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Thus the proportion to invest in each security is

The expected return on the portfolio is

The variance of the return on the portfolio is6

4See Appendix C at the end of this chapter for a description of solution techniques for systems of simultaneous equations.

5In the case of Lintnerian short sales, simply scale so that j=i

6The variance of the portfolio could have been determined in another way. Recall that X is the ratio of the excess return on the optimum portfolio divided by the variance of the optimum portfolio. Thus

Also recall that z. = XX, so that 2Z{. = X2X. = X. Earlier we determined that 2Z. = X = 18/63. Equating these two equations and solving for u2p yields the value presented above.

Figure 6.2 The efficient set with riskless lending and borrowing.

The efficient set is a straight line with an intercept at the risk-free rate of 5% and a slope equal to the ratio of excess return to standard deviation. (See Figure 6.2.) There are standard computer packages for the solution of a system of simultaneous equations. Appendix C at the end of this chapter presents some methods of solving them when the number of securities involved is limited so that hand calculations are reasonable.

SHORT SALES ALLOWED: NO RISKLESS LENDING AND BORROWING

When the investor does not wish to make the assumption that he can borrow and lend at the riskless rate of interest, the solution developed in the last section must be modified. However, much of the analysis can still be utilized. Consider Figure 6.3. The riskless lending and borrowing rate of 5% led to the selection of portfolio B. If the riskless lending and borrowing rate had been 4%, the investor would invest in portfolio A. If the investor's lending and borrowing rate was 6%, the investor would select portfolio C. These observations suggest the following procedure. Assume that a riskless lending and borrowing rate exists and find the optimum portfolio. Then assume that a different riskless lending and borrowing rate exists and find the optimum portfolio that corresponds to this second rate. Continue changing the assumed riskless rate until the full efficient frontier is determined.7 In Appendix D we present a general solution to this problem. We show that the optimal proportion to invest in any security is simply a linear function of Rp. Furthermore, since the entire efficient frontier can be constructed as a combination of any two portfolios that lie along it, the identification of the characteristics of the optimal portfolio for any two arbitrary values of Rp is sufficient to trace out the total efficient frontier.

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