Why Short Returns Differ from Negated Long Returns
Summary
The note explains why a short position’s cumulative return cannot be calculated by simply reversing the signs of an asset’s daily returns and compounding them. The example compares a short AAPL position held over a period of rising prices with the result from negating daily returns, showing that the two calculations produce substantially different losses.
The key distinction is that a short position loses or gains the opposite dollar amount of the underlying asset’s price move, while percentage returns are measured against the position’s changing value. After losses, the short’s capital base is smaller, so an equal further price rise creates a larger percentage loss. The explanation uses a two-period example to illustrate this compounding effect and notes that short ETFs must rebalance to maintain their target daily exposure. The discussion is conceptual: it does not account for borrowing costs, margin rules, dividends, or other real-world short-position cash flows.
Key ideas
- Negating an asset’s periodic returns does not give the cumulative return on a short position.
- A short position experiences dollar gains and losses opposite to the underlying asset’s price changes.
- Percentage returns depend on the short position’s changing value, so repeated adverse moves compound differently.
- Leveraged or inverse ETFs rebalance to maintain a daily exposure multiple, which affects long-term performance.
Tags
Full text
# Compounding negative returns?
# Compounding negative returns?
Assume you're short AAPL. Say you sold short at 24.28 as of 2016-01-01 and cover 2021-09-13 on 149.74. So your return is (24.28-149.74)/24.28 or -517%.
If you get the daily returns of AAPL and just reverse the sign, then do a cumulative return, you instead get -90.2%. Obviously the correct number is -517% since as your short underperforms the position gets bigger since you need more cash to cover your position. So is it wrong to use cumulative returns in this context?
```
import pandas as pd
price_data = yf.download(['AAPL', 'SPY'], '2010-01-01')
y = price_data.loc['2016-01-01':]['Adj Close']['AAPL'].pct_change().dropna()
np.exp(np.log(1 + y.values).cumsum())[-1] - 1
```
## Answer by D Stanley (score 1)
https://quant.stackexchange.com/a/67899
Negating periodic returns for a long position is not equivalent to the cumulative returns on a short position. Short positions have opposite absolute gains, not relative gains.
Take this example:
I start with a position of \$100. If the instrument gains \$5 (a 5% return), a long position makes \$5 and ends with a value of $105. If I have a short position, I lose \$5 and have a end total of \$95. If, on the second day, the instrument gains another \$5, the long position gains another \$5, which is a 4.7% return (5/105), but the short position loses \$5, which is a 5.26% loss (5/95).
The difference is that the starting value is different between long and short positions. So if an instrument goes up in value twice, the long position will have a smaller "return" than a short position, because the starting value after the first period will be higher (e.g. 105 vs 95 in the example).
Or, mathematically, if $r > 0$:
$\begin{aligned} (1+r)(1+r) - 1 &= (1 + 2r + r^2) - 1\\ &= 2r + r^2 \\ &> 2r - r^2 \\ &> 1 - (1 - 2r + r^2) \\ &> 1 - (1-r)(1-r) \\ \end{aligned}$
It's one of the reasons that short ETFs are not recommended for long-term investing, since they must be rebalanced in order to match the proper multiple of daily returns, and losses can have a higher magnitude than the equivalent positive return.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.