N² – n: why shorting is mathematically cursed

Recall the levered silver flows post where we see the quick math of levered ETFs. For a fund to maintain its mandated exposure, the amount of $$ worth of reference asset they need to trade at the close of the business day is:

x(x - 1) * percent change in the reference asset * prior day AUM

where x = leverage factor

examples of x:
x=2 double long 
x=-1 inverse ETF
x= 3 triple long
x= -2 double inverse

This isn’t just a levered ETF thing. The -1 leverage factor is exactly the same as just a vanilla short position. It’s a sneaky reason why the shorting is mathematically challenged.

The easiest way to think of this as an individual investor is to imagine you have an account value of $100. The account is holding $100 in cash, but it’s the proceeds from shorting a $100 stock (assume you don’t need any excess margin to maintain the short). If the stock falls to $50, your account value is now $150 (your cash + $50 mark-to-market profit on the short). You earned a 50% return on a 50% drop in the stock.

Now what?

If the stock falls another 50%, you make $25.

$25/$150 = 16.7%

If you want to maintain the same exposure so that you make 50% on your account on that second 50% drop, you would have needed to short more shares at $50.

How many more dollars’ worth of stock?

-1 (-1 -1) x -50% x $100 = -$100

You needed to sell an additional $100 worth of stock or 2 more shares at $50. Then on that last leg down, you would have made $25 on 3 shares total or $75.

$75 profit /$150 account value = 50% return

Learn more:

🔗 The difficulty with shorting and inverse positions.