how do you formally prove that the lowest variance a well-diversified portfolio can have is the market risk?

Answer 1

I don't know enough math to do a formal proof, but it can also be explained conceptually.

As more and more securities are added to a portfolio, the unsystematic risk of the existing securities are offset with the securities being added with covariances to the portfolio of less than 1. Each new security addition like this lowers the total variance of a portfolio.

The systematic risk (the covariance between each fund and the market) is something that all securities will typically have in common. With unsystematic risk, those factors that cause the risk are not held in common.

The only way to lower risk is to invest in securities with low or negative Betas. This also lowers the potential reward, though. In this way, your risk could be potentially lower or higher than the market risk.

You can theoretically make your own portfolio according to the risk/reward traits that you would like to have using the "market portfolio" and the risk-free asset (like Treasury bills are considered). You cannot change the risk/reward ratio in a diversified portfolio like this, though. This is what you may be referring to when you say "market risk."

Investopedia offers an explanation of the concept as well at http://www.investopedia.com/articles/06/MPT.asp.

Answer 2

If your portfolio has two assets one with a variance of +a and the other -a, with market risk as r then your total risk =r+a-a=r the market risk, which is the proof.