# Correlation sensitivity in multivariate $t$-copula for portfolio VaR of electricity futures using Kendall's tau-$b$ correlation matrix

My t-copula model captures the daily dollar returns of a portfolio of approximately 400 assets. I am curious if there's a generally accepted way to quantify the sensitivity of portfolio movements with respect to the underlying correlation matrix. My first instinct is to try a discrete approximation, such that If C is my correlation matrix, and X is my current returns:

$$\frac{dX}{dC} \sim [X(C + 0.0001) - X(C - 0.0001)] / 0.0002$$

Is this a valid approach? Your help is greatly appreciated!

EDIT: Forgot the outer parentheses on the numerator.

## Further Edits from own comments

• I applied a univariate t CDF with 3df to a multivariate t distribution in Python with 3df. Then, having gotten values on (0, 1) I applied the respective inverse probability transform for each of the data to rescale to the original level. I then applied those returns to the previous closing prices and multiplied by the notional tied to the series, and summed the result. My idea in the above was to quantify correlation risk. I chose 0.01% somewhat arbitrarily, but the idea is: how can I perform a correlation / dependency risk attribution?
• I meant to add: the copulas are based on Kendall's tau-b of individual asset returns. I thought the rank correlation part was implied.
• I’m just interested in separating correlation risk from price risk and (as I’m dealing with electricity futures) generation / volume risk
• My question was specifically about correlation risk sensitivity and attribution. I didn’t see a reason to get into VaR. But, since we’re now into that rabbit hole, my VaR simulations are in line with data we’ve seen from our trading desks over the past 500 trading days and across ten books.
• As for correlation selection: copulas feature nonlinear transformations, so a rank correlation matrix is necessary, and Kendall's tau handles ties (which I have) better than Spearman does.
• I’m not optimizing anything. I’m just modeling a VaR for a portfolio of correlated assets. I know from the literature that these are best described using a multivariate t copula with v=3. I estimate the empirical probability distributions of my observed returns. I get my rank correlation matrix. I want to measure my correlation risk and correlation sensitivity.
• What does $dX/dC$ mean when $X$ is a vector and $C$ is a matrix? What does $C + 0.0001$ mean? There are ways to assign meaning, but what is it you are implying. How are you using a t-copula (a multivariate cummulative distribution with uniform marginals) to get "current returns". – RRL Oct 8 '20 at 0:48
• I am trying to determine how much can my PnL move for a small change in the correlations. I already have my university distributions and have selected three degrees of freedom for the t copula. The correlation matrix is based on Kendall’s tau-b. Am I misunderstanding something? – CasusBelli Oct 8 '20 at 1:05
• How did you derive your $\sim$ formula. It looks like it came from nowhere since it definitely isn't approximate or equivalent to $dX/dC$ like how you say it is since it still contains $C$ (differentiation w.r.t. a variable, here $C$, should effectively eliminate that variable). Just as important, what is $0.0002$? Overall though, your set-up has completely excluded the copula, which is based on rank transforms of $X$, not $X$ itself. Your set-up also ignores portfolio weights even though your question is about portfolio sensitivity, which $X$ has to be weighted for, not used directly – develarist Oct 8 '20 at 4:44
• I applied a univariate t CDF with 3df to a multivariate t distribution in Python with 3df. Then, having gotten values on (0, 1) I applied the respective inverse probability transform for each of the data to rescale to the original level. I then applied those returns to the previous closing prices and multiplied by the notional tied to the series, and summed the result. My idea in the above was to quantify correlation risk. I chose 0.01% somewhat arbitrarily, but the idea is: how can I perform a correlation / dependency risk attribution? – CasusBelli Oct 8 '20 at 12:18
• I meant to add: the copulas are based on Kendall's tau-b of individual asset returns. I thought the rank correlation part was implied. :) – CasusBelli Oct 8 '20 at 13:06

## 1 Answer

Instead of thinking "at the margin", I've opted to conduct an attribution of sorts, by running the copula with the empirical Kendall's tau-b correlation matrix and again with a zero matrix. The difference between the two scenarios represents correlation risk.

• margin, attribution? What are you talking about – develarist Oct 16 '20 at 23:20
• The whole point of the exercise is to understand the impact that my correlation structure has on my portfolio VaR. So, if my correlations were to increase marginally, how would my portfolio move? That was the first approach. Alternatively, if I have a portfolio with nonzero correlation matrix and one with zero correlation matrix, assuming all else equal, the VaR discrepancy is my correlation risk. That’s all. – CasusBelli Oct 17 '20 at 1:18
• still have no idea how you're constructing portfolios. it's now the first time you've mentioned VaR. are you minimizing portfolio VaR? how is this being done with a copula in your code? and within all this, what does it mean to "run the copula with Kendall's tau correlated and then non-correlated matrix". In order to generate asset returns? I know Kendall's tau (don't think it has any edge over Pearson's correlation here) but "run the copula" could mean many different things. I don't think you really know what you're doing at all. what references or code are you even basing your work on – develarist Oct 17 '20 at 3:26
• Thank you for your opinion. My question was specifically about correlation risk sensitivity and attribution. I didn’t see a reason to get into VaR. But, since we’re now into that rabbit hole, my VaR simulations are in line with data we’ve seen from our trading desks over the past 500 trading days and across ten books, so while I appreciate your input, please refrain from insulting others when you are the ignorant one. As for correlation selection: copulas feature nonlinear transformations, so a rank correlation matrix is necessary, and tau handles ties (which I have) better than Spearman does. – CasusBelli Oct 17 '20 at 3:37
• you haven't explained if the copulas are being used to generate asset returns or form portfolios. any talk of a correlation matrix, it doesn't matter if it's Pearsons, Spearman or Kendall's tau, it should only be used for generating asset returns when framed how you did (first uncorrelated matrix, then correlated matrix). from the very beginning of the question, there has been little explanation of the procedure you're running, let alone where the first formula $dX/dC$, especially its denominator, even came from – develarist Oct 17 '20 at 3:40