Efficient Portfolios in a Market with a Risk-Free Asset

Abstract

The chapter continues the study of a financial market with a risk-free asset. It provides formulas for the expected return and the variance of the return on an efficient portfolio and shows how to represent the efficient frontier for the market with a risk-free asset through equations in the $∖sigma$-$m$ plane and in the $∖sigmaˆ{2}$-$m$ plane. The chapter introduces the notion of the tangency portfolio, examines conditions under which it exists and derives a formula for it. A discussion of the properties of the tangency portfolio is followed by a geometric illustration explaining the term “tangency.” The highlight of the chapter is the notion of the Sharpe ratio and the evaluation of the Sharpe ratio for the tangency portfolio. The chapter concludes with Tobin’s mutual fund theorem, which is formulated and proved.