The Beta-Zeta Risk Architecture of the Mean-Variance Parabola

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 riskBeta risk associated with average market returnsAverage market return and zeta risk related to the cross-sectional standard deviation of returnsCross-sectional standard deviation of returns of all assets (or market return dispersion)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 architectureBeta risk sensitivitybeta-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 parabolaParabola 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?General stock market index In this chapter, we provide U.S. stock return evidence to answer these questions. In forthcoming analyses, we sort stocks into beta quintilesBeta loadingsbeta quintiles and zeta quintilesZeta coefficientzeta 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 symmetryAxis of symmetry of the parabola. Also, the CRSP market index lies in the vicinity of the axis of symmetryAxis of symmetry in the middle of the parabola, not near the empirical efficient frontierEmpirical efficient frontier determined by beta risk and zeta risk. Many researchers believe that the CRSP indexCRSP 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 portfolioProxy market portfolio in the CAPMCapital Asset Pricing Model (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 CAPMCapital Asset Pricing Model (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 riskBeta risk and zeta risk coefficients estimated by means of the empirical ZCAPMEmpirical 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.