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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