A portfolio is a combination of securities. Portfolio analysis provides a framework by which the investors can identify optimal portfolios from a universe of infinite portfolios. Investors need to estimate the expected return and variance of each security under consideration for inclusion in the portfolio along with all the covariances between securities. With these measures the investors proceed to calculate the expected return and risk of alternative portfolios to evaluate their desirability and derive a set of efficient portfolios.
wi = percentage of investor’s funds invested in security i
wj = percentage of investor’s funds invested in ...view middle of the document...
Expected return and variance of a portfolio consisting of n securities are given by
[pic] = [pic]
for i ( j
where (ij equals variance when i = j and covariance when i ( j.
In matrix notation
[pic] = [pic][pic]
(2p = [pic] [pic][pic]
where diagonal elements are variances and off-diagonal elements are covariances. There are n variances and n(n - 1)/2 unique covariances. Total number of inputs needed to calculate risk and return for an n-security portfolio is n(n + 3)/2 which comprises n expected returns, n variances, and n(n - 1)/2 unique covariances. In a portfolio of a large number of securities, portfolio risk is dominated by covariance terms. That is, a security’s variance contributes a source of risk that can be diversified away by forming a large enough portfolio.
The risk of the portfolio can be thought of as a weighted average of the contribution of total risk of the individual securities to the portfolio risk. The weights applied to each security are wi’s. Thus the contribution of an individual security to portfolio risk depends upon a security’s own variance and its covariance with other securities in the portfolio. Using variance as a measure of risk, we can rewrite the expression for portfolio variance as follows:
(2p = [pic][Total risk of security i in the portfolio]
i ( j
The two terms in the square brackets represent total risk of security i in the portfolio. In a well-diversified portfolio, the first term, [pic], will be small and relatively unimportant. The next term is a weighted sum of the security’s covariance with all other securities in the portfolio. In a diversified portfolio, this term will not be small, since the sum over j = 1 to n (where j ( i) will be almost 1. In a well-diversified portfolio, the major risk a security adds to the portfolio is the covariance that the security has with the rest of the portfolio.
The portfolio selection technique developed by Markowitz is known as Markowitz diversification. Markowitz diversification involves combining securities with less than perfect positive correlation in order to reduce portfolio risk without sacrificing any of the portfolio’s return. It addressees the impact of choosing portfolio weights optimally upon the diversification risk. It takes advantage of the fact that risk can be reduced further by the management of portfolio weights. It improves upon naive or simple diversification which allocates equal amounts of wealth to each security in the portfolio. In general, the lower the correlations (or equivalently, covariances) between the securities in the portfolio, the less risky the portfolio will be. This will be true regardless of how risky the securities in the portfolio are when considered in isolation.
Impact of Correlation on Portfolio Variance
Let us examine...