## Short Sales Allowed With Riskless Lending And Borrowing

The derivation of the efficient set when short sales are allowed and there is a riskless lending and borrowing rate is the simplest case we can consider. From Chapter 5 we know that the existence of a riskless lending and borrowing rate implies that there is a single portfolio of risky assets that is preferred to all other portfolios. Furthermore, in return standard deviation space, this portfolio plots on the ray connecting the riskless asset and a risky portfolio that lies furthest in the counterclockwise direction. For example, in Figure 6.1 the portfolio on the ray Rp—B is preferred to all other portfolios of risky assets. The efficient frontier is the entire length of the ray extending through Rp and B. Different points along the ray RF—B represent different amounts of borrowing and/or lending in combination with the optimum portfolio of risky assets portfolio B.

An equivalent way of identifying the ray RF—B is to recognize that it is the ray with the greatest slope. Recall that the slope of the line connecting a riskless asset and risky portfolio is the expected return on the portfolio minus the risk-free rate divided by the standard deviation of the return on the portfolio. Thus, the efficient set is determined by finding that portfolio with the greatest ratio of excess return (expected return minus risk-free rate) to standard deviation that satisfies the constraint that the sum of the proportions invested in the assets equals 1. In equation form we have: maximize the objective function

This is a constrained maximization problem. There are standard solution techniques available for solving it. For example, it can be solved by the method of Lagrangian multipliers. There is an alternative. The constraint could be substituted into the objective function and the objective function maximized as in an unconstrained problem. This latter procedure will be followed below. We can write RF as RF times 1. Thus we have

Making this substitution in the objective function and stating the expected return and standard deviation of return in the general form, derived in Chapter 4, yields

'Lintner [25] has advocated an alternative definition of short sales, one that is more realistic. He assumes correctly that when an investor sells stock short, cash is not received but rather is held as collateral. Furthermore, the investor must put up an additional amount of cash equal to the amount of stock he or she sells short. The investor generally does not receive any compensation (interest) on these funds. However, if the investor is a broker-dealer, interest can be earned on both the money put up and the money received from the short sale of securities. As shown in Appendix A, this leads to the constraint, 2IX;1 = 1, and leaves all other equations unchanged.