Building the Global Minimum Variance Portfolio G

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 testsOut-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 ZCAPMZCAPM, 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 ZCAPMZCAPM in addition to extensive empirical tests that prove its validity as an asset pricing modelAsset 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 $$R_{at}$$ R at , and cross-sectional market return dispersionCross-sectional market return dispersion, or $$\sigma _{at}$$ σ at , can be computed using G—that is, $$R_{at}$$ R at is replaced by $$R_{Gt}$$ R Gt , and $$\sigma _{at}$$ σ at is replaced by $$\sigma _{Gt}$$ σ Gt . Since G has no zeta riskZeta risk, it is an excellent choice to use among all portfolios along the axis of symmetryAxis of symmetry of the parabola. It only has beta riskBeta 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.