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We have already seen (Chapter 5) that the expected return from portfolio A is simply the sum of the products of the proportion invested in each stock and the expected return on each stock. We have also seen that the Beta on a portfolio is simply the sum of the product of the proportion invested in each stock times the Beta on each stock. Now consider a portfolio C made up of one half of portfolio A and one half of portfolio B. From the facts stated earlier, the expected return on this portfolio is 11 and its Beta is 1.2. These three potential investments are plotted in Figure 13.3. Notice they lie on a straight line. This is no accident. All portfolios composed of different fractions of investments A and B will lie along a straight line in expected return Beta space.5

Now hypothesize a new investment D that has a return of 13% and a Beta of 1.2. Such an investment cannot exist for very long. All decisions are made in terms of risk and return. This portfolio offers a higher return and the same risk as portfolio C. Hence, it would pay

5If we let X stand for the fraction of funds invested in portfolio A, then the equation for return is

Solving the second equation for X and substituting in the first equation, we see that we are left with an equation of the form

Rp = a + bfip or the equation of a straight line.

Beta

Figure 13.3 Combinations of portfolios.

all investors to sell C short and buy D. Similarly, if a security were to exist with a return of 8% and a Beta of 1.2 (designated by £), it would pay arbitragers to step in and buy port-lulio C while selling security E short. Such arbitrage would take place until C, D, and E .ill yielded the same return. This is just another illustration of the adage that two things that .ire equivalent cannot sell at different prices. We can demonstrate the arbitrage discussed earlier in a slightly more formal manner. Let us return to the arbitrage between portfolios ("and D. An investor could sell $100 worth of portfolio C short and with the $100 buy portfolio D. If the investor were to do so, the characteristics of this arbitraged portfolio nould be as follows:

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