## P

Figure 14.2 The zero Beta capital asset pricing line.

if investors could lend and borrow unlimited amounts of funds at the rate R'p they would hold the market portfolio.

The investor who could lend and borrow at the riskless rate R'p would face an investment opportunity set as depicted in Figure 14.3. To solve for optimal proportions, he or she would face a set of simultaneous equations directly analogous to Equation (13.4). One such equation is5

Note that in the equation the X.'s are market proportions because R'f is defined as that value of the riskless rate that causes investors to hold the market portfolio.

In the previous chapter we showed that the term in parentheses in the left-hand side of Equation (14.2) was simply the covariance between the return on security j and the return on the market portfolio. Thus, Equation (14.2) can be written as a. cov(*,i?M) =

The expected return on the market portfolio is a weighted average of the expected return on individual securities. Since Equation (14.3) holds for each security, it must also hold for the market. Thus,

Figure 14.3 The opportunity set with rate Rp.

5These equations are first-order conditions and must hold for the tangency point of any line drawn from the vertical axis and the efficient frontier.

But cov(RmRm) is the variance of M so that rm = K +

Substituting the expression for X into Equation (14.3) and rearranging yields