I. The Risk and Return of Securities
Markowitz's great insight was that the relevant information about securities can be
summarized by three measures: the mean return (taken as the arithmetic mean), the standard deviation of the returns and the correlation with other assets' returns. The mean and the standard deviation can be used to plot the relative risk and return of any selection of securities. Consider six asset classes:
(Courtesy Ibbotson Associates)
II. Portfolios of Assets
Typically, the answer to the investment problem is not the selection of one asset above all others, but the construction of a portfolio of assets, i.e. diversification across a number of different securities. The key to diversification is the correlation across securities. Recall from data analysis and statistics that the correlation coefficient is a value between -1 and 1, and measures the degree of co-movement between two random variables, in this case stock returns. It is calculated as:
III. More Securities and More Diversification
Now consider what will happen as you put more assets into the portfolio. Take the special case in which the correlation between all assets is zero, and all of them have the same risk. You will find that you can reduce the standard deviation of the portfolio by mixing across several assets rather than just two. Each point represents an equally-weighted combination of assets; from a single stock to two, to three, to thirty, and more. Notice that, after 30 stocks, diversification is mostly achieved. There are enormous gains to diversification beyond one or two stocks.
(Courtesy Campbell Harvey)
IV. Markowitz and the First Efficient Frontier
The first efficient frontier was created by Harry Markowitz, using a handful of stocks from the New York Stock Exchange. Here it is, reproduced from his book Portfolio Selection Cowles Monograph 16, Yale University Press, 1959. It has a line going to the origin, because Markowitz was interested in the effects of combining risky assets with a riskless asset: cash.
Notice, too, that it is tipped on its side. The convention of STD on the X axis is developed later.
V. An Actual Efficient Frontier Today
This figure is an efficient frontier created from historical inputs for U.S. and international assets over the period 1970 through 3/1995, using the Ibbotson EnCorr Optimizer program.
(Courtesy Ibbotson Associates)
(Courtesy Campbell Harvey)
In this special case, the new efficient frontier is a ray, extending from Rf to the point of tangency (M) with the "risky-asset" efficient frontier, and then beyond. This line is called the Capital Market Line (CML). It is actually a set of investable portfolios, if you were able to borrow and lend at the riskless rate! All portfolios between Rf and M are portfolios composed of treasury bills and M, while all portfolios to the right of M are generated by BORROWING at the riskless rate Rf and investing the proceeds into M.
VII. Summary
The Markowitz model was a brilliant innovation in the science of portfolio selection. With almost a disarming slight-of-hand, Markowitz showed us that all the information needed to choose the best portfolio for any given level of risk is contained in three simple statistics: mean, standard deviation and correlation. It suddenly appeared that you didn't even need any fundamental information about the firm! The model requires no information about dividend policy, earnings, market share, strategy, quality of management -- nothing about the myriad of things with which Wall Street analysts concern themselves! In short, Harry Markowitz fundamentally altered how investment decisions were made. Virtually every major portfolio manager today consults an optimization program. They may not follow its recommendations exactly, but they use it to evaluate basic risk and return trade-offs.
Why doesn't everyone use the Markowitz model to solve their investment problems? The answer again lies in the statistics. The historical mean return may be a poor estimate of the future mean return. As you increase the number of securities, you increase the number of correlations you must estimate -- and you must estimate them CORRECTLY to obtain the right answer. In fact, with more than 1,500 stocks on the NYSE, one is certain to find correlations that are widely inaccurate. Unfortunately, the model does not deal well with incorrect inputs. That is why it is best applied to allocation decisions across asset classes, for which the number of correlations is low, and the summary statistics are well estimated.