# Volatility and weights of a portfolio whose value is negative

How do you calculate the one day standard deviation (in dollars) for a portfolio that is short $30,000? How do you calculate the weightings to use? I already have the necessary covariance matrix. • Do you seek the minimum variance portfolio? In general, you ought to know the weights of your portfolio (unless you want to find them by optimization techniques). The portfolio variance is just$w^T \Sigma w$where$w$are your weights and$\Sigma$the covariance matrix – Kevin Sep 24 '19 at 14:11 • @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? – Alex Sep 24 '19 at 14:17
• 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. – Kevin Sep 24 '19 at 14:25
• 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). – demully Sep 24 '19 at 23:36

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.