Financial Management I

Module 2 Notes

William N. Goetzmann

Class II


The Geography of the Efficient Frontier

In the previous class, we saw how the risk and the return for investments may be characterized by measures of central tendency and measures of variation, i.e. mean and standard deviaiton. In fact, statistics are the foundations of modern finance, and virtually all the innovations of the past thirty years, called "Modern Portfolio Theory," have been based upon statistical models. Because of this, it is useful to review what a statistic is, and how it relates to theinvestment problem. In general, a statistic is a function that reduces a large amount of information to a small amount. For instance, the average is a single number that summarizes the typical "location" of a set of numbers. Statistics boil down a lot of information to a few useful numbers -- as such, they ignore a great deal. Before modern portfolio theory, the decision about whether to include a security in a portfolio was based principally upon fundamental analysis of the firm, its financial statements and its dividend policy. Finance professor Harry Markowitz began a revolution by suggesting that the value of a security to an investor might best be evaluated by its mean, its standard deviation, and its correlation to other securities in the portfolio. This audacious suggestion amounted to ignoring a lot of information about the firm -- its earnings, its dividend policy, its captial structure, its market, its competitors -- and calculating a few simple statistics. In this class, we will follow Markowitz' lead and see where the technology of modern portfolio theory will take us.

I. The Risk and return of securities

I. Two dimensions of investor choice



Some investors prefer D, some prefer E, some prefer A. No songle asset dominates. In fact, using historical risk and return data, we can look at investor choices over major U.S. asset classes. The axes plot monthly stadard deviation of total returns, and average monthly returns over the period 1926 through 1995. Notice that small stocks, provide the highest return, but with the highest risk. Which asset class would you choose it invest your money? Is there any single asset class that dominates the rest?

(Courtesy Ibbotson Associates)



II. Portfolios of Assets

The dynamics of stocks
Correlation, Covariance and Boom and Bust cycles



The standard deviation of any portfolio of A and B Can be expressed as:




What if the correlation of A&B = 0 ?
What if the correlation of A&B = 1 ?
What if the correlation of A&B = -1 ?

Negative Weights, and shorting stock

III. More Securities and More Diversification



III. The Effects of Diversification

IV. Features of the Efficient Frontier
The Efficient Set:

picture courtesy of Campbell Harvey
  • minimum level of risk for each level of return
  • maximum level of return for each level of risk
  • Minimum Variance Portfolio
  • Maximum Return Portfolio
  • The Inefficient Frontier

  • V. Markowitz and the First Efficient Frontier
    [FRONTIER]

    VI. The Efficient Frontier with the Riskless Asset

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    YALE School of Management