# The difficulty with shorting and inverse positions

The brutal arithmetic behind shorting

**Shorting is hard**

Shorting assets is intrinsically difficult because

- while your position goes against you it gets bigger
- and when you win your position is getting smaller

Consider the impact of a $1mm fund that is designed to mimic a $1mm short in stock X.

**X down 50% scenario**

The fund earns 50% return. So now the fund has $1.5mm aum and the short is only $500k. For the fund to match the return of X going forward the fund must now triple its position.

- Note this requires selling into a declining market (negative gamma)
- The fund must keep its initial Position/AUM ratio constant. So initially this was 1:1 but then became 1/3 which is why it needs to triple the position

**X up 50% scenario**

If the stock increased 50% the fund loses 50%. Its AUM is $500k and the short is $1.5mm. The fund must cover much of the short.

- The fund must buy in a rallying market (negative gamma).
- The new position/AUM ratio is 3:1 so the fund must buy back $1mm or 2/3 of its position so that its AUM is $500k and its position is $500k. In this case the fund is insolvent.

**Inverse ETFs and ETNs**

The above dynamic is also how an inverse ETF or ETN work. The ETN must match the inverse return of a reference asset. So if all the AUM is exposed to the asset then we calculate the fund PositionSize/AUM.

NAV = AUM / Shares Outstanding

**The down case**

- As the reference asset moves lower the fund must sell more of it to maintain the PositionNotional/AUM ratio. In this case, as the reference asset moved lower, the fund AUM increased due to profits while its position size decreased as the price of the reference asset declined. The fund must sell enough of the asset to rebalance the initial PositionNotional/AUM ratio. Selling into a declining market. This ensures the ensuing percentage move in the reference asset corresponds to the percentage change in NAV.
- Redemptions are stabilizing as they require the position rebalance to be smaller as the AUM declines and the reference asset is purchased

**The up case**

- As the reference asset rallies the fund must cover its notionally increasing short. PositionSize is increasing while AUM declines, so the reference asset must be purchased to reduce the position size and again normalize the notional/AUM ratio.
- In this case, redemptions are de-stabilizing as they reduce AUM which further moves away from its initial value and the redemption also prompts an in-kind purchase of the already appreciated reference asset.

**In sum:**

- For inverse etfs to maintain a constant exposure in return space to their reference asset they must rebalance such that the dollar size of the underlying position is a fixed ratio to the AUM.
- The inverse nature means that the AUM and position size are always moving in opposite directions requiring constant rebalance (negative gamma). This creates a downward drift to the product NAVs.
- As the reference asset rallies, position size gets bigger and AUM drops due to losses. As reference asset falls, position size shrinks while AUM increase due to profits.
- Redemptions can stabilize rebalance requirements in declines and exacerbate rebalance quantities in rallies as redemptions reduce shares outstanding and in turn AUM while in both cases triggering the fund’s need to buy the reference asset which again is stabilizing after declines but not after rallies. In other words, profit-taking is stabilizing while puking is de-stabilizing.
- I extend this explanation to levered funds here.