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:
The previous equation actually presents a big problem.
If a portfolio consists of only two stocks, we are required to estimate two variance terms and on covariance term (parameters). This will only require, given the above equation, 30 days of price data. If the portfolio consists 10 stocks then we are required to estimate 10 variances and 45 covariance terms, in total 55 parameters. For that we would require 110 days of data. This is little of one quarter's price data and is not a problem at all. The problem starts to compound as the portfolio gets bigger and bigger (a very real life scenario).
What if we have 100 stocks in the portfolio, then we need to estimate 100 variances and 4,950 covariance terms, in total 5,050 parameters. This would require 1,010 days of price data or in other words if we assume 252 trading days in a year then this would mean that the portfolio manager needs roughly 4 years of price data. Again this is not so difficult as most liquid and large cap stocks will have 4 years of trading history.
For a 1000 stock portfolio one would need 10,010 days of price data or 39.72 years worth of trading history. Even most of the large cap stocks may not have that long a history.
Now what if a large fund manager, like our client, holds 5000 stocks in a portfolio? Then in the covariance matrix there would be 5,000 variances and 12,497,500 (12.497 million) covariance terms (i.e. in all around 12.5 million parameters). This would require 50,010 days of price data or roughly 198 years of trading history!
This presents a big problem. How to estimate the covariance matrix in an accurate and meaningful way? And that begs another question. Even if we have long trading histories for stocks how could we account for the changing variances and covariances, i.e. how do we take care of a changing covariance matrix of a portfolio?
Thankfully, there are models and methods to deal with the above problem, which of course involves a bit of detailed matrix algebra. The reader may be interested in reading up further on this topic, which is quite intriguing.
Note: The reader is referred to an excellent book called Optimal Trading Strategies by Robert Kissell and Morton Glantz for more on the above subject.
Disclaimer
"Risk Latte uses proprietary and non-proprietary mathematical and empirical models to measure the volatility and estimate the direction of the market. There is no guarantee of any particular outcome happening and readers must exercise caution while interpreting the conclusions of this article. Risk Latte Company is not a registered stock broker or an SFC registered entity and readers must take advise from their financial advisors, stock brokers, research analysts and bankers while making any buying or selling decisions. Risk Latte Company is not in the business of making stock or asset forecasts whether explicitly or implicitly and shall not be responsible for and/or liable for any losses arising out of any trading decisions based on the above article."
  
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