Volatility, Correlation and the Variance-Covariance Matrix
Oct 10, 2007
Volatility and correlation are two important metrics of financial risk. Volatility is the "volume" risk and correlation is the "shape" risk. If an asset, say Dollar-Yen has an annualized volatility of 8% and if it increases to 10% then the volume of the risk has increased. Simply put, volatility measures the quantum of the risk. On the other hand, if one asset, say Dollar-Yen, has a correlation of 0.85 with another asset, say, the shares of Vodafone, then this correlation will manifest as an elliptical surface on the scatter plot of Dollar-Yen returns and return of Vodafone shares. If now the correlation changes to -0.65 then the shape of that ellipse will be altered significantly. If the correlation becomes 1.0 then the ellipse will collapse into a straight line. If the correlation changes to -1.0 (minus one) then the straight line will be inverted.
Variance is the square of the volatility. And covariance is the product of two volatilities - between two assets - and the correlation between them. If the volatility of Dollar-Yen is 8%, the volatility of Vodafone shares is 12% and the correlation between these two assets is 0.85 then the covariance is: Covar(1,2) = vol(1)*vol(2)*correl(1,2) = 0.00816.
Covariance, simply put, measures both the quantum of movement of two assets as well as their directional change. It captures both the "volume" (quantum of) risk as well as the "shape" risk.
In finance, we work with correlation matrices and variance-covariance matrices all the time. Traders usually like to work with correlation matrices, and in fact, using correlation matrix makes intuitive sense. If you are told that the correlation between two assets is 0.75 then you can immediately infer that if one is moving in one particular direction (going up or down) then the other is also moving in the same direction with a scaling factor of 0.75. Correlation immediately signifies some kind of causal relationship in our minds. On the other hand a variance-covariance matrix (or covariance matrix, for short) does not immediately make sense to us. If you are told that the covariance between two assets is 0.00135, you may struggle for a while to figure out what this means. This is the reason traders generally don't like this measure; they are more comfortable working with correlation numbers and correlation matrices. On the other hand, risk managers prefer to work with co-variances and covariance matrices.
For a three asset case, say, if we are working with Dollar-Yen, Nikkei Futures and Gold and if say, Dollar-Yen has a 0.75 correlation with Nikkei futures and 0.25 correlation with Gold and if Nikkei futures have 0.45 correlation with gold then we will write the three by three correlation matrix (for Dollar-Yen, Nikkei futures and Gold) as:
Correlation matrix is symmetrical in the sense that the half of the matrix below the diagonal is a mirror image of the matrix above the diagonal. All elements of the diagonal are one, signifying the correlation of an asset with itself.
Further, if, say, the volatility of Dollar-Yen is 8%, that of Nikkei futures is 17% and that of Gold is 12% then the variance covariance matrix of these three assets will be given by:
Covariance matrix is also a symmetrical matrix in the sense that the lower half of the matrix (below the diagonal) is a mirror image of the upper half of the matrix. Along the diagonal we have the variances (square of the volatilities) of the assets and off diagonal elements are covariances (the product of two volatilities times the correlation).
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