# Covariance matrix for historical series w/ different start and end dates

I am trying to compute the variance-covariance matrix of my portfolio composed by some shares of different companies. I would select a time horizon of two years but for some shares of one company I don't have prices older than one year because the company was not already quoted in the market. What is the best thing to do? Am I forced to select a time horizon in which I have prices for all shares(one year)? In this case, is an historical series of only one year acceptable or is it too short?

• Does this answer your question? How do I adjust a correlation matrix whose elements are generated from different market regimes? Nov 9, 2020 at 19:57
• And there are other questions in this vein for ex. quant.stackexchange.com/questions/21666/… Nov 9, 2020 at 20:02
• I meant that for example for AMZN I have prices of the last 2 years, but for MSFT only of the last year because Microsoft has been quoted in the market only one year ago (for example), so my number of prices of MSFT is the half of the number of prices of AMZN Nov 9, 2020 at 23:16
• I don't have problem of dates from different markets Nov 9, 2020 at 23:18

## 1 Answer

I don't have a survey, but I think most people who look at the total returns (price and dividends) of stocks, look at 3-5 years of history. Depending on what you intend to do with your covariance matrix, 1 or 2 years of daily history may be too little.

It's quite normal that if you look at 3-5 years of history, some series will have some short gaps. There are many approaches for dealing with such gaps. (You can pretend that all the return happened at the end of the gap, or happened gradually during the gap, etc.)

But if a stock really does not have history long enough to be in your universe, then there's nothing you can do to make up the missing data. If the criterion for your portfolio is that a candidate stock must have $$n$$ years of history, then you just can't have a stock only with $$n/2$$ years.

The best way around it is not to use pairwise covariance of your stock, but to identify a few factors (such as French-Fama). Then even if you have a stock with a short history (because or recent IPO or corporate action), then you can estimate its beta's to the factors without messing up your other stocks.

• my goal is to minimize the portfolio variance. in order to compute it I need the covariance matrix. Nov 9, 2020 at 23:17
• A stock whose history is too short would not be in the universe of possible stocks from which you select a portfolio with minimum variance. Nov 9, 2020 at 23:20