Professional Investment Portfolio Management

The holy grail of professional investment management is the construction of efficient portfolios. Professor Harry Markowitz won the 1990 Nobel Prize in Economic Sciences for groundbreaking work on the theory of portfolio allocation under uncertainty. Markowitz’s 1952 seminal paper entitled “Portfolio Selection” and later 1959 book Portfolio Selection: Efficient Diversification of Investments laid the foundation for modern portfolio investment. His main contribution was that diversification can reduce the risk of a portfolio of different assets without decreasing its return. This innovation enables a manager to increase the return per unit risk of a portfolio. What investor does not want lower risk, all else the same? In this respect, diversification provides (in his words) a “free lunch” for investors. In this chapter, we review how diversification can decrease investment risk. Using diversification, Markowitz developed the mean-variance investment parabola. The parabola is mathematically derived using the returns on assets, their total risks, and the correlations of returns between assets. Importantly, the upper boundary of this parabola is the efficient frontier with portfolios that earn the highest rate of return for each level of total risk. Investors should seek portfolios on the efficient frontier based on their risk preference. No portfolios exist above the efficient frontier. Portfolios below the efficient frontier are inefficient in that they have lower return per unit risk and therefore are not desired (except by short sellers seeking to profit on their falling prices and returns). Professional investment managers seek efficient portfolios to earn alpha. Alpha is a measure of manager skill that says that an investment portfolio outperformed a chosen benchmark portfolio, the average performance of some group of managers, or specific competitor investment firms. Clearly, if a manager can construct efficient portfolios, they can earn positive alpha, benefit their clients, and outperform other investment managers. The big question is: How can an investment manager build efficient portfolios? One answer is to use the mathematical and statistical methods developed by Markowitz. Unfortunately, after many years since his publications, researchers and practitioners have been unable to successfully choose stocks (for example) to build efficient portfolios. One problem ironically lies in the math of efficient portfolios. It is necessary to use sophisticated statistical techniques that cause difficulties in the estimation of efficient portfolios. A second and even larger problem is that efficient portfolios constructed using historical data do not stay efficient in the next period. That is, ex post efficient portfolios from past data are not ex ante efficient in the future. The reason is that risk is not controlled. In this book, unlike so many before who keep trying to make the mathematics of efficient portfolios work, we take a different approach. We employ asset pricing methods to control risk and thereby ensure better ex ante performance for efficient portfolios in the future. To do this, our focus will be on the ZCAPM, a new asset pricing model published in a recent book by Kolari et al. (A new capital asset pricing model: Theory and evidence. Palgrave Macmillan, New York, 2021). The ZCAPM is derived from the mean-variance investment parabola of Markowitz. It turns out that the investment parabola has an architecture that is determined by risks measured in the ZCAPM. Using these risks, a manager can choose assets to construct relatively efficient portfolios; alternatively, any portfolio within the parabola can be constructed if desired. We demonstrate these new portfolio methods in this book. We next review the pioneering work of Markowitz. Key concepts and notations important to later chapters are introduced. Subsequently, we discuss potential statistical problems in building efficient portfolios. Given these problems, we review new insights about the mean-variance investment parabola—importantly, these insights motivate our new approach for finding efficient portfolios. We overview intuitive concepts about controlling risk in portfolios using an asset pricing model. The forthcoming chapters provide the stepping stones for building efficient portfolios. To demonstrate our investment approach, we document U.S. stock market evidence spanning over 50 years. Our results convincingly show that efficient portfolios can be constructed by controlling risk. Hence, we advocate that investment managers utilize our risk control methods to boost portfolio performance.

Asset pricing models seek to value securities and other assets based on their risk. If the risk of an asset can be accurately measured, its rate of return can be estimated. Following the basic finance axiom that higher risk implies higher returns, asset pricing models are central to the question of valuation. The field of asset pricing models as a branch of financial economics has its origins in the now famous Capital Asset Pricing Model (CAPM) by William Sharpe (Journal of Finance 46:209–237, 1964). The CAPM builds upon Harry Markowitz’s (Journal of Finance 7:77–91, 1952, Portfolio selection: Efficient diversification of investments. Wiley, New York, 1959) mean-variance investment parabola covered in the last chapter. Together, Markowitz and Sharpe shared the Nobel Prize in Economics in 1990 (along with Merton Miller). A number of other researchers are credited for contemporaneously creating similar versions of the CAPM, including Jack Treynor (Market value, time, and risk, 1961, Toward a theory of market value of risky assets, 1962), John Lintner (1965), and Jan Mossin (Econometrica 34:768–783, 1966). The CAPM is a revolutionary model derived in a general market equilibrium setting that introduced a new measure of risk known as beta risk (or \(\beta\)) related to the market factor. This breakthrough model stimulated a tremendous amount of research and professional applications that continues today. Unfortunately, early empirical tests of the CAPM using U.S stock returns were weaker than expected. The statistical relation between stock returns and beta risk was relatively flat. In an attempt to better adapt the CAPM to real world evidence and account for this flat relationship, Fischer Black (Journal of Business 45: 444–454, 1972) proposed the zero-beta CAPM with two factors: (1) a market factor plus (2) an orthogonal zero-beta portfolio factor. Also, other researchers proposed further changes to the CAPM in an effort to bolster its applicability to real world markets. In this chapter we review the CAPM, zero-beta CAPM, and other CAPM-based models. Our interest is to review the foundation of theoretical asset pricing models. In forthcoming Chapters 4 and 5, we build upon this foundation by discussing the theoretical and empirical ZCAPM, respectively. The ZCAPM will be used in later chapters to build diversified, efficient investment portfolios.

As discussed in Chapter 2, early empirical evidence on the CAPM by Sharpe (Journal of Finance 19: 425–442, 1964) and others using the market model to perform tests using U.S. stock returns was disappointing. The Security Market Line (SML) relating market beta risk to average stock returns was flatter with a higher intercept (or alpha) than expected. In an attempt to fix the problem, Black (Journal of Business 45:444–454, 1972) proposed the zero-beta CAPM with the zero-beta portfolio return replacing the riskless rate in the CAPM. Merton (Econometrica 41:867–887, 1973) advanced the intertemporal CAPM (ICAPM) to allow multiple periods and multiple state variables as potential asset pricing factors. Extending Merton’s pathbreaking work, a number of CAPM variants appeared in the literature that sought to enrich the model with real world assumptions and variables.

Departing from the general equilibrium CAPM models, Ross (Journal of Economic Theory 13:341–360, 1976) developed the arbitrage pricing theory (APT). A very different model with no market factor, the APT simply said that arbitragers price assets using multiple long/short, zero-investment portfolios. The upshot was the later creation of multifactor models by Fama and French (Journal of Finance 47:427–465, 1992; Journal of Financial Economics 33:3–56, 1993). Supplanting the CAPM, these authors: (1) presented evidence that the CAPM did not work; and proposed the three-factor model that augments the market factor with size and value factors. These new factors are long/short, zero-investment portfolio returns that have become known as multifactors. The success of the three-factor model in terms of better fitting stock return data was impressive. Indeed, it was so impressive that other researchers began to propose a long list of multifactors.

A new problem in asset pricing arose. With so many factors, Cochrane (Journal of Finance 56:1047–1108, 2011) humorously observed that a “factor zoo” existed. Which multifactors should be used in asset pricing models? What is the theoretical justification for all these factors? Asset pricing was becoming bogged down in a quagmire of factors and their myriad risks. The promising simple risk concepts of total risk by Markowitz (Journal of Finance 7:77–91, 1952; Portfolio selection: Efficient diversification of investments. Wiley, New York, NY, 1959) and beta risk by Sharpe had become complicated by a growing number of long/short multifactors. Recently, to deal with this problem, some researchers have begun using machine learning models to let the data tell us what the factors are, as opposed to researcher judgment and discretion.

In this chapter, we review the APT, multifactor models, and machine learning models. At the time of this writing, multifactor models were very popular in academic research and investment practice. But as the universe of multifactors expands, it is becoming increasingly unclear how to implement them in the real world.

A recent book by Kolari, Liu, and Huang (KLH) (A new model of capital asset prices: Theory and evidence. Palgrave Macmillan, Cham, Switzerland, 2021) proposed a new capital asset pricing model named the ZCAPM. The book is based on research work by the authors over the past decade, including Liu et al. (Financial Management Association 2012 conference, 2012), Liu (Doctoral dissertation, 2013), and Liu et al. (Southern Finance Association conference, 2020). Some recent publications on the ZCAPM are Kolari et al. (Journal of Risk and Financial Management 15:1–23, 2022a), Kolari et al. (Journal of International Financial Markets, Institutions, and Money 79:101607, 2022b), Kolari et al. (Western Economic Association International 2023 conference, 2023), Kolari and Pynnonen (Investment valuation and asset pricing: Models and methods. Palgrave Macmillan, Cham, Switzerland, 2023), and Kolari et al. (Working paper available on SSRN, Texas A&M University, 2024). Moreover, the authors worked with the pension investment division at the Teachers Retirement System of Texas (TRS) to manage about $100 million using the model from 2012 to 2014. This practical experience was instrumental in helping us to develop innovative portfolio investment methods. Derived as a special case of Black’s (Journal of Business 45: 444–454, 1972) zero-beta CAPM, the theoretical ZCAPM is a new CAPM variant. Like the CAPM, it contains a market factor, but rather than being associated with the beta risk of the market portfolio M, it captures beta risk related to the average market return of all assets in the market. KLH posited that the latter average market return lies somewhere along the axis of symmetry of the Markowitz (Journal of Finance 7:77–91, 1952, Portfolio selection: Efficient diversification of investments. John Wiley & Sons, New York, NY, 1959) mean-variance investment parabola. Hence, beta risk is related to the overall movement of the entire investment parabola up and down over time. Using the investment parabola, KLH mathematically derived a second return dispersion factor based on the cross-sectional standard deviation of returns of all assets in the market (e.g., on any given day). A concept not previously known, return dispersion largely determines the width or span of the investment parabola. As this width changes over time, asset returns can be positively or negatively affected depending on whether they are located in the upper or lower half of the parabola. Importantly, they showed that this return dispersion factor yields a new bi-directional risk dubbed zeta risk, which can be either positive or negative for any asset with respect to return dispersion changes. As the parabola moves over time, average market returns and market return dispersion affect asset returns via their sensitivity to these changes as measured by beta risk and zeta risk, respectively. The novel ZCAPM is unlike any previous theoretical or empirical model. Consequently, it cannot be empirically modeled using standard ordinary least squares (OLS) regression analyses used by virtually all other asset pricing models. Departing from previous models, the empirical ZCAPM (counterpart of the theoretical ZCAPM) was developed using expectation-maximization (EM) regression methods. The EM algorithm allows for the latent or unobserved positive or negative sign of zeta risk associated with market return dispersion. It computes a probability of the sign (direction) of zeta risk that is critical in measuring its association with return dispersion. No previous asset pricing models use EM regression in their empirical estimation. With the empirical ZCAPM in hand, KLH proceeded to test the model using U.S. stock returns over more than 50 years. The results were stunning—the ZCAPM virtually always outperformed a number of popular multifactor models in standard out-of-sample Fama and MacBeth (1973) cross-sectional tests. Recall from Chapter 3 that Fama and French (Journal of Financial Economics 128:234–252, 2018) recommended that the validity of asset pricing models and their factors should be evaluated using out-of-sample tests and theoretical justification. The ZCAPM passes both of these validation tests. The tables had turned. Now multifactor models were less successful in explaining average stock returns than the CAPM-based ZCAPM. Previously, multifactor models were justified by their superior explanatory power in empirical tests of stock returns. U.S. test results in KLH were confirmed in other countries by Kolari et al. (Journal ofInternational Financial Markets, Institutions, and Money 79:101607, 2022b), who investigated stock returns in Canada, France, Germany, Japan, and the United Kingdom. In repeated tests in different sample periods and for different test asset portfolios as well as individual stocks, zeta risk in the ZCAPM is very significantly priced in cross-sectional tests. Indeed, the significance of zeta risk consistently surpasses those of widely accepted factors in multifactor models. Adding further evidence, in a recent working paper by Kolari et al. (Testing for missing asset pricing factors, 2023), the authors showed that zeta risk premiums almost completely explain Jensen’s alpha (\(\alpha\)) coefficients in the market model version of the CAPM. Thus, no missing factors remain after taking into account return dispersion and related zeta risk. The long search for alpha in empirical asset pricing studies, and with it the search for multifactors to explain alpha, is over. The CAPM is not dead but alive and well in the form of the ZCAPM. Surprisingly, the ZCAPM suggests that multifactors are rough measures of return dispersion and therefore related to zeta risk. Hence, by implication, all multifactor models are related to the CAPM to some extent. Akin to the old Keynesian economics saying, KLH (p. 270) inferred that: “We are all Sharpians now.” This chapter provides an overview of the theoretical ZCAPM. It is important to understand the theory underlying this new model before going to the empirical ZCAPM in the next chapter. We later use the ZCAPM to build investment portfolios using beta and zeta risk. By controlling risk, as we will show in forthcoming chapters, we can create relatively efficient portfolios that well outperform general market indexes in terms of returns per unit risk.

As discussed in the previous chapter, the theoretical ZCAPM by Kolari, Liu, and Huang (KLH) (A new model of capital asset prices: Theory and Evidence. Palgrave Macmillan, Cham, Switzerland, 2021) contains beta risk related to average market returns and zeta risk associated with the cross-sectional return dispersion of all assets in the market. Zeta risk takes into account sensitivity to market return dispersion that can be positive or negative in sign. That is, asset returns can increase or decrease depending on the sign of zeta risk. In the real world, the positive or negative direction of zeta risk is unknown. KLH provide an empirical form of the ZCAPM that solves the problem of the direction of zeta risk. Using the expectation-maximization (EM) algorithm, an estimate of the probability that zeta risk is positive or negative for asset can be obtained from historical daily returns within (for example) the past year. Based on this probability, a mixture of two-factor models can be specified. One model corresponds to a positive zeta risk, and the second model has negative zeta risk. Which model is operative for an asset depends on the probability estimate that gives the direction of zeta risk. It turns out that the EM algorithm is well-known in the hard sciences and widely used in regression analyses. In finance it has been utilized but is not widely known. Even so, it is commonly used in our everyday lives. For example, weather mapping technology that forecasts the direction that a storm will move in the next hour often utilizes the EM algorithm. In this chapter, we give an overview of the empirical ZCAPM. For readers seeking details of the procedure, we recommend referring to KLH’s book. As we will see, with some simple modifications, the theoretical ZCAPM can be adapted to readily available empirical data in the financial markets. To justify the use of the empirical ZCAPM for the creation of efficient investment portfolios in forthcoming chapters, we review some earlier tests of the model. These analyses compare the empirical ZCAPM to other well-known asset pricing models discussed in Chapter 3. The results are convincing. In standard out-of-sample cross-sectional regression tests, the empirical ZCAPM has very strong goodness-of-fit as measured by \(R^2\) estimates. Also, zeta risk is highly significant in these cross-sectional tests; indeed, zeta risk dominates all of the prominent factors in the literature in terms of being significantly priced in the cross section of average stock returns. These and other results suggest that the empirical ZCAPM is a valid asset pricing model. The fact that the empirical ZCAPM, which is a CAPM theory-based model, consistently outperforms multifactor models (at times by large margins) is quite a reversal in the field of asset pricing. Multifactor models have gained popularity over time due to better explaining stock returns than the market model version of the CAPM. However, the ZCAPM is a CAPM model that outperforms the multifactor models. Interestingly, KLH have argued that long/short, zero-investment factors in multifactor models are actually rough measures of cross-sectional return dispersion. As such, they are akin to the ZCAPM that incorporates total return dispersion as a new asset pricing factor. In effect, total return dispersion subsumes all of the multifactors which become redundant as they capture only different slices of return dispersion. Paradoxically, multifactor models are cousins of the ZCAPM so to speak, and by implication linked to the CAPM. Rather than the CAPM being dead, it is alive and well in the form of the ZCAPM. Also, through an unexpected twist, the CAPM is even alive to some extent in the form of multifactor models!

The true market portfolio of the CAPM on the efficient frontier (located at the tangent point of the line extending from the riskless rate) is not observable in the real world. Sharpe received the Nobel Prize in Economics for developing the CAPM and proposed that portfolio performance can be compared by computing their excess returns divided by the standard deviation of returns (or total risk). This so-called Sharpe ratio is one of the most widely used measures of portfolio efficiency. However, it does not take into account beta risk in the CAPM. In this regard, Gibbons et al. (Econometrica 57:1121–1152, 1989) (GRS) proposed a test of whether any particular portfolio is ex ante mean-variance efficient in the context of the market model version of the CAPM. More specifically, they modified the Hotelling \(T^2\) test to take into account whether a portfolio lies on the efficient frontier. While the Sharpe ratio and GRS test are prominent in academic studies on portfolio performance, practitioners have developed other metrics to evaluate portfolio performance. One such popular measure is drawdowns, which is tantamount to value at risk (VaR). If a portfolio exhibits high periodic drawdowns, it is necessary to carefully evaluate the potential gains in the long run that may serve to counterbalance this risk.

As shown in Chapter 1, in the context of the infinite number of portfolios on the boundary of the mean-variance efficient frontier of Markowitz (Journal of Finance 7: 77–91, 1952, Portfolio selection: Efficient diversification of investments. New York, NY: Wiley, 1959) in Fig. 1.3, the global minimum variance portfolio G is unique. Efficient portfolios on the efficient frontier have minimum risk for any given return. However, portfolio G is comprised of securities with weights that are independent of predicted or forecasted expected returns of the securities. Thus, it does not require any expected return inputs. Research evidence on constructing efficient portfolios has been discouraging. Many times the out-of-sample performance of efficient portfolios has been unable to beat a simple, equal-weighted portfolio of stocks. By contrast, because only the variance-covariance matrix is needed, researchers have been able to create G portfolios that outperform equal-weighted portfolios in out-of-sample tests. This success attracted interest of researchers and practitioners over the years in building G portfolios. For example, in 2008 MSCI introduced Global Minimum Volatility Indices, which are based on global stocks in the MSCI World Index. Other low volatility equity portfolios are available nowadays, including the S&P 500 Minimum Volatility Index, Vanguard Global Minimum Volatility Fund Investor Shares, and others (See discussion in Feldman [Building minimum variance portfolios with low risk, low drawdowns and strong results, 2016] on STOXX Minimum Variance Indices.). They have proven to have relatively lower beta and volatility characteristics over time than their corresponding equity market indexes. We like to think that the G portfolio “pins” the mean-variance parabola in return/risk space. Without G, it is not possible to graphically locate the parabola in the Markowitz mean-variance framework. Portfolio G changes the level of the entire parabola as it moves over time. As G’s return increases (decreases), the general level of returns of assets within and on the boundary of the parabola tends to increase (decrease), all else the same. Also, as G’s variance increases (decreases), the parabola shifts right (left) to depict a generally higher (lower) variance among all assets’ returns. In the context of the ZCAPM, which we reviewed in Chapters 4 and 5 (See the recent book by Kolari et al. [A new model of capital asset prices: Theory and evidence. Cham, Switzerland: Palgrave Macmillan, 2021], which contains the theoretical derivation of the ZCAPM in addition to extensive empirical tests that prove its validity as an asset pricing model. For further information on the ZCAPM, see Liu et al. [Financial Management Association 2012 conference, 2012], Liu [A new asset pricing model based on the Zero-Beta CAPM: Theory and evidence, Doctoral dissertation, Texas A&M University, 2013, Return dispersion and the cross-section of stock returns, Palm Springs, CA (October): Presentation at the annual meetings of the Southern Finance Association, 2020], Kolari et al. [Journal of Risk and Financial Management, 2022, Journal of International Financial Markets, Institutions, and Money 79:101607, 2022, Testing for missing asset pricing factors, San Diego, CA: Paper presented at the Western Economic Association International, 2023], and Kolari and Pynnonen [Investment valuation and asset pricing: Models and methods. Cham, Switzerland: Palgrave Macmillan, 2023]), G plays a particularly important role. The average market return, or \(R_{at}\), and cross-sectional market return dispersion, or \(\sigma _{at}\), can be computed using G—that is, \(R_{at}\) is replaced by \(R_{Gt}\), and \(\sigma _{at}\) is replaced by \(\sigma _{Gt}\). Since G has no zeta risk, it is an excellent choice to use among all portfolios along the axis of symmetry of the parabola. It only has beta risk as defined in the ZCAPM based on average market returns. Unlike previous studies that build G portfolios, we utilize the ZCAPM to achieve this task. In this chapter, we briefly review earlier studies on the subject of estimating G portfolios. Subsequently, extending this literature, we review our novel ZCAPM approach to building the G portfolio. Empirical evidence using U.S. stock returns is presented to demonstrate the out-of-sample performance of our G portfolio. We form the G portfolio and then compute its return for all returns in the next out-of-sample month, which represents an investable strategy of an actual investor. In forthcoming chapters, this G portfolio is an essential building block in the construction of long portfolios that trace out a mean-variance investment parabola, long/short portfolios, and combined investment strategies with both long and long/short portfolios. The latter combined portfolios represent net long portfolios that form a relatively efficient frontier that well outperforms the CRSP market index.

It was long believed that well-known aggregate stock market indexes, such as the S&P 500 index and overall CRSP stock market index, were efficient portfolios. Stock market indexes utilize either equal-weighting or value-weighting of individual stocks in their portfolios. However, Kolari, Liu, and Huang (A new model of capital asset prices: Theory and evidence, Palgrave Macmillan, Cham, Switzerlnd, 2021) have found that these commonly used stock market indexes are inefficient. More efficient, highly diversified portfolios can be constructed using market stock price information that outperform such market indexes. The key to enhancing diversification is to estimate optimized weights for individual stocks in a portfolio that enables higher returns per unit risk. Tests by the authors using historical series of U.S. stock prices repeatedly demonstrated that commonly used stock market indexes are less efficient than portfolios composed of almost the same stocks. Long-run tests were performed based on stock prices over the past 50 years to verify that the results were robust and could be trusted. According to Markowitz’s (Journal of Finance 7:77-91, 1952; Portfolio selection: Efficient diversification of investments, Wiley, New York, NY 1959) Nobel Award winning investment theory, assuming that asset prices fully reflect all available information to investors, no portfolios or individual assets can outperform or lie above the efficient frontier over extended periods of time. If assets are priced correctly by the market, this inference must hold. For investment horizons longer than three or four years, efficient portfolios developed by the techniques provided in this chapter can provide investors with higher risk-adjusted stock returns than possible from other investment strategies. Our highly diversified portfolios using common stocks are well-suited to institutional investors. For example, pension funds can earn higher returns on long-only stock portfolios, thereby enabling them to better meet the needs of pensioners relying on those returns to help pay retirement costs of living. Not only can efficient stock portfolios boost retirement incomes of pensioners, but mutual funds, hedge funds, insurance companies, securities firms, and other investors can benefit too. Of course, short portfolios can be created using the same optimized weighting methods. As such, long-short investment strategies can be created. In this chapter we build net long portfolios whose weights sum to one for all stocks. We utilize U.S. stock returns from July 1964 to December 2022 to build efficient portfolios. To build relatively efficient portfolios, optimized weights are created for individual stocks based on estimating the empirical ZCAPM. Using Chapter 7’s G portfolio returns and cross-sectional standard deviation of returns (or market return dispersion) as factors, we estimate the ZCAPM with one year of daily stock returns. Optimal weights are computed from the ZCAPM, a number of portfolios are formed with different financially engineered levels of risk, and the out-of-sample return performance of all portfolios is computed in the next month. This process is rolled forward one month at a time to construct a series of relatively efficient portfolios that are rebalanced each month. We plot the constructed net long portfolios in mean-variance return space and compare their performance to the CRSP market index, which is widely regarded as an efficient portfolio. Our results demonstrate that the CRSP index is oftentimes far from efficient and, as predicted by ZCAPM theory, tends to be located in the vicinity of the axis of symmetry of the parabola far away from the efficient frontier. Sharpe ratios verify the greater efficiency of ZCAPM-based portfolios compared to the CRSP index, especially so for our higher risk net long portfolios. Since most investors cannot consistently outperform general market indexes over periods of three to five years, we conclude that the ZCAPM asset pricing model is a powerful investment tool. Our investment technology can be used by professional institutional investors to create efficient portfolios that are highly diversified and outperform well-known stock market indexes. Importantly, portfolios can be engineered for different risk levels to coincide with the unique risk preferences of different investors. The ability to control risk is a major advantage of our machine-based ZCAPM portfolios. Managers can match up their investment clients with customized risk levels to meet their needs. Subsequent return performance is consistent with these risk levels—that is, lower (higher) risk will result in lower (higher) long-run returns. In this regard, our investment techniques are intended for long-run investors with three-to-five investment time horizons at a minimum. They are not intended as a short-run trading strategy. A surprising finding in our net long portfolios is the outsized performance of high risk portfolios. There appears to be important information in the tails of the cross-sectional distributions of returns in the market that can be employed to build these high-performing portfolios. We discuss this interesting finding in more detail in forthcoming empirical results. The next section gives background discussion. We then review our empirical methods. Subsequently, we present empirical results for U.S. stock returns. As we will see, the performance of our efficient portfolios is remarkable by any existing standard of measurement.

In Chapter 8, we created net long portfolios (with weights that add up to one) comprised of the global minimum variance portfolio G plus a number of long-short portfolios. Portfolio G was developed in Chapter 7 using the ZCAPM, a new asset pricing model developed by Kolari et al. (A new model of capital asset prices: Theory and evidence. Palgrave Macmillan, Cham, Switzerland, 2021). The long-short portfolios are based on different levels of zeta risk (associated with market return dispersion) as estimated by the ZCAPM. The returns and risks of these net long portfolios traced out an empirically efficient frontier that looked like the theoretical efficient frontier of Markowitz’s (Journal of Finance 7:77–91, 1952, Portfolio selection: Efficient diversification of investments. John Wiley & Sons, New York, NY, 1959) mean-variance investment parabola. Strikingly, in the analysis period from July 1964 to December 2022, some net long portfolios yielded out-of-sample average returns in the next month of 30% or more. Subperiod analyses confirmed these findings with some decrease in average returns in the second half of the analysis period due to general market conditions. Sharpe ratios further corroborated the results that tended to increase with the zeta risk of the net long portfolios. Referring to graphs of average stock returns and total risk, the CRSP market index was located in the vicinity of the axis of symmetry of the mean-variance parabola. The result is consistent with ZCAPM theory, which hypothesizes that average market returns lie on the axis of symmetry of the parabola. Thus, ZCAPM theory is supported. Additional evidence showed that Sharpe ratios of net long portfolios well exceed the CRSP index and tend to increase as zeta risk in the long-short portfolios increase. These findings further support the ZCAPM, which hypothesizes that zeta risk related to market return dispersion can be used to boost average portfolio returns. Although the performance of our ZCAPM-based net long portfolios was outstanding by any standards for well-diversified portfolios, a natural question that arises is: What are the risk characteristics of the portfolios? Investment managers and their clients need to have realistic expectations not only about the returns but risks of their investments. In this way, the return/risk preferences of investors can be aligned by managers to meet their needs. In this chapter we apply risk metrics discussed in Chapter 6 to the net long portfolios, including the Gibbons, Ross, and Shanken (GRS) (Econometrica 57: 1121–1152, 1989) test for efficient portfolios involving the Sharpe ratio, value at risk (VaR), and drawdowns. As we will see, most net long portfolios have good risk characteristics that recommend their real world implementation for investors. The next section covers GRS tests of the net long portfolios. Subsequent sections address VaRs and drawdowns, respectively.

In this chapter we demonstrate how to use the ZCAPM asset pricing model to build high-performing long only efficient portfolios. We begin by showing the results for zeta risk sorted portfolios. Recall that in Chapters 8 and 9 we used zeta risk sorted portfolios to form long-short zeta risk portfolios that were added to the global minimum variance portfolio G to build net long portfolios with both short and long positions. Additionally, we report the results of beta risk sorted portfolios using the empirical ZCAPM. In general, the empirical asset pricing literature has documented many times that there is no relation between beta risk and average stock returns. However, previous literature employs the CAPM market model to estimate beta wherein the CRSP market index or S&P 500 market index are normally used to proxy the market portfolio. Fama and French (Journal of Finance 47: 427–465, 1992; Journal of Financial Economics 33: 3–56, 1993; Journal of Finance 50: 131–156, 1995; Journal of Finance 51: 1947–1958, 1996) have proven that the CAPM fails using this renowned empirical model. However, as already mentioned in earlier chapters, beta risk in the empirical ZCAPM is different as it is estimated by using the average market return, not a proxy for the market portfolio. As discussed there, a good proxy for the average market return is the G portfolio. G has no zeta risk and therefore is a good portfolio to measure beta risk related to average market returns. Our empirical results based on U.S. stock returns are extraordinary. Zeta risk portfolios trace out the shape of a theoretical Markowitz (Journal of Finance 7: 77–91, 1952; Portfolio selection: Efficient diversification of investments. Wiley, New York, NY, 1959) mean-variance investment parabola, which we refer to as the empirical parabola. The G portfolio is located at the leftmost minimum variance location in risk/return space. The CRSP market index lies in the interior of the empirical parabola in the vicinity of the axis of symmetry as predicted by ZCAPM theory. Thus, the CRSP index is far from efficient and represents a very poor proxy for the market portfolio. In this respect, Roll (Journal of Financial Economics 4: 129–176, 1977) has asserted that, without a good proxy for the market portfolio, the CAPM cannot be tested. We infer that tests by Fama and French and others that have used the CRSP index to proxy the efficient market portfolio to conclude the CAPM is dead are misleading. Interestingly, with the exception of very high idiosyncratic risk stocks (i.e., high residual errors or high standard deviations of returns), our out-of-sample results for beta risk portfolios estimated via the empirical ZCAPM and portfolio G yield a positive relation with average stock returns. Also, a number of our beta risk portfolios outperform the CRSP market index by substantial amounts. Since the CRSP index and S&P 500 index are very highly correlated and have similar average return performance over time, these findings are notable. Our beta risk results stand in stark contrast to the vast research on the failure of CRSP-based CAPM beta risk. Given the fact that G-based ZCAPM zeta risk as well as beta risk are related to average stock returns, our zeta and beta sorted portfolios should be of keen interest to the professional investment community. Surely stock market investors could benefit from these portfolios by outperforming general market indexes that are so popular among passive index investors. Moreover, zeta and beta risk levels can be controlled to generate portfolio returns coincident with the risk preferences of investors. In this way, professional managers can customize portfolios to meet the risk/return needs of different investors. The next section provides a review of our empirical methods. The subsequently section reports and discusses the empirical results for long only stock returns. The last section summarizes the chapter.

Markowitz (Journal of Finance 7:77–91, 1952; Portfolio selection: Efficient diversification of investments, John Wiley & Sons, New York, NY, 1959) proposed the celebrated mean-variance investment parabola that is foundational to modern investment management and was awarded the 1990 Nobel Memorial Prize in Economic Sciences for this achievement. The upper boundary of the parabola is the efficient frontier with portfolios earning the highest possible return per unit total risk. The lower boundary is comprised of inefficient portfolios with the lowest possible return per unit total risk. This familiar theoretical picture of the investment opportunity set available to investors in asset markets is known by all students of finance and investment professionals. Normally individual assets and portfolios inside the parabola are drawn as points located in return/risk space but little or no explanation about their risk characteristics is provided. But what if a well-defined risk structure determines the locations of assets within the investment parabola? In this chapter, we show that U.S. stock portfolios are located within the parabola according to their risk structures. This new architecture was discovered by Kolari, Liu, and Zhang (A new model of capital asset prices: Theory and evidence, Palgrave Macmillan, Cham, Switzerland, 2021) in their recent book. As reviewed in Chapter 4 of this book, they proposed a new asset pricing model dubbed the ZCAPM that is comprised of beta risk associated with average market returns and zeta risk related to the cross-sectional standard deviation of returns of all assets (or market return dispersion). The authors showed that stocks within the parabola can be mapped into a grid with beta risk and zeta risk coordinates. Here we extend their analyses with more in-depth analyses of stock portfolios. Importantly, our main interest is in the investment implications of the parabola’s beta-zeta risk architecture. How does beta risk affect portfolio performance? What about zeta risk? How do beta risk and zeta risk work together to affect performance? Can we use this parabola architecture to evaluate the past performance of investment portfolios? In terms of professional investment management, is there a way to use this architecture to boost stock portfolio returns to outperform general stock market indexes? In this chapter, we provide U.S. stock return evidence to answer these questions. In forthcoming analyses, we sort stocks into beta quintiles and zeta quintiles. In this way, we can better understand how these risks impact stock returns and risks. We use the minimum variance portfolio G (see Chapter 7) and the CRSP market index as reference points to help define the mean-variance parabola as well as evaluate our stock portfolio results. Consistent with ZCAPM theory, G has the smallest risk of all our stock portfolios and lies approximately at the leftmost point on the axis of symmetry of the parabola. Also, the CRSP market index lies in the vicinity of the axis of symmetry in the middle of the parabola, not near the empirical efficient frontier determined by beta risk and zeta risk. Many researchers believe that the CRSP index should be a relatively efficient portfolio compared to other real world portfolios. But our evidence contradicts this commonly held belief. Since the CRSP index is clearly an inefficient portfolio, it is a poor proxy for the market portfolio in the CAPM. Consequently, tests of the CAPM using the CRSP index are invalid; as Roll (Journal of Financial Economics 4: 129–176, 1977) has asserted, the CAPM cannot be tested with an inefficient market portfolio. In the next section, we describe our empirical methods. The subsequent section gives details of the out-of-sample portfolio return results for stocks sorted into portfolios based on their beta risk and zeta risk coefficients estimated by means of the empirical ZCAPM (see Chapter 5). Results for the entire analysis period July 1964 to December 2022 are reported in addition to results for subperiods and for portfolios after dropping high idiosyncratic risk stocks.

Most workers invest money from their monthly paychecks into a pension fund. Major tax benefits of tax deductions and tax shields encourage people to save money through their pension fund investments. The money invested in pension funds is typically placed with different mutual funds. For example, an employer could allow employees to invest in mutual funds offered by Fidelity Investments, Vanguard Asset Management, iShares, USB Group, State Street Global Advisors, Morgan Stanley, JP Morgan Chase, Allianz Group, Capital Group, Goldman Sachs, BlackRock, American Funds, etc. Also, non-pension investors can buy mutual funds but do not receive the tax benefits of pension accounts. As of this writing, there are over 7,000 mutual funds managing over $28 trillion in investment funds in the U.S. Open-end funds make available unit shares of ownership on demand. Closed-end funds have a limited number of outstanding unit shares that can be bought in the market. Mutual funds offer a wide variety of long only portfolios to investors, including equity, money market funds, bonds, real estate, commodities, target date funds, etc. In this chapter, we will focus on equity mutual funds, which are popular among long-run pension investors. A major problem facing employees and others that invest in mutual funds is: How do I choose from the thousands of available funds? This problem harkens back to Markowitz (Journal of Finance 7:77-91, 1952; Portfolio selection: Efficient diversification of investments, Wiley, New York, 1959), who proposed that investors should choose portfolios with the highest return per unit total risk on the efficient frontier. Mutual funds are essentially portfolios that lie somewhere within the mean-variance parabola. In the context of the ZCAPM by Kolari et al. (A new model of capital asset prices: Theory and evidence. Palgrave Macmillan, Cham, Switzerland, 2021), they all have beta risk and zeta risk characteristics that determine their location within the parabola. Thus, we can apply our ZCAPM-based investment methods to assess the return and risk performance of mutual funds. Historical evidence has shown that passive general stock market indexes like the S&P 500 index outperform over 90% of professional money managers over longer periods of time beyond three to five years. Due to these well-known performance results, many mutual fund investors select passive market indexes with relatively low management costs compared to actively managed funds. In previous chapters, we showed that general stock market indexes, such as the CRSP market index, lie approximately on the axis of symmetry in the middle of Markowitz’s mean-variance parabola. As demonstrated in Chapters 8 and 9, net long portfolios based on the G portfolios and long-short portfolios with different levels of ZCAPM zeta risk well outperformed the CRSP index. Chapter 10 applied ZCAPM investment methods to the construction of long only stock portfolios. Again zeta risk sorted portfolios well outperformed the CRSP index. Much more efficient portfolios can be constructed using individual stocks in optimally weighted net long portfolios. Can the ZCAPM be applied to mutual fund investments in the same way as individual stocks in earlier chapters? That is, can mutual funds be combined into portfolios based on beta and zeta risk in the ZCAPM boost performance relative to general stock market indexes? In this chapter, we apply the ZCAPM to mutual fund investments. Only mutual funds specializing in publicly traded equities that are actively managed are selected (i.e., no passive equity funds). Mutual funds are formed into portfolios containing 10 funds. The mutual funds are rebalanced monthly over the analysis period from January 2000 to June 2022. Out-of-sample returns are computed in the next month to be consistent with real work investor behavior. Mutual funds are sorted into portfolios with different levels of beta risk and zeta risk. Do average returns of mutual fund portfolios increase as beta risk and zeta risk increase? How do their return and risk profiles compare to the CRSP market index? To the minimum variance portfolio G that we developed in Chapter 7? Can the ZCAPM help investors to boost their mutual fund returns? These and other questions are addressed here. In the next section, we overview our empirical methods. The following section presents our findings for U.S. mutual funds. The last section gives a summary.

Quantitative methods are increasingly becoming an important ingredient in professional investment practice. Markowitz (Journal of Finance 7:77–91, 1952, Portfolio selection: Efficient diversification of investments. John Wiley & Sons, New York, NY, 1959) laid the foundation of modern investment management with the invention of the mean-variance investment parabola. During the time of this writing, Nobel Prize laureate Harry Markowitz passed away on June 22, 2023. His landmark diversification concepts, which are grounded in mathematics and statistics, forever changed investment practice. Professional managers are careful to make sure that their portfolios contain securities or other assets that together reduce their risk. He once remarked that diversification is the only free lunch in the market. In other words, investors can use diversification to reduce the risk of their investment portfolios. He advised that investors should seek to maximize returns per unit risk, not returns per se. In this book, we have sought to build upon the foundation that Markowitz established. At the center of our analyses is the ZCAPM of Kolari, Liu, and Huang. The ZCAPM is derived from Markowitz’s mean-variance investment parabola in combination with Black’s (Journal of Business 45:444–454, 1972) zero-beta CAPM. Unfortunately, the dream of building portfolios that lie on the efficient frontier of the parabola has eluded academic and professional researchers. It turns out that problems with estimating expected returns in the future for securities as well as the covariance matrix for a large number of securities have been more difficult than anticipated. Departing from this traditional portfolio approach, we break new ground by taking a different approach to building high performing stock portfolios—namely, we apply the ZCAPM asset pricing model. As documented in this book, our efforts have paid off by creating portfolios that form an empirical investment parabola consistent with the theoretical parabola of Markowitz. In their 2021 book, Kolari, Liu, and Huang documented extensive evidence to prove that the empirical ZCAPM dominates multifactor models in standard, out-of-sample Fama and MacBeth (Journal of Political Economy 81:607–636, 1973) cross-sectional regression tests. Given the remarkable empirical success of the ZCAPM, the authors performed a limited set of experiments to show that the ZCAPM could be applied to portfolio construction. Using the model, relatively high performing stock portfolios were built. In the present book, we substantially expand these portfolio experiments. Extending their previous analyses, we began by building a proxy for the global minimum variance portfolio G that pins the mean-variance parabola in return/risk space. We then employed portfolio G to compute the two factors for the empirical ZCAPM: (1) average market returns related to beta risk; and (2) cross-sectional return dispersion associated with zeta risk. Stocks were sorted on beta and zeta risks to form portfolios. Portfolios were built by optimally weighting the individual stocks. Finally, one-month-ahead returns for the portfolios were computed on an out-of-sample basis to be consistent with real world investor experience. Portfolio performance was evaluated in detail, including average annual returns, Sharpe ratios, and various risk metrics. In general, our findings showed that empirically efficient portfolios can be produced that well outperform general market indexes, such as the CRSP index and S &P 500 index. Since most professional managers cannot consistently beat these general market indexes over time (e.g., three-to-five-year periods of time), our results are noteworthy. Also, we applied the ZCAPM to the problem of building high-performing equity mutual fund portfolios. In sum, our portfolio results showed that investors would benefit from machine-made stock and equity mutual fund portfolios based on the ZCAPM. The authors encourage cooperation with the professional investment community to implement these portfolios in the real world.