Calculating Long-Short Portfolio Returns with Changing Weights
Summary
The document examines daily returns for a portfolio holding one stock long and another short, with cash proceeds from the short sale included in net asset value. It shows that the portfolio return for a later day should be calculated from the change in total NAV relative to the prior day’s NAV. In the example, this reproduces the reported return from the two listed NAV values. The discussion also points out that portfolio weights change as asset prices and NAV change; weights from the initial date should not be reused for every subsequent period.
A further answer cautions that long-short portfolios may have negative values. Discrete returns can represent a transition from positive to negative NAV, but once NAV is negative, a positive return can correspond to a more negative portfolio value. Continuously compounded returns cannot describe that transition, and the difference of two lognormal asset values need not itself be lognormal. The example is illustrative; handling financing, transaction costs, rebalancing, and changing exposures would require additional assumptions.
Key ideas
- Calculate each discrete portfolio return from consecutive NAV values.
- Portfolio weights change as asset values and total NAV change.
- A long-short portfolio can have negative NAV, which complicates return interpretation.
- Continuously compounded returns cannot capture a transition from positive to negative portfolio value.
- The example does not address financing costs, transaction costs, or rebalancing rules.
Tags
Full text
# calculate portfolio return with one long position and one short position
# calculate portfolio return with one long position and one short position
I was trying to learn how to work out the performance of a portfolio where you are long one stock and short another.
I found an example below. The NAV is calculated by adding the value of the long stock (100) plus the cash line (58.16) and minus the short stock (58.16). Guessing this cash line is the amount raised by shorting stock XYZ?
```
Long Short
Stock Stock Cash NAV Return %
ABC XYZ
Price Value Price Value
1st Jan 12.11 100 17.83 58.16 58.16 100
2nd Jan 11.84 97.79 17.5 57.08 58.16 98.87 -1.1317
3rd Jan 11.62 95.96 17.03 55.55 58.16 98.57 -0.3046
```
My question is though, is how to replicate those returns but when given weights & shares instead. For example,
Lets say you are long 8.25754 shares in ABC & short 3.26173 shares in XYZ.
The returns below are based on the prices above.
```
ABC XYZ
Return % Return %
1st Jan
2nd Jan -2.21 -1.85
3rd Jan -1.88 -2.69
```
So I am able to calculate the return for the 2nd of Jan by doing the following,
```
Assume starting NAV of 100,
weight of ABC is 100%
weight of XYZ is 58.16%
So total return is (-2.21 * 1) - (-1.88 * 0.5816) = -1.13%
```
But whatever I do I can't get a return for the 3rd Jan of -0.3046%. How do I calculate this value?
## Answer by ZRH (score 1, accepted)
https://quant.stackexchange.com/a/44561
While it is of course possible to apply standard definitions of returns, one needs to bear in mind that a long/short portfolio may end up having a net negative value. Thus:
i) You cannot use continuously compounded returns. Starting out with a positive portfolio value, continuously compounded returns can never take you to a negative portfolio value whatever large negative returns you consider.
ii) Discrete single-period returns: $R=(V_{T+1}-V_T)/V_T$. With discrete returns, it is possible to go from positive portfolio values to negative portfolio values. However, once the portfolio is in the regime of negative values, positive returns mean (somewhat paradoxically) that portfolio values go more negative.
When attempting to use continous returns, the problem is basically that the difference of two lognormal distributions cannot be described by a lognormal distribution, as it exhibits negative values too.
## Answer by Sanjay (score 1)
https://quant.stackexchange.com/a/44555
What seems to be your problem? Which calculations do you do that will not give you a decent answer?
Your portfolio value is `NAV = 98.87` and the next day it is: `NAV=98,57`.
$$ r=\frac{NAV_1-NAV_0}{NAV_0}=\frac{98.87-98.57}{98.57} = -0.0030=-0,30\% $$
Also, be aware that you weights are not $1$ and $0.5816$ anymore but $$ w_{ABC}= 97.79/98.57 $$ $$ w_{XYZ}= 57.08/98.57 $$
(I would personally have defined $w$ such that $w_{XYZ}=- 57.08/98.57$, but that does not mater)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.