On the sensitivity of the Black capital asset pricing model to the market portfolio

Abstract

We show that Black Capital Asset Pricing Model (Black CAPM) is extremely sensitive to the choice of the market portfolio and becomes unstable as market portfolios approach the Global Minimum-Variance portfolio. When market portfolios approach the minimum-variance portfolio, the expected return on the zero beta asset approaches negative infinity and its variance increases rapidly. Moreover, expected return on a fixed portfolio becomes indefinite (i.e., takes infinitely many values), and betas of all portfolios approach one. Unlike the Sharpe–Lintner CAPM, the market risk premium in the Black CAPM always has a positive minimum, while beta may have a negative minimum value, dependent on the underlying covariance matrix.