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- BuffaloFan32
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Calculating marginal risk from a covariance matrix
I have a portfolio of assets (positive and negative weights) and a corresponding covariance matrix. I know I can estimate the VaR at the 95 percentile by doing -1.645*sqrt(wNw) but now I am interested in estimating each assets contribution to the risk. Using historical data in excel, I would use the slope() function but right now I only have a covariance matrix. I am trying to remember back to my college days and take the derivative of the portfolio with respect to each asset but I can't figure it out.
Re: Calculating marginal risk from a covariance matrix
Using the chain rule and the symmetry of c, if I haven't made a mistake:
[$]\frac{\partial}{\partial w_i} (\sum_{j,k} w_j w_k c_{jk})^{1/2} = \frac{\sum_j w_j c_{ij}}{ (\sum_{j,k} w_j w_k c_{jk})^{1/2}}[$]
[$]\frac{\partial}{\partial w_i} (\sum_{j,k} w_j w_k c_{jk})^{1/2} = \frac{\sum_j w_j c_{ij}}{ (\sum_{j,k} w_j w_k c_{jk})^{1/2}}[$]
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- BuffaloFan32
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Re: Calculating marginal risk from a covariance matrix
Thanks again Alan. If I understand correctly, the denominator is just the variance of the portfolio. The numerator is the sum of each asset's weight times its covariance with the asset in question.
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- BuffaloFan32
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Re: Calculating marginal risk from a covariance matrix
Hmmmmmmm...I must be doing something wrong. I multiplied my vector of weights by the first column in my covariance matrix and then divided that product by the portfolio variance. Then, I repeated that for each position in the portfolio. The risk contributions I am seeing are tiny and do not sum to 100%.
Re: Calculating marginal risk from a covariance matrix
The denominator of the expression I posted is the portfolio's std dev.
Note: I was just answering what I thought was your question about a derivative of the portfolio std dev. But, now, I suspect that you don't really want that derivative, but instead the decomposition explained on pgs 11-12 here
Note: I was just answering what I thought was your question about a derivative of the portfolio std dev. But, now, I suspect that you don't really want that derivative, but instead the decomposition explained on pgs 11-12 here
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- BuffaloFan32
- Posts: 10
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Re: Calculating marginal risk from a covariance matrix
Yes, that was exactly what I was looking for. thank you!