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.
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.
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.