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There are as many covariances as there are couples of variables. But the covariance or the correlation between any pair of variables is identical to the covariance and the correlation between the same pair of variables in reverse order. The above formula shows that, with m variables, we have m variances terms and only (m - l)2 covariances and correlations, a total of m(m - 1) + m=m2 terms. If we arrange these variances and covariances into a square table, the table is also symmetric.

The variance-co variance table has mxm terms: it is square and symmetric. The off-diagonal terms above and below the diagonal are identical. Such table is a matrix with m rows and m columns. All matrices in the text are designated by bold letters. The current matrix is a variance-covariance matrix and is shown here.

In the above example of two random variables, the variance-covariance matrix is as shown here.

Matrix Notations


The conventions are that bold letters designate matrices and vectors. The essentials of matrix algebra are reminded in the text. For any matrix, we specify the number of rows and the number of columns. For example, a matrix M with n rows and m columns is a (n x m) matrix. The first number is always number of rows and the second number is always the number of columns. Unless otherwise specified, a vector is a set of m terms in a column: It is a matrix of 1 column and m rows, or a (1 x m) matrix. We add an additional notation, M1, which is the transpose of M, where the transpose of an n x m matrix is obtained by swapping rows and columns.

Variance-Covariance Matrix of Portfolio Returns

A variance-covariance matrix is a square symmetric matrix of variances and covariances of m variables, such as stock returns. The asset return is y.. We use the subscript / for asset /and there are m such assets. The variances are along the diagonal and the covariances are off-diagonal terms. The relation between variances, covariances and correlation coefficients hold. Next, we specify the notations for variances and covariances for all m assets:

O: standard deviation of asset / return a: standard deviation of asset j return

Cov(y, y): covariance between two asset returns, equal to poo = o V(y.) a.2: variance of asset i return.

For all i ^ j, Cov(y., y) = 9tpPj= o = Cov(y, y) = 9jPpt = o since the variance-covariance and correlation matrices are symmetric. The variance-covariance matrix is an (m x m) matrix E:

An alternate form, using correlation coefficients, is another mxm matrix, identical to above, but specifying covariances as a function of correlation coefficients and variances:

Consider the above example of two stocks. Their correlation matrix is simple.

Their variance-covariance matrix can be written as the product:

Next, we move to more realistic examples that apply to real portfolios, for calculating the volatility of a weighted portfolio of assets, such as stocks. The difference with that above, where we used unweighted data, is that we need to explicitly introduce the weights of each asset within the portfolio.

  • [1] See appendix for some common matrix definitions and formulas.
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