Prove that variance of a portfolio cannot exceed variance of individual assets?

Question

When reading on Markowitz's portfolio theory, I stumbled across the fact that in a market with two risky assets, if no short selling is not allowed, the variance of a portfolio consisting of the two assets cannot exceed the variances of the risky assets individually. That is:$${\\sigma _p}^2 \\le \\max \\{ {\\sigma _A}^2,{\\sigma _B}^2\\} $$Where A and B are two different assets.Could you kindly prove this statement, and possible provide some intuition for why this is the case?

Answer 1

Portfolio theory states that there are diversification (risk reduction) benefits where there is more than one risky asset. The principal outcome is that a combination of the two risks assets will result in a portfolio risk that is less than the riskiest asset, not necessarily both assets. There are many factors that will determine the portfolio risk. The equation is as follows:

Risk_p = √ (weight₁² x variance₁ + weight₂² x variance₂ + 2 x weight₁ x weight₂ x correlation₁₂ x sd₁ x sd₂)

The critical determinants of portfolio risk are:

1. Individual asset risk
2. Weight of each asset
3. Correlation

In a two asset portfolio with individual risk of A = 10% and B = 20%, the minimum variance portfolio (MVP) when correlation is perfect (1), is by placing all of your assets in A because there are no benefits to diversification.

If the correlation is less than 1, then placing some asset B could reduce portfolio risk. For example, when correlation is 0.3, then just 10% in B will reduce portfolio risk to 9.8%, which is less than the risk of A, thus there are benefits to diversification.

The answer to question is a minimization problem that states some combination of A and B will result in a MVP for a given return. Thus, for higher levels of given return, the portfolio result does not necessarily have a result that is lower risk than both assets. The risk minimization process for a given return, creates the efficient frontier: the MVP for a given return, which are said to be optimal or efficient.

In summary, the assumption that asset A and B will give a portfolio risk that is lower than each individual asset risk requires more information: the correlation between the assets and the target level of return.

A direct answer to your question is that the correlation will determine the ability of the two asset portfolio to achieve lower risk than the each asset individually. Any value of the correlation less than 1 will reduce risk, where at some lower level of correlation (e.g. 0.3), the portfolio risk will be less than each individual asset risk.