Problems in Estimating the Covariance Matrix of a Portfolio
Team Latte
Jul 18, 2005

Recently, during a consulting assignment for a portfolio Manager, we were asked by a trainee as to how to estimate a covariance matrix of stocks (a portfolio of stocks) in $/share from price history. Also, how many days data should be included to estimate accurately a $/share covariance matrix.

Both the above questions could seem trivial and may just boil down to an exercise in elementary statistics. However, in practice they may not be that trivial. Besides, a covariance matrix of a portfolio of stocks expressed in $/share is of immense practical help to an equity trader and a fund manager and he would need to have a covariance matrix estimated on a daily basis not only make trading decisions but also to decompose risk and calculate the residual risks of his trading strategy.

We will deal with the second question in this column. Of course by doing so, we presuppose that the reader already knows what a covariance matrix is and how it is computed. If there are n stocks in a portfolio then the covariance matrix VCV (variance-covariance matrix) will have n(n+1)/2 unique parameters. Now, from our knowledge of elementary statistics we know that there should be at least 20 data points to have statistically meaningful results. Therefore, there should be at least 20*n(n+1)/2 data points to make a meaningful observation on the portfolio of n stocks.

Elementary algebra tells us that if K is the number of historical observations, then in order to solve for each parameter using historical data the following condition needs to be satisfied:

Now over a t day period there would be t*n data observations. Therefore, the number of days required to accurately estimate the covariance matrix (VCV) for n stocks is given by the equation:

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