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0.251 25.1%.
Thus, by forming the two-asset portfolio you will have the same expected
return, 13 percent, but with a standard deviation of only 25.1 percent ver-
sus 30 percent with the one-asset portfolio. So, because the stocks were not
perfectly positively correlated, diversi¬cation has reduced your risk.
The numbers would change if the two stocks had different expected
returns and standard deviations, or if we invested different amounts in each
of them, or if the correlation coef¬cient were different from 0.4. Still, the
bottom line conclusion is that, provided the stocks are not perfectly posi-
1.0, then diversi¬cation will be bene¬cial.11
tively correlated, with 1,2
If a two-stock portfolio is better than a one-stock portfolio, would it be
better to continue diversifying, forming a portfolio with more and more
stocks? The answer is yes, at least up to some fairly large number of stocks.
Figure 28-1 shows the relation between the number of stocks in a portfolio
and the portfolio™s risk for average NYSE stocks. Note that as the number
of stocks in the portfolio increases, the total amount of risk decreases, but
at a lower and lower rate, and it approaches a lower limit. This lower limit
is called the market risk inherent in stocks, and no amount of diversi¬cation

11
The standard deviation of a portfolio consisting of n assets with standard deviations i, weights wi and pairwise cor-
relations i,j is given by this equation:



B ia ja
n n
(28-13)
wiwj i j ij.
p
1 1



Here 1.0. Note that this reduces to the two asset formula given above if n 2.
i,j
28-14 Basic Financial Tools: A Review
Chapter 28



Effects of Portfolio Size on Portfolio Risk for Average Stocks
Figure 28-1
Portfolio Risk,σ p
(%)
35



30


Diversifiable Risk
25



σM = 20.1




15 Portfolio's
Risk:
Declines Portfolio's
as Stocks Market Risk:
Are Added The Risk That Remains,
10 Even in Large Portfolios



5



0
1 10 20 30 40 2,000+
Number of Stocks
in the Portfolio




can eliminate it. On the other hand, the risk of the portfolio in excess of the
market risk is called diversi¬able risk, and, as the graph shows, investors can
reduce or even eliminate it by holding more and more stocks. It is not shown
in the graph, but diversi¬cation among average stocks would not affect the
portfolio™s expected return”expected return would remain constant, but
risk would decline as shown in the graph.
Investors who do not like risk are called risk averse, and such investors will
choose to hold portfolios that consist of many stocks rather than only a few
so as to eliminate the diversi¬able risk in their portfolios. In developing the
relationship between risk and return, we will assume that investors are risk
averse, which implies that they will not hold portfolios that still have diversi-
¬able risk. Instead, they will diversify their portfolios until only market risk
remains. These resulting portfolios are called well-diversi¬ed portfolios.12

12
If one selected relatively risky stocks, then the lower limit in Figure 28-1 would plot above the one shown for aver-
age stocks, and if the portfolios were formed from low-risk stocks, the lower limit would plot below the one we show.
Similarly, if the expected returns on the added stocks differed from that of the portfolio, the expected return on the port-
folio would be affected. Still, even if one wants to hold an especially high-risk, high-return portfolio, or a low risk and
return portfolio, diversi¬cation will be bene¬cial.
Note too that holding more stocks involves more commissions and administrative costs. As a result, individual
investors must balance these additional costs against the gains from diversi¬cation, and consequently most individuals
limit their stocks to no more than 30 to 50. Also, note that if an individual does not have enough capital to diversify
ef¬ciently, then he or she can (and should) invest in a mutual fund.
28-15
Risk and Return


The Capital Asset Pricing Model
Some investors have no tolerance whatever for risk, so they choose to invest
all of their money in riskless Treasury bonds and receive a real return of
about 3.5 percent.13 Most investors, however, choose to bear at least some
risk in exchange for an expected return that is higher than the risk-free rate.
Suppose a particular investor is willing to accept a certain amount of risk in
hope of realizing a higher rate of return. Assuming the investor is rational,
he or she will choose the portfolio that provides the highest expected return
for the given level of risk. This portfolio is by de¬nition an optimal portfo-
lio because it has the highest possible return for a given level of risk. But how
can investors identify optimal portfolios?
One of the implications of the Capital Asset Pricing Model (CAPM) as
discussed in Chapters 2 and 3 is that optimal behavior by investors calls for
splitting their investments between the market portfolio, M, and a risk-free
investment. The market portfolio consists of all risky assets, held in propor-
tion to their market values. For our purposes, we consider an investment in
long-term U.S. Treasury bonds to be a risk-free investment. The market
ˆ
portfolio has an expected return of rM and a standard deviation of M, and
the risk-free investment has a guaranteed return of rRF. The expected return
on a portfolio with weight wM invested in M, and weight wRF (which equals
1.0 wM) in the risk-free asset, is

rp wM rM (1 wM) rRF.
ˆ ˆ (28-11a)

Because a risk-free investment has zero standard deviation, the correlation
term in Equation 28-12 is zero, hence the standard deviation of the portfo-
lio reduces to

wM M. (28-12a)
p


These relationships show that by assigning different weights to M and to the
risk-free asset, we will form portfolios with different expected returns and
standard deviations. Combining Equations 28-11a and 28-12a, and elimi-
nating wM, we obtain this relationship between an optimal portfolio™s return
and its standard deviation:


a b
rRF
rM
ˆ (28-14)
rRF p.
rp
ˆ
M


This equation is called the Capital Market Line (CML), and a graph of this
relationship between risk and return is shown in Figure 28-2.
The CML shows the expected return that investors can expect at each risk
level, assuming that they behave optimally by splitting their investments
between the market portfolio and the risk-free asset. Note that the expected
return on the market is greater than the risk-free rate, hence the CML is
upward sloping. This means that investors who would like a portfolio with
a higher rate of return must be willing to accept more risk as measured by

13
Indexed T-bonds are essentially riskless, and they currently provide a real return of about 3.5 percent. This expected
nominal return is 3.5 percent plus expected in¬‚ation.
28-16 Basic Financial Tools: A Review
Chapter 28



The Capital Market Line (CML)
Figure 28-2
Expected Rate
of Return, r p

r M “ r RF
σp
CML = r RF + σM

rM


r RF



σM Risk, σ p
0




the standard deviation. Thus, investors who are willing to accept more risk
are rewarded with higher expected returns as compensation for bearing this
additional risk.
ˆ
For example, suppose that rRF 10%, rM 15%, and M 20%. Under
these conditions, a portfolio consisting of 50 percent in the risk-free asset and
50 percent in the market portfolio will have an expected return of 12.5 per-
cent and a standard deviation of 10 percent. Varying the portfolio weights
from 0 to 1.0 traces out the CML. Points on the CML to the right of the mar-
ˆ
ket portfolio (rM) can be obtained by putting portfolio weights on M greater
than 1.0. This implies borrowing at the risk-free rate and then investing this
extra money, along with the initial capital, in the market portfolio.

Beta
If investors are rational and thus hold only optimal portfolios (that is,
portfolios that have only market risk and are on the CML), then the only
type of risk associated with an individual stock that is relevant is the risk the
stock adds to the portfolio. Refer to Figure 28-2 and note that investors
should be interested in how much an additional stock moves the entire port-
folio up or down the CML, not in how risky the individual stock would be
if it were held in isolation. This is because some of the risk inherent in any
individual stock can be eliminated by holding it in combination with all the
other stocks in the portfolio. Chapters 2 and 3 show that the correct mea-
sure of an individual stock™s contribution to the risk of a well-diversi¬ed
portfolio is its beta coef¬cient, or simply beta, which is calculated as follows:

i,M i M i,M i
Beta of stock i bi . (28-15)
o2 M
M



Here i,M is the correlation coef¬cient between Stock i and the market.
By de¬nition, the market portfolio has a beta of 1.0. Adding a stock with
a beta of 1.0 to the market portfolio will not change the portfolio™s overall
risk. Adding a stock with a beta of less than 1.0 will reduce the portfolio™s
risk, hence reduce its expected rate of return as shown in Figure 28-2.
Adding a stock with a beta greater than 1.0 will increase the portfolio™s risk
and expected return. Intuitively, you can think of a stock™s beta as a measure
28-17
Risk and Return


of how closely it moves with the market. A stock with a beta greater than 1.0
will tend to move up and down with the market, but with wider swings. A
stock with a beta close to zero will tend to move independently of the market.
The CAPM shows the relationship between the risk that a stock con-
tributes to a portfolio and the return that it must provide. The required rate
of return on a stock is related to its beta by this formula:

Required rate of return on Stock i ri rRF bi(rM rRF). (28-16)


For given values of rRF and rM, the graph of ri versus bi is called the Security
Market Line (SML). The SML shows the relationship between the required
rate of return on a stock, its riskiness as measured by beta, and the required
rate of return on the market. Figure 28-3 shows the SML and required rates
of return for a low beta and a high beta stock. Note that the required rate
of return on a stock is in excess of the risk-free rate, and it increases with
beta. The extra return associated with higher betas is called the risk pre-
mium, and the risk premium on a given stock is equal to bi(rM rRF). The
term (rM rRF), which is called the market risk premium, or RPM, amounts
to the extra return an investor requires for bearing the market™s risk.
The SML graph differs signi¬cantly from the CML graph. As Figure 28-2
shows, the CML de¬nes the relationship between total risk, as measured by
the standard deviation, and the expected rate of return for portfolios that are

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