Developing and Defining a Well Managed Portfolio - A Primer on Modern Portfolio Theory


Edward A. Moses, Ph.D., J. Clay Singleton, Ph.D., and Stewart A. Marshall III, Esq.

Authors' Note: Modern Portfolio Theory has become a customary tool used by investment professionals and, as such, constitutes an industry standard that investment decision makers cannot ignore. This academic theory has become the bedrock of investment practice. We have elected to publish three articles in consecutive editions of Wealth Strategies Journal to provide its readership with an understanding of Modern Portfolio Theory and the application of this theory to pertinent issues surrounding the administration and formulation of portfolios. Sequential publication eliminates the need to redevelop Modern Portfolio Theory and other concepts in each article. Wealth Strategies Journal readers will have the option of reviewing earlier articles to clarify any points of interest in subsequent articles. This first article will provide a foundation for understanding the underpinnings of Modern Portfolio Theory. The articles to follow in this series are: "Using a Portfolio's Required Return to Develop an Appropriate Risk Level," and "Determining the Appropriate Withdrawal Rate for a Portfolio: The Crossover Rate." I. Introduction An investment decision maker (IDM) can be broadly defined as anyone undertaking responsibilities associated with providing investment advice or implementation. IDMs can include, among others, fiduciaries, trustees, investment advisers, family office directors, portfolio managers, and individuals managing personal assets. One of the most important tasks undertaken by the IDM is portfolio design. Design means assembling and maintaining a portfolio of assets with a risk tolerance suitable for the purposes, terms, distribution requirements, and other conditions of the portfolio. Modern Portfolio Theory (MPT) guides the IDM in constructing and managing a portfolio that provides the highest expected return at an appropriate level of risk tolerance. Further, MPT requires the IDM to develop return and risk expectations for all potential portfolio assets - an exercise that always plays a role, explicitly or implicitly, in investment management. Finally, MPT focuses on the whole portfolio, not the individual assets because a portfolio is different than the sum of its constituent parts. The principles of MPT, therefore, can help the IDM develop a well managed portfolio.¹ Section II of this article serves as a primer on MPT and demonstrates how the IDM can use MPT as a tool for managing the asset allocation of a portfolio. We also emphasize the importance of business judgment in portfolio construction and the trade-off between strictly MPT portfolios and other desirable characteristics such as liquidity and diversification. Section III concludes with a summary of why the IDM should use MPT as a guide for portfolio construction. II. A Primer on Modern Portfolio Theory A. Asset Allocation. Most IDMs know that asset allocation probably affects portfolio risk and return more than any other investment decision. Intuitively, a portfolio that is 70% stocks and 30% bonds is more risky and should hold the promise of a higher return than a portfolio with 30% stocks and 70% bonds. MPT quantifies this intuition by constructing portfolios that provide the highest expected return for every achievable level of expected risk. By the same token MPT can find the portfolio with the lowest expected risk for every possible expected return. Because no one can invest in the past, MPT must involve forecasts and expectations. The benefits of MPT, however, do not depend on accurate forecasts. Even if the IDM's expected return and risk turn out to be less than prescient, MPT provides a flexible platform that can help construct suitable asset allocations. B. Efficient Portfolios. Portfolios that have the lowest expected risk for a given level of expected return are termed efficient portfolios. MPT is a mathematical algorithm analogous to a milling machine. Raising and lowering the expected return is like changing a manufacturing specification and causes MPT to produce portfolios that differ in their asset allocation and, therefore, their risk. A low expected return might produce our 30-70 portfolio while the 70-30 portfolio, with its commensurately higher expected return and risk, might be the result when higher returns and risk are appropriate. The ability to find the lowest expected risk portfolio with a pre-specified level of expected return is one of the characteristics that makes MPT an attractive tool for the IDM. C. Basic Building Blocks and Diversification. The basic ingredients for MPT are expected return, expected risk, and the correlation between the expected returns for each pair of assets. Expected return for a portfolio is the weighted average of the expected returns of the constituent assets where the weights are the fraction of the portfolio's value allocated to each asset. Risk is represented by the standard deviation - a statistic that evaluates how volatile the returns are over time. Correlation is another statistic measuring the tendency of the assets' returns in the portfolio to offset each other. The idea is fairly simple: if the portfolio consists of two assets in equal measure and the assets mirror each such that when the return of one goes up the other goes down, the variation of the combination is very small. As a purely hypothetical example consider a portfolio of 50% stocks and 50% bonds and assume that the returns of stocks and bonds always move in opposite directions. This portfolio will be less volatile than a portfolio of either 100% stocks or 100% bonds. MPT embodies the old saying: "Don't put all your eggs in one basket" and the correlation statistic measures the impact of diversification. MPT also reminds us that it is not the risk of individual assets that matter as much as the risk of the combination of assets. D. Modern Portfolio Theory's Building Blocks Illustrated. The following simplified example illustrates the rationale behind using correlation to quantify diversification. Assume we have two stocks - an airline (AIR) and an oil company (OIL). Their stock prices will react very differently to a change in oil prices. When oil prices rise, OIL benefits and AIR suffers. Assume that the IDM has just these two stocks at her disposal and that she develops expected returns, risks, and correlations shown in Figure 1.² The IDM might have conducted her own independent research, consulted expert analysts, studied historical patterns or some combination of these techniques to arrive at these estimates. An equally weighted portfolio would produce an expected return of 12.5%, exactly half way between 10 and 15%. The standard deviation of this portfolio, however, would be 14%, less than the risk of either OIL or AIR, because the less than lock-step correlation mitigates some of the risk. The hypothetical performance of the equally weighted portfolio is depicted in Figure 2. Notice that OIL varies more than AIR with a higher return while the MPT portfolio tracks between the two stocks with a lower volatility. Figure 2 - Diversification Illustrated The MPT algorithm allows an IDM to dial up or down the expected return (and expected risk) and construct portfolios with different return-risk trade-offs as Figure 3 illustrates. The next section describes the conditions under which MPT portfolios are the best portfolios. E. MPT Portfolios are the Best Portfolios.³ Figure 3 suggests that MPT can generate any number of portfolios with different asset allocations. The IDM will choose only from among the best of these - in the sense that only those that produce the highest expected return for each level of risk constitute the best. This set of best portfolios constructed by MPT constitutes the efficient frontier. To illustrate this concept a more reasonable set of potential assets is introduced in Figure 4.⁴ Between 1972 and 2006 small (market capitalization) stocks domiciled in the US had the highest return followed by foreign stocks and US large cap stocks. The final ingredient in MPT is the correlations, shown in Figure 5. These historical correlations reveal that not all asset classes have marched to the same drummer. US T-bills have had low correlations with other asset classes which make them prime candidates for diversifying stock and bond portfolios. Likewise, foreign stocks have had low correlations with other asset classes and high diversification potential.⁵ These seven asset classes can be formed into many MPT portfolios, as illustrated in Figure 6. The portfolios, depicted in Figure 6 as boxes, represent a fraction of the possible portfolios that could be formed by varying the asset allocation among the seven asset classes in Figure 4. Only those portfolios that offer the highest expected return for each level of expected risk are efficient MPT portfolios and lie on the efficient frontier line, depicted in Figure 6 as a solid line and duplicated in Figure 7. In this sense these portfolios are the best of all possible portfolios. They may not, however, be suitable or appropriate, as we explain below. To clarify the exposition, Figure 7 duplicates Figure 6 then eliminates the inefficient portfolios depicted in Figure 6, leaving only the efficient frontier (the line) and the constituent asset classes (boxes). The most and least risky asset classes, US Small Stocks and US T-bills, anchor opposite ends of the risk-return spectrum. The other asset classes, taken individually, are not efficient until combined into portfolios. A sample of three MPT efficient portfolios (Conservative, Moderate and Aggressive) are labeled in Figure 7 and their asset allocations are shown in Figure 8. These portfolios illustrate the power of MPT. By allowing the IDM to match the investor's tolerance for risk and preference for return, the IDM can create suitable portfolios that have the highest level of expected return at a specified level of risk.⁶ F. MPT Portfolios are Best but May Not Be Suitable. Figure 8 also shows some of the drawbacks to MPT. Notice that three asset classes, US Large Stocks, Real Estate, and US Corporate Bonds, receive no allocation in any of the three sample portfolios. MPT is a mechanical algorithm that produces efficient portfolios without regard to any criteria other than expected return, standard deviation, and correlation. These allocations are efficient but do not necessarily reflect the IDM's investment experience, knowledge, or judgment. The asset classes missing from these efficient portfolios were part of the original set because either the investor wanted to hold these assets or because the IDM believed they were appropriate. Portfolios near the efficient frontier will often be more suitable than strictly efficient portfolios. Choosing nearly efficient portfolios is a matter of business judgment. By sacrificing some return, the IDM can often find portfolios that are more diversified and more suitable with the same amount of risk. As an example consider the moderate portfolio shown in Figure 8. If the IDM determines the risk level to be appropriate and considers it important to include US Large Stocks, Real Estate, and US Corporate Bonds, many nearly efficient portfolios are available. Figure 9 shows two of these alternatives. The two sample alternative portfolios in Figure 9, also constructed using MPT, maintain the same level of risk (standard deviation = 12.0%) with slightly less expected return (12.95% and 12.50%, respectively) than the original efficient Moderate Portfolio. The difference is in the asset allocation. Instead of excluding US Large Stocks, Real Estate, and US Corporate Bonds, these portfolios include these asset classes in increasing amounts as the expected return is lowered, first to 12.95% and then to 12.50%. These alternative portfolios have the same amount of risk and trade a small amount of expected return for better diversification. Only the most academic adherents of MPT would insist on strictly efficient portfolios. Common sense argues that professional judgment balances the mathematical attraction of efficient portfolios with real-world characteristics MPT is not designed to consider - like diversification, liquidity, investment horizon and all the other factors that make investment management as much an art as a science. MPT can, however, be a valuable tool for the IDM. III. Conclusions: The Investment Decision Maker and Modern Portfolio Theory IDMs are not required by law to employ the principles of MPT in portfolio design or portfolio reallocation. The IDM can rely solely on business judgment derived from experience in making asset allocation decisions. The question naturally arises: If a valuable tool such as MPT is available to the IDM, why would he or she not use it to assist in the initial construction of a portfolio or to assess an existing portfolio for possible changes? The answer to this question is that an IDM should use MPT because it combines mathematical rigor with professional judgment. The rigor comes in part from the need to forecast the inputs for MPT analysis. These inputs are as follows: • The expected return of the individual assets or asset classes. • The proportion of the portfolio to be allocated to the individual assets or asset classes. • The risk as measured by the standard deviation of the expected return of the individual assets or asset classes. And finally, • The correlation of the expected returns of each of the individual assets or asset classes with each other. The inputs MPT requires should always be gathered in some fashion by the IDM but a formal MPT analysis focuses attention on these key variables. MPT also uses precise algorithms to construct portfolios that are not influenced by emotion or fads. Finally, and most importantly, MPT enables professional judgment to temper mathematical rigor. The output from MPT analysis allows the IDM to identify: • An efficient frontier providing the composition of portfolios with the highest level of expected return at every level of risk. • The portfolios on the efficient frontier that correspond to the investor's risk tolerance. • A suitable portfolio near, but not necessarily on, the efficient frontier reflecting the IDM's investment experience, knowledge, judgment and desire for increased diversification. In summary, the principles of MPT can assist an IDM in developing a well managed portfolio or assessing an existing portfolio for possible changes. ¹ While the usefulness of MPT in portfolio formulation has been well documented by academics and investment practitioners, it is interesting to note the role MPT has played in trust law. Some form of the Uniform Prudent Investor Act has been passed in most every jurisdiction in the United States. The Act is based on the Prudent Investor Rule as more thoroughly developed in the Restatement (Third) of Trusts, Prudent Investor Rule (1992). In turn, the Rule is based upon a large body of academic work that has come to be known as MPT. Professor Harry M. Markowitz first espoused the principles of MPT based upon research he began in 1950 while a Ph.D. candidate in economics at the University of Chicago. Forty years later, he was awarded the Nobel Prize in Economic Sciences for his part in developing MPT. For a full development of the nexus of MPT and the Prudent Investor Rule see Edward A. Moses, J. Clay Singleton, and Stewart A. Marshall, III, "Modern Portfolio Theory and the Prudent Investor Act", ACTEC Journal, Vol. 30 (2004) p. 165-175. ²Correlation is an index, measured from -1 to +1, that indicates the degree to which returns from two assets move together (approaching +1) or in opposite directions (approaching -1). ³We explain the sense in which these portfolios are best in Section F. ⁴While not all investment professionals design portfolios at the asset class level, our experience suggests this is a common and sensible approach. If the IDM elects to design the portfolio at the asset class level, individually selected assets such as common stocks, real estate, fixed income securities, etc., can be assigned to a particular asset class. For example, common stocks can be classified as large capitalization growth, large capitalization value, mid capitalization growth, mid capitalization value, small capitalization growth and small capitalization value. Fixed income assets can be classified according to their maturities and other types of assets can be assigned to appropriate asset classes. With this approach the IDM can employ commercially available data bases and software to identify the efficient frontier based on asset classes, a portfolio on the efficient frontier at the appropriate risk tolerance, and the location of a proposed or existing portfolio relative to the efficient frontier. ⁵Past correlations may not be a reliable guide to future correlations. Some experts believe that correlations are increasing, especially among countries where currencies are coordinated. Whether the correlation between stocks and bonds is increasing is more speculative. Most practitioners use historical data as a guide to future correlations. ⁶ The next article in this series, "Using a Portfolio's Required Return to Develop an Appropriate Risk Level," describes a technique for determining an investor's risk preference.