Seventh Edition - The Addison-Wesley Series in Economics Phần 3 pot

p
2
ϭ probability of occurrence return 2 ϭϭ.33
R
2
ϭ return in state 2 ϭ 8% ϭ 0.08
Thus:
R
e
ϭ (0.67)(0.12) ϩ (0.33)(0.08) ϭ 0.1068 ϭ 10.68%
The degree of risk or uncertainty of an asset’s returns also affects the demand for the
asset. Consider two assets, stock in Fly-by-Night Airlines and stock in Feet-on-the-
Ground Bus Company. Suppose that Fly-by-Night stock has a return of 15% half of
the time and 5% the other half of the time, making its expected return 10%, while
stock in Feet-on-the-Ground has a fixed return of 10%. Fly-by-Night stock has uncer-
tainty associated with its returns and so has greater risk than stock in Feet-on-the-
Ground, whose return is a sure thing.
To see this more formally, we can use a measure of risk called the standard devi-
ation. The standard deviation of returns on an asset is calculated as follows. First cal-
culate the expected return, R
e
; then subtract the expected return from each return to
get a deviation; then square each deviation and multiply it by the probability of occur-
rence of that outcome; finally, add up all these weighted squared deviations and take
the square root. The formula for the standard deviation, ␴, is thus:
␴ ϭ
(2)
The higher the standard deviation, ␴, the greater the risk of an asset.
EXAMPLE 2: Standard Deviation
What is the standard deviation of the returns on the Fly-by-Night Airlines stock and Feet-

on-the-Ground Bus Company, with the same return outcomes and probabilities
described above? Of these two stocks, which is riskier?
Solution
Fly-by-Night Airlines has a standard deviation of returns of 5%.
␴ ϭ
R
e
ϭ p
1
R
1
ϩ p
2
R
2
where
p
1
ϭ probability of occurrence of return 1 ϭϭ0.50
R
1
ϭ return in state 1 ϭ 15% ϭ 0.15
p
2
ϭ probability of occurrence of return 2 ϭϭ0.50
R
2
ϭ return in state 2 ϭ 5% ϭ 0.05
R
e

ϭ expected return ϭ (0.50)(0.15) ϩ (0.50)(0.05) ϭ 0.10
1
2
1
2
͙
p
1
(R
1
Ϫ R
e
)
2
ϩ p
2
(R
2
Ϫ R
e
)
2
͙
p
1
(R
1
Ϫ R
e
)

2
ϩ p
2
(R
2
Ϫ R
e
)
2
ϩ . . . ϩ p
n
(R
n
Ϫ R
e
)
2
Calculating
Standard Deviation
of Returns
1
3
Models of Asset Pricing
2
Thus:
␴ ϭ
␴ ϭ ϭ 0.05 ϭ 5%
Feet-on-the-Ground Bus Company has a standard deviation of returns of 0%.
␴ ϭ
R

e
ϭ p
1
R
1
where
p
1
ϭ probability of occurrence of return 1 ϭ 1.0
R
1
ϭ return in state 1 ϭ 10% ϭ 0.10
R
e
ϭ expected return ϭ (1.0)(0.10) ϭ 0.10
Thus:
ϭ
Clearly, Fly-by-Night Airlines is a riskier stock, because its standard deviation of
returns of 5% is higher than the zero standard deviation of returns for Feet-on-the-
Ground Bus Company, which has a certain return.
Benefits of Diversification
Our discussion of the theory of asset demand indicates that most people like to avoid
risk; that is, they are risk-averse. Why, then, do many investors hold many risky assets
rather than just one? Doesn’t holding many risky assets expose the investor to more
risk?
The old warning about not putting all your eggs in one basket holds the key to
the answer: Because holding many risky assets (called diversification) reduces the over-
all risk an investor faces, diversification is beneficial. To see why this is so, let’s look
at some specific examples of how an investor fares on his investments when he is
holding two risky securities.

Consider two assets: common stock of Frivolous Luxuries, Inc., and common
stock of Bad Times Products, Unlimited. When the economy is strong, which we’ll
assume is one-half of the time, Frivolous Luxuries has high sales and the return on
the stock is 15%; when the economy is weak, the other half of the time, sales are low
and the return on the stock is 5%. On the other hand, suppose that Bad Times
Products thrives when the economy is weak, so that its stock has a return of 15%, but
it earns less when the economy is strong and has a return on the stock of 5%. Since
both these stocks have an expected return of 15% half the time and 5% the other half
of the time, both have an expected return of 10%. However, both stocks carry a fair
amount of risk, because there is uncertainty about their actual returns.
Suppose, however, that instead of buying one stock or the other, Irving the
Investor puts half his savings in Frivolous Luxuries stock and the other half in Bad
͙
0 ϭ 0 ϭ 0%
␴ϭ
͙
(1.0
)
(0.10 Ϫ 0.10
)
2
͙
p
1
(R
1
Ϫ R
e
)
2

͙
(0.50
)
(0.0025
)
ϩ (0.50
)
(0.0025
)
ϭ
͙
0.0025
͙
(0.50
)
(0.15 Ϫ 0.10
)
2
ϩ (0.50
)
(0.05 Ϫ 0.10
)
2
Appendix 1 to Chapter 5
3
Times Products stock. When the economy is strong, Frivolous Luxuries stock has a
return of 15%, while Bad Times Products has a return of 5%. The result is that Irving
earns a return of 10% (the average of 5% and 15%) on his holdings of the two stocks.
When the economy is weak, Frivolous Luxuries has a return of only 5% and Bad Times
Products has a return of 15%, so Irving still earns a return of 10% regardless of whether

the economy is strong or weak. Irving is better off from this strategy of diversification
because his expected return is 10%, the same as from holding either Frivolous
Luxuries or Bad Times Products alone, and yet he is not exposed to any risk.
Although the case we have described demonstrates the benefits of diversification,
it is somewhat unrealistic. It is quite hard to find two securities with the characteristic
that when the return of one is high, the return of the other is always low.
1
In the real
world, we are more likely to find at best returns on securities that are independent of
each other; that is, when one is high, the other is just as likely to be high as to be low.
Suppose that both securities have an expected return of 10%, with a return of 5%
half the time and 15% the other half of the time. Sometimes both securities will earn
the higher return and sometimes both will earn the lower return. In this case if Irving
holds equal amounts of each security, he will on average earn the same return as if he
had just put all his savings into one of these securities. However, because the returns
on these two securities are independent, it is just as likely that when one earns the
high 15% return, the other earns the low 5% return and vice versa, giving Irving a
return of 10% (equal to the expected return). Because Irving is more likely to earn
what he expected to earn when he holds both securities instead of just one, we can
see that Irving has again reduced his risk through diversification.
2
The one case in which Irving will not benefit from diversification occurs when the
returns on the two securities move perfectly together. In this case, when the first secu-
rity has a return of 15%, the other also has a return of 15% and holding both securi-
ties results in a return of 15%. When the first security has a return of 5%, the other
has a return of 5% and holding both results in a return of 5%. The result of diversi-
fying by holding both securities is a return of 15% half of the time and 5% the other
half of the time, which is exactly the same set of returns that are earned by holding
only one of the securities. Consequently, diversification in this case does not lead to
any reduction of risk.

The examples we have just examined illustrate the following important points
about diversification:
1. Diversification is almost always beneficial to the risk-averse investor since it
reduces risk unless returns on securities move perfectly together (which is an
extremely rare occurrence).
2. The less the returns on two securities move together, the more benefit (risk reduc-
tion) there is from diversification.
Models of Asset Pricing
1
Such a case is described by saying that the returns on the two securities are perfectly negatively correlated.
2
We can also see that diversification in the example above leads to lower risk by examining the standard devi-
ation of returns when Irving diversifies and when he doesn’t. The standard deviation of returns if Irving holds
only one of the two securities is . When Irving holds
equal amounts of each security, there is a probability of
1
/
4
that he will earn 5% on both (for a total return of
5%), a probability of
1
/
4
that he will earn 15% on both (for a total return of 15%), and a probability of
1
/
2
that
he will earn 15% on one and 5% on the other (for a total return of 10%). The standard deviation of returns when
Irving diversifies is thus .

Since the standard deviation of returns when Irving diversifies is lower than when he holds only one security,
we can see that diversification has reduced risk.
͙
0.25 ϫ (15% Ϫ 10%
)
2
ϩ 0.25 ϫ (5% Ϫ 10%
)
2
ϩ 0.5 ϫ (10% Ϫ 10%
)
2
ϭ 3.5%
͙
0.5 ϫ (15% Ϫ 10%
)
2
ϩ 0.5 ϫ (5% Ϫ 10%
)
2
ϭ 5%
4
Diversification and Beta
In the previous section, we demonstrated the benefits of diversification. Here, we
examine diversification and the relationship between risk and returns in more detail.
As a result, we obtain an understanding of two basic theories of asset pricing: the cap-
ital asset pricing model (CAPM) and arbitrage pricing theory (APT).
We start our analysis by considering a portfolio of n assets whose return is:
R
p

ϭ x
1
R
1
ϩ x
2
R
2
ϩ
…
ϩ x
n
R
n
(3)
where R
p
ϭ the return on the portfolio of n assets
R
i
ϭ the return on asset i
x
i
ϭ the proportion of the portfolio held in asset i
The expected return on this portfolio, E(R
p
), equals
E(R
p
) ϭ E(x

1
R
1
) ϩ E(x
2
R
2
) ϩ
…
ϩ E(x
n
R
n
)
ϭ x
1
E(R
1
) ϩ x
2
E(R
2
) ϩ
…
ϩ x
n
E(R
n
) (4)
An appropriate measure of the risk for this portfolio is the standard deviation of the

portfolio’s return (␴
p
) or its squared value, the variance of the portfolio’s return (␴
p
2
),
which can be written as:
␴
p
2
ϭ E[R
p
Ϫ E(R
p
)]
2
ϭ E[{x
1
R
1
ϩ
…
ϩ x
n
R
n
} Ϫ {x
1
E(R
1

) ϩ
…
ϩ x
n
E(R
n
)}]
2
ϭ E[x
1
{R
1
Ϫ E(R
1
)} ϩ
…
ϩ x
n
{R
n
Ϫ E(R
n
)}]
2
This expression can be rewritten as:
␴
p
2
ϭ E[{x
1

[R
1
Ϫ E(R
1
)] ϩ
…
ϩ x
n
[R
n
Ϫ E(R
n
)]} ϫ {R
p
Ϫ E(R
p
)}]
ϭ x
1
E[{R
1
Ϫ E(R
1
)} ϫ {R
p
Ϫ E(R
p
)}] ϩ
…
ϩ x

n
E[{R
n
Ϫ E(R
n
)} ϫ {R
p
Ϫ E(R
p
)}]
This gives us the following expression for the variance for the portfolio’s return:
␴
p
2
ϭ x
1
␴
1p
ϩ x
2
␴
2p
ϩ x
n
␴
np
(5)
where
␴
ip

ϭ the covariance of the return on asset i
with the portfolio’s return ϭ E[{R
i
Ϫ E(R
i
)} ϫ {R
p
Ϫ E(R
p
)}]
Equation 5 tells us that the contribution to risk of asset i to the portfolio is x
i
␴
ip
.
By dividing this contribution to risk by the total portfolio risk (␴
p
2
), we have the pro-
portionate contribution of asset i to the portfolio risk:
x
i
␴
ip
/␴
p
2
The ratio ␴
ip
/␴

p
2
tells us about the sensitivity of asset i’s return to the portfolio’s return.
The higher the ratio is, the more the value of the asset moves with changes in the
Appendix 1 to Chapter 5
5
value of the portfolio, and the more asset i contributes to portfolio risk. Our algebraic
manipulations have thus led to the following important conclusion: The marginal
contribution of an asset to the risk of a portfolio depends not on the risk of the asset
in isolation, but rather on the sensitivity of that asset’s return to changes in the
value of the portfolio.
If the total of all risky assets in the market is included in the portfolio, then it is
called the market portfolio. If we suppose that the portfolio, p, is the market portfolio,
m, then the ratio ␴
im
/␴
m
2
is called the asset i’s beta, that is:
␤
i
ϭ␴
im
/␴
m
2
(6)
where
␤
i

ϭ the beta of asset i
An asset’s beta then is a measure of the asset’s marginal contribution to the risk of the
market portfolio. A higher beta means that an asset’s return is more sensitive to
changes in the value of the market portfolio and that the asset contributes more to the
risk of the portfolio.
Another way to understand beta is to recognize that the return on asset i can be
considered as being made up of two components—one that moves with the market’s
return (R
m
) and the other a random factor with an expected value of zero that is
unique to the asset (⑀
i
) and so is uncorrelated with the market return:
R
i
ϭ␣
i
ϩ␤
i
R
m
ϩ⑀
i
(7)
The expected return of asset i can then be written as:
E(R
i
) ϭ␣
i
ϩ␤

i
E(R
m
)
It is easy to show that ␤
i
in the above expression is the beta of asset i we defined before
by calculating the covariance of asset i’s return with the market return using the two
equations above:
␴
im
ϭ E[{R
i
Ϫ E(R
i
)} ϫ {R
m
Ϫ E(R
m
)}] ϭ E[{␤
i
[R
m
Ϫ E(R
m
)] ϩ⑀
i
}
ϫ {R
m

Ϫ E(R
m
)}]
However, since ⑀
i
is uncorrelated with R
m
, E[{⑀
i
} ϫ {R
m
Ϫ E(R
m
)}] ϭ 0. Therefore,
␴
im
ϭ␤
i
␴
m
2
Dividing through by ␴
m
2
gives us the following expression for ␤
i
:
␤
i
ϭ␴

im
/␴
m
2
which is the same definition for beta we found in Equation 6.
The reason for demonstrating that the ␤
i
in Equation 7 is the same as the one we
defined before is that Equation 7 provides better intuition about how an asset’s beta
measures its sensitivity to changes in the market return. Equation 7 tells us that when
Models of Asset Pricing
6
the beta of an asset is 1.0, it’s return on average increases by 1 percentage point when
the market return increases by 1 percentage point; when the beta is 2.0, the asset’s
return increases by 2 percentage points when the market return increases by 1 per-
centage point; and when the beta is 0.5, the asset’s return only increases by 0.5 per-
centage point on average when the market return increases by 1 percentage point.
Equation 7 also tells us that we can get estimates of beta by comparing the aver-
age return on an asset with the average market return. For those of you who know a
little econometrics, this estimate of beta is just an ordinary least squares regression of
the asset’s return on the market return. Indeed, the formula for the ordinary least
squares estimate of ␤
i
ϭ␴
im
/␴
m
2
is exactly the same as the definition of ␤
i

earlier.
Systematic and Nonsystematic Risk
We can derive another important idea about the riskiness of an asset using Equation
7. The variance of asset i’s return can be calculated from Equation 7 as:
␴
i
2
ϭ E[R
i
Ϫ E(R
i
)]
2
ϭ E{␤
i
[R
m
Ϫ E(R
m
)} ϩ⑀
i
]
2
and since ⑀
i
is uncorrelated with market return:
␴
i
2
ϭ␤

i
2
␴
m
2
ϩ␴
⑀
2
The total variance of the asset’s return can thus be broken up into a component that
is related to market risk, ␤
i
2
␴
m
2
, and a component that is unique to the asset, ␴
⑀
2
. The
␤
i
2
␴
m
2
component related to market risk is referred to as systematic risk and the ␴
⑀
2
component unique to the asset is called nonsystematic risk. We can thus write the total
risk of an asset as being made up of systematic risk and nonsystematic risk:

Total Asset Risk ϭ Systematic Risk ϩ Nonsystematic Risk (8)
Systematic and nonsystematic risk each have another feature that makes the dis-
tinction between these two types of risk important. Systematic risk is the part of an
asset’s risk that cannot be eliminated by holding the asset as part of a diversified port-
folio, whereas nonsystematic risk is the part of an asset’s risk that can be eliminated
in a diversified portfolio. Understanding these features of systematic and nonsystem-
atic risk leads to the following important conclusion: The risk of a well-diversified
portfolio depends only on the systematic risk of the assets in the portfolio.
We can see that this conclusion is true by considering a portfolio of n assets, each
of which has the same weight on the portfolio of (1/n). Using Equation 7, the return
on this portfolio is:
which can be rewritten as:
R
p
ϭ␣ϩ␤R
m
ϩ 1͞n
)
͚
n
iϭ1
⑀
i
R
p
ϭ (1͞n
)
͚
n
iϭ1

␣
i
ϩ (1͞n
)
͚
n
iϭ1
␤
i
R
m
ϩ (1͞n
)
͚
n
iϭ1
⑀
i
Appendix 1 to Chapter 5
7
where
ϭ the average of the ␣
i
’s ϭ
ϭ the average of the ␤
i
’s ϭ
If the portfolio is well diversified so that the ⑀
i
’s are uncorrelated with each other, then

using this fact and the fact that all the ⑀
i
’s are uncorrelated with the market return, the
variance of the portfolio’s return is calculated as:
(average varience of ⑀
i
)
As n gets large the second term, (1/n)(average variance of ⑀
i
), becomes very small, so
that a well-diversified portfolio has a risk of , which is only related to system-
atic risk. As the previous conclusion indicated, nonsystematic risk can be eliminated
in a well-diversified portfolio. This reasoning also tells us that the risk of a well-diversified
portfolio is greater than the risk of the market portfolio if the average beta of the assets
in the portfolio is greater than one; however, the portfolio’s risk is less than the mar-
ket portfolio if the average beta of the assets is less than one.
The Capital Asset Pricing Model (CAPM)
We can now use the ideas we developed about systematic and nonsystematic risk and
betas to derive one of the most widely used models of asset pricing—the capital asset
pricing model (CAPM) developed by William Sharpe, John Litner, and Jack Treynor.
Each cross in Figure 1 shows the standard deviation and expected return for each
risky asset. By putting different proportions of these assets into portfolios, we can gen-
erate a standard deviation and expected return for each of the portfolios using
Equations 4 and 5. The shaded area in the figure shows these combinations of stan-
dard deviation and expected return for these portfolios. Since risk-averse investors
always prefer to have higher expected return and lower standard deviation of the
return, the most attractive standard deviation-expected return combinations are the
ones that lie along the heavy line, which is called the efficient portfolio frontier. These
are the standard deviation-expected return combinations risk-averse investors would
always prefer.

The capital asset pricing model assumes that investors can borrow and lend as
much as they want at a risk-free rate of interest, R
f
. By lending at the risk-free rate, the
investor earns an expected return of R
f
and his investment has a zero standard devia-
tion because it is risk-free. The standard deviation-expected return combination for
this risk-free investment is marked as point A in Figure 1. Suppose an investor decides
to put half of his total wealth in the risk-free loan and the other half in the portfolio on
the efficient portfolio frontier with a standard deviation-expected return combination
marked as point M in the figure. Using Equation 4, you should be able to verify that
the expected return on this new portfolio is halfway between R
f
and E(R
m
); that is,
[R
f
ϩ E(R
m
)]/2. Similarly, because the covariance between the risk-free return and the
return on portfolio M must necessarily be zero, since there is no uncertainty about the
␤
2
␴
2
m
␴
2

p
ϭ␤
2
␴
2
m
ϩ (1͞n
)
(1͞n
)
͚
n
iϭ1
␣
i
␤
(1͞n
)
͚
n
iϭ1
␣
i
␣
Models of Asset Pricing
8
return on the risk-free loan, you should also be able to verify, using Equation 5, that
the standard deviation of the return on the new portfolio is halfway between zero and
␴
m

, that is, (1/2)␴
m
. The standard deviation-expected return combination for this new
portfolio is marked as point B in the figure, and as you can see it lies on the line
between point A and point M. Similarly, if an investor borrows the total amount of her
wealth at the risk-free rate R
f
and invests the proceeds plus her wealth (that is, twice
her wealth) in portfolio M, then the standard deviation of this new portfolio will be
twice the standard deviation of return on portfolio M, 2␴
m
. On the other hand, using
Equation 4, the expected return on this new portfolio is E(R
m
) plus E(R
m
) Ϫ R
f
, which
equals 2E(R
m
) Ϫ R
f
. This standard deviation-expected return combination is plotted
as point C in the figure.
You should now be able to see that both point B and point C are on the line con-
necting point A and point M. Indeed, by choosing different amounts of borrowing
and lending, an investor can form a portfolio with a standard deviation-expected
return combination that lies anywhere on the line connecting points A and M. You
may have noticed that point M has been chosen so that the line connecting points A

and M is tangent to the efficient portfolio frontier. The reason for choosing point M
in this way is that it leads to standard deviation-expected return combinations along
the line that are the most desirable for a risk-averse investor. This line can be thought
of as the opportunity locus, which shows the best combinations of standard deviations
and expected returns available to the investor.
The capital asset pricing model makes another assumption: All investors have the
same assessment of the expected returns and standard deviations of all assets. In this
case, portfolio M is the same for all investors. Thus when all investors’ holdings of
portfolio M are added together, they must equal all of the risky assets in the market,
Appendix 1 to Chapter 5
FIGURE 1 Risk Expected
Return Trade-off
The crosses show the combination
of standard deviation and expected
return for each risky asset. The
efficient portfolio frontier indicates
the most preferable standard
deviation-expected return combi-
nations that can be achieved by
putting risky assets into portfolios.
By borrowing and lending at the
risk-free rate and investing in port-
folio M, the investor can obtain
standard deviation-expected return
combinations that lie along the
line connecting A, B, M, and C.
This line, the opportunity locus,
contains the best combinations of
standard deviations and expected
returns available to the investor;

hence the opportunity locus shows
the trade-off between expected
returns and risk for the investor.
Expected
Return
E
(
R
)
2E(R
m
) — R
f
E(R
m
)
R
f

+
E(R
m
)
2
R
f
Efficient
Portfolio
Frontier
Opportunity

Locus
1/2␴
m
␴
m
2␴
m
A
B
M
C
Standard Deviation of Retuns ␴
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
9
which is just the market portfolio. The assumption that all investors have the same
assessment of risk and return for all assets thus means that portfolio M is the market

portfolio.Therefore, the R
m
and ␴
m
in Figure 1 are identical to the market return, R
m
,
and the standard deviation of this return, ␴
m
, referred to earlier in this appendix.
The conclusion that the market portfolio and portfolio M are one and the same
means that the opportunity locus in Figure 1 can be thought of as showing the trade-
off between expected returns and increased risk for the investor. This trade-off is
given by the slope of the opportunity locus, E(R
m
) Ϫ R
f
, and it tells us that when an
investor is willing to increase the risk of his portfolio by ␴
m
, then he can earn an addi-
tional expected return of E(R
m
) Ϫ R
f
. The market price of a unit of market risk, ␴
m
,
is E(R
m

) Ϫ R
f
. E(R
m
) Ϫ R
f
is therefore referred to as the market price of risk.
We now know that market price of risk is E(R
m
) Ϫ R
f
and we also have learned
that an asset’s beta tells us about systematic risk, because it is the marginal contribu-
tion of that asset to a portfolio’s risk. Therefore the amount an asset’s expected return
exceeds the risk-free rate, E(R
i
) Ϫ R
f
, should equal the market price of risk times the
marginal contribution of that asset to portfolio risk, [E(R
m
) Ϫ R
f
]␤
i
. This reasoning
yields the CAPM asset pricing relationship:
E(R
i
) ϭ R

f
ϩ␤
i
[E(R
m
) Ϫ R
f
] (9)
This CAPM asset pricing equation is represented by the upward sloping line in Figure 2,
which is called the security market line. It tells us the expected return that the market
sets for a security given its beta. For example, it tells us that if a security has a beta of
1.0 so that its marginal contribution to a portfolio’s risk is the same as the market
portfolio, then it should be priced to have the same expected return as the market
portfolio, E(R
m
).
Models of Asset Pricing
FIGURE 2 Security Market Line
The security market line derived
from the capital asset pricing
model describes the relationship
between an asset’s beta and its
expected return.
S
T
Expected
Return
E
(
R

)
Security
Market
Line
E(R
m
)
R
f
0.5 1.0 Beta ␤
10
To see that securities should be priced so that their expected return-beta combi-
nation should lie on the security market line, consider a security like S in Figure 2,
which is below the security market line. If an investor makes an investment in which
half is put into the market portfolio and half into a risk-free loan, then the beta of this
investment will be 0.5, the same as security S. However, this investment will have an
expected return on the security market line, which is greater than that for security S.
Hence investors will not want to hold security S and its current price will fall, thus
raising its expected return until it equals the amount indicated on the security mar-
ket line. On the other hand, suppose there is a security like T which has a beta of 0.5
but whose expected return is above the security market line. By including this secu-
rity in a well-diversified portfolio with other assets with a beta of 0.5, none of which
can have an expected return less than that indicated by the security line (as we have
shown), investors can obtain a portfolio with a higher expected return than that
obtained by putting half into a risk-free loan and half into the market portfolio. This
would mean that all investors would want to hold more of security T, and so its price
would rise, thus lowering its expected return until it equaled the amount indicated on
the security market line.
The capital asset pricing model formalizes the following important idea: An asset
should be priced so that is has a higher expected return not when it has a greater

risk in isolation, but rather when its systematic risk is greater.
Arbitrage Pricing Theory
Although the capital asset pricing model has proved to be very useful in practice,
deriving it does require the adoption of some unrealistic assumptions; for example,
the assumption that investors can borrow and lend freely at the risk-free rate, or the
assumption that all investors have the same assessment of expected returns and stan-
dard deviations of returns for all assets. An important alternative to the capital asset
pricing model is the arbitrage pricing theory (APT) developed by Stephen Ross of
M.I.T.
In contrast to CAPM, which has only one source of systematic risk, the market
return, APT takes the view that there can be several sources of systematic risk in the
economy that cannot be eliminated through diversification. These sources of risk can
be thought of as factors that may be related to such items as inflation, aggregate out-
put, default risk premiums, and/or the term structure of interest rates. The return on
an asset i can thus be written as being made up of components that move with these
factors and a random component that is unique to the asset (⑀
i
):
R
i
ϭ␤
i
1
(factor 1) ϩ␤
i
2
(factor 2) ϩ
…
ϩ␤
i

k
(factor k) ϩ⑀
i
(10)
Since there are k factors, this model is called a k-factor model. The ␤
i
1
,
…
, ␤
i
k
describe
the sensitivity of the asset i’s return to each of these factors.
Just as in the capital asset pricing model, these systematic sources of risk should
be priced. The market price for each factor j can be thought of as E(R
factor j
) Ϫ R
f
, and
hence the expected return on a security can be written as:
E(R
i
) ϭ R
f
ϩ␤
i
1
[E(R
factor 1

) Ϫ R
f
] ϩ
…
ϩ␤
i
k
[E(R
factor k
) Ϫ R
f
] (11)
Appendix 1 to Chapter 5
11
This asset pricing equation indicates that all the securities should have the same mar-
ket price for the risk contributed by each factor. If the expected return for a security
were above the amount indicated by the APT pricing equation, then it would provide
a higher expected return than a portfolio of other securities with the same average
sensitivity to each factor. Hence investors would want to hold more of this security
and its price would rise until the expected return fell to the value indicated by the
APT pricing equation. On the other hand, if the security’s expected return were less
than the amount indicated by the APT pricing equation, then no one would want to
hold this security, because a higher expected return could be obtained with a portfo-
lio of securities with the same average sensitivity to each factor. As a result, the price
of the security would fall until its expected return rose to the value indicated by the
APT equation.
As this brief outline of arbitrage pricing theory indicates, the theory supports a
basic conclusion from the capital asset pricing model: An asset should be priced so
that it has a higher expected return not when it has a greater risk in isolation, but
rather when its systematic risk is greater. There is still substantial controversy about

whether a variant of the capital asset pricing model or the arbitrage pricing theory is
a better description of reality. At the present time, both frameworks are considered
valuable tools for understanding how risk affects the prices of assets.
Models of Asset Pricing
12
Both models of interest-rate determination in Chapter 4 make use of an asset market
approach in which supply and demand are always considered in terms of stocks of assets
(amounts at a given point in time). The asset market approach is useful in understand-
ing not only why interest rates fluctuate but also how any asset’s price is determined.
One asset that has fascinated people for thousands of years is gold. It has been a
driving force in history: The conquest of the Americas by Europeans was to a great
extent the result of the quest for gold, to cite just one example. The fascination with
gold continues to the present day, and developments in the gold market are followed
closely by financial analysts and the media. This appendix shows how the asset mar-
ket approach can be applied to understanding the behavior of commodity markets, in
particular the gold market. (The analysis in this appendix can also be used to under-
stand behavior in many other asset markets.)
Supply and Demand in the Gold Market
The analysis of a commodity market, such as the gold market, proceeds in a similar
fashion to the analysis of the bond market by examining the supply of and demand
for the commodity. We again use our analysis of the determinants of asset demand to
obtain a demand curve for gold, which shows the relationship between the quantity
of gold demanded and the price when all other economic variables are held constant.
To derive the relationship between the quantity of gold demanded and its price, we
again recognize that an important determinant of the quantity demanded is its
expected return:
where R
e
ϭ expected return
P

t
ϭ price of gold today
P
e
t ϩ 1
ϭ expected price of gold next year
g
e
ϭ expected capital gain
In deriving the demand curve, we hold all other variables constant, particularly
the expected price of gold next year P
e
tϩ1
. With a given value of the expected price of
gold next year P
e
tϩ1
, a lower price of gold today P
t
means that there will be a greater
R
e
ϭ
P
e
tϩ1
Ϫ P
t
P
t

ϭ g
e
Demand Curve
Applying the Asset Market
Approach to a Commodity
Market: The Case of Gold
appendix 2
to chapter
5
1
appreciation in the price of gold over the coming year. The result is that a lower price
of gold today implies a higher expected capital gain over the coming year and hence
a higher expected return: R
e
ϭ (P
e
tϩ1
Ϫ P
t
)/P
t
. Thus because the price of gold today
(which for simplicity we will denote as P) is lower, the expected return on gold is
higher, and the quantity demanded is higher. Consequently, the demand curve G
d
1
slopes downward in Figure 1.
To derive the supply curve, expressing the relationship between the quantity supplied
and the price, we again assume that all other economic variables are held constant. A
higher price of gold will induce producers to mine for extra gold and also possibly

induce governments to sell some of their gold stocks to the public, thus increasing the
quantity supplied. Hence the supply curve G
s
1
in Figure 1 slopes upward. Notice that
the supply curve in the figure is drawn to be very steep. The reason for this is that the
actual amount of gold produced in any year is only a tiny fraction of the outstanding
stock of gold that has been accumulated over hundreds of years. Thus the increase in
the quantity of the gold supplied in response to a higher price is only a small fraction
of the stock of gold, resulting in a very steep supply curve.
Market equilibrium in the gold market occurs when the quantity of gold demanded
equals the quantity of gold supplied:
G
d
ϭ G
s
With the initial demand and supply curves of G
d
1
and G
s
1
, equilibrium occurs
at point 1, where these curves intersect at a gold price of P
1
. At a price above this
Market
Equilibrium
Supply Curve
Applying the Asset Market Approach to a Commodity Market: The Case of Gold

FIGURE 1 A Change in the
Equilibrium Price of Gold
When the demand curve shifts right-
ward from G
1
d
to G
2
d
—say, because
expected inflation rises—equilibrium
moves from point 1 to point 2, and
the equilibrium price of gold rises
from P
1
to P
2
.
1

P
2

P
1
Price of Gold
P

G
d

1

G
d
2

G
s
1
Quantity of Gold
G
2
2
equilibrium, the amount of gold supplied exceeds the amount demanded, and this
condition of excess supply leads to a decline in the gold price until it reaches P
1
, the
equilibrium price. Similarly, if the price is below P
1
, there is excess demand for gold,
which drives the price upward until it settles at the equilibrium price P
1
.
Changes in the Equilibrium Price of Gold
Changes in the equilibrium price of gold occur when there is a shift in either the sup-
ply curve or the demand curve; that is, when the quantity demanded or supplied
changes at each given price of gold in response to a change in some factor other than
today’s gold price.
Our analysis of the determinants of asset demand in the chapter provides the factors
that shift the demand curve for gold: wealth, expected return on gold relative to alter-

native assets, riskiness of gold relative to alternative assets, and liquidity of gold rela-
tive to alternative assets. The analysis of how changes in each of these factors shift the
demand curve for gold is the same as that found in the chapter.
When wealth rises, at a given price of gold, the quantity demanded increases, and
the demand curve shifts to the right, as in Figure 1. When the expected return on gold
relative to other assets rises—either because speculators think that the future price of
gold will be higher or because the expected return on other assets declines—gold
becomes more desirable; the quantity demanded therefore increases at any given price
of gold, and the demand curve shifts to the right, as in Figure 1. When the relative
riskiness of gold declines, either because gold prices become less volatile or because
returns on other assets become more volatile, gold becomes more desirable, the quan-
tity demanded at every given price rises, and the demand curve again shifts to the
right. When the gold market becomes relatively more liquid and gold therefore
becomes more desirable, the quantity demanded at any given price rises, and the
demand curve also shifts to the right, as in Figure 1.
The supply curve for gold shifts when there are changes in technology that make gold
mining more efficient or when governments at any given price of gold decide to
increase sales of their holdings of gold. In these cases, the quantity of gold supplied
at any given price increases, and the supply curve shifts to the right.
Study Guide To give yourself practice with supply and demand analysis in the gold market, see if
you can analyze what happens to the price of gold for the following situations,
remembering that all other things are held constant: 1) Interest rates rise, 2) the gold
market becomes more liquid, 3) the volatility of gold prices increases, 4) the stock
market is expected to turn bullish in the near future, 5) investors suddenly become
fearful that there will be a collapse in real estate prices, and 6) Russia sells a lot of gold
in the open market to raise hard currency to finance expenditures.
Shifts in the
Supply Curve for
Gold
Shift in the

Demand Curve
for Gold
Appendix 2 to Chapter 5
3
Applying the Asset Market Approach to a Commodity Market: The Case of Gold
Changes in the Equilibrium Price of Gold Due to a Rise in
Expected Inflation
Application
To illustrate how changes in the equilibrium price of gold occur when sup-
ply and demand curves shift, let’s look at what happens when there is a
change in expected inflation.
Suppose that expected inflation is 5% and the initial supply and demand
curves are at G
s
1
and G
d
1
so that the equilibrium price of gold is at P
1
in Figure
1. If expected inflation now rises to 10%, prices of goods and commodities
next year will be expected to be higher than they otherwise would have been,
and the price of gold next year P
e
tϩ1
will also be expected to be higher than
otherwise. Now at any given price of gold today, gold is expected to have a
greater rate of appreciation over the coming year and hence a higher expected
capital gain and return. The greater expected return means that the quantity

of gold demanded increases at any given price, thus shifting the demand
curve from G
d
1
to G
d
2
. Equilibrium therefore moves from point 1 to point 2,
and the price of gold rises from P
1
to P
2
.
By using a supply and demand diagram like that in Figure 1, you should
be able to see that if the expected rate of inflation falls, the price of gold today
will also fall. We thus reach the following conclusion: The price of gold
should be positively related to the expected inflation rate.
Because the gold market responds immediately to any changes in
expected inflation, it is considered a good barometer of the trend of inflation
in the future. Indeed, Alan Greenspan, the chairman of the Board of
Governors of the Federal Reserve System, at one point advocated using the
price of gold as an indicator of inflationary pressures in the economy. Not
surprisingly, then, the gold market is followed closely by financial analysts
and monetary policymakers.
4
120
PREVIEW
In our supply and demand analysis of interest-rate behavior in Chapter 5, we exam-
ined the determination of just one interest rate. Yet we saw earlier that there are enor-
mous numbers of bonds on which the interest rates can and do differ. In this chapter,

we complete the interest-rate picture by examining the relationship of the various
interest rates to one another. Understanding why they differ from bond to bond can
help businesses, banks, insurance companies, and private investors decide which
bonds to purchase as investments and which ones to sell.
We first look at why bonds with the same term to maturity have different inter-
est rates. The relationship among these interest rates is called the risk structure of
interest rates, although risk, liquidity, and income tax rules all play a role in deter-
mining the risk structure. A bond’s term to maturity also affects its interest rate, and
the relationship among interest rates on bonds with different terms to maturity is
called the term structure of interest rates. In this chapter, we examine the sources
and causes of fluctuations in interest rates relative to one another and look at a num-
ber of theories that explain these fluctuations.
Risk Structure of Interest Rates
Figure 1 shows the yields to maturity for several categories of long-term bonds from
1919 to 2002. It shows us two important features of interest-rate behavior for bonds
of the same maturity: Interest rates on different categories of bonds differ from one
another in any given year, and the spread (or difference) between the interest rates
varies over time. The interest rates on municipal bonds, for example, are above those
on U.S. government (Treasury) bonds in the late 1930s but lower thereafter. In addi-
tion, the spread between the interest rates on Baa corporate bonds (riskier than Aaa
corporate bonds) and U.S. government bonds is very large during the Great
Depression years 1930–1933, is smaller during the 1940s–1960s, and then widens
again afterwards. What factors are responsible for these phenomena?
One attribute of a bond that influences its interest rate is its risk of default, which
occurs when the issuer of the bond is unable or unwilling to make interest payments
when promised or pay off the face value when the bond matures. A corporation suf-
fering big losses, such as Chrysler Corporation did in the 1970s, might be more likely
Default Risk
Chapter
The Risk and Term Structure

of Interest Rates
6
to suspend interest payments on its bonds.
1
The default risk on its bonds would
therefore be quite high. By contrast, U.S. Treasury bonds have usually been consid-
ered to have no default risk because the federal government can always increase taxes
to pay off its obligations. Bonds like these with no default risk are called default-free
bonds. (However, during the budget negotiations in Congress in 1995 and 1996, the
Republicans threatened to let Treasury bonds default, and this had an impact on the
bond market, as one application following this section indicates.) The spread between
the interest rates on bonds with default risk and default-free bonds, called the risk
premium, indicates how much additional interest people must earn in order to be
willing to hold that risky bond. Our supply and demand analysis of the bond market
in Chapter 5 can be used to explain why a bond with default risk always has a posi-
tive risk premium and why the higher the default risk is, the larger the risk premium
will be.
To examine the effect of default risk on interest rates, let us look at the supply and
demand diagrams for the default-free (U.S. Treasury) and corporate long-term bond
markets in Figure 2. To make the diagrams somewhat easier to read, let’s assume that
initially corporate bonds have the same default risk as U.S. Treasury bonds. In this
case, these two bonds have the same attributes (identical risk and maturity); their
equilibrium prices and interest rates will initially be equal (P
c
1
ϭ P
T
1
and i
c

1
ϭ i
T
1
),
and the risk premium on corporate bonds (i
c
1
Ϫ i
T
1
) will be zero.
CHAPTER 6
The Risk and Term Structure of Interest Rates
121
FIGURE 1 Long-Term Bond Yields, 1919–2002
Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941–1970; Federal Reserve: www.federalreserve.gov/releases/h15/data/.
16
14
12
10
8
6
4
2
0
1950 1960 1970 1980 1990 2000
State and Local Government
(Municipal)
U.S. Government

Long-Term Bonds
Corporate Baa Bonds
Annual Yield (%)
Corporate Aaa Bonds
19401930
1920
1
Chrysler did not default on its loans in this period, but it would have were it not for a government bailout plan
intended to preserve jobs, which in effect provided Chrysler with funds that were used to pay off creditors.
www.federalreserve.gov
/Releases/h15/update/
The Federal Reserve reports the
returns on different quality
bonds. Look at the bottom of
the listing of interest rates for
AAA and BBB rated bonds.
Study Guide Two exercises will help you gain a better understanding of the risk structure:
1. Put yourself in the shoes of an investor—see how your purchase decision would
be affected by changes in risk and liquidity.
2. Practice drawing the appropriate shifts in the supply and demand curves when
risk and liquidity change. For example, see if you can draw the appropriate shifts
in the supply and demand curves when, in contrast to the examples in the text,
a corporate bond has a decline in default risk or an improvement in its liquidity.
If the possibility of a default increases because a corporation begins to suffer large
losses, the default risk on corporate bonds will increase, and the expected return on
these bonds will decrease. In addition, the corporate bond’s return will be more
uncertain as well. The theory of asset demand predicts that because the expected
return on the corporate bond falls relative to the expected return on the default-free
Treasury bond while its relative riskiness rises, the corporate bond is less desirable
(holding everything else equal), and demand for it will fall. The demand curve for

corporate bonds in panel (a) of Figure 2 then shifts to the left, from D
c
1
to D
c
2
.
At the same time, the expected return on default-free Treasury bonds increases
relative to the expected return on corporate bonds, while their relative riskiness
122 PART II
Financial Markets
FIGURE 2 Response to an Increase in Default Risk on Corporate Bonds
An increase in default risk on corporate bonds shifts the demand curve from D
c
1
to D
c
2
. Simultaneously, it shifts the demand curve for
Treasury bonds from D
T
1
to D
T
2
. The equilibrium price for corporate bonds (left axis) falls from P
c
1
to P
c

2
, and the equilibrium interest rate
on corporate bonds (right axis) rises from i
c
1
to i
c
2
. In the Treasury market, the equilibrium bond price rises from P
T
1
to P
T
2
, and the equilib-
rium interest rate falls from i
T
1
to i
T
2
. The brace indicates the difference between i
c
2
and i
T
2
, the risk premium on corporate bonds. (Note: P
and i increase in opposite directions. P on the left vertical axis increases as we go up the axis, while i on the right vertical axis increases as
we go down the axis.)

Quantity of Corporate Bonds
Quantity of Treasury Bonds
Price of Bonds,
P
(
P
increases
↑
)
Interest Rate,
i
(
i
increases )
↑
Interest Rate,
i
(
i
increases )
↑
Price of Bonds,
P
(
P
increases
↑
)
(a) Corporate bond market
(b) Default-free (U.S. Treasury) bond market

Risk
Premium
P
c
2
P
c
1
S
c
D
c
1
D
c
2
i
c
1
P
T
2
P
T
1
i
c
2
S
T

D
T
1
D
T
2
i
T
1
i
T
2
i
T
2
declines. The Treasury bonds thus become more desirable, and demand rises, as
shown in panel (b) by the rightward shift in the demand curve for these bonds from
D
T
1
to D
T
2
.
As we can see in Figure 2, the equilibrium price for corporate bonds (left axis)
falls from P
c
1
to P
c

2
, and since the bond price is negatively related to the interest rate,
the equilibrium interest rate on corporate bonds (right axis) rises from i
c
1
to i
c
2
. At the
same time, however, the equilibrium price for the Treasury bonds rises from P
T
1
to P
T
2
,
and the equilibrium interest rate falls from i
T
1
to i
T
2
. The spread between the interest
rates on corporate and default-free bonds—that is, the risk premium on corporate
bonds—has risen from zero to i
c
2
Ϫ i
T
2

. We can now conclude that a bond with
default risk will always have a positive risk premium, and an increase in its default
risk will raise the risk premium.
Because default risk is so important to the size of the risk premium, purchasers
of bonds need to know whether a corporation is likely to default on its bonds. Two
major investment advisory firms, Moody’s Investors Service and Standard and Poor’s
Corporation, provide default risk information by rating the quality of corporate and
municipal bonds in terms of the probability of default. The ratings and their descrip-
tion are contained in Table 1. Bonds with relatively low risk of default are called
investment-grade securities and have a rating of Baa (or BBB) and above. Bonds with
CHAPTER 6
The Risk and Term Structure of Interest Rates
123
Rating
Standard Examples of Corporations with
Moody’s and Poor’s Descriptions Bonds Outstanding in 2003
Aaa AAA Highest quality General Electric, Pfizer Inc.,
(lowest default risk) North Carolina State,
Mobil Oil
Aa AA High quality Wal-Mart, McDonald’s,
Credit Suisse First Boston
A A Upper medium grade Hewlett-Packard,
Anheuser-Busch,
Ford, Household Finance
Baa BBB Medium grade Motorola, Albertson’s, Pennzoil,
Weyerhaeuser Co.,
Tommy Hilfiger
Ba BB Lower medium grade Royal Caribbean, Levi Strauss
B B Speculative Rite Aid, Northwest Airlines Inc.,
Six Flags

Caa CCC, CC Poor (high default risk) Revlon, United Airlines
Ca C Highly speculative US Airways, Polaroid
C D Lowest grade Enron, Oakwood Homes
Table 1 Bond Ratings by Moody’s and Standard and Poor’s
ratings below Baa (or BBB) have higher default risk and have been aptly dubbed
speculative-grade or junk bonds. Because these bonds always have higher interest
rates than investment-grade securities, they are also referred to as high-yield bonds.
Next let’s look back at Figure 1 and see if we can explain the relationship between
interest rates on corporate and U.S. Treasury bonds. Corporate bonds always have
higher interest rates than U.S. Treasury bonds because they always have some risk of
default, whereas U.S. Treasury bonds do not. Because Baa-rated corporate bonds have
a greater default risk than the higher-rated Aaa bonds, their risk premium is greater,
and the Baa rate therefore always exceeds the Aaa rate. We can use the same analysis
to explain the huge jump in the risk premium on Baa corporate bond rates during the
Great Depression years 1930–1933 and the rise in the risk premium after 1970 (see
Figure 1). The depression period saw a very high rate of business failures and defaults.
As we would expect, these factors led to a substantial increase in default risk for bonds
issued by vulnerable corporations, and the risk premium for Baa bonds reached
unprecedentedly high levels. Since 1970, we have again seen higher levels of business
failures and defaults, although they were still well below Great Depression levels.
Again, as expected, default risks and risk premiums for corporate bonds rose, widen-
ing the spread between interest rates on corporate bonds and Treasury bonds.
124 PART II
Financial Markets
The Enron Bankruptcy and the Baa-Aaa Spread
Application
In December 2001, the Enron Corporation, a firm specializing in trading in the
energy market, and once the seventh-largest corporation in the United States,
was forced to declare bankruptcy after it became clear that it had used shady
accounting to hide its financial problems. (The Enron bankruptcy, the largest

ever in the United States, will be discussed further in Chapter 8.) Because of the
scale of the bankruptcy and the questions it raised about the quality of the infor-
mation in accounting statements, the Enron collapse had a major impact on the
corporate bond market. Let’s see how our supply and demand analysis explains
the behavior of the spread between interest rates on lower quality (Baa-rated) and
highest quality (Aaa-rated) corporate bonds in the aftermath of the Enron failure.
As a consequence of the Enron bankruptcy, many investors began to
doubt the financial health of corporations with lower credit ratings such as
Baa. The increase in default risk for Baa bonds made them less desirable at
any given interest rate, decreased the quantity demanded, and shifted the
demand curve for Baa bonds to the left. As shown in panel (a) of Figure 2,
the interest rate on Baa bonds should have risen, which is indeed what hap-
pened. Interest rates on Baa bonds rose by 24 basis points (0.24 percentage
points) from 7.81% in November 2001 to 8.05% in December 2001. But the
increase in the perceived default risk for Baa bonds after the Enron bank-
ruptcy made the highest quality (Aaa) bonds relatively more attractive and
shifted the demand curve for these securities to the right—an outcome
described by some analysts as a “flight to quality.” Just as our analysis predicts
in Figure 2, interest rates on Aaa bonds fell by 20 basis points, from 6.97%
in November to 6.77% in December. The overall outcome was that the
spread between interest rates on Baa and Aaa bonds rose by 44 basis points
from 0.84% before the bankruptcy to 1.28% afterward.
Another attribute of a bond that influences its interest rate is its liquidity. As we
learned in Chapter 4, a liquid asset is one that can be quickly and cheaply converted
into cash if the need arises. The more liquid an asset is, the more desirable it is (hold-
ing everything else constant). U.S. Treasury bonds are the most liquid of all long-term
bonds, because they are so widely traded that they are the easiest to sell quickly and
the cost of selling them is low. Corporate bonds are not as liquid, because fewer bonds
for any one corporation are traded; thus it can be costly to sell these bonds in an
emergency, because it might be hard to find buyers quickly.

How does the reduced liquidity of the corporate bonds affect their interest rates
relative to the interest rate on Treasury bonds? We can use supply and demand analy-
sis with the same figure that was used to analyze the effect of default risk, Figure 2,
to show that the lower liquidity of corporate bonds relative to Treasury bonds
increases the spread between the interest rates on these two bonds. Let us start the
analysis by assuming that initially corporate and Treasury bonds are equally liquid
and all their other attributes are the same. As shown in Figure 2, their equilibrium
prices and interest rates will initially be equal: P
c
1
ϭ P
T
1
and i
c
1
ϭ i
T
1
. If the corporate
bond becomes less liquid than the Treasury bond because it is less widely traded, then
(as the theory of asset demand indicates) its demand will fall, shifting its demand
curve from D
c
1
to D
c
2
as in panel (a). The Treasury bond now becomes relatively more
liquid in comparison with the corporate bond, so its demand curve shifts rightward

from D
T
1
to D
T
2
as in panel (b). The shifts in the curves in Figure 2 show that the price
of the less liquid corporate bond falls and its interest rate rises, while the price of the
more liquid Treasury bond rises and its interest rate falls.
The result is that the spread between the interest rates on the two bond types has
risen. Therefore, the differences between interest rates on corporate bonds and
Treasury bonds (that is, the risk premiums) reflect not only the corporate bond’s
default risk but its liquidity, too. This is why a risk premium is more accurately a “risk
and liquidity premium,” but convention dictates that it is called a risk premium.
Returning to Figure 1, we are still left with one puzzle—the behavior of municipal
bond rates. Municipal bonds are certainly not default-free: State and local govern-
ments have defaulted on the municipal bonds they have issued in the past, particu-
larly during the Great Depression and even more recently in the case of Orange
County, California, in 1994 (more on this in Chapter 13). Also, municipal bonds are
not as liquid as U.S. Treasury bonds.
Why is it, then, that these bonds have had lower interest rates than U.S. Treasury
bonds for at least 40 years, as indicated in Figure 1? The explanation lies in the fact
that interest payments on municipal bonds are exempt from federal income taxes, a
factor that has the same effect on the demand for municipal bonds as an increase in
their expected return.
Let us imagine that you have a high enough income to put you in the 35% income
tax bracket, where for every extra dollar of income you have to pay 35 cents to the gov-
ernment. If you own a $1,000-face-value U.S. Treasury bond that sells for $1,000 and
has a coupon payment of $100, you get to keep only $65 of the payment after taxes.
Although the bond has a 10% interest rate, you actually earn only 6.5% after taxes.

Suppose, however, that you put your savings into a $1,000-face-value municipal
bond that sells for $1,000 and pays only $80 in coupon payments. Its interest rate is
only 8%, but because it is a tax-exempt security, you pay no taxes on the $80 coupon
payment, so you earn 8% after taxes. Clearly, you earn more on the municipal bond
Income Tax
Considerations
Liquidity
CHAPTER 6
The Risk and Term Structure of Interest Rates
125
after taxes, so you are willing to hold the riskier and less liquid municipal bond even
though it has a lower interest rate than the U.S. Treasury bond. (This was not true
before World War II, when the tax-exempt status of municipal bonds did not convey
much of an advantage because income tax rates were extremely low.)
Another way of understanding why municipal bonds have lower interest rates than
Treasury bonds is to use the supply and demand analysis displayed in Figure 3. We
assume that municipal and Treasury bonds have identical attributes and so have the
same bond prices and interest rates as drawn in the figure: P
m
1
ϭ P
T
1
and i
m
1
ϭ i
T
1
. Once

the municipal bonds are given a tax advantage that raises their after-tax expected return
relative to Treasury bonds and makes them more desirable, demand for them rises, and
their demand curve shifts to the right, from D
m
1
to D
m
2
. The result is that their equilib-
rium bond price rises from P
m
1
to P
m
2
, and their equilibrium interest rate falls from i
m
1
to
i
m
2
. By contrast, Treasury bonds have now become less desirable relative to municipal
bonds; demand for Treasury bonds decreases, and D
T
1
shifts to D
T
2
. The Treasury bond

price falls from P
T
1
to P
T
2
, and the interest rate rises from i
T
1
to i
T
2
. The resulting lower
interest rates for municipal bonds and higher interest rates for Treasury bonds explains
why municipal bonds can have interest rates below those of Treasury bonds.
2
126 PART II
Financial Markets
FIGURE 3 Interest Rates on Municipal and Treasury Bonds
When the municipal bond is given tax-free status, demand for the municipal bond shifts rightward from D
m
1
to D
m
2
and demand
for the Treasury bond shifts leftward from D
T
1
to D

T
2
. The equilibrium price of the municipal bond (left axis) rises from P
m
1
to
P
m
2
, so its interest rate (right axis) falls from i
m
1
to i
m
2
, while the equilibrium price of the Treasury bond falls from P
T
1
to P
T
2
and
its interest rate rises from i
T
1
to i
T
2
. The result is that municipal bonds end up with lower interest rates than those on Treasury
bonds. (Note: P and i increase in opposite directions. P on the left vertical axis increases as we go up the axis, while i on the

right vertical axis increases as we go down the axis.)
Quantity of Treasury Bonds
Quantity of Municipal Bonds
Price of Bonds,
P
(
P
increases
↑
)
Interest Rate,
i
(
i
increases )
↑
Price of Bonds,
P
(
P
increases
↑
)
(a) Market for municipal bonds
(b) Market for Treasury bonds
Interest Rate,
i
(
i
increases )

↑
P
m
1
P
m
2
S
m
D
m
1
i
m
1
i
m
2
D
m
2
P
T
2
P
T
1
S
T
D

T
1
D
T
2
i
T
1
i
T
2
2
In contrast to corporate bonds, Treasury bonds are exempt from state and local income taxes. Using the analy-
sis in the text, you should be able to show that this feature of Treasury bonds provides an additional reason why
interest rates on corporate bonds are higher than those on Treasury bonds.
The risk structure of interest rates (the relationship among interest rates on bonds
with the same maturity) is explained by three factors: default risk, liquidity, and the
income tax treatment of the bond’s interest payments. As a bond’s default risk
increases, the risk premium on that bond (the spread between its interest rate and the
interest rate on a default-free Treasury bond) rises. The greater liquidity of Treasury
bonds also explains why their interest rates are lower than interest rates on less liquid
bonds. If a bond has a favorable tax treatment, as do municipal bonds, whose inter-
est payments are exempt from federal income taxes, its interest rate will be lower.
Summary
Term Structure of Interest Rates
We have seen how risk, liquidity, and tax considerations (collectively embedded in the
risk structure) can influence interest rates. Another factor that influences the interest
rate on a bond is its term to maturity: Bonds with identical risk, liquidity, and tax
characteristics may have different interest rates because the time remaining to matu-
rity is different. A plot of the yields on bonds with differing terms to maturity but the

same risk, liquidity, and tax considerations is called a yield curve, and it describes the
term structure of interest rates for particular types of bonds, such as government
bonds. The “Following the Financial News” box shows several yield curves for
Treasury securities that were published in the Wall Street Journal. Yield curves can be
classified as upward-sloping, flat, and downward-sloping (the last sort is often
referred to as an inverted yield curve). When yield curves slope upward, as in the
“Following the Financial News” box, the long-term interest rates are above the short-
term interest rates; when yield curves are flat, short- and long-term interest rates are
the same; and when yield curves are inverted, long-term interest rates are below
short-term interest rates. Yield curves can also have more complicated shapes in
which they first slope up and then down, or vice versa. Why do we usually see
CHAPTER 6
The Risk and Term Structure of Interest Rates
127
Effects of the Bush Tax Cut on Bond Interest Rates
Application
The Bush tax cut passed in 2001 scheduled a reduction of the top income tax
bracket from 39% to 35% over a ten-year period. What is the effect of this
income tax decrease on interest rates in the municipal bond market relative
to those in the Treasury bond market?
Our supply and demand analysis provides the answer. A decreased income
tax rate for rich people means that the after-tax expected return on tax-free
municipal bonds relative to that on Treasury bonds is lower, because the
interest on Treasury bonds is now taxed at a lower rate. Because municipal
bonds now become less desirable, their demand decreases, shifting the
demand curve to the left, which lowers their price and raises their interest
rate. Conversely, the lower income tax rate makes Treasury bonds more desir-
able; this change shifts their demand curve to the right, raises their price, and
lowers their interest rates.
Our analysis thus shows that the Bush tax cut raises the interest rates on

municipal bonds relative to interest rates on Treasury bonds.
upward slopes of the yield curve as in the “Following the Financial News” box but
sometimes other shapes?
Besides explaining why yield curves take on different shapes at different times, a
good theory of the term structure of interest rates must explain the following three
important empirical facts:
1. As we see in Figure 4, interest rates on bonds of different maturities move
together over time.
2. When short-term interest rates are low, yield curves are more likely to have an
upward slope; when short-term interest rates are high, yield curves are more
likely to slope downward and be inverted.
3. Yield curves almost always slope upward, as in the “Following the Financial
News” box.
Three theories have been put forward to explain the term structure of interest
rates; that is, the relationship among interest rates on bonds of different maturities
reflected in yield curve patterns: (1) the expectations theory, (2) the segmented mar-
kets theory, and (3) the liquidity premium theory, each of which is described in the
following sections. The expectations theory does a good job of explaining the first two
facts on our list, but not the third. The segmented markets theory can explain fact 3
but not the other two facts, which are well explained by the expectations theory.
Because each theory explains facts that the other cannot, a natural way to seek a bet-
ter understanding of the term structure is to combine features of both theories, which
leads us to the liquidity premium theory, which can explain all three facts.
If the liquidity premium theory does a better job of explaining the facts and is
hence the most widely accepted theory, why do we spend time discussing the other
two theories? There are two reasons. First, the ideas in these two theories provide the
128 PART II
Financial Markets
Following the Financial News
The Wall Street Journal publishes a daily plot of the yield

curves for Treasury securities, an example of which is
presented here. It is typically found on page 2 of the
“Money and Investing” section.
The numbers on the vertical axis indicate the interest
rate for the Treasury security, with the maturity given by
the numbers on the horizontal axis. For example, the
yield curve marked “Yesterday” indicates that the interest
rate on the three-month Treasury bill yesterday was
1.25%, while the one-year bill had an interest rate of
1.35% and the ten-year bond had an interest rate of
4.0%. As you can see, the yield curves in the plot have the
typical upward slope.
Source: Wall Street Journal, Wednesday, January 22, 2003, p. C2.
Yield Curves
www.ratecurve.com/yc2.html
Check out today’s yield curve.
Treasury Yield Curve
Yield to maturity of current bills,
notes and bonds.
Source: Reuters
1
1.0
2.0
3.0
4.0
5.0%
3 6 2 5 10 30
mos. yrs. maturity
Yesterday
1 month ago

1 year ago