Chapter 8 Risk and Rates of Return 249- A portfolio consisting of low-beta stocks will also have a low beta because the
beta of a portfolio is a weighted average of its individual securities’ betas,
found using this equation:
bp " w 1 b 1 $ w 2 b 2 $... $ wNbN
" ∑
i" 1N
wib N. 8-5Here bp is the beta of the portfolio, and it shows how volatile the portfolio is rela-
tive to the market; wi is the fraction of the portfolio invested in the ith stock; and
bi is the beta coef" cient of the ith stock. To illustrate, if an investor holds a
$100,000 portfolio consisting of $33,333.33 invested in each of three stocks and if
each of the stocks has a beta of 0.70, the portfolio’s beta will be bp $ 0.70:
bp = 0.333(0.70) + 0.333(0.70) + 0.333(0.70) = 0.70.
Such a portfolio would be less risky than the market, so it should experience
relatively narrow price swings and have relatively small rate-of-return! uctu-
ations. In terms of Figure 8-7, the slope of its regression line would be 0.70,
which is less than that for a portfolio of average stocks.
Now suppose one of the existing stocks is sold and replaced by a stock
with bi $ 2.00. This action will increase the portfolio’s beta from bp1 $ 0.70 to
bp2 $ 1.13:
bp2 " 0.333(0.70) $ 0.333(0.70) $ 0.333(2.00) " 1.13.
Had a stock with bi $ 0.20 been added, the portfolio’s beta would have declined
from 0.70 to 0.53. Adding a low-beta stock would therefore reduce the portfo-
lio’s riskiness. Consequently, changing the stocks in a portfolio can change the
riskiness of that portfolio.
- Because a stock’s beta coef" cient determines how the stock affects the riski-
ness of a diversi" ed portfolio, beta is, in theory, the most relevant measure of
a stock’s risk.
SEL
F^ TEST Explain the following statement: An asset held as part of a portfolio is gener-
ally less risky than the same asset held in isolation.
What is meant by perfect positive correlation, perfect negative correlation, and
zero correlation?
In general, can the riskiness of a portfolio be reduced to zero by increasing
the number of stocks in the portfolio? Explain.
What is an average-risk stock? What is the beta of such a stock?
Why is it argued that beta is the best measure of a stock’s risk?
If you plotted a particular stock’s returns versus those on the S&P 500 Index
over the past # ve years, what would the slope of the regression line indicate
about the stock’s risk?
An investor has a two-stock portfolio with $25,000 invested in Stock X and
$50,000 invested in Stock Y. X’s beta is 1.50, and Y’s beta is 0.60. What is the
beta of the investor’s portfolio? (0.90)