# 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? 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. Feb 4 '18 at 14:48

• Right, so $r_p = 1 + r_a - r_b$, then $\sigma_p^2 = \sigma_a^2 + \sigma_b^2 - 2\sigma_a\sigma_b\rho_{AB}$? 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. Jan 14 '18 at 10:26