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