-The part of the frontier that lies above the MVP is called the efficient frontier. For every portfolio on the efficient frontier, there is an inefficient portfolio with the same standard deviation and lower expected return directly below it.-Thus, of the initial feasible area, we are left only with the northwest edge as the efficient frontier. In general, as we keep adding more and more assets, the efficient frontier will move west. To see this in a spreadsheet using 5 risky assets and the risk-free asset, look at the “Several Assets Dynamic Model” I have provided for you. It is this optimization problem that a young graduate student called Harry Markowitz set up and solved (not for 3 assets, but for N assets), at the University of Chicago in 1951. This was the dissertation he submitted for his Ph.D. 39 years later, he would win the Nobel Prize for being the first to think in mathematical terms about risk and return. This seminal paper “Portfolio Selection” was published in the March 1952 issue of the Journal of Finance.In fact, his work was so radically mathematical for a paper in investments, that Milton Friedman, who was on his graduate committee (who would win the Nobel himself in 1976), said, “Harry, I don’t see anything wrong with the math here, but I have a problem. This isn’t a dissertation in economics, and we can’t give you a Ph.D. in economics for a dissertation that’s not economics. It’s not mathematics, it’s not economics, it’s not even business administration.” Needless to say, he did get his Ph.D. in economics. It is in honor of Markowitz that all the stuff we are studying is called “Markowitz Portfolio Theory”.-We will now examine an important result called two-fund separation. This result will be very useful.12
Two fund separation: Allportfolios on the mean-variance efficient frontier can be formed as a weighted average of anytwo portfolios on the efficient frontier.-Two fund separation has dramatic implications. According to this result, two mutual funds would be enough for all investors. There would be no need for investing in individual stocks separately; every investor could invest in a combination of these two funds (portfolios). But which two funds?Diversification re-examined-So far, we have seen that by adding more and more assets, we can get more and more diversification and reduce portfolio variance and standard deviation. Can we ever eliminate all portfolio variance? In other words, can we reduce the portfolio variance to zero? -Let us start from our general formula for portfolio variance (our variance/covariance matrix):222111().( )2...( .)NNNpiiijijiijjirwrw w Cov r rσσ====+���E5F%%% %-Since we are considering a portfolio of many assets, assume 1iwN=, assume 22( )irσσ=%, and ( ,)covijCov r r=% %. Note that we have N variance terms and N2-N covariance terms.