Diversification With An Equally-weighted Portfolio
The expected return on a portfolio is a weighted average of the expected returns on each individual asset:
where is the proportion of the investor's total invested wealth in asset .
The variance of the portfolio return is given by:
Inserting in the expression for :
Rearranging:
where is the variance on asset and is the covariance between assets and . In an equally-weighted portfolio, .
The portfolio variance then becomes:
Where is the average of the convariances for . Simplifying we obtain
As the number of assets grows we get the asymptotic formula:
Thus, in an equally-weighted portfolio, the portfolio variance tends to the average of covariances between securities as the number of securities becomes arbitrarily large.
Read more about this topic: Diversification (finance)