposite. No inequality required.

The problem for strategies that are continually and systematically levered (other than +1x) is that they have to rebalance. Else they will no longer be levered the same way after day 1 as on entry.

If I'm 1x short, then at the end of day 1, I have assets of 99 and a market delta of -101. I've become 1.0202x short. To stay 1x short, I have to buy back 2, to make me assets 99, exposure -99 on day 2. Your "inequality" fails to account for this adjustment. The "missing P&L" here is the P&L of the counterparty on day 2. He's up 1% on a position of 2 equals your missing 20bps in the example above.

The usual problem here isn't that returns can disappear into the ether (for the reasons above). It's more normally represented as a temporal mismatch, that's frequently un(der)appreciated.

Consider an ABC future. It has a 50% chance of rising of falling 10% each month.

So that's a:

• 25% chance of up&up = 121
• 50% chance of up&down = 99
• 25% chance of down&down = 81

NPV = 100, but quite clearly in the long-run, it's expected CAGR is negative, because 1.1^0.5 * 0.9^0.5 <1.

To have an expected CAGR of zero, you have to believe that there is an equal chance of a 10% gain and a 9.0909% loss (ie 1/(1+10%)). Flip that fair coin infinitely often, and E() = 1.

Except: 25% * 121 + 50% * 100 + 25% * 82.6 = 1.009. A geometrically fair coin is arithmetically favourable; while an arithmetically fair coin is geometrically unfavourable. Which is your real inequality. The difference between the two averages is (with the inevitable Gaussian caveat) half sigma squared. Hence the term "variance drag".

Except for the purposes of shorting, selling an index future is doing this arithmetically, while buying an inverse ETF is doing it geometrically. They're related, and directionally consistent; but not quite the same thing.

hope this makes sense.