You should pursue it on an in- finitely large scale until the return discrepancy between the two portfolios disappears. Well-diversified portfolios with equal betas must have equal expected returns in market equilibrium, or arbitrage opportunities exist.

What about portfolios with different betas? We show now that their risk premiums must be proportional to beta. To see why, consider Figure 11.3. Suppose that the risk-free rate is 4% and that well-diversified portfolio, C, with a beta of .5, has an expected return of 6%. Portfolio C plots below the line from the risk-free asset to Portfolio A. Consider, therefore, a new portfolio, D, composed of half of Portfolio A and half of the risk-free asset. Portfolio Ds beta will be (1⁄2 0 1⁄2 1.0) .5, and its expected return will be (1⁄2 4 1⁄2 10) 7%. Now Portfolio D has an equal beta but a greater expected return than Port- folio C. From our analysis in the previous paragraph we know that this constitutes an arbitrage opportunity.

We conclude that, to preclude arbitrage opportunities, the expected return on all well-di-versified portfolios must lie on the straight line from the risk-free asset in Figure 11.3. The equation of this line will dictate the expected return on all well-diversified portfolios.

Notice in Figure 11.3 that risk premiums are indeed proportional to portfolio betas. The risk premium is depicted by the vertical arrow, which measures the distance between the risk-free rate and the expected return on the portfolio. The risk premium is zero for 0, and rises in direct proportion to .

More formally, suppose that two well-diversified portfolios are combined into a zero-beta portfolio, Z, by choosing the weights shown in Table 11.4. The weights of the two as- sets in portfolio Z sum to 1, and the portfolio beta is zero:

III. Equilibrium In Capital Markets

11. Arbitrage Pricing Theory The McGraw−Hill

Companies, 2001

328 PART III Equilibrium in Capital Markets

Figure 11.3

An arbitrage opportunity.

Expected return (%) A 10