# 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?

• Can you please give some context. As it stands there is no problem. For instance take A to be 2 units of some stock C and B to be 1 unit of stock C. If you go long A and short B you have C and there is no problem despite A and B being perfectly correlated. Which book is it that you are quoting and which page? – Borun Chowdhury Feb 4 '18 at 8:35
• I was quoting from an interview question, not a book. I think the question was quite vaguely worded, so apologies for the slightly vague question. The question also said stock A was a UK stock and B was an Australian stock, but I think that was just a red herring. I think the question assumed an initial net position of zero. I think the down vote on this question was rather harsh. I want to contribute to the community and have given a reasonable description of the question as I remember it. – Ronnie268 Feb 4 '18 at 14:48

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.

• Right, so $r_p = 1 + r_a - r_b$, then $\sigma_p^2 = \sigma_a^2 + \sigma_b^2 - 2\sigma_a\sigma_b\rho_{AB}$? – Ronnie268 Jan 12 '18 at 20:38
• Almost. $r_p = r_a - r_b$ assuming the cash position returns 0. Your formula for variance is correct. – Tim Wilding Jan 14 '18 at 10:26
• Adding a long, riskless position also makes sense from a real-world perspective. In order to put together such a portfolio, you need to have some initial cash. You then use that cash to borrow securities to sell (for your short leg), and you use those proceeds to finance your long position. But at the end of the day, you still have the initial cash in the bank (or posted as margin). – Frank Fingerman Feb 3 '18 at 14:03
• Why are you assuming the value of the portfolio is zero? The assets being correlated is usually a statement about returns being correlated and prices themselves can be vastly different. – Borun Chowdhury Feb 4 '18 at 8:31

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.