Risk Measurement for a Zero-Net Long-Short Portfolio
Summary
The discussion distinguishes portfolio volatility from portfolio return measurement for a long position in one stock and an equal short position in another. A covariance-based volatility formula can produce a small value when the assets have similar volatility and move closely together, but the portfolio’s net initial value is zero, so a conventional return expressed as a fraction of that value is undefined.
One proposed workaround is to include a long position in riskless cash, giving the portfolio a nonzero value against which returns can be calculated. The discussion also warns that an observed high correlation may not persist out of sample. Correlation can be supported by structural links, yet even those relationships can break; selecting pairs because of unusually high measured correlation can also lead to an optimistic in-sample risk estimate. The exchange offers conceptual guidance rather than a detailed procedure for forecasting future correlation or choosing a cash allocation.
Key ideas
- A zero-net portfolio has no initial value to use as the denominator for conventional percentage returns.
- A covariance-based volatility calculation can still describe fluctuations in the long-short position.
- Adding a riskless cash holding gives the portfolio a nonzero value for return calculations.
- High in-sample correlation may weaken, raising realized risk beyond the historical estimate.
Tags
Full text
# Standard deviation of a long-short portfolio with net position zero
# Standard deviation of a long-short portfolio with net position zero
I've come across the following question and I'm slightly stuck in answering it:
> Suppose you have a two-stock portfolio that is long one stock of asset A, and short one stock of asset B, with A and B strongly correlated. Normally, you calculate the risk of the portfolio by calculating the standard deviation of historic returns (or similar). What is the problem in this instance and how could you resolve it?
My first thought is that the porfolio standard deviation is small, because of the net zero position
$$\sigma_P = \sqrt{\sigma_A^2+\sigma_B^2-2\sigma_A\sigma_B\rho_{AB}} $$
If $\rho_{AB} = 1-\epsilon$ then $\sigma_P = \sqrt{(\sigma_A-\sigma_B)^2+2\epsilon\sigma_A\sigma_B} $, which can get pretty small if A and B have similar risk.
Has anyone any better way of describing the problem/resolving it?
## Answer by Tim Wilding (score 2, accepted)
https://quant.stackexchange.com/a/37694
I think the question refers to a rather simpler problem. It is difficult to calculate portfolio returns when the net value of the portfolio is 0. The concept of return involves the change in value expressed as a proportion of the original value. If the original value is zero, then there is no way to calculate returns!
The simplest way to resolve this would be to add a long, riskless cash position to the two-stock portfolio. You can then use your proposed formulae for calculating the risk of the overall portfolio.
## Answer by Frank Fingerman (score 0)
https://quant.stackexchange.com/a/38056
In addition to the answer posted above, you also have the question of whether the correlation will remain as high as it looks in-sample. Sometimes you should expect this to be true for structural reasons (e.g. BRK.A and BRK.B should always be 99+% correlated), though these can sometimes break down (CHF was pegged to the EUR, until it wasn't). More generally, if you've selected the pair for having high correlation, you should expect some reversion to the mean, which means your out-of-sample risk will be higher than as measured in-sample.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.