When a risk-free asset exists, the relation between the relevant risk of an investment and its expected return can be derived directly from equation (5.2). Specifically, equation (5.3) is obtained by moving the covariance of the investment with the tangency portfolio to the right-hand side of equation (5.2):
var(RT)
(For simplicity in notation, we have dropped the i subscript.)
Equation (5.3) describes the relation between the expected return of an investment and a measure of its risk. In this case, the relevant measure of risk is the covariance between the returns of the tangency portfolio and the investment.16 Example 5.6 illustrates how to apply equation (5.3).
15Discounting means dividing by a power of the sum: one plus a rate of return. It is formally defined in Chapter 9. Chapter 11 explains how to derive this rate of return in great detail.
16Section 5.8 discusses how this equation is altered by the absence of a risk-free asset in the economy.
Chapter 5 Mean-Variance Analysis and the Capital Asset Pricing Model 147
Example 5.6: Implementing the Risk-Return Equation
The risk-free return is 8 percent. The return of General Motors stock has a covariance with the return of the tangency portfolio that is 50 percent larger than the corresponding covariance for Disney stock. The expected return of Disney stock is 12 percent per year. What is the expected return of General Motors stock?
Answer: The risk premium of Disney stock is 4 percent per year (the expected return minus the risk-free return: 12 percent less 8 percent). General Motors' risk premium must be 50 percent larger or 6 percent per year. Adding the risk-free return to this number yields 14 percent, the expected return of General Motors.
Betas
The first factor in the product on the right-hand of equation (5.3) is commonly referred to as beta and typically denoted by the Greek letter ft; that is p = cov(r, Rt) var(RT)
This notation is used because the right-hand of equation (5.3) also happens to be the formula for the slope coefficient in a regression, which commonly uses ft to denote the slope. With this notation, equation (5.3) becomes:
The Securities Market Line versus the Mean-Standard Deviation Diagram. Panel A of Exhibit 5.5 plots the familiar mean-standard deviation diagram. For the same financial market, panel B to the right of panel A plots what is commonly known as the securities market line. The securities market line is a line relating two important attributes for all the investments in the securities market. In equation (5.4), it is the graphical representation of mean versus beta. The four securities singled out in both panels are the same securities in both diagrams. The tangency portfolio is also the same portfolio in both panels.
Note in panel B that the tangency portfolio has a beta of 1 because the numerator and denominator of the ratio used to compute its beta are identical. The risk-free asset necessarily has a beta of zero; being constant, its return cannot covary with anything. Each portfolio on the capital market line (see panel A), a weighted average of the tan-gency portfolio and the risk-free asset, has a location on the securities market line found by taking the same weighted average of the points corresponding to the tangency portfolio and the risk-free asset.17 What is special about the securities market line, however, is that all investments in panel A lie on the line: both the efficient portfolios on the capital market line and the dominated investments to the right of the capital market line.
Exhibit 5.5 purposely places the two graphs side by side to illustrate the critical distinction between the securities market line and the mean-standard deviation diagram. The difference between the graphs in panels A and B is reflected on the horizontal axis. Panel A shows the standard deviation on this axis while panel B shows the beta with the return of the tangency portfolio, which is proportional to the marginal variance. Thus, while investments with the same mean can have different standard deviations, as seen
17For this weighting of two portfolios, think of the risk-free asset as a portfolio with a weight of 1 on the risk-free asset and 0 on all the other assets.
Part II Valuing Financial Assets
Exhibit 5.5 The Relation between the Mean-Standard Deviation Diagram and a Mean-Beta Diagram
Panel A
Panel B
Mean return
Tangency portfolio
Security 1
.Security 3
Security 4
Capital market line
.Security 3
Security 4
Capital market line
Standard deviation of return
Mean return
Tangency portfolio
X .^Security 1 Portfolio Af T
Security 2
'Security 3 ' Security 4
Securities Market line
Beta with return of the tangency portfolio
in panel A, they must have the same beta, as seen in panel B. For example, in panel A, all of the points on the grey line to the right of point T, labeled "3 = 1," are portfolios with the same beta as the tangency portfolio. In panel B, all of these portfoliosâ€” even though they are distinct in terms of their portfolio weights and standard deviationsâ€”plot at exactly the same point as the tangency portfolio. For the same reason, all points on the grey horizontal line to the right of the risk-free asset in the mean-standard deviation diagram, designated "3 = 0" are portfolios with a beta of 0 even though they have positive and differing standard deviations. In the mean-beta diagram, which graphs the securities market line in panel B, these portfolios plot at the same
point as rf.
Portfolio Betas. An important property of beta is found in Result 5.4.19
Result 5.4 The beta of a portfolio is a portfolio-weighted average of the betas of its individual secu rities; that is
18Because mean return and beta plot on a straight line (see panel B), all investments with the same mean have the same beta and all investments with the same beta have the same mean.
19Because 3 is merely the covariance of security i with the tangency portfolio divided by a constant, Result 5.4 is a direct extension of Result 4.5 in Chapter 4: the covariance of the return of a portfolio with the return of a stock is the portfolio-weighted average of the covariances of the investments in the portfolio with the stock return.
Chapter 5 Mean-Variance Analysis and the Capital Asset Pricing Model 149
where
Thus, a portfolio that is 75 percent invested in a stock with a beta of 1.2 and 25 percent invested in a stock with a beta of 0.8 has a beta of 1, since
Contrasting Betas and Covariances. Note that betas and covariances are essentially the same measure of marginal variance. Beta is simply the covariance divided by the same constant for every stock. For historical reasons as well as the ease of estimation with regression, beta has become the more popular scaling of marginal variance. In principle, however, both are equally good as measures of marginal risk.
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