An Introduction to Investment TheoryWilliam N. GoetzmannYALE School of Management Chapter II: The Geography of the Efficient FrontierIn the previous chapter, we saw how the risk and return of investments may be characterized by measures of central tendency and measures of variation, i.e. mean and standard deviation. In fact, statistics are the foundations of modern finance, and virtually all the financial innovations of the past thirty years, broadly termed "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 the investment 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 capital structure, its market, its competitors -- and calculating a few simple statistics. In this chapter, we will follow Markowitz' lead and see where the technology of modern portfolio theory takes us.
(Courtesy Ibbotson Associates) This figure was constructed using historical risk and return data on Small Stocks, S&P stocks, Corporate and Government Bonds, and an international stock index called MSCI, or Morgan Stanley Capital International World Portfolio. The figure shows the difficulty an investor faces about which asset to choose. The axes plot annual standard deviation of total returns, and average annual returns over the period 1970 through 3/1995. Notice that small stocks provide the highest return, but with the highest risk. In which asset class would you choose to invest your money? Is there any single asset class that dominates the rest? Notice that an investor who prefers a low risk strategy would choose T-Bills, while an investor who does not care about risk would choose small stocks. There is no one security that is best for ALL investors.
Where the sigma AB is the covariance of the two securities. Here is how to use correlation in the context of portfolio construction. Consider two securities, A and B. Security A has a mean of 10% and an STD of 15%. Security B has a mean of 20% and an STD of 30%. We can calculate the standard deviation of a portfolio composed of different mixtures of A & B using this equation: The mean return is not as complicated. It is a simple weighted average of the means of the two assets: mean p = W A RA + W B R B. Notice that a portfolio will typically have a weight of one, so usually, W A + W B = 1. In other words, a mixture of 20% of the MORE RISKY SECURITY actually decreases the volatility of the portfolio! This is a remarkable result. It means you can reduce risk and increase return by diversifying across assets. (Courtesy Campbell Harvey)
(Courtesy Campbell Harvey) If you allow yourself to vary the portfolio weights, rather than keeping them equal, the benefits are even greater, however the mathematics is more challenging. You not only have to calculate the STD of the mixture between A&B, but the STD of every conceivable mixture of the securities. None-the-less, If you did so, you would find that there is a set of portfolios which provide the lowest level of risk for each level of return, and the highest level of return for each level of risk. By considering all combinations of assets, a special set of portfolios stand out -- this set is called the efficient frontier. The efficient frontier, shown in blue, is the set of dominant portfolios, at least from the perspective of a risk averse investor. For ANY level of risk, the efficient frontier identifies a point that is the highest returning portfolio in its risk class. By the same token, for any level of return, the frontier identifies the lowest risk portfolio in that return class. Notice that it extends from the maximum return portfolio (actually a single asset) to the minimum variance portfolio. The efficient frontier has a portfolio for everyone -- there are an infinite number of points in the set, corresponding to the infinite variation in investor preferences for risk. The area called the feasible set represents all feasible combinations of assets. There are no assets that fall outside of the feasible set.
Notice, too, that it is tipped on its side. The convention of STD on the X axis is developed later.
(Courtesy Ibbotson Associates) This is state-of-the-art portfolio selection technology, however it is still based upon Markowitz's original optimization program. There are some basic features to remember:
VI. The Efficient Frontier with the Riskless Asset T-Bills are often taken to be riskless assets, and their return is indicated as Rf, the risk-free rate. Once you allow the riskless asset to be combined into a portfolio, the efficient frontier can change. Since it is riskless, it has no correlation to other securities. Thus it provides no diverisfication, per se. It does provide an opportunity to have a low-risk portfolio, however. This picture is a diagram of the efficient frontier composed of ALL the risky assets in the economy, as well as the riskless asset. (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.
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