I am trying to see how diversified a portfolio is, using the portfolio holdings weights and their return streams.
For example, let's say that the portfolio has $3$ stocks, so I will have the $3x1$ W (weight) matrix and $3x3$ correlation matrix C.
The following matrix multiplication $W^T * C * W$ will result in a single number that tells you how diversified the portfolio is. The bigger the number, the less diversified and vice versa.
One challenge is that the stocks may have different time periods. For example, stock A may have $3$ months of data, stock B may be $5$ months of data and stock C may have 1 year of data. Using the naive, standard correlation function will use the common date ranges between the stock pairs and calculate correlations.
This may be a problem if the date ranges are extremely different (e.g. stock A may be $1$ month of data, while stock B and C may have 5 years of data), which will skew the result number.
Is there a name for this phenomenon? What kinds of techniques are out there to solve this kind of problem?