Figure 132a

Portfolio expected return (E(RP))

Portfolio beta

The Reward-to-Risk Ratio What is the slope of the straight line in Figure 13.2A? As always, the slope of a straight line is equal to "the rise over the run." In this case, as we move out of the risk-free asset into Asset A, the beta increases from zero to 1.6 (a "run" of 1.6). At the same time, the expected return goes from 8 percent to 20 percent, a "rise" of 12 percent. The slope of the line is thus 12%/1.6 = 7.5%.

Notice that the slope of our line is just the risk premium on Asset A, E(RA) - Rf, divided by Asset A's beta, (3A:

What this tells us is that Asset A offers a reward-to-risk ratio of 7.5 percent.2 In other words, Asset A has a risk premium of 7.50 percent per "unit" of systematic risk.

The Basic Argument Now suppose we consider a second asset, Asset B. This asset has a beta of 1.2 and an expected return of 16 percent. Which investment is better, Asset A or Asset B? You might think that, once again, we really cannot say—some investors might prefer A; some investors might prefer B. Actually, however, we can say: A is better because, as we will demonstrate, B offers inadequate compensation for its level of systematic risk, at least, relative to A.

To begin, we calculate different combinations of expected returns and betas for portfolios of Asset B and a risk-free asset, just as we did for Asset A. For example, if we put 25 percent in Asset B and the remaining 75 percent in the risk-free asset, the portfolio's expected return will be:

2This ratio is sometimes called the Treynor index, after one of its originators.

Ross et al.: Fundamentals I V. Risk and Return I 13. Return, Risk, and the I I © The McGraw-Hill of Corporate Finance, Sixth Security Market Line Companies, 2002

Edition, Alternate Edition

436 PART FIVE Risk and Return

E(RP) = .25 X E(RS) + (1 - .25) X Rf = .25 X 16% + .75 X 8% = 10%

Similarly, the beta on the portfolio, (P, would be:

Some other possibilities are as follows:

Percentage of Portfolio in Asset B

Portfolio Expected Return

Portfolio Beta

0%

8%

0 0

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