Morningstar defines the overall size of a stock fund as the geometric mean of the market capitalization for all of the stocks it owns. It's calculated by raising the market capitalization of each stock to a power equal to that stock's stake in the fund's portfolio. The resulting numbers are multiplied together to produce the geometric mean of the market caps of the stocks in the fund's portfolio, which is reported as geometric average cap.
Calculating the Geometric Average Capitalization for a Portfolio
The formula for geometric average capitalization of a portfolio is:
Where:
GACp = geometric average capitalization of an overall portfolio p
Cx = capitalization of stock or fund x
n = number of common stocks, funds, and equivalents in the portfolio p
Wxp = weight of stock or fund x in the overall stock portfolio p **
Example 1
Take a portfolio of three equally weighted stocks, with market capitalizations of $2 billion, $10 billion, and $25 billion:
Geometric Avg Cap = ($2 billion^1/3) * ($10 billion^1/3) * ($25 billion^1/3) = $7.94 billion
Example 2
Suppose the above three-stock portfolio is actually a mutual fund, and represents 50% of an investor’s stock portfolio, which also includes a 25% holding in a $20 billion stock and a 25% holding in a $30 billion stock.
Geometric Avg Cap = ($7.94^1/2)*($20^1/4)*($30^1/4) = $13.94 billion
Note that this is the exact same number that results from “breaking open” the mutual fund and calculating the geometric average using the underlying holdings:
($2^1/6) * ($10^1/6) * ($25^1/6)*($20^1/4)*($30^1/4) = $13.94 billion
Hence, geometric averages for funds and stocks can be combined together to create the overall geometric average for a portfolio.
Note: The calculation has to be based on the fund’s weighting as part of the overall stock portfolio. If a fund represents 10% of the overall portfolio but has only a 75% weighting in stocks, that 10% figure must be adjust to represent the fund’s actual % weighting of the stock portion of the overall portfolio.
Example
Consider a portfolio with a 33.33% stakes in:
a $2 billion stock,
a $10 billion stock,
a mutual fund with a geometric average market cap of $25 billion.
Assuming a 100% stock allocation in the fund:
Geometric Avg Cap = ($2 billion^1/3) * ($10 billion^1/3) * ($25 billion^1/3) = $7.94 billion
However, if the mutual fund were a balanced fund, with only 50% in stocks, we'd need to account for the non-stock portion and calculate the geometric average based on the how each security contributed to the stock portion of the overall portfolio.
In this case, the two stocks each represent 40% of the overall stock portfolio and the fund contributes 20% to the overall stock portfolio.
($2 billion^2/5) * ($10 billion^2/5) * ($25 billion^1/5) = $6.31 billion