How do you calculate the one day standard deviation (in dollars) for a portfolio that is short $30,000? How do you calculate the weightings to use? I already have the necessary covariance matrix.
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$\begingroup$ Do you seek the minimum variance portfolio? In general, you ought to know the weights of your portfolio (unless you want to find them by optimization techniques). The portfolio variance is just $w^T \Sigma w$ where $w$ are your weights and $\Sigma$ the covariance matrix $\endgroup$– KevinSep 24, 2019 at 14:11
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$\begingroup$ @KeSchn - but what are the weights when all is negative? And once I have the portfolio variance, do I multiply by $30000 to get the expected one day move? $\endgroup$– AlexSep 24, 2019 at 14:17
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1$\begingroup$ Your portfolio consists of $N$ assets. Some may be long positions, some may be short position. Long and short means for the weight simply that $w_i$ is positive or negative. So you compute the portfolio variance via $w^T\Sigma w$ and then you can compute the standard deviation by taking the square root. $\endgroup$– KevinSep 24, 2019 at 14:25
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$\begingroup$ The variance of the portfolio should be no different whether you were 30k long or 30k short that portfolio. If variance isn't indifferent to direction, you have a spreadsheet or coding bug to correct. And if it is, it will always be positive; in which case, sigma is root variance.(irrespective of direction). $\endgroup$– demullySep 24, 2019 at 23:36
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2 Answers
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The sigma is the same short as if you were long.
Imagine you held exactly the opposite portfolio. It stands to reason that volatility of holding both is net zero; and they're -100% correlated. It therefore stands to reason that you have to have the same sigma (for the same value) for them to cancel out thus, as they must.
Alternatively, just look at the variance, which will be a sign-indifferent positive, the same as for -30k as for +30k.
Where the signs do start to matter is when you have longs and shorts in the portfolio, and you want to start to attribute the portfolio risk amongst its constituents. Then you can indeed have assets with a negative stdev; because their interactions with other constituents represent a net reduction in (expected!!!) volatility.
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Your weights, including cash, should sum to 1. Divide the positions by the portfolio net asset value to get the weights. For example, a \$100 portfolio with a \$50 short position would have \$150 in cash so the weights would be -0.5 and 1.5 for the stock and cash respectively.
